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Development of a prototype scanner for pulsed ultrasound computed tomography McFarland, Sheila J. 2000

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DEVELOPMENT OF A PROTOTYPE SCANNER FOR PULSED ULTRASOUND COMPUTED TOMOGRAPHY By Sheila J. McFarland B.Sc. (Hon. Physics), University of Regina, 1991 M.Sc. (Physics), University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 2000 © Sheila J. McFarland , 2000 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of Br i t i sh Columbia 6224 Agricul tura l Road Vancouver, Canada V 6 T 1Z1 Date: A B S T R A C T A prototype scanner for pulsed ultrasound computed tomography ( U C T ) has been built and tested wi th the aim of developing new imaging techniques that hold potential for the improved early detection of breast cancer. Two reconstruction algorithms were tested for their ability to produce quantitative 2D cross-sections of 7 K ( r ) and reflectivity, R(r), where 7«(r) = ^ ^ (0.1) J 2 ( r ) = 7 « ( r ) - 7 p ( r ) (0.2) and / \ P ( r ) - PO / n Q \ >(r) = -pJrT m K(T) and p(r) are the compressibility and density throughout the image space, while n0 and po describe water. jp is subsequently calculated from 7* and R. A Direct Fourier Method ( D F M ) that ignores attenuation was tested both as is and wi th the addition of an approximate attenuation correction based on the average distance ultrasound travels through tissue and water. A n Algebraic Reconstruction Technique ( A R T ) was also de-veloped, which iteratively solves for jK and R subject to rigorous attenuation correction. In computer simulation, the D F M produced valid 7 K and jp cross-sections only in the absence of attenuation and when |r | < 0.05|r-j|, where r and are the position vectors of any scatter point and the source or detector, respectively. Points separately by 0.07 m m were resolved in simulation, and point amplitudes were reconstructed quantitatively to wi th in 97 ± 3% and 95 ± 4% of their actual values for 7 K and 7 P , respectively. Beyond |r | < 0.05|r-j|, Gaussian point-spread-functions (PSF's) were distorted into low amplitude halos. The approximate attenuation correction failed in simulation. Experimentally, the D F M reconstructed images of a cylindrical tissue phantom with negative 7* and R and a 3 m m diameter. The resulting 7« and R cross-sections were primari ly negative with an object diameter of 2.94 ± 0.06 m m and 1.94 ± 0.04 m m for i i 7« and R, respectively. A valid 7 P could not be calculated due to the narrow R function. Experimentally, point-spread-functions were severely distorted by errors of only 1-2% in the measurement of |r,|. In simulation, valid 7* and 7 P cross-sections were reconstructed wi th the A R T for | r | < 0.3|rj | and in the presence of strong attenuation. Points separated by 0.05 m m were resolved in simulation, and point amplitudes were reconstructed quantitatively to wi th in 100 ± 14% and 99 ± 4% of their actual values for jK and 7^, respectively. Strong attenuation was corrected to wi thin an error of 0.2%. Convergence was not possible for images of more than 20 x 20 pixels due to ill-conditioning of the system. These theoretical results indicate, though, that the method holds promise given the application of conditioning algorithms. i i i T A B L E O F C O N T E N T S Abstract ii List of Figures viii List of Tables xi Acknowledgements xii Glossary xiii 1 Introduction 1 1.1 A i m of this Work - Hypothesis 1 1.2 Thesis Highlights 2 1.3 Organization of this Thesis 5 1.4 Medical Motivat ion 6 1.5 Ultrasound as an Imaging Probe 10 1.6 Some Basic Sound and Ultrasound Concepts 13 1.7 Current Use of Ultrasound in Breast Cancer Imaging 18 1.7.1 B-Scan Imaging 18 1.7.2 3D Ultrasound 23 1.8 Computerized Tomographic Imaging 24 1.8.1 Diffracting Versus Nondiffracting Sources 26 1.8.2 Ultrasound Computed Tomography 28 2 Theoretical Background 34 2.1 Nonplanar Sources and Pulsed Ultrasound 35 2.2 Min ima l ly Attenuating Objects 38 2.2.1 The Physics Basis 38 2.2.2 The Reconstruction Algor i thm 45 2.2.3 Corrected Green's Function Propagator 53 2.2.4 Use of the Fast Fourier Transform 53 iv 2.2.5 The Fourier Transform Sign Convention 56 2.2.6 Behavior of the Exact Propagator 57 2.2.7 Theoretical Resolution 58 2.2.8 Interpolation to a Square G r i d 63 2.2.9 A Note on Lowpass Fil ter ing and Truncation 67 2.3 Process Behind the Reconstruction of Each View 70 2.4 Discussion of Attenuation Effects 74 3 Apparatus 79 3.1 The Sources 81 3.2 The Hydrophone 82 3.3 Stepper Motor Apparatus 89 3.4 Water Tank Design . . 93 3.5 The Electronics 97 3.5.1 The Pulser-Receiver 97 3.5.2 Components Beyond The Hydrophone 97 4 Experimental Methods 101 4.1 Approximate Correction for Attenuation 101 4.2 A R T wi th Attenuated Propagators 106 4.2.1 The Attenuated Propagator 107 4.2.2 ps in Presence of Attenuation 108 4.2.3 Derivation of the A R T 112 4.2.4 Iterative Solution 116 4.3 Necessary Noise Removal 116 4.3.1 Analysis of Noise in the Signal 117 4.4 Conventional Denoising Techniques 121 4.5 Wavelet Theory 126 4.5.1 Wavelet Analysis and the Wavelet Transform 129 4.5.2 Process of Wavelet Denoising 130 4.5.3 Types of Wavelets Used 133 4.6 Interpolation Methods Necessary for Use of F F T 134 4.6.1 M a t l a b ™ Interpolation 137 4.6.2 Unified Frequency-Domain Image Reconstruction 139 4.6.3 Sinc-based Interpolation 141 4.6.4 Comparison of the Interpolation Methods 141 5 Results 144 5.1 Effect of Various Assumptions 146 5.1.1 The Assumption that VSrj « ikhjSTj 146 5.1.2 The Assumption that VSTj « ikhjSrj 152 v 5.1.3 The Effect of Use of an Approximate STj 158 5.2 General Notes Regarding the Computer Simulations 163 5.2.1 Simulation of Object and Scatter 163 5.2.2 Points Regarding Analysis 164 5.2.3 Discussion of Appl ied Fil ter ing 165 5.3 Tests of the Method by Blackledge et al in Computer Simulation 167 5.3.1 The Effect of Fi l ter ing and Data Truncation 172 5.3.2 The Effect of Scatterer Location 173 5.3.3 The Effect of the Number of Views 177 5.3.4 The Effect of Cartesian G r i d Size in Fourier Space 178 5.3.5 The Effect of Sample Rate 181 5.3.6 The Effect of Dispersion 181 5.3.7 Reconstruction of a Square Block 186 5.4 Effect of Attenuation and the Approximate Correction 189 5.5 Comparison of Fourier and Wavelet Denoising Methods 194 5.5.1 Reconstruction with Noise-free and Noisy Da ta 196 5.5.2 Prel iminary Results of Noise Fil ter ing 198 5.6 Experimental Tests of the Method by Blackledge et al 204 5.6.1 Temperature Dependent Speed of Sound 204 5.6.2 Introducing A General Source 204 5.6.3 Experimental Work wi th Tissue Phantoms 211 5.7 Tests of the A R T in Computer Simulation 225 5.7.1 Choice of Iterative Method 227 5.7.2 Attenuation Correction and Choice of x ° 228 5.7.3 Reconstruction of Points 234 5.7.4 Reconstruction of a Square Block 239 6 Conclusions and Future Work 241 6.1 General Remarks 241 6.2 K e y Future Work 243 6.2.1 Apparatus Improvements 243 6.2.2 Improved Digi ta l Signal Processing 245 6.2.3 Improvement of Method by Blackledge et al 245 6.2.4 Improvement of the Algebraic Reconstruction Technique 248 6.3 Addi t iona l Future Work . . . 249 6.3.1 Tests wi th Better Phantoms 249 6.3.2 Stacking of Planar Views to Create 3D Images 250 6.4 F i n a l Remarks 250 Bibliography 251 v i A Detailed Derivation of Method by Blackledge et al 258 A . l The Wave Equation 258 A . 2 Fourier Domain and Born Approximation 261 A . 3 Introducing the Pulsed Line Source 262 A . 4 Derivation of Equation 2.16 264 A . 5 Simplification of Equation A.38 267 A . 6 Substitution for rs and r^ 270 B Derivation of V S 276 B . l Derivation of VS(fc |r-r. , . | ) 276 B . 2 Derivation of VS(Jfe |r/-r |) 278 C Improved Derivation of Ps 280 C. l Nonattenuated Propagators Revisited 281 C.2 Attenuated Propagators Revisited 282 D Phantom Construction 284 E Iterative Methods 286 E . l Conjugate Gradient Method 286 E.2 Jacobi Iterative Technique 288 E.3 Gauss-Seidel Iterative Technique 289 E.4 S U R Iterative Technique 290 v i i L I S T O F F I G U R E S 1.1 Continuous Waves and Pulses 15 1.2 Simple Schematic of B-Scan Imaging 21 1.3 Schematic of 3D Ultrasound Imaging 24 1.4 Example of 3D Ultrasound Obstetric Image 25 1.5 Illustration of Tomographic Projections for Nondiffracting Sources . . . . 27 1.6 Illustration of Reflection Tomography Method 31 2.1 U C T Experiment Geometry 44 2.2 Illustrating the Projection Slice Theorem 48 2.3 Illustration of Projections in C W U C T 50 2.4 Illustration of Data Recorded at Opposite ipa 51 2.5 (u, v) Coordinates of ipe A long One Radia l Slice 52 2.6 Error in Approximate Green's Function Suggested by Blackledge et al . . 54 2.7 Comparison of the Exact and Approximate Green's Function Propagators 55 2.8 Simulated Incident Pulsed Fie ld 59 2.9 Pulse Propagation 60 2.10 Reduction of wv-Spatial Extent Due to Interpolation G r i d 63 2.11 Illustration of Scatter that is Ignored wi th Small Image Sizes 66 2.12 Illustration of Equivalent Fi l ter ing 69 2.13 Illustration of Isochrones 71 2.14 Illustration of Several Isochrones 73 2.15 Fourier Da ta and Corresponding Image for 1 View 75 3.1 Schematic of Prototype Scanner 80 3.2 Schematic of S P R H - B - a Hydrophone 83 3.3 Closeup of Hydrophone T i p 84 3.4 Frequency Response of the SPRH-B-0500 Hydrophone 86 3.5 Characteristics of the Hydrophone 88 3.6 Illustration of the Scanner Traction Drive 91 3.7 Categories of Detected Sound 94 3.8 T O F Profile of Detected Signal 96 v i i i 3.9 Frequency Response of the S E A A D B 1 7 Preamplifier 99 3.10 Frequency Response of Pulser-Receiver 100 4.1 Effect of Approximate Attenuation Correction 106 4.2 Illustration of dw, dt, | V d _ | and \Vdt\ in Simulation I l l 4.3 G r i d for A R T Development 113 4.4 T ime Profile of Typica l Signals 118 4.5 Fourier Spectra of Typica l Noise Signals 120 4.6 Ampli tude Histograms of Typical Noise Signals 122 4.7 Ampl i tude Histogram of Noise after L P F 124 4.8 Ampli tude Histogram of Noise after L P F and H P F 125 4.9 Illustration of the Moving Average Fil ter 127 4.10 Examples of Wel l -Known Wavelet Functions 128 4.11 Diagram Illustrating Radia l G r i d Point Density 135 4.12 Radia l G r i d Point Density V s r 136 4.13 Illustration of Interpolation Errors 138 4.14 Comparison of Interpolation Methods 142 5.1 Q i , Q2 and Q3 as a Function of ^r 1 wi th no Attenuation 151 5.2 Q i as a Function of ^ with Attenuation 156 5.3 Q2 and Q3 as a Function of ^ wi th Attenuation 157 5.4 Effect of Approximate 5 r . 159 5.5 Error in T O F Along Diagonal 160 5.6 Error in T O F Due to Isochrone Warping 162 5.7 Example of a Scaled Image 170 5.8 Profile of 7 P A long the a;-axis 171 5.9 7 K Results for Cases 1 and 3 174 5.10 Effect of Scatterer Location 176 5.11 Effect of Number of Views 177 5.12 Effect of Number of Cartesian G r i d Points 180 5.13 Normalization Factor Versus Sampling Frequency 182 5.14 Effect of Dispersion on Scatter Pulse Shape 183 5.15 Effect of Dispersion on Image Reconstruction 185 5.16 Reconstruction of 7* for a Square Block 187 5.17 Reconstruction of jp for a Square Block 188 5.18 Effect of Attenuation for a Single Point 190 5.19 G r i d Reconstruction in the Absence of Attenuation 192 5.20 Effect of Scaled Attenuation in Tissue 193 5.21 G r i d Reconstruction in the Absence of Attenuation 195 5.22 Effect of Scatterer Location 196 5.23 Noise-free and Noisy Reconstructions 197 ix 5.24 Daubechies 9 and Moving Average Fil ter ing with L P F / 0 = 2.6 M H z . . 199 5.25 Non-Smoothed and Smoothed Wavelet Filtered Data 200 5.26 Comparison of Moving Average, Sym8 and Daub9 Fi l ter ing 202 5.27 Speed of Sound Versus Water Temperature 205 5.28 Propagated Versus Measured Pulse at Point B 210 5.29 Comparison of (pinu)bs and (pinitjss 2 1 2 5.30 Apparatus for Measuring Speed of Sound in G G - T E M 213 5.31 Density of Water Versus Temperature 215 5.32 Scatter Da ta for a G G - T E M Phantom 217 5.33 Daub20 Denoising 219 5.34 R and 7« for a G G - T E M Phantom 221 5.35 Effect of a on 7« 223 5.36 Analysis of a Line Source 224 5.37 A R T Result for a G r i d with Custom x° 231 5.38 A R T Result for a G r i d in the Absence of Attenuation 232 5.39 A R T Result for Point at (0,0) m m 235 5.40 A R T Result for Point at (-0.7,0) m m 238 5.41 A R T Result for a Square Block 240 A . l U C T Experiment Geometry 266 x L I S T O F T A B L E S 1.1 Statistics Regarding Breast Cancer 7 1.2 Age-Related Five-Year Survival Rates for Women with Breast Cancer . . 8 1.3 Differences between Ultrasound and X-rays 12 1.4 Speed of Sound in Some Biological Materials 16 2.1 Sample Theoretical Image Resolutions 62 2.2 Example Image Sizes 65 2.3 Ampli tude Attenuation in Water 77 4.1 Error in Scattering Ampli tude with No Attenuation Correction 105 5.1 Representative Data in the Graph of Q i vs ^ wi th No Attenuation . . . 152 5.2 7 P and jK Values for Different Tissues 152 5.3 Representative Data in the Graph of Q i vs ^ with Attenuation 158 5.4 Power Sum Residuals wi th L P F /„ = 2.6 M H z 201 5.5 Power Sum Residuals with L P F /„ = 6.6 M H z 203 5.6 Power Sum Residuals with L P F / 0 = 9.0 M H z 203 5.7 Comparison of Constant and Varied x° 230 5.8 Background and £ Values - A R T for Point at the Origin 236 5.9 P ixe l Average Over 0.19 m m Disc for a Square Block using the A R T . . . 239 x i A C K N O W L E D G E M E N T S The completion of a P h . D . thesis project is a multifaceted endeavor that is touched by many people. From providing technical support to offering encouragement, numerous individuals and organizations have influenced my work and are deserving of my thanks. I would like to thank my supervisor, Dr . Ca lum MacAulay, for his technical advice and faith in my work. I feel that the experience of being his graduate student has shaped me into a confident professional capable of independent analysis and project management. M y thanks goes out to M r . Jerry Posakony for his technical support and encouragement. I would also like to thank Drs. Frank Podd and Inaki Schlaberg for providing me wi th their specialized line sources. M y committee members, Drs. A lex Mackay, Branko Palcic, Doug Beder, and Greg Smith, also deserve thanks for their time and expert advice. In particular, I 'd like to acknowledge Greg for his encouragement throughout the years. A s well, I would like to thank our summer student Brock Wilson for doing a great job. M o r a l support has been a crucial ingredient of my work, and for this I would like to thank my friends and extended family, my mother Edna, my father David , and my brother Jeff. In particular my mother's continual encouragement has been key in leading me to where I am today. I would like to thank my husband Mohammad Kermani , who provided a sounding board for me and encouraged me throughout the years. I would like to acknowledge Dr . Jeff Young for making my years of teaching lab classes at U B C an enjoyable learning experience. I would also like to thank M s . Frances Lasser for mentoring me in my entrepreneurial business assistantship in the Technology Development Office of the B C Cancer Agency. This project has received the generous support of several organizations. I have been fortunate to be a Research Student of the National Cancer Institute of Canada supported wi th funds provided by the Terry Fox Run . Throughout my doctoral training, I was also supported by funds provided by the National Science and Engineering Research Counci l of Canada and the University of Br i t i sh Columbia. Addi t iona l funding for this project was kindly provided by both the Br i t i sh Columbia Science Counci l and the Canadian Breast Cancer Foundation. Travel funds for the presentations of results were provided by the B C Cancer Foundation, the National Cancer Institute of Canada, and the Ultrasonics International '99 Symposium. x i i G L O S S A R Y V a r i a b l e s r , the positional vector corresponding to any scatter point Tj, the positional vector corresponding to either the source or the detector rs, the positional vector corresponding to the source rd, the positional vector corresponding to the detector hj, the unit vector corresponding to Tj ns, the unit vector corresponding to r s n,*, the unit vector corresponding to rd x, unit vector along the x-axis y , unit vector along the y-axis (u, v), coordinates in the Fourier space of the image r , radi i of the data circles in the Fourier space of an image TVnaz, the maximum distance from the origin of data in the Fourier domain of the image r 0 , r-dependent lowpass filter cutoff TT, truncation l imit in terms of r X, the angles of the data rays in the Fourier space of an image Tmax, the maximum radius of the data circles in the Fourier space of an image r , radius, where applicable ts, time of flight of ultrasound from source to scatter point td, time of flight of ultrasound from scatter point to detector a, distance from the origin to either the source or detector c, general term for the speed of sound cw, the speed of sound in water Ct, the speed of sound in tissue / , the frequency of sound fNyq, the Nyquist frequency fD, the frequency at which ultrasound data is digitized fs, sampling frequency, equal to fD fi, lower band l imit frequency f2, upper band l imit frequency Hi, 2^, the angular frequency band limits of the incident field x i i i A(u}), the frequency spectrum of the incident field wo, central angular frequency of Ricker wavelet /o, central frequency of Ricker wavelet A, the wavelength of sound u, the angular frequency of sound = 2nf k, the wave number of sound, which equals j k, the angular wave number of sound = ^ ri, index of refraction vc, amplitude of cosine component of narrow band signal vs, amplitude of sine component of narrow band signal Po(r, u)), the incident insonifying field, as a function of position and UJ pa, the scattered field wi th simplifying assumptions pes, exact scattered field p(r, t), the total ultrasound pressure field expressed a function of both position and time p(r, OJ), the total ultrasound pressure field expressed a function of position and OJ HQ, the Hankel function of the first k ind YT, the Ricker wavelet <7(r|r-j, k), 2D Green's function describing sound propagation as a function of r and k g, the Green's function multiplied by attenuation terms S, the r and k dependent term in the approximate 2D Green's function propagator 5 , the function S multiplied by attenuation terms a, multiplier term in the approximate 2D Green's function propagator a, incorrect a parameter from the paper by Blackledge et al a and ft, where applicable, coefficients in the approximate attenuation correction ipeiVsik), projection data in the image Fourier space for a given source angle, ips Nv, number of projections or views K0, the compressibility of water po, the density of water K(T), the compressibility of the object being imaged, as a function of position p{r), the density of the object being imaged, as a function of position 7 K (r), the compressibility gamma function 7 P (r) , the density gamma function / , the image being reconstructed x iv R, reflectivity function $ , multiplier term in front of jp in the image function 9, the relative angle between the incident beam direction and the detector angle <ps, the angle of the source (pa, the angle of the detector $'phase, Fourier phase angle £, image normalization factor PR, percentage ratio matrix e, the error in an iterative technique ATTW, attenuation due to water A T T t , attenuation due to tissue Xt, coefficient in A T T t Xw, coefficient in A T T ^ dt, the distance that a particular ultrasound wave travels through tissue dw, the distance that a particular ultrasound wave travels through water dt, the average distance that al l ultrasound waves travel through tissue dw, the average distance that al l ultrasound waves travel through water Z, acoustic impedance P<?(t), projection data at some angle, 9 t, independent variable along the projection i(x, y), tissue object function in general tomography experiment j3, received echo amplitude in reflection tomography CR, intensity reflection coefficient M<2, mean of the power in the data Mn, mean of the power in the noise signal B, bulk modulus V, volume p, pressure Tg, glass transition temperature T c , Curie temperature Vp, dc poling voltage q, the angle of incidence of a sound wave on a transducer face relative to the normal (j), the irradiation angle in x-ray tomography xv d, density of points in the Fourier domain of the image A c r o n y m s A D C , analogue-to-digital converter A R T , algebraic reconstruction technique B F , background field C T , computed tomography C W , continuous wave D F T , Discrete Fourier Transform D T , diffraction tomography F O V , field of view F F T , Fast Fourier Transform F T , Fourier Transform F W H M , full-width-half-maximum H V L , half-value layer I F T , Inverse Fourier Transform L P F , lowpass filter M R I , magnetic resonance imaging P S F , point spread function P Z T , Lead-Zirconate-Titanate R T , reflection tomography S D , scatter data SS, secondary scatter S A , scattering amplitude S A o p p P , scattering amplitudes corrected for attenuation wi th the approximate approach S A B , scattering amplitudes not corrected for attenuation SAR, scattering amplitudes corrected rigorously through path tracing S N R , signal-to-noise ratio T O F , time of flight T S , scatter off the tank U C T , ultrasound computed tomography x v i U F R , unified frequency domain reconstruction Definitions Thermography A breast thermogram is a pictorial representation of the infrared radiation of the skin over the breast, motivated by the fact that breast cancers can elevate the skin tempera-ture of the affected breast. MRI M R I makes use of the magnetic properties of protons in hydrogen. The protons act like magnetic dipoles, which align with strong magnetic fields. Appl ica t ion of a radio fre-quency (RF) pulse destroys this alignment. Upon cessation of the R F pulse, the protons realign wi th the magnetic field and an R F signal is emitted, which is used to create the M R image. Transillumination Visual izat ion of the translucency of the breast, obtained by transmission of infrared light through the tissue. Benign and malignant diseases can lead to changes in shades of colors and to distortions of vascular patterns. Electrical Impedance Imaging This form of imaging seeks to provide an image of the electrical impedance (resistance to the transmittance of faint electrical signals) of tissue. x v i i C H A P T E R 1 I N T R O D U C T I O N 1.1 A I M O F THIS W O R K - H Y P O T H E S I S A prototype scanner for pulsed ultrasound computed tomography ( U C T ) has been built and tested wi th the aim of developing new imaging techniques that hold potential for the improved early detection of breast cancer. This waterbath based system incorporates two reconstruction algorithms based on the solution of the Chernov wave equation subject to the Born approximation. The algorithms attempt to reconstruct quantitative 2D cross-sections of compressibility and density for use in tissue characterization. The term quantitative indicates that the brightness of any given pixel is directly proportional to the value at that location of the tissue parameter in question. The first algorithm produces a reconstruction through the inversion of data in the 2D Fourier domain of the image (Direct Fourier Method introduced by Blackledge et al). The second algorithm is an algebraic reconstruction technique ( A R T ) that produces images based on the iterative solution of a matr ix system built upon knowledge of the scattered field and ultrasound propagation. In order to be of potential use in the early detection of breast cancer, an imaging system must reconstruct quantitative images with resolution on the order of 2 m m [69]. This means that the system must be able to detect two separate point objects that are 2 m m apart. It was thus hypothesized in this thesis that the reconstruction algorithms incorporated in the pulsed U C T scanner could be demonstrated both in simulation and experimentally to have image resolution of at least 2 mm. It was hypothesized that the resultant 7* and 7 P images would indeed be quantitative. Furthermore, given the non-1 Chapter 1. Introduction 2 periodic nature of pulsed U C T data, it was hypothesized that digital signal processing (DSP) techniques from the relatively young field of wavelet denoising could be applied to improve the data upon which the image reconstruction is based. The overall aim of this thesis was to determine if indeed these hypotheses hold for the techniques and algorithms incorporated into the prototype pulsed U C T scanner. This thesis was furthermore a proof of concept study to identify the future development that is necessary for the creation of a second generation scanner that can be evaluated in a clinical setting. It should be noted that image resolution is only one piece of the puzzle wi th respect to the evaluation of ultrasound scanners. A complete system development program, which is beyond the scope of this thesis, involves system evaluation in terms of potential lesion detectability, which is a complex issue affected by contrast, clutter and noise, as well as resolution. Noise refers to random processes that arise from thermal and electronic noise in the transducer and the electronics of the system. Clut ter is also termed structural noise. This can take the form of either echoes from objects that are too small to be resolved by the system, which is termed speckle, or echoes from larger objects that are not of interest, such as vessel walls in the case of Doppler blood flow imaging and ribs in the case of breast imaging. Speckle imparts noise into the images, while echoes from larger structures can degrade image contrast i f they are indistinguishable from the signals of interest. This thesis concentrated on evaluating image resolution capabilities under carefully controlled conditions in both simulation and experiment. 1.2 T H E S I S H I G H L I G H T S This thesis outlines the study of two ultrasound computed tomography algorithms in a prototype U C T scanner. In addition, this work involved the study of methods for data interpolation, noise removal, and the iterative solution of linear systems, in support of the reconstruction algorithm development and testing. The prototype scanner Chapter 1. Introduction 3 employs pulsed ultrasound fields, which is a relatively novel approach both theoretically and experimentally. Fewer than five papers have been written to develop theoretical algorithms for pulsed U C T . W i t h regard to experimental scanner development, only one other group appears to be working in pulsed U C T , and this group is currently exploring non-destructive evaluation ( N D E ) in industry [78]. This first part of this thesis focusses on extensive theoretical tests of the method by Blackledge et al for the reconstruction of 2D functions of density and compressibility for non-attenuating objects using pulsed U C T [10]. Specifically the functions reconstructed are K 0 and po are the compressibility and density of the background fluid, which is generally water. Note that the 7K(r) definition differs from that of jp(r) in that the background pa-rameter, rather than the tissue parameter, is the denominator. These particular gamma functions in turn result from the solution of the wave equation to be discussed in Chapter 2. In reality, 7 K and the reflectivity function, R = jK — jp, are reconstructed, and ^ p is calculated from these images. In simulation, the method by Blackledge et al was tested as is for non-attenuating objects. It was also tested for attenuating objects upon the ad-dit ion of an approximate attenuation correction based on the average distance traveled by ultrasound through tissue and water in a given experiment. Prel iminary experimental tests were also performed to determine the potential viabil i ty of the approach. The second part of this thesis involves an attempt to develop a rigorous reconstruction algorithm that includes a correction for frequency-dependent attenuation, which is a major stumbling block in pulsed U C T . The algorithm developed includes what is herein called an "attenuated propagator," which is a function that describes the propagation of ultrasound in the presence of attenuation due to water and tissue. This propagator was P(r) ~ Po (1.1) Chapter 1. Introduction 4 incorporated into an algebraic reconstruction technique ( A R T ) intended to determine 7«(r) and 7 p (r) of the object being investigated. The method was found to be very computationally intensive, as most A R T routines are. Regarding prototype system design and results, the following key results of simula-tions and experiments were obtained: • In experimental tests, the method by Blackledge et al was dramatically effected by errors as low as 1-2% in the measurement of the distance from the source or detector to the origin, given by \TJ\. • The method by Blackledge et al was able to reconstruct a 3 m m wide tissue phantom with negative 7«, and R. The resulting jK and R cross-sections indicated object functions that were primarily negative. The F W H M of the jK image was 2 . 9 4 ± 0 . 0 6 mm, while that for the R image was 1.94 ± 0.04 mm. • Wavelet denoising with the Daubechies 9 and Daubechies 20 wavelet families offered no improvement of data compared to lowpass filtering. • In simulation, the method by Blackledge et al produced narrow, Gaussian yK and jp point-spread-functions (PSF's) for points in the absence of attenuation whose dis-tance from the origin, | r | , was less than 5% of | r j | . The maximum image resolution for this l imited case was approximately 0.07 mm. • This method unfortunately cannot produce proper P S F ' s for points wi th > 5%. It is also not possible to add a rigorous attenuating correction to this algorithm, and the approximate correction failed. • In simulation, the algebraic reconstruction technique produced consistently shaped, narrow jK and jp P S F ' s for points with varying ^ (tested up to 30%) in both the absence and presence of attenuation. The maximum image resolution was approximately 0.05 mm. • The A R T is capable in simulation of rigorously correcting for attenuation such that images suffer a change in power of less than 0.15% when comparing results in the absence and presence of attenuation. • The algebraic reconstruction technique has difficulty converging on an image solu-tion for jK and 7 P cross-sections of more than 20 x 20 pixels. These results together wi th others outlined in Chapter 5 provide the basis for additional research discussed in Chapter 6, primarily in the refinement of iterative methods. Chapter 1. Introduction 5 1.3 O R G A N I Z A T I O N O F THIS T H E S I S This thesis is composed of six chapters. The first chapter includes background infor-mation that describes the medical motivation behind this thesis. It also describes current techniques in ultrasound imaging. Computed tomography is explained in brief and the chapter concludes wi th an outline of developments in ultrasound tomography, together wi th the advantages and disadvantages of the different U C T methods. Chapter 2 provides extensive theoretical background. The first part of the thesis explains the motivation behind the use of pulsed ultrasound and describes the physics model that is the basis behind both the method by Blackledge et al as well as the A R T . The remainder of the chapter outlines pertinent details of the derivation of the algorithm by Blackledge et al and describes the effect of attenuation in the tissue-water system. Chapter 3 provides details of the prototype scanner that was built for experimental U C T tests. The different sources and the hydrophone detector are described, along wi th the stepper motor apparatus that was designed to move the sources and hydrophone automatically in very small steps. Principles behind the design of the water tank are outlined. The chapter concludes wi th a description of the electronics chosen for the prototype scanner. The fourth chapter describes various experimental methods that were incorporated into this work. The chapter begins with a development of the approximate attenuation correction as well as the A R T with attenuated propagators. Chapter 4 continues wi th a discussion of necessary noise removal and digital signal processing (DSP) techniques based on both conventional Fourier methods as well as wavelet analysis. The chapter ends wi th a comparison of various data interpolation methods for potential use in the method by Blackledge et al. Chapter 5 presents numerous results of jK and jp image reconstruction using the algorithms outlined in this thesis. The method by Blackledge et al was extensively tested in computer simulation to determine its capabilities. Various situations were examined, Chapter 1. Introduction 6 including the presence or absence of frequency-dependent attenuation and dispersion and the presence or absence of noise. Bo th Fourier-based and wavelet-based denoising techniques were studied. A tissue phantom experiment was also performed using the method by Blackledge et al to gain additional understanding of its potential. Chapter 5 also includes several key preliminary studies of the A R T in simulation. The image resolution and quantitative capabilities of the algorithm were studied. A s well, tests were done to determine the abili ty of the algorithm to reconstruct points wi th varying locations and in both the absence and presence of attenuation. Chapter 6 summarizes the findings of this thesis and discusses possibilities for future work. The thesis concludes wi th five appendices. The first appendix includes a detailed derivation of the Blackledge method, which augments both Chapter 2 as well as the outline contained in Reference [10]. Appendix B adds to this wi th a derivation of VS, an approximate term that is used in the mathematical description of ultrasound wave propagation. The third appendix derives a more rigorous expression for VS in both non-attenuating and attenuating media. Appendix D outlines phantom construction, and Appendix E discusses iterative algorithms for potential use in the algebraic reconstruction technique. 1.4 M E D I C A L M O T I V A T I O N Breast cancer is a widespread disease that afflicts women worldwide. In a 1985 survey of 18 cancer types in 24 regions of the world, breast cancer was shown to be the most common malignancy in the women, accounting for 19.1% of al l such cancers [67]. Today in Canada and the United States, breast cancer is the most common cancer in women and the second leading cause of cancer death among women [65, 63, 76]. Table 1.1 illustrates that tens of thousands of women in North America are afflicted wi th invasive breast cancer each year, and that thousands more die each year as a result. More than 1 in 10 women who live to age 80 w i l l develop the disease, and more than 24% w i l l die Chapter 1. Introduction 7 Country Estimated New Cases in 1999 Estimated Deaths in 1999 U S Canada 175000 18700 43300 5400 Table 1.1: The above data illustrate the statistics for breast cancer in Nor th American women. Listed are the 1999 Canadian and US incidence and mortality estimates [1, 63]. as a result [1, 65, 63]. It is also the number one killer of Nor th American women aged 35-55, resulting in the most lost years of life among this group [65, 63]. The incidence of breast cancer is expected to increase as the world population ages, wi th estimates of more than one mil l ion new cases worldwide per annum by the year 2000 [59]. It is of interest to note that although the disease is primarily a women's affliction, there are in fact approximately 1300 new cases of breast cancer among men annually in the U S , and a further 400 deaths [5]. Most women who develop breast cancer have no risk factors to aid in early detection and treatment [63]. A s such, the best approach towards controlling this disease is through the development of new technologies for improved early detection and diagnosis to allow for earlier treatment [31]. Detection is the ability to find suspicious breast features, while diagnosis is the abili ty to distinguish which features indicate malignancy. In recent years, early detection has proven key in increasing the survival rates of women diagnosed wi th breast cancer. Between 1950 and the late 1980's, the overall breast cancer mortality was relatively stable and over 40% [1]. However, since 1989 the death rate has decreased an average of 1.8% per year due to earlier detection and diagnosis and improved treatments [1]. Table 1.2 outlines the current age-related five-year survival rates for U S women. It is alarming that the survival rate for women decreases as the age at cancer diagnosis increases. Researchers speculate the reason to be that younger women may have tumors that are more aggressive and less responsive to hormonal therapies. It is also alarming that regardless of age, the five-year survival rate is no better than 86%. However, as Chapter 1. Introduction 8 Age (years) Percentage of US Women (%) < 45 81 45-64 85 > 65 86 Table 1.2: The above data illustrate the age-related five-year survival rates in the U S for women diagnosed wi th breast cancer [1, 63]. more breast cancers are diagnosed at earlier stages, the death rates are expected to decline. A s such, scientists and engineers continually work to develop new technologies for improved accuracy and reliability in breast cancer detection. The common goal is to develop systems that produce high resolution images of breast structural features as well as quantitative parameters that correlate to the pathological state of the tissue. X-ray mammography is currently the only imaging technique routinely used in breast cancer screening. It has had much success in early breast cancer detection, and indeed has proven more useful than any other breast imaging technique to date. There is recent evidence that regular mammographic screening may reduce the chances of dying from breast cancer by 17% for women in their forties, and by 30% for women between the ages of 50 and 69. However, even though the field of mammography is continually advancing, the imaging technique continues to suffer from the difficulty of maximizing both test sensitivity and specificity. High sensitivity means that a test for a disease can identify nearly a l l afflicted patients, resulting in few false negative results. If the test also has high specificity, it is further able to properly exclude patients that do not have the disease; thus high specificity results in few false positive results. There remains a significant rate of false-negatives and false-positives in x-ray mam-mography. W i t h regard to sensitivity, several examples are quoted in the literature. For instance, approximately 25% of breast tumors in women in their forties evade de-tection by mammography [63]. There is also a high false-negative rate among woman less than age thirty-five [7, 61]. The poor sensitivity is mostly due to poor imaging of Chapter 1. Introduction 9 dense tissue by mammography and failure to recognize subtle early signs of breast abnor-mality [1, 63]. The statistics regarding specificity indicate significantly more problems. Screening mammography cannot yet provide a definite diagnosis, and every person wi th a positive screening test must undergo a biopsy to determine i f in fact cancer exists. The biopsy is currently the "gold standard" in breast cancer diagnosis. However, studies have indicated that 68-87% of biopsies yield negative results [52]. Part of the problem is that mammograms poorly differentiate solid tumors from benign cysts, which occur in more than 50% of women [7, 61]. The high rate of unnecessary biopsies not only causes undue stress and discomfort for millions of women the world over, but it also presents a significant drain of health care dollars. To illustrate this point, consider the example of a country wi th 25 mil l ion women of the age at which screening is recommended. Assume that 50% of these women actually have an x-ray screening examination. If 5% of the mammograms performed annually are positive, breast lesions in an estimated 625 thousand women would require further evaluation each year. This example further assumes a low estimate that 50% of these women undergo biopsy. O n average each procedure costs US$2000, which is calculated based on the costs of the conventional techniques of needle biopsy and surgical biopsy. The final estimated cost of biopsies annually would be US$625 mil l ion. Efforts to improve the specificity and sensitivity of breast cancer screening and thus eliminate the biopsy of benign breast lesions have led to several technological develop-ments. These include improved x-ray mammographic techniques and interpretation, ther-mography, transillumination, electrical impedance imaging, magnetic resonance imaging ( M R I ) , conventional ultrasound, and 3D ultrasound. Brief descriptions of these meth-ods can be found in the Glossary, and further information can be found in Reference [28]. Concurrent wi th this has also been the development of new micro-biopsy proce-dures such as fine needle aspiration and computer-directed cutt ing needle biopsy, also described in the Glossary. These technologies, however, s t i l l involve extensive use of * Chapter 1. Introduction 10 hospital resources, as well as stress and discomfort to patients. 1.5 ULTRASOUND AS AN IMAGING PROBE This thesis investigates the potential for a new ultrasound computed tomography method that employs pulsed ultrasound fields. Ultrasound imaging is a useful and non-invasive technique in which high frequency sound (1-10 M H z ) is transmitted through the body. The first use of diagnostic ultrasound was reported in the 1940's by Dr . K a r l Dussik, a psychiatrist who was at the hospital in B a d Ischl, Aus t r ia . He attempted to locate brain tumors by using two transducers, one to insonify the tissue and the other to detect the transmitted sound waves and hence measure the transmission of ultrasound through the head. Routine use of diagnostic ultrasound imaging began in the 1950's. It was during this time that academic interest began in the field of breast sonography. Notable studies were those by W i l d , Neal and Reid, who used pulse-echo techniques to study benign and malignant breast diseases, and those by Howry, Bliss and Scott, who employed a 2 M H z compound scanning system to study the breast [41, 42, 84, 85, 86]. Currently ultrasound has many uses in medicine, with primary emphasis on obstet-rics and women's healthcare (invitro fertilization, guided breast biopsy, gynecology). Applicat ions in obstetrics allow for the investigation of fetal development. Today nearly every fetus undergoes examination wi th ultrasound at least once. The uses of ultrasound in obstetrics range widely, from determining pregnancy date by baby size and detecting multiple fetuses, to detecting certain defects, such as those related to the heart. A major application of ultrasound in obstetrics is biometry, which is the measurement of various fetal dimensions. Ultrasound imaging in gynecology has been made possible by the development of intravaginal imaging, which uses an endocavitary transducer probe. Applicat ions in gynecology include the monitoring of the dynamics of follicle growth, endometrium de-velopment, and fibroid growth. Endocavitary probes are also used in other fields of Chapter 1. Introduction 11 medicine. For instance, intrarectal ultrasonography of the prostate is used to diagnose benign prosthetic hypertrophy and prostate malignancies. The combination of laparo-scopic ultrasonography and laparoscopy also allows urologists and surgeons to perform direct biopsies more carefully under ultrasonographic guidance. In cardiology, ultrasound is used for the real-time dynamic imaging of the movement of cardiac structures (heart valves and ventricular walls) during the cardiac cycle. It is possible to monitor arterial diameters and the displacement of arterial walls during the cardiac cycle, which ultimately provides tissue characterization information regarding arterial wall stiffness and elasticity. Doppler imaging is an important application of ultrasound that is used for the non-invasive measurement of blood flow. The Doppler effect is a change in the wavelength of scattered ultrasound that is dependent upon the movement of scatterers relative to the transducer probe. The resulting change in wavelength is used to analyze blood flow. The most important recent development in Doppler ultrasound is the advent of color flow imaging which indicates flow direction as well as rate. Flows directed towards the probe are presented in shades of red, while those directed away from the probe are seen in shades of blue. Attempts have been made to develop transcranial Doppler for the imaging of brain arteries, but M R I remains superior for brain imaging. Regarding health effects, there have been no reports l inking i l l effects wi th ultrasound exposures well below those used in ablation procedures. The peak intensity of the u l -trasound field used for ablation is on the order of 1 k W / c m 2 , whereas the field intensity used in imaging is generally on the order of 10 m W / c m 2 . The official statement of the American Institute of Ultrasound in Medicine ( A I U M ) , included in their statement on Safety in Training and Research that was approved March 1993, is that during the history of the use of clinical ultrasound there have been no confirmed reports of adverse biolog-ical effects on patients resulting from insonification wi th field intensities less than 100 m W / c m 2 [2]. Furthermore, the A I U M makes the additional statement that insonification Chapter 1. Introduction 12 Diagnostic Ultrasound X-rays wave type longitudinal mechanical waves electromagnetic waves transmission media elastic medium none required wavelength in tissue ~ 1 . 5 x l 0 " 3 - 7.5 x l(r 5 ) m ~3x( io - 9 - nr 1 1 ) m resolution ~ 0.15 - 0.75 m m ~ 100 fj,m mode of generation stressing the medium accelerating electric charges velocity in tissue depends on the medium order of 1.5 mm/fxs depends on the medium order of ~ ± 2 .998x lO 8 m/ s n = index of refraction (~1.33) similar waves seismic, acoustic radio, light, microwaves Table 1.3: Differences between Ultrasound and X-rays. in the low M H z frequency range for up to 500 seconds is safe at even higher intensities when the energy per unit area is less than 50 J / c m 2 [51]. For comparison, the ultrasound fields used in the experimental work presented in this thesis had a corresponding energy per unit area of 50 / / J / c m 2 . Ultrasound energy is very different from x-ray energy, as can be seen in Table 1.3. In particular, the wavelength of ultrasound is several orders of magnitude larger that than of the x-rays used in screening mammography. A consideration of wave theory alone indicates that the resolution is approximately 0.5A, where A is the wavelength of the ultrasound or x-rays. However, ultrasound imaging is l imited to resolutions on the order of 0.15-0.75 m m due to scattering, refraction and attenuation. The most current technologies i n digi tal x-ray mammography have a resolution l imit on the order of 100 / /m due to x-ray detector capabilities. However, although x-ray mammography can resolve much smaller objects than ultrasound, 0.15-0.75 m m resolution is sufficient and useful for the detection of early breast cancer [46, 69]. Indeed, i t is difficult to biopsy questionable breast masses that are smaller than 2 m m in diameter [69]. Thus, although ultrasound imaging cannot image microscopic features such as calcifications, it s t i l l has the potential Chapter 1. Introduction 13 to find lesions at an early enough stage to be of use in breast cancer detection. Ultrasound does have some advantages over x-ray mammography. In particular, mam-mography cannot distinguish cysts from other breast diseases, while ultrasound can do so wi th accuracy rates of better than 96% [7, 61]. Furthermore, quantitative ultrasound imaging, such as tomography, can provide not only structural/density information, as does mammography, but also information regarding tissue compressibility and angular scattering cross-section. Measurements of angular scattering cross-section, also termed directivity, indicate how much ultrasound is scattered into various directions from a par-ticular region of tissue, which may be useful for tissue characterization. Knowledge of tissue compressibility in particular may be useful for the early detection of breast cancer. Palpat ion is routinely used for the detection of breast tumors close to the skin surface. Hence, compressibility imaging is a natural extension of the physical examination of pal-pable lesions. Palpat ion results are dependent upon the elastic properties of tissue, and the only imaging modality that makes use of elastic waves as a probe is ultrasound. Thus, ultrasound can be used to investigate compressibility, while imaging modalities that use ionizing radiation or magnetic fields cannot. The use of ultrasound in screening programs is hampered by poor image resolution due to noise in the data and the complex physics of the tissue/sound interaction. How-ever, given the resolution that can ultimately be achieved wi th continued research, it is certainly possible that ultrasound imaging, and in particular U C T , w i l l be able to detect early cancers on the order of | m m in diameter. 1.6 S O M E BASIC SOUND AND ULTRASOUND CONCEPTS Sound is defined as a periodic disturbance in either the density of a fluid or in the elastic strain of a solid. The disturbance is generated by a vibrat ing object, and it results in molecular motion within the medium. Sound waves are longitudinal, which means that the molecular motion is in the same direction as the propagation of wave energy. This Chapter 1. Introduction 14 is in contrast to transverse waves, such as water waves, for which molecular motion is mainly perpendicular to the propagation of wave energy. Ultrasound is merely sound wi th a frequency, / , greater than 20 K H z , which is about the upper l imit of human hearing. Often the amplitude of a sound wave throughout space is referred to as a field. Sound can be described as either continuous wave ( C W ) or pulsed. C W simply means that the sound wave is made up of mostly one frequency. Hence the shape of the disturbance field is sinusoidal, as illustrated in Figure 1.1.A. In contrast, a pulse is a single, transient disturbance that travels through space. A n example of a pulse is illustrated in Figure l . l . B . Sound waves have a wavelength, A, and a speed, c, which are related through frequency by the equation c = / A (1.2) A sound wave can also be described by its wave number k, which is the number of complete wavelengths that can fit into a unit distance. This parameter is given by ~k = \ (1-3) Alternatively, the angular wave number, k, is often referred to, where The speed of sound, c, is dependent upon both the inertial property and the elastic property of the propagation medium. Specifically, the inertial property is density and the elastic property is the bulk modulus, B, and the speed of sound can be expressed as m (1-5) P The bulk modulus is the parameter that describes the extent to which an element of the medium changes in volume, V, as the pressure, p, applied to that element is changed. The bulk modulus is given by " A V ' V ™ Chapter 1. Introduction 15 Example of C W Ultrasound 2, . . , , 1.5 -1.5 28 29 30 31 32 X (mm) (A) Example of a Pulse 40 45 50 55 60 x (mm) (B) Figure 1.1: Figure A illustrates an example of continuous wave ultrasound, while Figure B shows an example of a pulse. Chapter 1. Introduction 16 M a t e r i a l S p e e d o f S o u n d ( m m / j K s ) A i r 0.33 Fat 1.46 Water (20°C) 1.48 Average Soft Tissue 1.54 Bra in 1.51 Liver 1.56 Kidney 1.56 Blood 1.57 Muscle 1.57 Bone 4.1 Table 1.4: Speed of sound in some biological materials [43, 51]. Equat ion 1.5 makes intuitive sense. The elastic property determines the abili ty of the propagation medium to store potential energy. This is important because as the sound wave passes through the medium, potential energy is alternately stored and released as the medium undergoes rarefaction and compression. If the medium is able to store more potential energy, there wi l l be a greater restoring force acting on its molecules, and the wave wi l l travel faster. The inertial property determines the abili ty of the medium to store the kinetic energy of the sound wave. If a medium is more dense, there is more mass per unit volume. A given unit volume can therefore store more kinetic energy and have a lower velocity than a lighter unit volume in a less dense medium, resulting in a lower speed of sound. In this thesis, the compressibility is considered rather than the bulk modulus. This is simply the inverse of the bulk modulus, and hence the equation for c can alternatively be written as c = - L (1.7) Table 1.4 illustrates the speed of sound in various media of different density and com-pressibility. Evidently, c varies by up to 8% throughout different soft tissues of the body. Chapter 1. Introduction 17 The speed of sound is also dependent upon temperature and frequency. Evidently, as the temperature rises, the density decreases and the speed of sound increases. Temperature change is not a problem with conventional diagnostic ultrasound, since body temperature is at a nearly constant value. However, tissue is often immersed in a water bath for U C T imaging. The temperature of the water must be maintained and monitored to ensure the quality of image reconstruction. The dependence of sound speed upon frequency is termed dispersion. This is generally not of concern in the clinical imaging of soft tissue, since dispersion is less than 1% over the frequency range of 1-20 M H z [43]. However, pulsed U C T analyzes data in a complex mathematical process that magnifies the error which results i f dispersion is not considered. This topic is discussed in detail in Section 5.3.6. The acoustic speed in a medium together with its density defines the acoustic impedance, Z, which is given by Z = pc (1.8) A change in acoustic impedance defines an interface, and ultrasound is reflected by in-terfaces wi th dimensions larger than the maximum wavelength in the source. The corre-sponding reflection coefficient, C R , that describes the reflected ultrasound beam intensity is the following: Zx cos(fl2) - Z2 cos(fli) 2 ( °* {Z1cos(62) + Z2cos(e1) ) ' ' ' where cos(#i) and cos(<92) are the angles of incidence and transmission relative to the normal vector of the interface. Evidently, larger differences in impedance result in more ultrasound reflection. A s an example, consider water, average soft tissue, and bone, which have acoustic impedances of 1 . 4 8 x l 0 - 6 , 1 .63x l0~ 6 and 7 . 8 x l 0 - 6 respectively [43]. The resulting intensity of the reflected ultrasound at a soft tissue/bone interface is 46% of the incident beam, while it is only 0.2% at an interface between soft tissue and water for normal incidence. Chapter 1. Introduction 18 1.7 C U R R E N T U S E O F U L T R A S O U N D IN B R E A S T C A N C E R I M A G I N G Due to physical constraints in most medical situations, it is often impossible to per-form ultrasound image reconstruction wi th sound energy that is scattered in directions other than directly back into the source transducer (termed pulse-echo or backscattered sound energy). For example, in ultrasound cardiovascular imaging, signals that are not directly backscattered are generally immeasurable due to very large acoustic impedance discontinuities at tissue-bone and air-tissue interfaces that result in significant scatter back into the body, as well as due to attenuation losses [47]. A s such, most medi-cal ultrasound imaging is done with information obtained from reflected sound energy. Breast tissue, however, lends itself to investigation from all angles and tends to have similar acoustic properties throughout its boundaries wi th relatively small differences at tissue interfaces [47]. A s such, breast cancer is the subject of research in novel imag-ing techniques such as ultrasound computed tomography. Commercially, however, the widespread availability of conventional ultrasound B-scanners has motivated the use of this imaging modality in breast imaging, primarily for guiding needle biopsies and for the differentiation of cysts from solid masses [33]. 1.7.1 B - S C A N I M A G I N G B-scan imaging is also referred to as pulse-echo imaging because only backscattered sound energy is used in the image reconstruction process. This type of imaging uses a simplified image reconstruction algorithm that is based on geometrical considerations only and which insonifies the tissue only at one angle for each image. This is in contrast to 3 D ultrasound, which reconstructs an image from data collected at various angles, and ultrasound computed tomography, which again uses several view angles and also incor-porates a detailed physics model of the sound-tissue interaction. Due to the simplified approach of B-scanning, the resulting images are generally incomplete and exhibit poor resolution [47]. For many years, however, this was the only type of ultrasound imag-Chapter 1. Introduction 19 ing that was commercially feasible. A s such, B-scanners have been widely accepted in healthcare practice. B-scan imaging does afford the flexibility to image a variety of tissues throughout the body. It is not necessary to gather information with transmitters and receivers positioned in a circle around the tissue. Rather, insonification and scatter detection are done from the same side as the instrument head is swept by the clinician, and a small beam of ultrasound insonifies the object. A n image is formed by displaying the backscattered signal as a function of both beam direction and sound time-of-flight ( T O F ) , or rather the time required for the sound energy to travel from the source, backscatter from the tissue and undergo detection. B-scan imaging provides qualitative tissue information and produces images that approximately delineate tissue interfaces. The patient is supine, and scans are recorded in different directions as the operator manually moves the ultrasound probe. Because only backscattered sound waves are recorded, image reconstruction is based on only a small portion of scattered sound. A s such, the images generally suffer from distortions and poor resolution compared to x-ray mammography and M R I [43]. B-scan image reconstruction attempts to determine an approximation to the reflec-t ivi ty function of the insonified object, which is defined by R = 7 « ( r ) - 7 „ ( r ) (1-10) Thus, i f the tissue compressibility and density change similarly in space, the image wi l l show no change. Note that the reflectivity function and the reflection coefficient are two entirely different, yet related, entities. First of a l l , R refers to scatter, while CR refers to reflection. Recall that reflection takes place when ultrasound insonifies acoustic impedance interfaces wi th dimensions greater than A. When the interface dimensions are smaller than A, scattering occurs. This redirected sound energy spreads out in al l directions about point scatterers in the object. Sound energy that happens to travel back to the source location from throughout the tissue is a measure of the reflectivity function Chapter 1. Introduction 20 of the object 1 . Similarly, ultrasound that is reflected from a large interface is a measure of the reflection coefficient of the interface. Essentially, reflected ultrasound is a subset of the full complement of sound that is backscattered from all parts of the tissue. If one were to integrate the backscatter that arises due to R for al l points along an interface curve only, CR could then be extracted. Due to significant support in industry for B-scanner development, the techniques have become very sophisticated over the years. Developments in transducer design, signal processing techniques and image analysis techniques have helped to raise conventional ultrasound scanning to what is generally a useful imaging modality. Furthermore, the extension of B-scanning to 3D imaging that has occurred in the past 5-7 years has vastly improved the images that are possible wi th reflection data. It is beyond the scope of this introduction to go into great detail regarding the theory behind B-scan imaging. Hence, only key elements w i l l be discussed and the reader is referred to References [29] and [45] for further information. A brief analysis of B-scan imaging is a useful introduction to wave mathematics and ultrasound principles. A s an example, consider an ultrasound beam that is l imited to a narrow region along a line wi thin the object being imaged, as illustrated in Figure 1.2. To simplify the explanation, attenuation due to tissue is not being considered here. The tissue gives rise to spherically-expanding waves from scatter points wi th in its boundaries. The insonification pulse is very short, and hence there is a direct relationship between when the reflected wave is detected and the distance, x, from which it originated in the object space. Using simple wave mathematics, the incident pulse can be written as where pt is the original pulse waveform and c is the average speed of sound in this exper-1The relation between backscatter and R is mathematically derived in Section 2.2.2 0 elsewhere Chapter 1. Introduction 21 Transducer Reflected field measured at transducer x=tc /2 Figure 1.2: This schematic illustrates the process by which B-scan imaging is performed. The ultrasound beam is l imited to a narrow region along a line, which is scanned across the tissue. iment. This equation describes a pulse traveling along the x-axis, which is perpendicular to the transducer face. A t an arbitrary point (x, y), some fraction of the incident field is scattered back to be detected by the transducer. The fraction of scatter is given by the reflectivity function, R, and the scatter field at the location (x, y = 0) wi l l be approximately given by tj>(x, y = 0)= R(x, y = 0)Pt(t - ^) (1.11) A s this scatter field, tp, travels back to the transducer, it w i l l be reduced in amplitude due to spherical spreading of the beam. It also acquires a further t ime delay of | to account for the additional travel time. Considering a 2D model of sound scatter, the Chapter 1. Introduction 22 energy in the field is decreased by a factor of - . Thus the field amplitude is decreased by a factor of and the final detected scatter field that is reflected from the location (x, y = 0) can be expressed as x 1 V>s(due to point x, y = 0) = R(x,y = 0) pt(t - 2-)-= (1.12) c \J x This term is then integrated over al l scatter points along the narrow beam line (the ar-axis), resulting in a total scattered field at any time t of tl>.(t) = [ R(x,y = 0)Pt(t-2-)^=dx (1.13) J c *\J x Further reduction of mathematics is possible i f the insonification pulse can be approxi-mated by an impulse, or delta function. The total scattered field can then be approxi-mated by iba(t) = / R(x, y = 0) 5{t - 2 - ) 4= dx (1.14) J C yJX M a k i n g a change of variable given by x = yields M) = lj R{^,y = Q)5{t-x)^=dx (1.15) V 2 = \/Ifl<f'* = 0) Evidently, in the case of impulse insonification there is a direct relation between the scatter field that is detected at time t and the tissue reflectivity function, R, at distance x = j . Substituting t = ^ yields an estimate of the reflectivity function given by R(x,y = 0) = ^ M ^ ) (1-16) c2 ' c Recall ing that the ultrasound field employed in B-scan imaging is a narrow beam that travels along a line, it is evident by Equation 1.16 that each position of the transducer maps out an estimate of the reflectivity function along the corresponding beam line. B y scanning the transducer across the object, a set of scan lines are built up to provide a Chapter 1. Introduction 23 picture of the total R. The resolution of the R image is dependent upon both the time duration of the incident pulse (range resolution) and the width of the ultrasound beam (lateral resolution). 1.7.2 3 D U L T R A S O U N D The basis behind this imaging modality is that B-scan images at various angles are placed together in a coregistration process to yield an image of the approximate reflec-t ivi ty function wi th higher resolution than in conventional B-scanning. In 3D ultrasound imaging, data are obtained by recording conventional 2D B-scans at various orientations, together wi th information about the position and orientation of the ultrasound scanning head for each corresponding scan. This general idea is illustrated in the schematic of F ig -ure 1.3, which illustrates 3 such B-scans being recorded at different orientations. Each 2D B-scan is acquired in the typical fashion with smooth sweeps of the probe, to yield a rel-atively planar view of the tissue. Mul t ip le sweeps are then compounded wi th knowledge of the position and orientation of the probe during each B-scan. A common coordinate system is maintained throughout the imaging process by connecting a transmitter to the ultrasound scanning head, which communicates the scan orientation to the ultrasound machine. Two examples of devices for this purpose are the Polhemus F a s t r a k ™ and the Ascension B i r d ™ sensor. Using data from non-parallel scan planes, tissue volume reconstructions can be created wi th algorithms for fitting surfaces to incomplete noisy interfaces. The resolution that is obtainable wi th current systems is on the order of 0.2 m m . A s is illustrated in the example in Figure 1.4, excellent images of tissue boundaries have been achieved v ia 3D ultrasound. Dur ing the past 5-10 years, the success of 3D imaging in academic research has moti-vated several imaging technology companies to develop commercial systems. Currently, these companies include: A T L , A L O K A , General Electric, Kre tz , Medison, Medison America , and Tomtec. However, although 3D ultrasound is gaining popularity for its clear Chapter 1. Introduction 24 Figure 1.3: Th i s schematic illustrates the general method- of acquiring 3D ultrasound data. pictures of gross tissue boundaries, it does not have the potential for use in quantitative imaging of tissue parameters such as density, compressibility, or scattering cross-section, a l l of which may be useful for early cancer detection. Rather, these backscatter images provide high resolution images of tissue reflectivity. Thus, 3D ultrasound cannot detect variations i n density and compressibility when these parameters are changing similarly in space. 1.8 C O M P U T E R I Z E D T O M O G R A P H I C I M A G I N G Tomography refers to the cross-sectional imaging of some physical parameter of an object using data that are collected by investigation of the object from various angles. Chapter 1. Introduction 25 Figure 1.4: This figure illustrates an excellent example of a 3D image of fetus at 32 weeks. This work was done by Dr . Bernard Benoit [9] In describing this type of imaging as cross-sectional, it is meant that 2D slices of a 3D object are reconstructed. Most medical imaging systems perform imaging in this way, and many bui ld up a 3D view of the object by stacking consecutive 2D slices [4, 47]. Da ta that is collected at each angle about the object is termed a projection. The data are acquired v ia two means: either through i l lumination wi th a probing field or through detection of emissions from a radioisotope administered to the patient. In the former situation, the probing field can take many forms, such as acoustic energy (ultrasound), electromagnetic energy (microwaves), ionizing radiation (x-rays), and magnetic energy (magnetic resonance imaging). When a probing field is employed, the data can be col-lected either through reflection of the field from the object or transmission of the field through it . The latter is accompanied by scattering. The term computerized is used to describe this type of imaging because the reconstruction of the image is based on a mathematical algorithm that combines the information from various angles to yield an Chapter 1. Introduction 26 image of some parameter. The method of reconstructing an object function from its projections was developed by Radon in 1917 [47]. However, use of the concept was minimal unti l the 1960's wi th . the invention of the x-ray C T scanner by Hounsfield. The advent of sophisticated recon-struction algorithms for x-ray C T quickly yielded images that were highly accurate in the sense that morphological details were unambiguous and in excellent agreement wi th anatomical features. The success of x-ray C T motivated the development of C T methods in nuclear medicine, magnetic resonance, ultrasound and microwaves. 1.8.1 D I F F R A C T I N G V E R S U S N O N D I F F R A C T I N G S O U R C E S X-ray C T , magnetic resonance imaging (MRI) and tomography wi th radioisotopes are a l l imaging modalities that employ nondiffracting energy sources. In other words, the probing or emitted energy travels in straight lines wi th very little scatter. The image reconstruction algorithms are based on models of straight line energy propagation. This is not the case wi th ultrasound and microwaves, which are classified as diffracting sources of energy. When an object is il luminated with a diffracting source, the energy waves are scattered in every direction from inhomogeneities that are much smaller than the wavelength of the waves. This is termed diffraction, and it must be incorporated into the model of tissue/energy interaction upon which the image reconstruction algorithms are based. In experiments wi th nondiffracting sources, any given data point in a tomographic projection at a particular angle is equal to the integral of the object parameter in ques-t ion along a line through the object. This type of imaging is also termed straight ray tomography. Examples of projections in straight ray tomography are illustrated in F i g -ure 1.5. The parallel projection is the simplest in concept, and Figure 1.5 illustrates how two projections of this type are built up for a typical tissue object function, i(x, y). The source is extended linearly, and the source field travels along the parallel arrow lines Chapter 1. Introduction 27 Figure 1.5: This figure illustrates how to construct two parallel tomographic projections for a nondiffracting source. Chapter 1. Introduction 28 shown. The integrals are performed along these lines to produce the projection data P*i(t) and P# 2(t) at angles 9X and 92, respectively. Similarly, other geometries can be used, such as fan beam, in which the source is a point instead of a linearly extended object. This is not the case with diffracting sources such as ultrasound and microwaves. The probing energy field does not travel along rays, and image projections do not represent integrals of an object function along straight line paths. The projections i n fact have an entirely different meaning, and the development of corresponding image reconstruction algorithms involves solving a wave equation, to be discussed in more detail in Section 2.2.1. 1.8.2 U L T R A S O U N D C O M P U T E D T O M O G R A P H Y It is desirable to produce highly resolved and accurate quantitative images of tissue using ultrasound, in which one can assign a direct correspondence between the brightness of the image pixels and the value of a particular tissue parameter. Ultrasound computed tomography has the potential to provide such images, and as such has become a growing field of research. U C T involves the automated scanning of tissue at many different angles using a computer-controlled setup. Thus, the breast is well suited for U C T as it allows inspection from all angles when the patient is lying prone. Various tomographic methods look at different aspects of the interacting ultrasound, such as how much sound is reflected or scattered and in what direction. Depending on the type of tomography performed, this imaging modality can provide information such as the variation of sound speed throughout the tissue, the compressibility of the tissue, and the amount of sound absorbed or scattered by the tissue. The goal is to use this information to detect cancer in the breast. A t the very least, U C T systems could potentially provide an adjunct to x-ray mam-mography. Al though the cost of building a system would not warrant its use simply in Chapter 1. Introduction 29 differentiating cysts from breast lesions, where conventional ultrasound scanners perform well, U C T could be of extensive use in tissue characterization for the early detection of breast cancer. This would have the effect of reducing the number of unnecessary and often disfiguring biopsies. Fewer women would have to wait to undergo a biopsy and to receive test results. The ultimate goal of U C T research, however, is the development of high resolution, noninvasive imaging systems for breast cancer screening in situations where x-ray mammography produces questionable results. Research by various investi-gators indicates that U C T has the potential to provide accurate tissue information that can be used for early breast cancer detection [22, 32, 35, 46, 53]. T H E E A R L Y Y E A R S Ultrasound tomography was first applied experimentally by J . F . Greenleaf and his colleagues at the Mayo Cl in ic in the mid 1970's [37]. These first algorithms were adopted from transmission x-ray C T . Hence, straight paths with no scattering were assumed as a model for ultrasound propagation. However, since the wavelength of the sound used in U C T is roughly the same size as the diameters and spacing of the tissue structures that scatter the sound waves, diffraction effects are in fact significant. Thus, these in i t ia l algorithms produced images that could not be used to reliably detect tumors [35]. R E F L E C T I O N T O M O G R A P H Y A N D D I F F R A C T I O N T O M O G R A P H Y These early attempts in the field indicated that tomography wi th a diffracting energy source requires an entirely different approach in terms of how the data projections are mathematically modeled. A s such, new reconstruction algorithms were developed that included the solution of the wave equation, which was used to model sound propagation more rigorously. Two imaging modalities that arose from this work are known as diffrac-t ion tomography (DT) and reflection tomography (RT) . Diffraction tomography is the term generally used to describe U C T algorithms that make use of sound that is scattered Chapter 1. Introduction 30 at angles other than 180°. Reflection tomography, on the other hand, is based only on backscattered ultrasound. Al though this is similar to B-scan imaging, the modalities are in fact very different due to the inclusion of wave mathematics in the R T algorithm. Reflection tomography is simple in terms of imaging geometry. It is not necessary to encircle the object being imaged with an annular array of transducers, since insonifi-cation and scatter detection are done on the same side with the same transducer. This can alternatively be viewed as a l imitation since only a portion of the ultrasound is in fact reflected from tissue interfaces directly back to the ultrasound scanning head. The method thus neglects a large percentage of sound intensity on the order of 99% that is scattered into al l other angles about the interface. Reflection tomography is also l imited in terms of what information it can yield. Diffraction tomography aims at providing images of certain object parameters through the solution of different wave equations. Examples of such parameters are density, compressibility and refractive index, n = ^ y , where cn is the speed of sound in water and c(r) is the spatially varying speed of sound in the region in question. The mathematics in reflection tomography reduce to an image of reflectivity only. M u c h of reflection tomography theory has been developed by Norton and Linzer. Reference [66] presents detailed derivations of their reconstruction algorithms employing point sources. Algori thms employing planar sources can be found in Reference [47]. A key component to the R T algorithms is the use of broadband pulses, which have a short wavelength on the order of 1 mm. A s wi l l become evident, this allows for the measurement of line integrals through the object reflectivity function. These integrals are not necessarily taken along a straight path. Consider the example illustrated in Figure 1.6, in which a single point transducer operating in pulse-echo mode insonifies an object wi th spherically divergent pulses. The received echo amplitude, ft, measured at time t, is the sum of al l reflections from scatter points a distance of tc from the transducer, where c is the average speed of sound in the media that are being investigated. This Chapter 1. Introduction 31 Figure 1.6: This figure illustrates the method of reflection tomography wi th point sources. localization of scatter points is made possible by the short wavelength of the ultrasound pulses. The reflected signals that add up to create /? are directly proportional to the reflectivity function at the corresponding scatter points. Hence, this type of tomography measures some function of R over circular arcs. In Figure 1.6, the measurement point wi th amplitude /? is the sum of a l l reflection contributions over the arc i n the incident ultrasound beam. It should be noted that there is of course some error inherent in this simple model due to variances in the speed of sound throughout the propagation media. Diffraction tomography is more general than reflection tomography in that the u l -trasound scatter recorded for image reconstruction is not limited to pulse-echo energy. Ultrasound propagation is again modeled wi th a wave equation. This type of imaging has been developed over the last 15 years by a number of groups [22, 30, 36, 53, 80]. The Chapter 1. Introduction 32 scattering process is analyzed in terms of the spatially dependent tissue density, p(r), and compressibility, K(T). From this information, different mathematical techniques can be used to compute p(r) and K(T) given knowledge of the sound waves scattered at many angles. Several groups have worked to develop various D T mathematical techniques, which are generally 2D approaches due to the complex nature of the physics involved [22, 36, 53, 80]. A s w i l l become evident in Section 2.2, the solution of the wave equation in diffraction tomography is a nonlinear problem. Consequently an exact analytical solution does not exist, and the search for a suitable approximate solution is a difficult task. There are two approaches to this problem in D T . The first involves the use of iterative techniques to solve the wave equation. However, such techniques are generally too computationally intensive for real-time imaging and have thus had l imited success [11, 16, 17]. A second approach and far more common approach involves perturbation methods that assume tissue is a weak scatterer of sound [10, 21, 32, 46, 53, 80, 88]. Ei ther the Born and Rytov approximations can be applied to simplify the mathematics and make the wave equation solvable. In the Born approximation, the interacting sound field is assumed to be equal to the incident field. In the Rytov approach, one assumes that the phase shift between the incident and scattered sound fields is very small. In short, the tissue is assumed to be a weak scatterer of sound, and the ultrasound field is not changed significantly by the scattering process. These techniques have the potential to provide useful results wi th lit t le computation time, which is an important aspect of any clinically-viable imaging application. The final key feature of both R T and D T algorithms is that ultrasound reflection . or scatter data recorded at different angles yields a subset of data points in the Fourier space of the image that is being reconstructed. B y taking data at more angles, the Fourier space is filled in unt i l enough points are known to determine the corresponding image. Chapter 1. Introduction 33 A L G E B R A I C R E C O N S T R U C T I O N T E C H N I Q U E S The algebraic reconstruction technique differs from the tomographic methods de-scribed thus far in that data from different projections are built up in the physical space of the image rather than in its Fourier space. A measure of the image is stored as a vector of unknowns, given by x, and a measure of the scatter data is stored in another vector, given by y. A model of the tissue-sound interaction physics is then used to de-vise a system of linear equations that relate x to y. These equations comprise a set of simultaneous linear equations of the form y = Ax (1.17) The elements of x are therefore unknown, and the aim is to devise a system wi th more equations than unknowns and then solve the system iteratively for x. Evidently, the success of the method depends on two factors: first, how well the physics model describes reality, and second, how well the system converges iteratively on a valid x. Although this method is conceptually simpler than the Fourier space methods de-scribed previously, algebraic reconstruction techniques are generally too slow for use in most applications. However, it does appear to hold promise in cases where data from only a l imited number of view angles are available. In experiments wi th small gelatin phantoms by Ladas and Devaney, the use of an A R T was found to be superior to con-ventional diffraction tomography methods when data were acquired at only 26 angles, instead of an optimal 200 views [53]. C H A P T E R 2 T H E O R E T I C A L B A C K G R O U N D The development of a diffraction tomography algorithm requires several key steps. First , a source and detector of ultrasound must be chosen and the behavior of each must be well known and described by a model. These transducers must also be feasible to use in experimental work. Second, there must be a detailed understanding of the physics that describes'how ultrasound propagates through tissue. This process is governed by a wave equation whose solution is nonlinear, thus adding to the difficulty of the model. Frequency dependent attenuation further complicates the problem. The thi rd key step involves choosing assumptions that render the problem linear while maintaining the usefulness of the model in describing at least some subset of imaging situations. In the fourth step, the model of sound propagation is used to develop an image reconstruction algorithm that combines scatter data from different projections into an image of a specific tissue parameter. In this thesis, the steps outlined have been taken to develop a prototype pulsed U C T system aimed at investigating the spatially varying density, p(r), and compressibility, K(T), of both minimally attenuating and attenuating objects immersed in a background fluid, which is water. The algorithms yield images of the functions 7P(r) and 7«( r )> g i v e n by p(r) - Po 7P(r) = P(r) where K 0 and Po are the compressibility and density of the water. A s was noted in Chapter 1, the denominator is spatially varying in the case of 7P(r) and constant in the 7K(r) 34 Chapter 2. Theoretical Background 35 definition, and these particular functions result from the solution of the wave equation, to be discussed in Section 2.2.1. The basis for this work is the paper by Blackledge et al [10]. After some modifications, the algorithm cited in this paper was used to test pulsed U C T for minimally attenuating objects. The corresponding physics model was also the point of departure for the development of an algebraic reconstruction technique that includes a frequency dependent attenuation correction. The theory behind these image reconstruction algorithms is discussed in the following sections. 2.1 N O N P L A N A R S O U R C E S A N D P U L S E D U L T R A S O U N D The theoretical basis for this thesis work assumes the use of individual line sources, while the majority of U C T methods to date have assumed that tissue is insonified wi th plane waves [21, 22, 32, 47, 48]. The use of planar sources is somewhat troublesome from a practical point of view because plane waves are difficult to create [32, 66]. A truly plane wave is emitted by an ideal plane wave transducer, which is flat in shape and large in extent compared to the object being insonified. Such a transducer does not exist to date, and hence the conventional method of creating a plane wave is through the use of an array of point sources. The transducers are excited by the same signal, and their individual spherical fields are superimposed to yield an approximation to a plane wave [47]. The characteristics of a plane wave detector are likewise approximated by summing a l l the signals that the point transducers see. The lines sources used in this work emit pulsed ultrasound fields, and this choice was made after an extensive literature review. The use of pulsed ultrasound is a relatively novel approach in the field of computed tomography. To date, experimental U C T systems have employed continuous wave ultrasound only, and the vast majority of diffraction tomography algorithms also assume C W ultrasound [10, 32, 53, 80, 88]. However, the phase shift ambiguity problem associated wi th the use of continuous-wave ultrasound has l imited the capabilities of these methods. In an imaging experiment, an ultrasound Chapter 2. Theoretical Background 36 wave undergoes a phase shift due to propagation through regions wi th different acoustic impedances. In continuous wave U C T , the phase shift of the scattered field at any location in question is measured v ia quadrature detection [21, 32, 47, 53, 80]. This processing decomposes a narrowband signal, v(t), wi th a center frequency of / 0 into the following sum: v(t) = vc(t) cos(2ir f0t) + vs(t) sm(2nf0t) (2.2) According to Fourier Theory, vc(t) and v3(t) are directly related to the real and imaginary parts of the Fourier Transform (FT) of vc, from which the phase can in turn be calculated. This is compared to the phase of the insonifying wave, and the phase shift is determined for use in image reconstruction. However, the measured phase shift could be in error by a factor of 2n due to the 27r modulo periodic nature of the ultrasound wave. Hence there is an ambiguity in determining the phase shift, and image reconstruction is based on erroneous data. Furthermore, this error becomes increasingly large as the object being imaged increases in size. Hence, experimental work in continuous wave U C T has generally been l imited to the investigation of objects that are on the order of 10 wavelengths in diameter (4-6 m m wide) [21, 32, 47, 53, 80]. The severity of the problem is illustrated in Reference [47], in which computer simulations concluded that cylinders immersed in water could be properly reconstructed only i f the phase change across the object was less than 0.87T. Given a transducer operating frequency of 3 M H z and a change in the index of refraction of 20% relative to water, proper reconstructions are l imited to objects that are less than 0.5 m m in radius [47]. Several phase-unwrapping techniques have been developed in an attempt to correct this problem, with l imited success [47]. In contrast, pulsed ultrasound computed tomography insonifies objects wi th nonpe-riodic transient fields. Quadrature detection is not used to measure phase information. Rather, the transient scattered field is detected and Fourier Transformed. There is also always a reference point on the ultrasound pulse which can be used to determine any shift-ing that occurs upon propagation through an object. Comparison of the scatter Fourier Chapter 2. Theoretical Background 37 spectrum wi th the incident field Fourier spectrum yields unambiguous phase information for use in image reconstruction. Pulsed U C T also has other advantages over C W imaging. For instance, wi th transient fields one can simultaneously collect multifrequency data since a pulse is composed of a spectrum of frequencies within some particular band l imit . Multifrequency data could yield more information about a tissue since a collection of sound waves wi th different frequencies interact simultaneously wi th the object. Also , the band l imits generally include higher frequencies than the single frequency used in C W U C T . Higher frequencies can reconstruct smaller features on the order of A / 2 , to wi thin the physical limitations of the algorithms being used. Finally, both tissue compressibility and density can be determined wi th the methods for pulsed tomography used in this thesis, while C W U C T generally reconstructs only one specific parameter. Despite these advantages, only a handful of groups have studied pulsed ultrasound computed tomography, and the focus has been theoretical rather than experimental. The major factor that has l imited research in this field is frequency-dependent attenuation [68]. Da ta in any U C T experiment must be corrected in magnitude for the attenuation that occurs as the sound waves travel through tissue. The attenuation of a particular sound wave or portion thereof is in turn dependent upon both the total distance it travels through regions of different attenuation coefficient, particularly tissue, as well as the frequencies found in the sound wave. Evidently, for C W ultrasound, the correction process is simpler since the wave is composed of mostly one frequency. For pulsed U C T , a spectrum of frequencies is used; the attenuation coefficient therefore depends on the frequency region under analysis. This thesis includes two approaches towards solving this problem. One approach affords only an approximate correction that is based on the average distance that ultrasound travels through tissue and water, respectively in a given experiment. The second approach is more rigorous and includes attenuated propagators that describe the propagation of ultrasound from the source to detector v i a a l l possible Chapter 2. Theoretical Background 38 scatter points wi th attenuation due to water and tissue included. These methods are outlined in detail in Sections 4.1 and 4.2, respectively. 2.2 M I N I M A L L Y A T T E N U A T I N G O B J E C T S The method by Blackledge et al for reconstructing images of minimal ly attenuating objects was the starting point for the work involved in this thesis. This method for producing images of compressibility and density using pulsed U C T is classified as a direct Fourier method. That is, information in the Fourier domain of the image is built up from projections at various angles unti l the space is characterized sufficiently well to make inversion possible and yield an image. This requires the use of methods for interpolating data in Fourier space from a radial grid to a Cartesian grid, which is discussed in Section 4.6. This section presents the highlights of the physics model development, while full details of the derivation can be found in Appendix A . The derivation makes reference to three key resources: the paper by Blackledge et al [10], the paper by Norton and Linzer [66], and Section 6.2 of the book by Morse and Ingard [62]. Minor errors were found in the work of Morse and Ingard, while several errors were found in the paper by Blackledge et al. The errors appear to be primarily typographical, since the final result is the same in both this thesis and the paper by Blackledge et al. To aid the reader in following the derivations, errors in the references have been noted. 2.2.1 T H E P H Y S I C S B A S I S In this reconstruction algorithm, the tissue being imaged is described by the Chernov Equation: V . ( ^ V p ( r > t ) ) = « ( r ) ^ p ( r , t ) (2.3) p(r, t) is the pressure field at any time, t, during the experiment and at any location, r, within the image region [10, 62, 66]. Note that this equation is expressed incorrectly in Chapter 2. Theoretical Background 39 the paper by Blackledge et al, wi th a negative sign on the right hand side. Since the 3D problem is difficult to solve, this theory considers 2D sound pressure fields only, which is a conventional approach in U C T [21, 32, 47, 53, 80]. Thus r = i x + yy (2.4) where x and y are unit vectors in the zy-plane. Equation 2.3 models tissue as a non-viscous, compressible fluid. Viscosity refers to a fluid's resistance to flow due to internal frictional forces. A more viscous fluid flows more slowly than a fluid wi th less viscosity due to increased internal friction. Al though certain fluids in the body are considered viscous, such as lymph fluid, the most prevalent human biological fluids have relatively low viscosity in the healthy body. For instance, it is well known that blood generally has low viscosity, but can become more viscous in the presence of certain disorders such as leukemia Polycythemia vera [92]. Furthermore, the constituents of the human body have a high water content, ranging from 50-65% (women 50-60%, men 60-65%), and water has a low viscosity at the temperatures in the human body [55]. Hence, the modeling of soft tissue as a nonviscous fluid of spatially varying p and K is a valid approach that has been applied by several groups [23, 32, 47, 66]. To both sides of Equation 2.3, the following term is added «oSp(M) ~ - V 2 p M ) (2.5) OT Po Note that the step cited here in the reference by Morse and Ingard is incorrect in that the time derivative includes a factor of p and thus has improper units. In Equat ion 2.5, the substitution cn = («oPo) - ^ is made, where c 0 is the speed of sound in the background fluid, and the expression is further reduced to yield an equivalent wave equation given by: V 2 p ( r , t ) - ^ P M ) = 2 ^ | L p ( r , t ) + V • ( 7 p V p ( r , t ) ) (2.6) In the paper by Blackledge et al, this equation is quoted incorrectly wi th a factor of -1 in the time derivative on the right hand side. Equation 2.6 is a time dependent wave Chapter 2. Theoretical Background 40 equation wi th a forcing term that is a function of 7«(r) and 7 P ( r ) . The Fourier Transform is then applied to both sides of this equation to yield V 2 p ( r , u) + k2p(r, w) = -h2-yK(r)p(r, u) + V • ( 7 p ( r ) V p ( r , u)) (2.7) where k = UJ/CQ is the angular wave number in the background fluid. Note that the expres-sion quoted in the chapter by Morse and Ingard is incorrect in that the V - (7 p ( r )Vp( r , u>)) term includes an extra negative sign. The solution to Equation 2.7 at any detector posi-t ion r = ra is given by p(rd, u) = p0(rd, u) + k2 g(r\rd, k)r/K(r)p(r, w)d 2 r (2.8) Jst2 - f g(r\id, k)V • (7 P ( r )Vp (r , w) )d 2 r Note that the signs preceding the two integrals in the paper by Blackledge et al are both opposite to what they should be. P0(TS,U) is the Fourier Transform of the incident ultrasound field at any detector position rd, and p ( r | r s , k ) is the 2D Green's function that describes wave propagation in two dimensions in the background fluid. For outgoing waves, the 2D Green's function in the absence of attenuation has the form g(r\rd,k) = -L-H10(k\v-vd\) (2.9) where HQ is the Hankel function of the first kind. Note that the solution quoted in the reference by Blackledge et al is incorrect by a factor of -1, which results in the wrong pulse shape upon propagation of the incident field. The Green's function determines the phase of the frequency components in the ultrasound wave. Phase in Fourier space determines the propagation time (and by extension the propagation distance) in physical space, so consequently the Green's function provides information regarding how far the pulse has propagated. Since attenuation is not included, the Green's function does not change the shape of the ultrasound pulse. Note that the Green's function is symmetric wi th respect to the interchange of r and rd. Stated equivalently, the field value measured by a detector at position rd due to a source located at r is the same as the value that Chapter 2. Theoretical Background 41 would be measured i f the positions of the source and detector were switched. This is termed the Principle of Reciprocity. In terms of variable notation, this principle dictates that g(r\rd,k) =g(rd\r,k). Equat ion 2.8 is nonlinear in that the functional form of the total field p is not only-being solved for, but it is also found within the integral sign. Thus, it is not possible to determine an exact analytical solution to Equation 2.8. A s discussed i n Section 1.8.2, two approaches are possible here. The less common approach uses iterative techniques to solve the equation in a computationally intensive manner. The difficulties associated wi th this approach have motivated most researchers to follow the perturbative Born or Rytov approaches, in which weak scattering is assumed in order to render Equat ion 2.8 solvable [10, 21, 32, 46, 53, 80, 88]. For the ini t ia l prototype development included in this thesis, the Born approximation has been used. It is applied at this point to yield p(rd,oj) = p0(rd,u)+k2 g(r\rd, k)~/K(r)p0{T, w)d 2 r - (2.10) [ g(r\rd, k)V • ( 7 p (r)Vp 0 ( r , u/))d 2r Note again that the signs preceding the two integrals in the paper by Blackledge et al are both opposite to what they should be, which is the error carried over from Equat ion 2.8. The algorithm also assumes that the insonifying pulse is created by a line source (recall the model is 2D), which is generally considered to be a field that is easily obtainable in the lab [10, 32]. The incident field at any given location, r, due to a source wi th position vector, r g , is written Po(r:uj) = A(oj)g(r\vs,k) (2.11) where A(UJ) is the amplitude spectrum of the incident pulse. O f particular note is that the insonifying field has zero phase at the source location. The phase of al l frequency components in the source then evolves in a periodic fashion between 0 and 27r as the field propagates away from the source. The incident pulse is also assumed to be band Chapter 2. Theoretical Background 42 l imited from Qx to Cl2 (between frequencies / i and / 2 ) . The source and detector are also assumed to be significantly far from any point of origin of ultrasound scatter, located at r in the image space. A s such, the following relation holds for rj equal to rs or r^: | f c | | p - r j | » l (2.12) for every angular wave number, k, for which 9l<|fc|<2* (2.13) Co c 0 This means that the object is placed in the far field region of both the source and detector. Equat ion 2.12 essentially has the effect of imposing a highpass filter in the temporal frequency space of the ultrasound signals. These assumptions allow the exact Green's function to be expanded in terms of a simpler function given by g(T\Tj,k) « aS (2.14) .exp(3i7r/4) a = " S = 2V27T e x p ( i f c | r - r j | ) ( f c | r - R J | ) 2 Note that the value of a quoted in the paper by Blackledge et al is incorrect by a factor of i, resulting in the wrong phase information. Next an expression for V S is required, whose derivation is outlined in Appendix B . The result is V S = ikhjS, for | f c | | r - r j | > 1 (2.15) (r - TJ) Combining al l the assumptions and simplifications yields an expression for Fourier Transform of the total ultrasound field, p ( r s , r 0 , u ; ) , at any detector point p{Td,Ts,u) * aA(oj)S(k\rd-Ts\) + a2A(oj)k2I (2.16) / = / S(k\r-rd\)(lK(r)-(nd-ns)lp(v))S(k\v-rs\)^ 2 r Chapter 2. Theoretical Background 43 Note that in the paper by Blackledge et al, the source and detector location vectors (referred to as r 0 and r s , respectively) have been accidentally flipped in the corresponding equation. A n expression for n^ • n s must now be derived, which requires the definition of an imaging geometry. Figure 2.1 illustrates the experimental U C T setup in which a source transducer wi th position vector rs insonifies an object located about the origin wi th a pulse that has an amplitude spectrum A(t). The scattered field is detected by a second transducer wi th position vector r^. Bo th the source and detector are situated at a distance a from the origin, and at angles of <ps and ipd, respectively. Note that <ps is always larger than (pa- The unit vectors n s and n^ can then be written as hs = xcos(<p5) + ysin(<p s) (2.17) n d = x c o s ( p d ) +ysin(v?d) Furthermore, ip„ and <pd are related by 0 = <Pd ~ <Ps + n (2.18) which leads to n , • hd = - cos(0) (2.19) Note that the paper by Blackledge et al quotes a value for this expression that is incorrect by a factor of -1 . Substituting this expression for the dot product into Equat ion 2.16 results in p(Td,r„u) « aA(oo)S(k\rd-rs\) + a2A(u)k2I (2.20) / = / 5 ( f c | r - r d | ) ( 7 l . ( r ) - rC08 ( f l ) 7 p ( r ) ) 5 ( f e | r - r . | ) d 2 r The first term in Equation 2.20 corresponds to the propagation of the incident field from the source to the detector, while the second term represents the scatter term. Essentially the scatter term is the sum of contributions from all scatter points in the Chapter 2. Theoretical Background 44 Figure 2.1: This schematic illustrates the U C T experiment geometry upon which the reconstruction algorithm is developed. Vectors r x and r 2 are two examples of r, which is the location vector of scatter points wi th in the image. Chapter 2. Theoretical Background 45 image space, each propagated first from the source to the scatter location and finally to the detector by the free space Green's function. Each scatter contribution is weighted by ( 7 « ( r ) + cos(0)7 p(r)). Use of the free space Green's function implies that second order scattering, in which the scatter contributions undergo additional scatter processes, is not considered. Also , attenuation is ignored in this model. Equat ion 2.20 can be further simplified by once again applying the far field assump-tion, this time equivalently stated as Note that this approximation results in a loss of phase information and ultimately is responsible for the relatively poor performance of the reconstruction algorithm. This w i l l be discussed in more detail in Section 5.1.3. The application of 2.22 results in a final solution of (2.21) where again Tj is equal to rs or rd. The analysis yields the following result: (2.22) p(rd,rs,oj) aA(oj)(S(k\Td-rs\) + aJ) exp(ifc(|r dl + [r.D) r (2.23) J ( M I ' . U * ^ exp(ikhd • r)(7 K (r) + cos(0)7 p(r)) exp(i/cn s • r ) d 2 r X 2.2.2 T H E R E C O N S T R U C T I O N A L G O R I T H M Substituting the expressions x cos(y?d) + y sin(^j) x cos(cps) + y sm((ps) (2.24) n and rd = and (2.25) = a n Chapter 2. Theoretical Background 46 into Equat ion 2.23 leads to an algorithm for reconstructing images of 7 P (r) and 7«(r ) . The analysis yields the following result for the Fourier Transform of the total ultrasound field detected at any point r s : p(rd,rs,u) * F1 + F2 xxjje(<ps,k) (2.26) where F l = aA(U) e M 2 i k a C°f\] (2.27) V ' (2»ifeacos(|))3 F2 = a2A(u)k exp(2ika) a ipe(<Ps,k) = / exp(-i(ux + vy))(-yK(r) + cos(0)7 p(r)) d x d y and 9 9 u = - 2 f c s i n ( - ) s i n ( - + y>,)) (2.28) 9 9 v = 2k s i n ( - ) c o s ( - + <p,) Note that the signs quoted for both u and v in the paper by Blackledge et al are opposite to what they should be. The first and second terms of Equat ion 2.26 correspond to the propagated incident and scattered fields, respectively. Equat ion 2.27 indicates that ipeifsi is the subset of points corresponding to data for angles 9 and ips in the Fourier Transform space of the image, / , given by / = 7 K (r) + cos(0) 7 p ( r ) (2.29) for a nonattenuating object. Equations 2.26 through 2.28 provide the basis of an algorithm that reconstructs this image. For a given experiment, 9 is fixed and as such the angle between the source and detector is fixed. This configuration is then stepped around the object being imaged so that the detector is placed at each angle in a set of <pd values. A given cpd corresponds to Chapter 2. Theoretical Background 47 a different view at which data are collected. tpei'Psjk) can be determined for each view by rearranging Equation 2.26 as follows: il)eWs,k) « — (2.30) The incident field is measured at each angle (pd wi th the object removed from the tank. Each resulting ip$((pa, k) provides a subset of data in the Fourier Transform space of the image, / . Eventually enough data are collected to render inversion possible to yield the image. Evidently, reconstruction wi th ultrasound data scattered at 9 = 90° yields an image of 7« (r) alone since 7 P ( r ) is weighted by cos(0). Reconstruction wi th backscatter data, for which 9 = 180°, yields an image of the reflectivity function, which has been previously defined in Equat ion 1.10 as # = 7 « ( r ) - 7 p ( r ) (2.31) These two experiments allow 7 P ( r ) to be solved for, completing the investigation of both 7K(r) and 7 p ( r ) . Indeed, any two angles, 9± and 0 2 , theoretically allow both gamma functions to be determined as follows: h = 7*(r) + cos(0 1 ) 7 p ( r ) (2-32) h = 7«(r ) + cos(0 2)7 P(r) 7«(r ) = ^ ( / i + / 2 - 7 p ( r ) ( c o s ( r 5 1 ) - c o s ( 0 2 ) ) ) 7 p W = cos(0i) - cos(0 2) However, as w i l l become evident in Section 2.2.7, the resolution of the images depends upon the value of 9. To use the term coined by Blackledge et al, the reconstruction algorithm leads to a Diffraction Slice Theorem, which is illustrated in Figure 2.2. This theorem determines the Fourier space coordinates that correspond to data recorded for each source angle, (ps, Chapter 2. Theoretical Background 48 in an experiment with a constant 9. Equation 2.28 is a parametric expression of u and v in terms of <ps and k, and i t thus describes a geometric relationship between w-space and (pak-sp&ce. For a given <ps, varying k between the band l imits of the pulse results in (u, v) coordinates that map out a linear slice in Fourier space. This slice passes through the origin at an angle X=il + <P.) (2-33) relative to the u-axis. The length of each slice is J = 2 ( ^ - ^ ) r i n ( | ) (2.34) Co Co Z Thus, ultrasound scatter data corresponding to the same temporal angular wave number, Chapter 2. Theoretical Background 49 k and a different view angle, <ps, is transformed into Fourier data on circles in the uv-space of the image. The radius of the corresponding circles is easily calculated from the expressions for u and v in Equation 2.28, to yield r = v V + v2 (2.35) 9 = 2fcsin(-) Alternatively, scatter data corresponding to the same view angle and increasing angular wave number are transformed into Fourier data on radial lines. A s more views are recorded, a donut-shaped region of data is mapped out in the Fourier space of the image. Upon collecting enough data, inversion can yield the image / = 7«(r ) +cos(f9)7 p(r) given by Equat ion 2.29. This reconstruction algorithm is consistent wi th that of continuous wave computed tomography, in which insonification at a particular view angle yields Fourier data on a semicircle that passes through the origin. As such, data corresponding to a particular frequency are transformed onto a circular arc in Fourier space. Examples of such projec-tions are illustrated in Figure 2.3. Evidently, as the angular wave number increases, the radius of curvature of the arc decreases, which is also consistent wi th the theory outlined in this thesis. In the case of pulsed U C T , the circular paths corresponding to each fre-quency are in fact full circles rather than semicircles, and instead of passing through the origin, they are concentric about the origin. The Diffraction Slice Theorem is also of interest due to its similarity to the Projection Slice Theorem from tomography with nondiffracting sources, such as x-ray C T . The latter theorem states that the Fourier Transform of a projection recorded at an angle <p maps a linear slice in the Fourier space of the corresponding image at the same angle </> [21, 47]. In the case of diffraction tomography, however, the angles in the physical and Fourier spaces are not equal. Furthermore, in straight-ray tomography, the Fourier space data are not subject to a low frequency cutoff. Another important difference between x-ray C T methods and this reconstruction algorithm is that the total angular range over which data Chapter 2. Theoretical Background 50 - i v(mm 1) k = 2 k ° \ Y \ \ k=17k0 k=3k0\ \\ k=4k 0 | \ k=5k0 | k=6kQ Figure 2.3: This figure illustrates examples of projections in C W ultrasound computed tomography. Ultrasound scatter recorded at a particular angle yields data in the image Fourier space that lie on semicircular arc that passes through the origin. The angular view determines the orientation of the arc and the radius of curvature is inversely proportional to the angular wave number. must be recorded i n the former may be restricted to 180° since the projection obtained is identical for irradiation at any angle, <f>, and the corresponding angle, <j> + 180° [47]. For pulsed U C T , this is only true of objects that possess two symmetrical hemispheres because the data recorded are based on T O F . This is illustrated wi th the example of a point object in Figure 2.4. Evidently, wi th the source and detector at 90° and 0°, respectively, the scattered ultrasound pulse wi l l arrive earlier in the detected signal than if the source and detector are situated at 270° and 180° instead, due to the smaller sound travel paths involved. Chapter 2. Theoretical Background 51 Figure 2.4: This plot illustrates that data recorded from opposite directions are not the same. Equations 2.26 through 2.28 for transforming projection information into 2D Fourier space information are consistent with the required properties of real images. According to Fourier Theory, an image described by a real function f(x, y) has a 2D Fourier Transform for which the following symmetry rule holds: F T ( - « , -v) = FT+(u, v) (2.36) The I D data, tpe(ips,k), that are deposited on a given radial slice corresponding to a source angle of (p~s is derived from Equat ion 2.26. Th i s equation is dependent upon Pe{<Psik)> which is the Fourier Transform of a real signal and thus obeys the following Chapter 2. Theoretical Background 52 Fourier Data Coordinates for A Given Angle -80' 1 1 ' 1 1 1 ' 1 -80 -60 -40 -20 0 20 40 60 80 u ( m m - 1 ) Figure 2.5: This plot illustrates the (u, v) coordinates of tpe data along the radial slice corresponding to one source angle equal to ips. symmetry in fc-space: Pe(<Ps,-k) =Pe(0s,k) (2.37) The final expression for i^e{'Ps,k) also obeys the same symmetry rule and thus M(Ps,-k) = ^(<pa,k) (2.38) The corresponding (u,v) coordinates at which the ifig(<ps,k) data are placed i n Fourier space are illustrated in Figure 2.5. Evidently, the 2D symmetry rule expressed in Equa-tion 2.36 holds for these data. The result is the same for any <p3, and therefore the reconstruction algorithm always derives Fourier data that corresponds to a real image. Chapter 2. Theoretical Background 53 2.2.3 C O R R E C T E D G R E E N ' S F U N C T I O N P R O P A G A T O R The simplified approximation to the Green's function propagator quoted in the work by Blackledge et al has incorrect phase, which in turn yields incorrect ultrasound wave propagation. The following section illustrates the correction that was made to obtain an approximate Green's function for the description of ultrasound propagation in the near field of a pulsed source wi th a finite bandwidth [10]. Recall that the far field assumptions described by Equations 2.12 and 2.13 allow the exact Green's Function to be expanded in terms of a simpler function given by # „ t ) . . y - ? » (2.39) ( fc | r -r j | )» The a parameter is imaginary. A n y error in a thus affects the phase of the wave compo-nents and adds error to the propagation distance of the pulse. The value of a quoted in [10] is = exp(3«r/4) However, this propagator does not describe the real and imaginary parts of the propagator properly, as illustrated in Figure 2.6. Rather, the correct value of a was found to be = ,exp(3nr_/4) A s illustrated in Figure 2.7, the real and imaginary parts of the corrected approximation to the Green's function agree with the exact Green's function propagator. Examples are shown for propagation through distances of 20 and 100 mm. 2.2.4 U S E O F T H E F A S T F O U R I E R T R A N S F O R M The paper by Blackledge et al suggests the use of the Fini te Fourier Transform (Finite F T ) in the reconstruction algorithm [10]. For a I D discrete function of time made up of N time samples, this transform is given by N FT(w) = 5^exp(-iwt i)/(t i)At (2.42) Chapter 2. Theoretical Background 54 Propagator Real Part • A = 100 mm k (mm-1) (A) Propagator Imag Part - A = 100 mm (B) Figure 2.6: This figure illustrates the discrepancy between the exact free space Green's function propagator and the approximate one quoted in Blackledge et al. Figures A and B illustrate the real part and imaginary parts of these functions, respectively, for a propagation distance of 100 mm. Chapter 2. Theoretical Background 55 Propagator Real Part - A = 20 mm Propagator Imag Part - A = 20 mm (A) Propagator Real Part - A = 100 mm E (B) Propagator Imag Part - A = 100 mm (C) (D) Figure 2.7: This figure compares the exact and approximate 2D propagators. Figures A and B illustrate the real and imaginary spectra for propagation through a distance of 20 m m , while Figures C and D illustrate the same for a distance of 100 m m . Evidently, there is good agreement between the exact form and the approximation. Chapter 2. Theoretical Background 56 N f(t) = YJexp(ta; i«)FT(a;J-)Aw i=i Essentially the Fini te F T is simply the discrete F T wi th limits on the summation. B y extension, the Fini te F T of a 2D function known at M and N discrete values of x and y, respectively, is given by M N FT(u,v) = £ {^expi-iiuxj + vy^fix^y^AxAy} (2.43) 3=1 k=l M N f(x,y) = X ! {J2exP( i(tLJx +  Vky)) F T ( u j , t ; f c ) A « A i ; } j = l k=l Equations 2.42 and 2.43 are Discrete Fourier Transforms ( D F T ' s ) . The calculation of Fourier data by use of these equations is a very slow and inefficient process. A far better approach is to use the Fast Fourier Transform ( F F T ) , which is the most computationally efficient implementation of the D F T . Furthermore, the F F T yields the same result as a l l other less efficient implementations of the D F T . In using the F F T to compute the Fini te Fourier Transform, however, the result of the forward and inverse transforms must be multiplied by factors of A z A y and A w At), respectively, since these are not included in the F F T . Furthermore, there are various conventions for F F T formulas, such as mult iplying or dividing by either 27T or the number of points in the F F T , depending on whether the inverse or forward transform is being computed. These must be taken into account when using a particular F F T instead of the Finite Fourier Transform in the reconstruction algorithm. 2.2.5 T H E F O U R I E R T R A N S F O R M S I G N C O N V E N T I O N The sign convention of the Fourier Transform used in wave propagation theory is op-posite to that which is conventionally used in engineering and most of physics [62]. Gen-erally the exponential functions in the forward and inverse I D transforms are exp(—iut) and exp(iui), respectively. In wave propagation theory, however, the signs of the expo-nential arguments are opposite to this convention. In fact, i f the wrong convention is Chapter 2. Theoretical Background 57 used, a non-physical situation occurs when propagating a pulse to any location. Upon inverse transforming the propagated Fourier spectrum to obtain the time-dependent field that a detector would see at that location, a pulse with a negative time-of-flight results. In order to apply this convention wi th a typical F F T routine, such as that available in the M a t l a b ™ programming environment, the following transformation must be applied to the output of the forward I D F F T : The frequencies corresponding to this output are calculated as per usual. In order to change the sign convention used in the inverse I D transform, the output remains the same, while the corresponding time vector, i is calculated as follows: where t is the original time vector calculated as per usual. 2.2.6 B E H A V I O R O F T H E E X A C T P R O P A G A T O R The behavior of the exact propagator was investigated to gain an understanding of the model. A simulation was performed in which the source was assumed to be a line transducer that produced a pulsed field wi th a Ricker wavelet time profile, which is defined by In Equat ion 2.46, time t ranges over both negative and positive values, wi th the simulation experiment starting at t = 0. Also, / 0 is the peak frequency of the wavelet. Ricker wavelets wi th OJQ equal to 6.0, 7.5 or 20 M H z have pulse lengths 1 in water at 20°C of 1 H e r e p u l s e l e n g t h is d e f i n e d as t h e d i s t a n c e b e t w e e n p o i n t s o n t h e p u l s e w h e r e t h e a m p l i t u d e is r e d u c e d t o < 0.1% o f t h e p u l s e m a x i m u m . F F T = rea l (FFT) - u m a g ( F F T ) (2.44) t -t (2.45) (2.46) Chapter 2. Theoretical Background 58 approximately 3.2, 2.6, and 1.0 mm. The pulse shape at t = 0 for CJ0 = 7.5 M H z is illustrated in Figure 2.8.A, while the real part of the corresponding Fourier spectrum is illustrated in Figure 2.8.B. Note that since the incident pulse is centered over t = 0, the imaginary Fourier spectrum is zero. Figure 2.9 illustrates the pulse shape as it is propagated through increasing distances. A n important feature to note is that the propagated pulse is unphysically larger than the incident pulse for propagation distances less than ~ 2 x 1 0 - 2 mm. This is due to the bad behavior of the propagator wi th small distances, in the form of a lack of frequency-dependent oscillations. A second feature to note is that the pulse height decreases wi th distance, which is expected due to the nearly |r — r^ | — i dependence of the Hankel function. The last interesting feature is the change in pulse shape from symmetrical to asymmetrical. This results from the nearly dependence of the Hankel function. 2.2.7 T H E O R E T I C A L R E S O L U T I O N When determining the image resolution that is possible wi th the algorithm by Black-ledge et al, the Nyquist Theorem must be considered because image reconstruction is dependent upon data that have been discretely sampled in Fourier space. The Nyquist Theorem states that when a function f(m) is sampled at intervals of A m , data in the Fourier domain can be determined for frequencies up to a maximum of '»» = 2Km ( 2 ' 4 7 ) where fNyq is known as the Nyquist frequency. Alternatively, this also means that i f data in Fourier space are known for frequencies / < fNyq, then the maximum resolution in physical space is A m — ( 2 ' 4 8 ) ^ JNyq Furthermore, these equations preclude that the Nyquist frequency is one half the fre-quency at which f(m) is sampled. In 2D and 3D systems, each dimension is analyzed separately in this manner. Chapter 2. Theoretical Background 5 9 Figure 2.8: This figure illustrates the incident pulsed field used in various computer simulations. Figure A shows the time profile, while Figure B illustrates the real part of the Fourier spectrum of the source. Chapter 2. Theoretical Background 60 P u l s e at D = 2x10" m m P u l s e at D = 2x10 m m P u l s e at D - 2x10 m m P u l s e at D - 2x10 m m P u l s e at D = 4 m m P u l s e at D = 8 m m -3 -0.01 Figure 2.9: This figure illustrates the pulse as it is propagated by the 2D Green's function. Chapter 2. Theoretical Background 61 According to these relations, the image resolution that is theoretically possible for minimal ly attenuating objects can be determined. Consider an experiment in which ultrasound scatter data are recorded at several angles, (fd, wi th the relative angle between the source and detector remaining constant at 9. The scattered field is detected by a transducer whose output signal is digitally sampled at a frequency of fo- According to the Nyquist Theorem, the Nyquist frequency of these data is fxyq = \ID- Furthermore, the maximum angular wave number considered in the reconstruction algorithm can be calculated as kmax = ~ (2.49) c where c is the speed of sound in the background medium. Recalling Equat ion 2.35 for the radi i of data rings in uv-sp&ce, the maximum radius that is possible is given by 9 Tmax = 2kmaxsin(-) (2.50) B y extension, both umax and vmax are equal to rmax; as such, the analysis w i l l be identical for both the x and y dimensions, u is the angular wave number along the first dimension in 2D Fourier space, and it is equal to 2nfx, where fx refers to spatial sampling frequency along the x-axis. Combining Equations 2.49 and 2.50, it is evident that the Fourier data of the image are known for select fx values up to and including 1 9 (fx)max = — kmax s in ( - ) (2-51) 7T Z = -r1 sm -7T C 2 Aga in according to the Nyquist Theorem, Aa; can be calculated to be A x = (2.52) " \Jx)max C AfNyq sin(f) The result is identical for the y dimension. Hence, the maximum image resolution along both the x and y directions is inversely proportional upon both s in ( | ) and the Nyquist frequency of the digitized scatter signal. Chapter 2. Theoretical Background 62 fNyq ( M H z ) Arc for 9 = 90° ( m m ) A x for 9 = 180° ( m m ) 1 0.52 0.37 2 0.26 0.19 4 0.13 0.093 8 0.065 0.046 Table 2.1: The above data illustrate sample image resolutions based on the theoretical calculations for minimally attenuating objects. To gain some understanding of the magnitude of these values, consider an experiment in which scatter is digitized at 20 M H z . The corresponding fNyq is 10 M H z . The resulting Arc values for c = 1.48 mm/ps and 9 equal to 90° and 180° are 0.052 m m and 0.037 mm, respectively. Results for different upper band limits scale up or down wi th the value of fNyq, as illustrated in Table 2.1. These values suggest that excellent image resolution is theoretically possible. However, this level of performance is not realistically attainable for tissues due to the application of the Born approximation and the far field assumptions in the development of the reconstruction algorithm, which was the approach taken to render the mathematics of the tissue/ultrasound system solvable. The results quoted assume that al l available Fourier data are used in the image reconstruction process. However, in order to apply the Fast Fourier Transform, the data must be interpolated onto a square grid in Fourier space, which reduces the image resolution by a factor of y/2. The reason for this is illustrated in Figure 2.10. Wi thout resorting to extrapolation into the unknown space beyond the largest Fourier data circle defined by radius r T O Q X , the interpolation grid is l imited to a width of Zy^Tmax (2-53) A s such, (fx)max is decreased by a factor of \/2, while Aa; increases by the same factor to 0.074 m m and 0.052 m m for 9 equal to 90° and 180°, respectively. Aga in , the results are identical for Ay. Chapter 2. Theoretical Background 6 3 Figure 2.10: This figure illustrates how the extent of wt>-space is reduced by the interpo-lation of data from a radial grid to a square grid. The lines represent the radially-situated data, while the diamonds represent the data situated on the corresponding square grid. 2.2.8 I N T E R P O L A T I O N T O A S Q U A R E G R I D If an interpolation in Fourier space is performed, the spacing of the square grid should be compatible wi th the spacing of the original radially situated data. A s such, this spacing should not be less than the largest distance between any two adjacent points in the original Fourier data. Since the data are on a radial grid, this distance corresponds to the maximum of either the radial spacing or the spacing along the data circle wi th the largest radius, r m a x . In order to calculate the radial spacing, A r , consider again the sample experiment described in Section 2.2.7. The digitized ultrasound scatter signal is Fourier Transformed. If there are N sample points in the digitized signal, then according to Fourier Theory the transform wi l l provide discrete data at frequencies of / = * ^ H where i = - y , - ( f - 1 ) , 0 , ( | - 1) (2.54) Chapter 2. Theoretical Background 64 Therefore, A / = ^ (2.55) and A k = ^fNm ( 2 5 6 ) cN According to the expression for T given by Equation 2.35, the corresponding radial spacing in wv-space is A T = 2AJfcsin(^) (2.57) = ^ A r s i n ( 2 } The calculation of the spacing along the largest data circle is much more straight-forward. B y simple geometry, the arc length between points on this circle is given by Tmax A<p d, where A<pd = — (2.58) Nv is the number of view angles at which projections are recorded, and A<pd is expressed in radians. Substituting for TMAX and kmax using Equations 2.50 and 2.49 yields the following result for the spacing along the largest Fourier data circle: A a r c = 5 % ^ (2.59) Nvc The final result for the interpolation grid spacing is then equal to the maximum of AT and A a r c . The ratio of the two spacings is Aa rc TTN A T NV (2.60) so the arc length spacing is the l imit ing parameter unti l Nv > irN, which is unlikely. Revisi t ing the experiment described in Section 2.2.7, it is further assumed that the transducers are a typical distance of d = 150 m m from the center of the tank. Recall that the frequency spectrum of the incident field is assumed to be negligible beyond Chapter 2. Theoretical Background 65 fNyq = 10 M H z , the speed of sound is c = 1.48 mm//xs and the sampling frequency is fc — 60 M H z . The average T O F of ultrasound scatter is then approximately 2d/c = 203/is. The corresponding number of time points, N, in the scatter signal at 60 M H z sampling is 12180. The radial spacing according to Equation 2.57 is then 0.0099 m m - 1 and 0 . 0 1 4 m m _ 1 for 9 equal to 90° and 180°, respectively. If the number of views is assumed to be 400, the arc spacing is a factor of 95 times larger wi th values of 0.94 m m - 1 and 1.33 m m - 1 , respectively. Hence, the interpolation grid spacing in this example is l imited by the value of Aarc . This analysis leads to an interesting result. According to Equations 2.50 and 2.53, for this example Tmax = 42 m m - 1 and 60 m m - 1 for 9 equal to 90° and 180°, respectively. W i t h the Fourier grid spacings calculated above, the corresponding grid size is 45 x 45 for both values of 9. Recalling that Ax and Ay are equal to 0.074 m m and 0.052 m m for 9 values of 90° and 180°, respectively, the resulting images w i l l be only 3.3 m m and 2.3 m m wide. The width is proportional to the number of 'view angles, and some values are summarized for 9 = 90° in Table 2.2. The reconstruction algorithm Nv fNyq (MHz) Grid Size Image Size (mm) 200 10 22 x 22 1.6 400 10 45 x 45 3.3 800 10 90 x 90 6.7 1600 10 180 x 180 13.3 3200 10 360 x 360 26.6 3200 5 360 x 360 37.7 6400 5 720 x 720 75.3 800 3 90 x 90 15.7 3200 3 360 x 360 62.8 Table 2.2: The above data illustrate example image sizes for two values of fNyq and various Nv. evidently zooms in on a portion of the object to potentially yield useful information. B y moving the center of the U C T system to another part of the object, a different close-Chapter 2. Theoretical Background 66 detector <; source ^ ,' / -secondary scatter point object scatter \ / point \ water scatter point image / tissue outline Figure 2.11: This plot illustrates the sources of scatter from points outside the image space that are ignored when the image size is very small . up picture would result. However, there exists a source of error that increases as the image size decreases in proportion to the total object size. A s illustrated in Figure 2.11, there is additional ultrasound scatter in the system that is not considered in the image reconstruction process. A significant amount of scatter originates from areas of the object that lie outside the image space but which are s t i l l insonified by the incident field. A portion of scatter also originates from the surrounding water. Al though this source of error exists even when the image space spreads beyond the object perimeter, the effect is more pronounced when all scatter points in water are ignored. Second order scattering Chapter 2. Theoretical Background 67 is neglected throughout the object and water regardless of the image size. W i t h i n the framework of the original method by Blackledge et al, the problem of neglected first order scatter can be alleviated by either reducing fNyq or increasing Nv. Alternatively, one can attempt to reduce the interpolation grid spacing to much less than the spacing between adjacent points on the radial grid. Results of this exercise are presented in Section 5.3.4. The discussion thus far has assumed that fjfyq is not more than 10 M H z . However, data sampling rates are in fact generally much higher than 20 M H z in practice. According to the Nyquist Theorem, the reconstruction of data from discrete time samples is possible only i f the sampling rate is at least 2/BW, where fsw is the band l imit of the data [56]. Accurate reconstructions require a sampling rate that is a higher multiple of JEW- A s such, data that are band limited at 10 M H z wi l l be sampled at ~60 M H z to boost accuracy. However, this results in a Nyquist frequency of 30 M H z and, by Equat ion 2.52, corresponding values of Aa; equal to 0.017 m m and 0.012 m m for 9 of 90° and 180°, respectively. This precludes that 2000x2000 pixels would be required to reconstruct an image that is a square of only 17 m m x 17 mm. This alone is an unmanageable size in terms of the interpolation from the radial grid and the subsequent inverse F F T . For instance, interpolating from a radial grid of only 100 views and 2000 radial points to a Cartesian grid of 800x800 was found to take more than 24 hours wi thin the M a t l a b ™ programming environment. 2.2.9 A N O T E O N L O W P A S S F I L T E R I N G A N D T R U N C A T I O N Recall from Section 2.2.1 that the ultrasound source is band l imited from fi to f2, outside of which its corresponding Fourier data are negligible and characterized by noise. A s such, a lowpass filter ( L P F ) should be applied to the source and the scatter data, as per the discussion to follow in Section 4.3.1. In this thesis, a conventional Butterworth lowpass filter was used. However, although this filtering has the effect of improving the look of the data, it is rendered ineffective in Equation 2.30 of the reconstruction Chapter 2. Theoretical Background 68 algorithm. Here the filtered scatter Fourier data are divided on a frequency-by-frequency basis by the filtered source spectrum, and since Butterworth filtering and division are both linear operations, the filtering effect is removed. Therefore, the data processing must be done in the 2D Fourier space of the image instead, after the division of the data by the source spectrum. Th i s analysis illustrates that any desired lowpass filtering must be performed along each ray of data in the Fourier domain of the image. Figures 2.12. A and 2.12.B illustrate this concept. The first figure illustrates the linear transformation from scatter data, Ps(rd> rs, LU), for a given view to a radial line of data, ipg((ps, k), in the Fourier domain of the image 2 . This was discussed in Section 2.2.2. This ray of data is a discrete function of radius, r , ih uv-spa.ce. The second figure illustrates that a /-dependent L P F for the scatter data can be similarly transformed into a r-dependent L P F for the radial data. This equivalent filter can then operate on each radial line of data in the image Fourier domain, resulting i n the desired lowpass filtering. / and r are related by Equat ion 2.35, and therefore the cutoff r 0 can be calculated from the cutoff / 0 by r 0 = ^ a s i n ( | ) . / 0 is determined by the spectrum of the source and scatter data, as per usual. Since the /-dependent and r-dependent lowpass filters are equivalent, they are discussed inter-changeably throughout this thesis. B y this discussion it is evident that / 2 has a corresponding l imi t r 2 in the Fourier domain of the image, and the lowpass filtering operation wi l l effectively nul l data beyond a radius of r 2 in uv-space. Information is, however, embodied in the fact that Fourier data beyond r 2 are negligible, and these data could potentially enhance the F F T inversion process that generates an image. However, retaining al l the data result in significantly more processing. For example, if the Nyquist frequency is 3 times larger than / 2 , then rmax is also 3 times larger than the band l imit T 2 , and 8 times more data are processed i f a l l data are retained. This requires significant computing power, particularly wi th 2 Recal l the definition of ip in Equation 2.26 Chapter 2. Theoretical Background 6 9 Data Four ier D o m a i n i m a g e Four ier D o m a i n (A) Filtering in Data Filtering in Image Fourier Doma in Four ier D o m a i n (B) Figure 2.12: Figure A illustrates the transformation from scatter Fourier data to a radial line of data in the Fourier domain of the image. Figure B illustrates the equivalence of lowpass filtering in the Fourier domain of the data and along radial lines in the Fourier domain of the image. Chapter 2. Theoretical Background 70 respect to the interpolation in Fourier space. In addition, it was indicated in Section 2.2.8 that i f a l l Fourier data are retained, the reconstruction of images wi th practical total widths is not possible given current computing power in the prototype scanner. A n alternative is to truncate most of the Fourier data that correspond to spatial frequencies beyond the r 2 region. This reduces the Nyquist frequency and increases A x for a more viable reconstruction routine. In testing the reconstruction algorithm in simulation, the truncation and non-truncation methods were compared i n Section 5.3.1. 2.3 P R O C E S S B E H I N D T H E R E C O N S T R U C T I O N O F E A C H V I E W This section analyzes what is represented by the data recorded at each angle. Recall that the detector records a voltage trace that is directly proportional to the ultrasound field reaching it as a function of time. The reconstruction algorithm subsequently deter-mines from what locations in space this sound energy scattered; it does this by analyzing time-of-flight information. Figure 2.13 illustrates the concept of T O F through a sketch of cylindrical pulse propagation and the detection of scatter from points immersed in water. A t any time ts after the excitation of the source, the ultrasound field in water is spread out along a fuzzy arc of the same width as the pulse and a distance ts • cw away from the source. Scatter is shown originating at Point A along this arc, which is a distance dA-^det away from the detector. Hence, this scatter signal requires a time of td = dA->det/cw to travel to the detector. The total T O F is therefore ts + td for scatter originating from Point A . However, this analysis leads to the conclusion that scatter originating along arcs through the tissue have the same T O F . Figure B illustrates a nearly linear arc of points that al l result in scatter with the same T O F equal to ts + td. A s such, each digitized time point of data wi th a given T O F could have originated from a set of scatter points on a corresponding curve of constant travel time, called an isochrone. To determine the shape of the isochrones, it is useful to first determine the shape of lines of equal scatter path length, with is the distance ultrasound travels in going from Chapter 2. Theoretical Background 71 Detector Isochrone Source Wavefront (A) Nearly linear arc of points that generate scatter with TOF =ts+td ts + 2AT from source . • ts + AT from source -ts from source -U - 2 A T from detector ^ - AT from detector from detector tut Detector Isochrones (nearly linear) Source Isochrones (nearly linear) I (B) Figure 2.13: Figure A illustrates the distance that an ultrasound wave travels in water before it is detected after scattering from an arbitrary point labelled A . Figure B illus-trates the locations of other points that result in scatter signals wi th the same T O F as scatter that originates from Point A . Chapter 2. Theoretical Background 72 the source to a scatter point and finally to the detector. One could compose this line on paper by connecting a string of a given length to two pins representing the source and defector. B y pull ing the string taut with a pen while drawing a curve, the result would be a curve of equal path length. Mathematically, this is an ellipse by definition wi th the source and detector located at the focii. B y extension, the center of the ellipse is situated at the middle of the line joining the source and detector. If the speed of sound varies little or not at al l along al l paths from the source or detector to every location on the ellipse, then the curve evidently also corresponds to an isochrone. If the speed of sound varies along different paths, as is true in the physical world, then the isochrone wi l l not be a perfect ellipse. This situation is mathematically complex and does not appear to be analyzed to date in the U C T literature. To illustrate the elliptical isochrones, consider an example in a Cartesian coordinate system that has been both shifted and rotated such that the ellipse center is at the origin and the focii are on the rc-axis. The focii must be located at the source and detector positions, rs and rd. This necessitates that the total distance from the source to any point (x, y) on the ellipse to the detector is a constant, C. Mathematical ly this can be stated as C = {(x- rsx)2 + (y- rsy)2}^ + {(x - rdx)2 + (y - rdy)2}^ (2.61) Also the equation of the ellipse is l = ( - ) 2 + ( f ) 2 (2-62) a b Equations 2.61 and 2.62 are difficult to solve for simultaneously, but this can be done numerically by cycling through different values of a and b unt i l C is constant for al l (x, y) on the ellipse. Figure 2.14 illustrates the partial curves of several isochrones for a typical source and detector geometry, wi th r s = (0, -100) m m and rd = ( -100, 0) mm. Note that the curves are nearly linear, particularly where they cross the perpendicular bisector of the line joining the source and detector. Chapter 2. Theoretical Background 73 Scatter Ellipses 60 40 20 0 E - 2 0 E, >• - 4 0 - 6 0 - 8 0 - 1 0 0 -120 - 1 0 0 - 5 0 0 50 x (mm) Figure 2.14: This figure illustrates the partial curves of several isochrones for a typical source and detector geometry, wi th r s = (0, —100) m m and = (—100, 0) m m . The circle represents the object that is being insonified. Note that the curves are nearly linear inside the object. Essentially there is one such isochrone for every digitized time point in the detected ultrasound field. If the imaging algorithm could reconstruct the data exactly without any simplifying approximations, i t would take the detected ultrasound contained i n each time point at a particular view angle and spread it evenly over its corresponding el l ipt ical isochrone. Sequential data points for a given view would yield sequential el l ipt ical curves in the image space. Analysis of al l data points for a given view would thus bui ld up a part ial image that consists of these curves. This analysis of the single view reconstruction raises two points. The first is that the imaging system has no resolution in the lateral dimension (at 90° to the perpendicular bisector of the line jo in ing the source and detector) since i t cannot distinguish any two points on a given isochrone. Each view extracts information in the axial dimension only Chapter 2. Theoretical Background 74 (along the perpendicular bisector of the line joining the source and detector). This is why numerous views must be combined to build up an accurate image, wi th each view defining the image along one direction. A s data are collected at more views, the image space is defined along more directions, which is the basic idea behind computed tomography. The second point to note is that in fact the reconstruction algorithm is based on several simplifying approximations which effectively "warp" the nearly linear (inside the insonified object) elliptical isochrones into straight lines that are parallel to the line joining the source and detector. This is evident in that each view yields a radial line of data in the 2D Fourier domain of the image. Radia l lines of data provide spatial information in the image only along that line, as is illustrated by the example in Figure 2.15. Figure A plots the Fourier data for one representative view, while the image corresponding to these data is plotted in Figure B . Note that Fourier data are shown in a zoom view, and that the 2D function contains only zeros beyond the viewing region up to u and v = 0.5 m m - 1 . More wi l l be said on this warping of the isochrones in Section 5.3 of the Results Chapter. 2.4 DISCUSSION OF ATTENUATION EFFECTS Attenuation has a significant effect on the imaging process and a correction for i t must be applied in order to obtain accurate and highly resolved images of attenuating media such as breast tissue. In order to achieve these images, a correction for attenuation due to tissue and water should be included in the reconstruction process. B o t h forms of attenuation are dependent upon two parameters that vary in any given insonification experiment. The first parameter is the angular wave number, or alternatively frequency, of the ultrasound field. When using pulsed fields, the source Fourier spectrum spans a range of frequencies, and attenuation increases with frequency. The second parameter is distance. Energy in the ultrasound field is absorbed by the propagation medium, and the greater the distance of travel, the more energy is absorbed. Attenuation results in a Chapter 2. Theoretical Background 75 FFT - View 1 - 6 0 - 4 0 - 2 0 0 20 4 0 60 x ( m m ) (B) Figure 2.15: Figure A plots the Fourier data for one representative view, while the image corresponding to these data is plotted in Figure B. Chapter 2. Theoretical Background 76 reduction of ultrasound field amplitude at every point in the pulse. Attenuation in tissue varies depending on composition. However, breast tissue falls under the category of soft tissue and is thus generally considered to have one average attenuation coefficient. This coefficient is a well-known empirical value, given by 1 d b (2.63) M H z • cm v ' which is quoted in numerous sources [43, 51, 87]. Given that frequency is / = kct/2ir, where ct is the average speed of sound in soft tissue, Equation 2.63 can be used to derive an expression for the attenuated ultrasound field amplitude in tissue, which is given by At = A0ATTt(dt,k) (2.64) ATTt(dt,k) = 10~xtdtk - O . l c t X t 4 0 7 r M H z - m m Here A0 is the ini t ia l field amplitude, k is the angular wave number of the ultrasound in units of m m - 1 , and ct is equal to 1.54 mm/ps . dt is the distance in millimeters that the ultrasound field travels through tissue. The attenuation coefficient for water is quite different and is quoted in Reference [87] as 0.00022 (2.65) db ' m m • M H z 2 Note that this expression is again based on empirical results. It can be easily derived through an analysis of the frequency dependence of half-value layer for water, which is the path length that reduces the intensity of the ultrasound beam to half of its original value. Substituting / = kc0/2n, the following relation emerges that predicts the attenuated amplitude of the ultrasound field in water as a function of dw and wave number: Aw = A0ATTw(dw, k) (2.66) ATTw(dw,k) = l 0 - * » d » f c 2 1.2 „2 -0.00022c 0 ! X w ~ 8 0 7 r 2 M H z 2 - m m Chapter 2. Theoretical Background 77 Frequency ( M H z ) dw(mm) A m p l i t u d e At tenua t ion (%) 1 50 0.13 4 50 2.0 10 50 11.9 1 100 0.25 4 100 3.9 10 100 22.4 1 150 0.38 4 150 5.9 10 150 31.6 Table 2.3: The above data are values of amplitude attenuation for ultrasound fields of different frequencies travelling through various distances of water. where c 0 , dw, and k are measured in units of mm//xs, mm, and m m - 1 , respectively. Reductions in ultrasound field amplitude as predicted by this relation are tabulated in Table 2.3 for various frequencies and values of dw. The attenuation coefficient in water is evidently dependent upon the square of fre-quency while that in tissue appears to be linear with frequency. It is important to keep in mind however, that the value quoted for tissue is only an average. In reality, the attenuation coefficient for tissue varies depending upon the type of tissue. For instance, that for muscle and tumour cells is actually dependent upon f 1 1 , while that for fatty tissue is dependent upon / 1 - 5 . The power that frequency is raised to varies due to tissue properties such as density, stiffness and viscosity [87]. Al though very litt le is currently understood about the mechanisms of ultrasound attenuation, the expressions for the at-tenuation coefficients have been explained to some extent by finite element analysis [87]. This essentially models the ultrasound propagation medium as an extensive system of damped, coupled oscillators. The type of damping applied in the analysis falls into three main categories: • Stiffness-proportional damping - dependent upon element stiffness (= Area x Elas-tic Modulus) and dependent upon f2. Chapter 2. Theoretical Background 78 • Mass-proportional damping - dependent upon element mass and independent of frequency. • Viscoelastic damping - dependent upon element fluid viscosity, and the correspond-ing attenuation can depend on frequency raised to any power from 0 to 2. Through finite element analysis, it has been determined that attenuation in water is exactly described by stiffness-proportional damping, while attenuation in tissue appears to be modelled primarily by viscoelastic damping [87]. C H A P T E R 3 A P P A R A T U S The U C T prototype scanner design is shown in Figure 3.1. The scanner design includes a P l e x i g l a s ™ water bath. The tissue being imaged is immersed in water because ultrasound is highly attenuated by air. The water couples the transducers to the object, thus allowing most of the ultrasound energy to travel freely into the object. The water bath houses two transducers, one to insonify the object and the other to detect scattered ultrasound. The transducers are supported by spindles that are individually rotated to allow for full circular motion of the source and detector around the tissue. It would be ideal to use an array of transducers that can each transmit and detect ultrasound as directed by the data acquisition system. However, the cost would be several times greater and is not warranted at the prototype stage. The two-transducer setup only adds to the data collection time and does not l imit the experiments that can be performed. Prototypes of similar design have been used successfully by other groups, and they permit a large range of experiments to be performed [10, 32, 53, 80, 88]. The source is controlled by a pulser-receiver module that applies a negative voltage spike to the leads of the transducer. Several different pulsed transducers were employed in the course of the prototype development in order to test the system. These included concave and convex cylindrical transducers, a focussed commercial source, and a line source, al l of which are described in Section 3.1. The detector transducer is a spot-poled reflector type hydrophone, which is discussed in Section 3.2. It converts the mechanical energy in the detected ultrasound field into a voltage signal. This signal is then input into an electronics setup composed of a preamplifier, the receiver (amplifier) portion of the pulser-receiver, and an analogue-to-digital converter ( A D C ) . The A D C digitizes 79 Chapter 3. Apparatus 80 Stepper Motor Unit G a g e s c o p e 6012 Board Panamet r ics 5 0 7 2 P R Pu lse r -Rece ive r Phantom Holder S E A Hydrophone and A D B 1 7 Preamplif ier Plexiglas™ Tank Dedicated P C for Data Acquis i t ion, S igna l P rocess ing and Image Reconstruct ion (? —<j\ I — 1 Figure 3.1: Schematic of the prototype pulsed U C T scanner. the amplified voltage signal to enable subsequent digital signal processing and image reconstruction. The A D C is a board that sits inside a 400 M H z Pent ium II P C wi th 192 M B of R A M , which controls the entire scanner setup. This computer performs sound wave generation, signal acquisition, signal processing, and image reconstruction. A l l data acquisition and data analysis software has been written in the M a t l a b ™ programming environment. Chapter 3. Apparatus 81 3.1 T H E S O U R C E S Recal l that the reconstruction method by Blackledge et al assumes the incident pulsed ultrasound field has the form of Po(r,w) = A(w)g(r\ra,u) (3.1) where g(r\Ts,u) is the 2D Green's function propagator and A(u) is the amplitude spec-t rum of the source. A s such, the ideal source for use with this imaging technique has zero phase at its location, r s , and its field spreads out cylindrically from this point in 2D. Attempts were therefore made to physically generate this source field through the use of several different transducers. The first attempts made use of cylindrical sources that produced a pulsed field that was laterally uniform over an angle of ± 3 0 ° . The pulse had a wavelength of approximately 2 mm. Two of these sources were kindly produced by M r . Jerry Posakony, a world expert in transducer technology who was formerly the manager of the Automat ion and Measurement Sciences Department at Batelle Pacific Northwestern Laboratories. These sources are not identical in that one has a convex face that produces a diverging field wi th a focal point behind the transducer face, while the other has a concave face that produces a field that converges at a focal point in front of the transducer. Unfortunately, both sources became corroded after a few experiments. This resulted in significant ringing that destroyed the concave source data signal. In the convex source the corrosion completely broke the connection to the piezoelectric element. Piezo Systems, Inc., of Cambridge, quoted U S $ 1 K - 2 K to reproduce the convex source, so it was decided at this time to pursue the experimental work wi th other available sources. Two pulsed line sources were kindly provided by Drs. Frank Podd and Inaki Schlaberg, who are wi th the University of Leeds Process Tomography Uni t which develops time-pf-flight tomography methods for non-destructive evaluation. A s described in Reference [78], these transducers each have an active element with dimensions 0.5 x 10 mm. The emission angle is approximately ± 3 5 ° , and the field strength wi th in this region falls off Chapter 3. Apparatus 82 wi th distance. However, these custom-made sources are incorporated into a specialized array in their current application. When used in the prototype U C T scanner, the sol-dered leads of the single transducer were in contact with water, which caused significant ringing and signal noise when backscatter mode was used. In addition, the sidescatter signal from these transducers was quite small and submersed in significant noise. The application of electronic sealant to the leads unfortunately d id not alleviate this problem sufficiently to allow for use of these sources in the scanner. However, it was possible to detect a signal 3-5 cm directly in front of the line source for use in an additional test of the source model embodied in the reconstruction algorithm by Blackledge et al. The results of this are presented in Section 5.6.3. Due to the above mentioned difficulties, the only source that was available for ex-perimental use was a Panametrics V326 transducer with a 5.0 M H z center frequency and 0.375 inch diameter flat and circular active element. Unlike the cylindrical source assumed in the reconstruction algorithm, this source produces a field wi th a strong main lobe on axis and weak side lobes. The side lobes were not of concern in the experimental work because only objects wi th small diameters less than 4 m m were investigated. The field was foccussed at ~ 25 mm. The near field was not investigated, nor were objects placed in this region for insonification, due to the erratic nature of the field according to transducer theory. Beyond the focus, the field strength decreased wi th distance due to beam spreading. The field had a full-width-half-maximum of roughly 10 ± 2 m m between the distances of 30 and 60 m m measured axially from the transducer face. The incorpo-ration of this source into the reconstruction algorithm is discussed in Section 5.6.2, and the corresponding experimental results are presented in Section 5.6.3. 3.2 T H E H Y D R O P H O N E The hydrophone in the prototype scanner, which is illustrated in Figure 3.2, is a Mode l SPRH-B-0500 spot-poled reflector type device that was built by Speciality Engineering Chapter 3. Apparatus 83 -RG174 C a b l e 0 . 0 9 5 0 -- 1 , 5 0 0 0 -F e n a l e SMC C o n n e c t o r 0 .2680 - C o b l e leng-th 8 ' -Figure 3.2: This figure illustrates the general design of the S P R H - B - a hydrophones (Copyright Speciality Engineering Associates). The width of the active element, a , is determined by the spot-poling process. A l l units are in inches. Associates ( S E A ) . This device outputs a voltage that is directly proportional to the pressure applied to it by an acoustic field, which is a characteristic termed piezoelectricity. The active element of the hydrophone is a disc that is 0.003 inch thick and made of the specialized ceramic known as Lead-Zirconate-Titanate ( P Z T ) . P Z T is very versatile in that its physical, chemical and piezoelectric characteristics can be tailored to specific applications. It is also chemically inert. The ceramic disk is encased in a flat stainless-steel t ip that is approximately 0.095 inches in diameter, which is in turn covered wi th a thin coating of a polymer called parylene to protect against corrosion due to water. The term "reflector type" in the name of the hydrophone model refers to the transducer design in which a ceramic element is backed by a material wi th a high acoustic impedance, which facilitates high sensitivity together with fast voltage rise times (ie: rapid reaction to the acoustic field) [8]. When a P Z T ceramic is manufactured, the many dipolar molecules that make up its composition are not aligned. Wi thout dipole alignment, the ceramic cannot exhibit piezoelectric properties and thus cannot detect acoustic fields. "Poling" is the process Chapter 3. Apparatus 84 Polymeric Insulating Bubble >3? PZTDisk Backing Coax Cable Poling Axis Figure 3.3: This figure is a closeup of the hydrophone t ip . It illustrates the P Z T disc and its backing, as well as the coax cable that carries the poling electrodes! of permanently aligning the dipoles by applying a large electrical potential across the ceramic. "Spot-poling" is a special process in which the activated region of the ceramic can be l imited to a tiny spot that is determined by the size of the electrodes that apply the electric field to the transducer. In the scanner hydrophone, this spot size is 500 microns. A small active area is desirable because the hydrophone wi l l then have a wider field of view. Figure 3.3 is a closeup view of the hydrophone t ip that clearly shows the P Z T disc and its backing, as well as the coax cable that carries the poling electrodes. The positive and negative electrodes are connected to the front (outside the hydrophone) and back (inside the hydrophone) sides of the P Z T disc, respectively. In the poling process, the P Z T disc and backing are placed in a dielectric oi l bath, which is then heated to above the P Z T glass transition temperature, Tg, but below the Curie temperature T c . A dc poling voltage, Vp, is then applied across the electrodes. W i t h the field s t i l l on, the temperature is slowly lowered to below Tg, and the field is subsequently removed. The applied electrical field is usually on the order of several k V / m m . After the poling process, the poling electrodes are Chapter 3. Apparatus 85 used to transmit the piezoelectric voltage that results from acoustic field detection. The dimension between the poling electrodes is called the poling axis, and the piezoelectric dipoles are aligned wi th this axis. After the P Z T active element has been poled, its dimensions wi l l change whenever it is insonified by an acoustic pressure field. Fields that compress the element along its poling axis (and therefore expand it along the perpendicular axis) w i l l cause the hydrophone to output a voltage with opposite polarity compared to Vp. Fields that compress the element perpendicular to its poling axis (and therefore expand it along the poling axis) w i l l result in voltage output that has the same polarity as compared to Vp. The induced voltage is directly proportional to the amount of compression/expansion to wi th in the operating limits of the hydrophone. For U C T experiments using pulses, it is desirable to have a hydrophone wi th a rela-tively flat response over the frequency band width of the source, which generally spans 100 K H z to 10 M H z . In other words, the hydrophone should output a similar voltage for a given amount of acoustic deformation at its face wi th little dependence upon frequency in the region of interest. If a detector response is not flat, then a frequency dependent correction must be applied to the spectrum of its output signal in order to determine the proper spectrum of the measured ultrasound field. The correction in turn introduces error into the data. S E A builds hydrophones wi th a relatively flat response by mounting the piezo-ceramic disk on a backing that has a matched or higher acoustic impedance, followed by termination into a high electrical impedance. A stainless steel backing was used in the hydrophone for the U C T scanner. Figure 3.4 illustrates the frequency response of the hydrophone, as calibrated by the manufacturer. The response is relatively flat unti l around 10 M H z , which is the first region in which the ceramic disk exhibits a resonance. This resonance occurs when the wavelength of the longitudinal motion set up in the ceramic is equal to twice the thickness of the ceramic disk [73]. After 10 M H z , sound energy is rapidly dissipated Chapter 3. Apparatus 86 H y d r o p h o n e R e s p o n e 0.8' 1 ' ' —' 0 5 10 15 20 F r e q u e n c y ( M H z ) Figure 3.4: This figure illustrates the frequency response of the SPRH-B-0500 hy-drophone, as calibrated by the manufacturer. through resonant losses in the ceramic. Below 10 M H z , a small dip of about -2 db is evident at 1.5 and 2.5 M H z . The manufacturer explains that this is due to radial resonances in the steel backing of the hydrophone [73]. A description of the hydrophone angular response requires application of the Acous-t ical Reciprocity Theorem, which states that the receiving characteristics of a transducer acting as a detector are identical to its transmitt ing characteristics while acting as a source [50]. Stated another way, the directivity function of an ultrasound source as a function of direction and frequency is identical to the sensitivity of that same transducer when i t is operated as a detector. Direct ivi ty simply refers to the amount of ultrasound that is radiated in any given direction about a transducer. Returning to the discussion of the hydrophone, the hydrophone inventor suggests that its angular response can be loosely described as the combination of the free baffle and r igid baffle models [73]. A free baffle is essentially a membrane stretched across a cylinder. The directivity function Chapter 3. Apparatus 87 of this type of transducer is directly proportional to cos(g), where q is the angle of in -cidence of the sound wave on the baffle relative to the normal [50]. O n the other hand, a r igid baffle is essentially a piston transducer, and it has a directivity function that is proportional to the following angular function [50]: D^2Jl(ka sin(g)) ka sin(g) Ji is the Bessel function of the first kind. The directivity function described by Equat ion 3.2 has a main amplitude lobe between the first two null values and side lobes beyond these values. This pattern results from constructive and destructive interference between sound waves that are generated by different regions.of the transducer face. W i t h actual piston transducers, the ultrasound field produced does not fall to zero but instead reaches some min imum value. If the hydrophone in the prototype scanner were operated as a source of ultrasound, its directivity function would be proportional to a function given by the combination of the two models [73]: 2 cos(g) Jyjka sin(g)) ka sin(<7) B y the Reciprocity Theorem, the receiving characteristics of the hydrophone are identical to the directivity function, resulting i n a theoretical sensitivity described by the expres-sion above. Figure 3.5.A illustrates the theoretical angular response for a frequency of 5.05 M H z (wave number of 21.5 m m - 1 in water at 20°C). The corresponding experimen-tal response was measured for a narrow frequency band of 5.0-5.1 M H z , and this result is plotted Figure 3.5.B. Note that both the theoretical and experimental plots illustrate a relative angular response, since Figure 3.5.A plots Equat ion 3.3 and the experimental waveform was amplified before processing. Evidently, there is good agreement between experiment and theory. The experimental response drops by no more than 75% of its maximum over an angle of ± 9 . 5 ° . Thus, i f the hydrophone were situated 200 m m away from an object, it could view objects with diameters of up to 65 m m wi th very l i t t le drop in response over the extent of the object due to angle. Chapter 3. Apparatus 88 Theoretical Hydrophone Angular Response - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 Angle (degrees) (A) Measured Hydrophone Angular Response - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 Angle (degrees) (B) Figure 3.5: This Figure illustrates characteristics of the S E A 0.5 m m diameter hy-drophone used in the prototype U C T scanner. Figure A illustrates the theoretical angular response for a frequency of 5.05 M H z (wave number of 21.5 m m - 1 in water at 20°C) , while the corresponding measured response for the narrow frequency band of 5.0-5.1 M H z is plotted in Figure B . Chapter 3. Apparatus 89 3.3 S T E P P E R M O T O R A P P A R A T U S It is desirable for a pulsed U C T scanner to be capable of sampling a scattered ultra-sound field at points that are very close together. Recall from Section 2.2.8 that image size is directly proportional to the number of views, Nv, and therefore inversely propor-t ional to the arc spacing between field sample points on the detector ring. Furthermore, Table 2.2 illustrated that values of Nv > 6400 are necessary for image widths that are comparable to the sizes of tissues that are imaged. Recall that i f the image width is much smaller than the tissue size, there is a source of error due to the significant amount of scatter that originates from tissue areas outside the image space but which is ignored in the image reconstruction process. For a typical source and detector distance of 200 m m and Nv = 6400, the arc spacing between consecutive placements of the source-detector configuration is only 0.196 mm. Note that fine transducer stepping is also a requirement in continuous wave U C T , but for a different reason. In this situation, image resolution is dependent upon how finely in space the scattered field is sampled while the source is kept stationary. If a typical 5 M H z source is used, its wavelength in water at 20°C is A = 0.296 mm, and the maximum possible image resolution is 0.5A = 0.148 m m [32]. In order to potentially achieve this resolution, the detector must record the scattered field at successive locations that are no more than 0.5A apart [32]. Given a typical detector distance of 200 mm, this would necessitate the recording of at least 8500 views. B y extension, image resolution of approximately 1 m m would necessitate the recording of 1260 views. The design of the prototype U C T scanner thus incorporated a mechanism that was aimed at allowing the movement of both the source and detector transducers in very small angular steps. Desired specifications were set at: • 8000 steps over 360° (step size ~ 2.7 minutes of an arc) • error of ± 5 % in each step (~ ± 8 seconds of an arc) Chapter 3. Apparatus 90 The suitability of gear mechanisms was init ial ly studied. Expert consultation from a custom gear house was sought, which was backed up by information in References [26] and [27] by Darle Dudley, who is a leader i n the field of gear design. It was determined that step sizes and accuracies on this order are generally beyond the realm of even most sophisticated, custom-made worm gears. This is primarily due to the fact that gear motion involves the sliding of pinion teeth against gear teeth. Therefore, clearance must be left to allow for lubrication, and this space limits the stepping accuracy. Custom-made precision gears have tooth position accuracies of about Aa; = 0.0003 inches [72]. A typical gear setup for the scanner would have been a 17-tooth pinion combined wi th a 187-tooth gear that has a diameter of 3.896 inches. Each component would have a tooth position error of Aa; for a combined error of 2 A x . Given a detector and source distance of 200 mm, this would translate to an error in the transducer positioning of approximately 0.062 m m . Unfortunately, this error is 32% of the desired step size at 200 m m for pulsed U C T . It was suggested that the best option was to incorporate a traction drive into the scanner. The apparatus, which is illustrated in Figure 3.6, consists of a small capstan and large drive ring that interact v ia friction between their smooth surfaces. Hence, they are also referred to as friction drives. Traction drives are used where low power and highly accurate positioning is required. The unit was designed to position the transducers in steps of 0.05° over a full 360° with good accuracy on the order of ± 1 0 — 15%. The 25 m m diameter capstan is connected to the stepper motor shaft, and it fits snugly against the 250 m m diameter drive ring. The motor moves the capstan, which turns the drive ring through frictional force. The drive ring is connected to a spindle inside the tank that holds a transducer. There are two transducers in the prototype system, so there are two drive ring/capstan pairs in the traction drive. The connections between the drive rings and the transducer spindles pass through the tank by way of a bushing mechanism with two concentric pieces, one for each drive ring. Al though the concentric bushing Chapter 3. Apparatus 91 To Transducer #1 •«-To Transducer #24-Spring Load Plates ^ Motor Plate Stepper Motor #2 Motor Plate ^Capstan #1 li I " Capstan #2 \ Stepper Motor #1 Portion of Water Tank / Bushing Drive Wheels Figure 3.6: This figure illustrates the traction drive in the prototype U C T scanner. mechanism has been used in other experimental U C T scanners, traction drives have not been used previously in ultrasound tomography [32]. However, these drives are common in precision positioning systems for large telescopes, such as those at the W . M . Keck Observatory i n Hawaii and at the K i t t Peak National Observatory in Ar i zona [90, 91]. Traction drives are used to enable precision movement in robotics. The interested reader is referred to Reference [89] for a list of related publications. Before the traction drive could be implemented, extensive calculations were required to determine i f the system would theoretically work. In brief, the main l imi ta t ion is that there must be sufficient frictional force at the contact zone of the capstan and drive r ing surfaces in order to transmit motion to the drive ring. The frictional force, Chapter 3. Apparatus 92 in turn, depends on both the coefficient of friction in the contact zone as well as the normal force applied to the drive ring by loading of the capstan. However, i f the normal force is too large, the metal surfaces of the drive ring and capstan wi l l experience plastic deformation, which leads to slippage of the traction drive. The study thus involves the analysis of rolling contact between two elastic solid cylinders, which is governed by the Hertz Theory of elastic contact. The key references used for the contact problem were Landau and Lifshitz [54], and the books by D a r l Dudley [26, 27]. The final result of the calculations indicated that a steel capstan combined wi th a steel drive ring would provide more than sufficient frictional force without plastic deformation. However, a major problem in the design of the drive was unfortunately discovered upon incorporating the traction drive into the U C T scanner. Recall that the drive rings are connected to the spindles through a concentric bushing mechanism. The outer drive ring is very easy to move and in fact can be rotated by the traction drive. In contrast, the inner drive ring is subject to substantial friction inside the bushing, which has increased over time due to corrosion inside the bushing. The force required to overcome this friction is significantly greater than what the current traction drive can provide. Given the costly nature of a new stepper motor apparatus, it was decided that a proof of concept study of the reconstruction algorithms would be conducted before additional funding is allocated towards improving the drive. A s such, it was decided to use cylindrical phantoms for the majority of tests. For these, the scattered field can be recorded for a few representative angles, and these data can then be copied for as many views as desired. This approach has been used by other U C T researchers wi th useful results [32]. In the case of imaging experiments wi th noncylindrical phantoms, the required number of views can be kept to a minimum by restricting the object size to a few mm. The drive rings can then be clamped together at a relative angle of 9, and data can be recorded for each view by manually rotating the apparatus. The error in positioning the source and detector that is associated wi th this method is estimated Chapter 3. Apparatus 93 to be ± 1 1 % . This error is relatively large due primarily to the accuracy wi th which the apparatus can be positioned manually. It should be noted that an attempt was made at holding the transducers stationary while rotating the phantom. A phantom holder was designed and carefully machined wi th the intent of accurately rotating the phantom with a stepper motor. However, the accuracy wi th which this apparatus held the phantom on center was not sufficient for use wi th this tomography method. Section 5.6.3 describes the large effect on image reconstruction that results from small errors in the measurement of the distance from the origin to the source or detector. The phantom holder d id not maintain enough accuracy in this distance to be of use. Because the traction drive ultimately did not operate as expected, the calculations involved in the Hertz Contact Problem have not been included in this thesis for purposes of brevity. A n y parties interested in the mathematics involved are referred to Citations [26, 27, 54]. The author of this thesis may be contacted for further information. 3.4 W A T E R T A N K D E S I G N The water bath was designed to ensure that reflections from the tank walls are eas-i ly separated from data through T O F analysis. A simulation of first-order ultrasound scattering in the absence of refraction was performed, in which a cyl indrical tissue 60 m m in radius was insonified by a pulsed line source that was band l imited from 1 to 10 M H z . A n analysis was done of sound waves in the following four categories, which are illustrated in Figure 3.7: • Scatter Data (SD) - Waves that were scattered from tissue and then detected by the hydrophone without further interaction. • Secondary Scatter (SS) - Waves that were scattered from tissue, followed by reflection off the tank and subsequent detection. Chapter 3. Apparatus 94 Figure 3.7: This schematic illustrates the four categories of detected sound that were considered in the simulation of scatter (scatter data (SD), secondary scatter (SS), tank scatter (TS) , and background field (BF) ) . The source transmits a cyl indr ical wave, some portions of which follow the dashed arrows shown. Addi t iona l arrows indicate example paths along which these wave portions scatter. • T a n k S c a t t e r ( T S ) - Waves that missed the tissue, but which were detected after scattering off the walls of the tank. • B a c k g r o u n d F i e l d ( B F ) - Waves that missed the tissue and reached the detector without scattering from any structures. The simulated source transmits a cylindrical wave, and in the absence of refraction, selected portions of its field follow the arrows shown radiating from the transducer. Addi t iona l arrows in Figure 3.7 illustrate selected paths along which these field portions Chapter 3. Apparatus 95 scatter. The analysis involved generating 1000 source field portions at random angles about the line transducer and tracing their paths as the field scattered in tissue or off the tank walls. Each scatter location was modelled as a cylindrical scatterer, which generates a scatter wave equally in al l directions. The times-of-flight corresponding to a l l possible travel paths in the four scatter categories were calculated and histogrammed. In doing this analysis, the frequency dependent field-of-view ( F O V ) of the hydrophone was considered. A transducer is said to have a F O V = ±77° at a certain frequency, / , i f it can detect an ultrasound field characterized by / when the corresponding wave fronts are incident upon its face at angles up to 77 to the normal. Recall from Section 3.2 that the hydrophone is modelled as the combination of a free and rigid baffle, wi th a sensitivity that is described by Equation 3.3. The nulls of this expression are determined by the directivity function of the rigid baffle, or piston transducer. The nulls defining the main lobe lie at an angle of ±77, where • ( \ 0 - 6 l A / Q A \ sm(?7) = (3.4) and a is the radius of the transducer. Most of the ultrasound energy detected by a piston transducer is contained in the main lobe, and the F O V is therefore typically defined as this angle. In the T O F analysis to determine tank size, sound waves wi th a given wavelength A were rejected i f the corresponding angle of incidence at the hydrophone was greater than ±77. Through T O F analysis, it is evident that the background field is not an issue in an imaging experiment. A t 9 — 180°, there is no background field at the detector location because a backscatter signal is being recorded. A t 9 = 90°, the background field arrives at the detector well before the scatter. W i t h the source and detector located a typical distance of 200 m m from the origin, the background field arrives at the detector wi th a T O F = ' ~ 190 ps. The first scatter signals for tissues with radii on the order of 5-7 cm, " on the other hand, arrive wi th a T O F = ~ 214 - 227 /is due to the increased distance the sound waves travel on the path from the source to the object and then to the detector. Chapter 3. Apparatus TOF Histogram for Different Types of Scatter 3500 96 0 100 200 300 400 500 600 700 800 900 1000 1100 TOF (microseconds) Figure 3.8: This figure illustrates the T O F histogram of scatter data (solid line - T O F = 160-220/zs), tank scatter (dotted line - T O F = 290-850>s) and secondary scatter (dashed line - T O F = 550-1000/is) for a typical situation in which the tank was 70 cm wide, \ra\ = 12.4 cm, | r d | — 15.7 cm, and the relative angle between the source and detector ranged from 0° — 90°. The tissue radius was 4 cm. In the simulation, tissues with radii varying from 3 to 7 cm were studied, together wi th typical source and detector distances between 12 and 20 cm. Since the type of U C T to be explored had not yet been decided when the tank was designed, the relative angle between the source and detector was allowed to vary between +90° and -90°, which covers both imaging setups for the methods presented in this thesis. W i t h respect to T O F , scatter data arrive first due to the smaller sound path lengths involved, followed by tank scatter and then secondary scatter. Figure 3.8 illustrates a typical T O F histogram of scatter data, tank scatter and secondary scatter for a simulation in which the tank size was 70 Chapter 3. Apparatus 97 cm x 70 cm, |r s | was 12.4 cm, |r d | was 15.7 cm, and the relative angle between the source and detector was varied between 0° and 90°. The tissue radius was 4 cm. A tank of width 70 cm was found to be optimal in terms of practical use and separation of data from tank scatter by T O F analysis. W i t h the broad range of source/detector distances and tissue sizes studied, the last data signal arrived at least 40 ps earlier than the first tank scatter signal. Secondary scatter generally arrived at the hydrophone about 400 ps after the scatter data were detected. Evidently the tank has a reverberation time, given by the time required for all scatter off the walls to be absorbed and die down in amplitude to the noise level of the amplifier. The tank was found to have a reverberation time of approximately 1150 microseconds, allowing for a new ultrasound pulse to be transmitted for data-acquisition rates up to 800 Hz . 3.5 T H E E L E C T R O N I C S 3.5.1 T H E P U L S E R - R E C E I V E R The U C T scanner includes a Panametrics model 5072PR pulser-receiver that is de-signed for use in thickness gauging, flaw detection, medical research, and materials char-acterization. The pulser section of the unit produces an electrical pulse that excites the piezoelectric source transducer to produce an ultrasound pulse. The excitation pulse is a negative impulse wi th an amplitude on the order of -100 V . The impulse amplitude can be varied by selecting the pulse energy to be 13, 26, 42 or 104 pJoules. The output impedance can be set at 8 specific values in the range of 15-500 Ohms, including 50 Ohms. The pulse rise time is ~ 5 ns at 50 Ohms output and minimum pulse energy. The unit has an External Trigger connection and a Sync Out signal. 3.5.2 C O M P O N E N T S B E Y O N D T H E H Y D R O P H O N E U p o n detection by the hydrophone, the ultrasound field signal is converted to a voltage signal, which is then input into an electronic setup composed of three units: Chapter 3. Apparatus 98 • A Model A D B 1 7 Specialty Engineering Associates preamplifier. • Receiver section of the 5072PR Panametrics pulser-receiver. • A Gagescope 6012 analogue-to-digital converter. The preamp is necessary because the hydrophone construction includes a high impedance RG174 output cable, which attenuates voltage signals significantly. This cable is thus only 20 cm long, and it is connected to the preamplifier, which has a gain that is typically 17 db into 50 ohms or 20 db into a high impedence. The preamp output is also connected to a low impedence coax cable, which can transmit voltage signals for long distances. The preamp has a high input impedance that facilitates good sensitivity as well as wide band width. It consists of a low noise, three-stage discrete transistor amplifier. The first stage is an F E T for high impedance, and the final stage is an emitter follower for dr iving 50 ohms at frequencies up to 25 M H z . A s illustrated in Figure 3.9, the gain varies from only 17.3 to 16.8 db between 1 and 10 M H z . The preamplifier is waterproofed by the manufacturer. Following preamplification, the signal enters the Panametrics pulser-receiver, which can apply a gain of -59 to +59 db. According to the manufacturer, this unit has a linear response wi th in the range of ± 1 volt. Furthermore, from 0.1 to 20 M H z , the receiver amplifier gain is flat to wi thin 2 db. Ga in curves for the 40 and 50 db amplifier settings are shown in Figure 3.10. The final component of the data acquisition system is the Gagescope 6012 A D C board wi th 12 bit operation and 1 M B of memory. This unit digitizes the voltage signal from the transducer so that the information can be input to algorithms for digital signal processing and image reconstruction. When the Gagescope 6012 is set to its lowest input range of ± 1 0 0 m V , the smallest voltage that can be detected is ± 0 . 0 5 m V . B y extension, successive memory values represent an additional voltage of ~ 0.05 m V , and the error associated wi th each location is therefore ~ 0.025 m V . When the pulser-receiver is set at a Chapter 3. Apparatus 99 Figure 3.9: Figures A and B illustrate a broad view and a closeup, respectively, of the frequency response of the Speciality Engineering Associates A D B 1 7 preamplifier. The gain is very flat and varies from only 17.3 to 16.8 db between 1 and 10 M H z . Chapter 3. Apparatus 100 Response of Panametrics 5072 Amp 601 1 . . 55h 35h 301 , , , 0 5 10 15 20 Frequency (MHz) Figure 3.10: Illustration of the frequency response of the amplifier port ion of the Pana-metrics 5072 pulser-receiver for the amplifier settings of 40db and 50db. The response is flat to wi thin 2 db from 0.1-20 M H z . gain of 40 db, the average magnitude of the points in a typical digitized ultrasound signal is on the order of 30 m V . Thus the percentage error in the signal voltage measurements is very low and on the order of ± 0 . 1 % . The Gagescope 6012 has an External Trigger input that is fed by the Sync Out signal of the Panametrics pulser-receiver. Hence, t ime zero is defined as the moment when the pulser excites the source transducer wi th a voltage impulse, and the A D C board begins recording the hydrophone output from this time onward. C H A P T E R 4 E X P E R I M E N T A L M E T H O D S It was necessary for several experimental methods to be incorporated into the pro-totype U C T scanner. A n approximate correction for frequency dependent attenuation was developed i n an attempt to obtain "first order" corrected images. A novel imaging algorithm was also developed, which is based on an algebraic reconstruction technique that includes the concept of an attenuated propagator. This is similar to the Green's function propagator, wi th the exception that factors are added that describe attenuation due to water and tissue. In addition, a preliminary study was done of conventional D S P methods for noise reduction. Finally, a study was done to compare several methods of interpolating Fourier data from a radial grid to a square grid. The method that imparted the least error was incorporated into the image reconstruction algorithms. The experi-mental methods studied or developed in this thesis are discussed in detail in the following sections. 4.1 A P P R O X I M A T E C O R R E C T I O N F O R A T T E N U A T I O N A n approximate attenuation correction was studied in an attempt to extend the method of Blackledge et al. For each new imaging experiment, a corresponding attenu-ation factor can be calculated that is dependent upon ultrasound wave number, k, the average distance ultrasound travels through tissue, dt, and the average distance that the waves travel through water, dw. Like the original reconstruction method, the approximate attenuation correction assumes that only first order scattering occurs. The correction method requires knowledge of an approximate tissue outline, which can be obtained from a simple backscatter image based upon T O F analysis. The tissue 101 Chapter 4. Experimental Methods 102 can be insonified at several view angles by a source that emits a pulse wi th a short length and narrow width. Some fraction of the pulse wi l l be reflected directly back to the transducer, which can be used to detect the backscatter signal. The T O F is simply where Ad is the distance between the transducer face and the tissue surface that lies directly in front of it. The transducer position is known for each angle, and therefore the tissue perimeter can be built up from the composite pieces determined from each view between 0° and 360°. A s described in Section 2.4, attenuation due to both water and tissue is dependent upon the distance that an ultrasound wave travels in each medium and the angular wave number being considered. If it is known for certain that a particular scatter signal travelled through exactly Dt m m of tissue and Dw m m of water, then the detected signal can be Fourier Transformed to yield the function FTd(k), and an attenuation correction coefficient, Q, can be applied to each Fourier component to yield new components FT c(fc) given by FT C(A;) = FTd(k)Q (4.2) Q = ATT71 (Dt,k) ATT'1 (Dw,k) where A T T t and ATT„, are defined by equations 2.64 and 2.66, respectively. However, the cylindrical source formalism embodied in the method by Blackledge et al makes it impossible to know dt and dw exactly, even for small time windows of the scatter signal. The analysis in Section 2.3 illustrated that pulses scattering from different points along ell iptical isochrones through the image space have the same time-of-flight. A s such, even a single point i n the detected ultrasound signal can have scatter subsignals that each travelled along paths wi th vastly different dt and dw. Thus it is generally quite difficult to correct for frequency and path-dependent attenuation. Chapter 4. Experimental Methods 103 It was hypothesized that an approximate attenuation correction could be developed by reducing the independent parameters to only frequency through the adoption of a single dt and dw for al l regions of the tissue. Fourier data that are approximately cor-rected, FTappr(k), could then be calculated from the original detected signal components, FTd(k), in the approach of Blackledge et al. The process involves the application of an approximate correction coefficient, Qappr that is the inverse of a rough attenuation coef-ficient, C, as shown in the following: FTappr(k) = FTd(k) Qappr = FTd(k) (4.3) C = A T T t ( d t = adt, k) x ATTw(dw = Bdw, k) = 10~Xt (Q^') f c x IO -*"* (P^w)k2 Recal l that Xt and Xw are defined according to Equations 2.64 and 2.66 in Section 2.4. The parameters a and ft were introduced in order to allow some degree of flexibility in developing a reasonably good approximate attenuation correction. A n attempt was furthermore made to chose good values of a and ft through the following analysis. The approximate correction was developed v ia a computer simulation of first order scattering in a 2D system in the absence of refraction. The source was assumed to be cylindrical , and i t radiated a short wave train (a few cycles of C W ultrasound) of a given frequency, / , in al l 2D directions about the transducer. Scatter from 200 random points in three 2D cylindrical tissues with radii of 30, 40 and 50 mm, respectively, was analyzed for a multitude of frequencies between 1 and 10 M H z . Assuming the tissue to be cylindrical was considered to be a reasonable approach since the breast would approximate a cylinder in the waterbath setting. The majority of the breast tissue would be situated wi thin a cylindrical cross-section, and hence most attenuation would occur within,this cylinder. A first order scatter event was defined as a unit fraction of a sound wave of a given frequency radiating from the source, traversing the tissue, scattering from a point and reaching the detector. The amplitudes of the scattered unit waves are termed scatter amplitudes (SA's) , and these were attenuated in simulation due to both Chapter 4. Experimental Methods 104 water and tissue. The scatter amplitude for each event was then corrected wi th two approaches. First , an approximate correction was applied to the detected scatter amplitudes, S A ^ , as per Equat ion 4.3 to yield S A a p p r : S A a p p p — SA^j Qappr (4*4) Secondly, a more accurate correction coefficient, Q R , was determined based on tracing wave paths from the source to the detector v ia each scatter point in the absence of refraction. A far more accurate dt and dw was thus obtained for each of the 200 first order scatter events. This yielded a rigorously corrected scatter amplitude, SA#, for each scatter event, given by: SAH = SAdQR (4.5) QR = ATT;1 (dt,k) ATT'1 (dw,k) where A; corresponded to the wave train in the event under consideration. SAappr and SAd were then compared to the more accurate SAR. Percentage errors were calculated for each scatter event as follows: I X - SAfll e = x 100% (4.6) | S A f l | where X was either SAappr or SAB-V i a the simulation, values of a and /3 were determined which minimized the error in the scattering amplitudes. These variables were the only experimental parameters, and al l remaining terms in Equation 4.3 follow from the empirical formulas for attenuation in tissue and water given by Equations 2.64 and 2.66. The simulation indicated that the S A error was minimized for al l tissue examples only when a — 1.5. Stated another way, the approximate attenuation correction performed best when the assumed distance of propagation through tissue was l.5a\ for every scatter point wi th in the tissue. In contrast, the optimal value of /3 varied directly and nonlinearly wi th tissue radius. The Chapter 4. Experimental Methods 105 R a n g e o f e for S A B (%) % o f S c a t t e r E v e n t s w i t h e i n R a n g e 0-50 6.2 50-100 4.0 100-500 4.4 500-1000 3.7 1000-50000 34.7 50000-100000 8.3 > 100000 38.7 Table 4.1: The above data illustrate the error in scattering amplitude wi th no attenuation correction applied. Each S A corresponds to a different scattering point and ultrasound frequency. best 6 values for the three tissue radii of 30, 40 and 50 m m were determined to be 1.9, 3.0 and 5.0, respectively. Different values of 6 resulted in a few S A error values that were greater than 100%. It is not surprising that the dw multiplier is generally quite large. The attenuation due to water is 1-2 orders of magnitude less than that due to tissue, and thus the water term in the correction is essentially acting a fine-tuning mechanism. The dw multiplier was found to dictate the percentage of S A errors that were greater than 100%. For instance, multipliers of 3 and 2.5 together wi th a tissue radius of 40 m m resulted in 0% and 0.15% of the S A errors being greater than 100%, respectively. The multiplier in front of dt affected the relative percentages of S A errors in the ranges of 0 - 50% and 50 - 100%. The results of an example simulation for a tissue radius of 40 m m are presented in Figure 4.1, where e is plotted for S A a p p r . O n this graph, one curve as a function of frequency is plotted for each of the scatter points. Table 4.1 presents values of e for S A B given the same scattering tissue. The approximate correction had a positive effect on the scatter amplitude errors. S A a p p r values for 100% of the scatter events had less than 100% error when compared to the rigorously corrected SA ' s . In the case of no attenuation correction, SAB for 89.8% of the scatter events had greater than 100% error compared to S A ^ . Furthermore, SAB values for 38.7% of the scatter events Chapter 4. Experimental Methods 106 Error with Approximate Correction f (MHz) Figure 4.1: This graph illustrates the percentage errors in scatter amplitude that result after the application of the approximate attenuation correction described by Equat ion 4.3. One curve as a function of frequency is plotted for every scatter point. had greater than 1 0 5 % error. Evidently, the approximate attenuation correction yields input data for image reconstruction that has much less error than wi th no correction at al l . 4.2 A R T W I T H A T T E N U A T E D P R O P A G A T O R S According to Equation 2.10 in Section 2.2.1, the scattered ultrasound field in two dimensions can be written as Ps(Td,Ts,u) = p(rd,Ts,u) -po{rd,Ts,u) (4.7) = A(u) k2 g(r\rd, fc)7K(r)#(r|rs, k)d2r -Chapter 4. Experimental Methods 107 Mu) f g(r\Td, fc) V • (7P(r)V#(r|r s, k))d2r after the application of the Born approximation and the introduction of a pulsed line source. Recall that the propagator, g(r\rj,k), describes both the reduction in field am-plitude due to geometrical spreading, as well as the phase change due to ultrasound propagation from r to Tj. It has the form g(r\rj,k) = -I-Hl0(k\r-rj\) (4.8) where HQ is the Hankel function of the first kind [81]. Equation 4.7 is a suitable starting point for the development of an algebraic reconstruction technique in which frequency dependent attenuation can be included. The resulting algorithm has the potential to reconstruct 7P(r) and 7K(r) for attenuating objects. 4.2.1 T H E A T T E N U A T E D P R O P A G A T O R If the distances the ultrasound wave travels through water and tissue are known to be dw(r, Tj) and dt(r,Tj), respectively, terms describing attenuation due to tissue and water along the path from r to r^ - can be expressed as A T T t = I0~xtdtk (4.9) ATTW = irrx»*"*a Recall that these expressions follow from Equations 2.64 and 2.66 in Section 2.4. The original propagator in Equation 4.8 can be transformed into an attenuated propagator of the form g{r\rj,k) = ATTw(dw(r,Tj),k)ATTt{dt{r,Tj),k)g(r\rj,k) (4.10) Recal l that in the nonattenuating case, the assumptions of far field imaging and use of a band l imited source enables the Green's function to be expanded as g(r\rjtk) = <*S (4.11) Chapter 4. Experimental Methods 108 s _ e x p ( i f c | r - r j | ) ( A ; | r - r j | ) l .exp(3nr/4) a = %-2V2TT Equat ion 4.11 can therefore be transformed into an approximate attenuated propagator of the form g{r\vj,k) = a 5 ( f c | r - r i | ) (4.12) - S ^ l r - r j - l ) - A T T M ( d U ) ( r , r i ) , A ; ) A T T t ( d t ( r , r i ) , f c ) 5 ( A ; | r - r J - | ) 4.2.2 P s IN P R E S E N C E O F A T T E N U A T I O N The expressions for g and S can be substituted into Equat ion 4.7 to derive an ex-pression for the spatially varying ps in the presence of frequency dependent attenuation. Using the simplified notation that STj — S(k\r — Tj\), the expression for the scattered field is Ps(vd,rs,oj) • = A(oo)a2 [ 5 r d { P 7 « ( r ) 5 r a - V • ( 7 p ( r ) V 5 r J } d 2 r (4.13) A l l that remains is the derivation of an expression for VSTj. The derivation of VS follows steps similar to those in the derivation of S in Appendix B , wi th the exception that terms due to attenuation in water and tissue are included. This exercise yields a new result that does not appear to have been previously cited. Firs t the operation of V on S is expanded to yield VS = VS ATTW A T T t + T l + T 2 (4.14) T l = S V { A T T „ , } A T T t T 2 = S ATTW V { A T T J (4.15) From the C . R . C . S t a n d a r d M a t h e m a t i c a l Tab le s , V ( 1 0 u ) can be writ ten V ( 1 0 u ) = ^ ^ - V u = 10u ln(10) Vu (4.16) du Chapter 4. Experimental Methods 109 Thus, VATTv = V{10~xM2} (4.17) = ATT„, • ln(10) • V(-Xw dw k2) = -xw k2 A T T ^ l n ( 1 0 ) V ^ Similarly, the expression for V A T T 4 is V A T T t = V{10-*tdtk} (4.18) = -Xtk ATTtln(lO)Vdt Substituting these expressions for T I and T2 into Equation 4.14 and recalling that VS = ikhjS (4.19) yields V S = SATTwATTt{ikhj-k\n{10)(xwkVdw-XtVdt)} (4.20) = S A T T ^ A T T * {ikAj — T3} Evidently, i f it can be shown that T 3 < ikn.j, then the attenuated propagator w i l l simply reduce to the original propagator multiplied by the attenuation coefficients in water and tissue. Term T 3 can be expanded as follows: T 3 = k\n(10)(xwkVdw-XtVdt) (4.21) , w-mw -0.00022c 2 - O . l c t = k\n(10){————r^—kVdw-———- Vdt} V ; 1807T2 M H z 2 • m m 407T M H z • m m - 3 = k2Vdw 1.4053 x 1 0 " 6 m m + Wdt 1.2255 x 10 where cw and ct are 1.48 m m / ^ s and 1.54 mm// is , respectively. Note that T 3 has the same units as the ikhj term of VS, as required. dw(r,Tj) and dt(r,Tj) are functions that are not known analytically but which must be determined numerically given r and Chapter 4. Experimental Methods 110 Tj. To a first order approximation, in which refraction is not considered, dw(r, Tj) and dt(r, Tj) are simply the distances travelled through water and tissue, respectively, along the straight-line path joining r and Tj. Calculation of these distances involves determining numerically i f in fact the straight-line path intersects tissue, and where these intersection points lie. In this thesis, this analysis was performed v ia a computer simulation that tracked the ultrasound paths. A 2D function of dw(r, Tj) and dt(r, Tj) was thus calculated numerically throughout sample image spaces, and from this, Vdw(r,Tj) and \Vdt(r,Tj)\ was also determined numerically. Results for r s = (150,0) mm, = (0,150) mm, 9 = 90°, and tissue radius equal to 40 m m are presented in Figure 4.2. Figures A and B of this composite illustrate the 2D dt and dw functions, respectively, while Figure C and D illustrate the | V d t | and \Vdw\ functions. Note that these graphs are intuitively correct since one expects the largest gradients to lie along the edges of the tissue "shadow" lines. Regardless of the values of 9, Tj and tissue radius, the range of minimum and maximum values for ^f- , ^jjf, and ^ always fell in the range of-11 to +11. A s can be seen in Figure 4.2, the maxima and minima occur in regions where ultrasound paths just begin to enter the tissue. Most value of the partial derivatives are in fact much smaller, wi th average values of and ^ lying in the range of 0.5 to 1.3. Note that a l l values quoted are unitless by definition since dt and dw have units of length. U p o n substituting Vdw = Vdw 10(x + y) into Equation 4.21, T 3 reduces to T 3 < « {k2 1.4053 x 1 0 - 5 m m + 1.2255 x 1 0 - 2 fc}(x + y) (4.22) Returning to Equat ion 4.20 VS " « S A T T ^ A T T t x (4.23) {ikhj - (k2 1.4053 x I O - 5 m m + 1.2255 x 1 0 - 2 A;)(x + y)} Given that < 4 2 m m _ 1 for a typical imaging experiment, the term in (x + y) is much smaller in magnitude than the imaginary component in the above expression. A s such, Chapter 4. Experimental Methods 111 Distance in T issue Distance in Water - 1 0 0 - 5 0 0 5 0 1 0 0 - 1 0 0 - 5 0 0 5 0 1 0 0 x (mm) x (mm) (C) (D) Figure 4.2: This figure illustrates the 2D functions of dt (A) , dw (B), \Vdt\ (C) and |Vd«,| (D) for a particular simulation in which r s = (150,0) mm, rd = (0,150) m m , 0 = 90°, and the tissue radius was 40 mm. Chapter 4. Experimental Methods 112 it follows that VS « A T T ^ A T T t ikhjS (4.24) « A T T ^ A T T j V ^ This yields the interesting result that the gradient of the attenuated propagator is ap-proximately equal to the original propagator in the absence of attenuation multiplied by the attenuation coefficients. However, it is important to note that this assumption implies that VSTj does not impart a phase shift when operating on an ultrasound field. However," the expression for VS must be examined in the context of Equat ion 4.13 to ensure that the term in (x + y) remains negligible after al l mathematical processing is complete. The approximation is addressed in more detail in Section 5.1.1. Results of a computer simulation analysis verifying the validity of this assumption are presented in Section 5.1.2. Substituting g for g, S for 5 , and VS for VS in Equation 4.7 allows an expression for the spatially varying P s in the presence of attenuation to be derived. The mathematical reduction proceeds as wi th no attenuation, as outlined in Equations A.23 through A.38 in Appendix A . The result is Ps(vd,rs,u) = a2A(u)k2 f oSrd(lK(r) + cos(6)LP(T))Srsd2r (4.25) which is similar to the result in the absence of attenuation, aside from the inclusion of the attenuated propagator, S. This expression can be made additionally accurate by substituting the exact attenuated propagator, g{r\rj, k), in the place of aSTj to arrive at Ps(vD,Ts,u) = A(u)k2 f p ( r | r d , A ; ) ( 7 K ( r ) + c o s ( ^ 7 p ( r ) ) ^ r | r s , f c ) d 2 r (4.26) 4.2.3 D E R I V A T I O N O F T H E A R T Although Equat ion 4.26 looks similar to the equation for the scattered ultrasound field from Chapter 2, it can not be reduced to the reconstruction algorithm by Blackledge Chapter 4. Experimental Methods 113 Figure 4.3: This figure illustrates the grid of pixels that is superimposed upon the image space for the purposes of A R T development. et al due to the presence of the nonanalytical functions dw(r,Tj) and dt(r,Tj). Instead, an Algebraic Reconstruction Technique can be derived from i t . In developing the A R T , a discrete coordinate system is first introduced as illustrated in Figure 4.3 by superimposing a square grid of pixels on an image space that includes the object being imaged as well as a small perimeter of background pixels. The grid size is M x N , and each cell has a location defined by r ^ . The grid represents a 2D image wi th elements defined by hj — 7/c( rij) + cos(0)7 p(rjj). It is assumed that the grid size is small enough such that the image is relatively constant throughout each pixel. A s described in Section 4.1, knowledge of the object outline can be obtained from a backscatter image taken wi th a Chapter 4. Experimental Methods 114 narrow beam source. Each cell is then labelled either wi thin or outside the object, and this information is mathematically represented in a spatial extent map. This map shall henceforth be termed an indicator function, T , after the work of Kaveh and Soumekh regarding the frequency domain interpolation problem [49]. Please note, however, that the work cited is unrelated to the current analysis. The indicator function has the simple form t 1, (xi,yj) G imaged object 0, elsewhere A n attenuated propagator, Sij, can be calculated for each pixel in the image. Equat ion 4.26 can then be discretized as follows: M N ps(rd,rs,oj) = A(u)k2Y^ Y,9{rij\Ts,k) Iijg^iAr^k) AxAy (A.21) »=i j=i A n alternative way of storing the grid and its associated variables is in the form of vectors. The grid is then stored as a vector of length L , and each cell has a location stored in the vector r^. The grid represents an image that is stored as a vector wi th elements defined by Ii = 7«(rj) + cos(0)7 P(rj), and each pixel of the image has a corresponding attenuated propagator, Si. The indicator function is also redefined as a vector given by T,-(r,) = 1, Tj E imaged object 0, elsewhere Since the input data are converted to a digital signal by an A D C , the wave numbers are in fact also discretized into Nk different values spanning both positive and negative fc-space. Corresponding to these are N f c angular frequencies, Wj. Using the notation g) = g(Tj\rs,ki)g(rj\Td,ki) (4.28) h = pS(Td,rs,ki) Ai = A(uji) = A(h) Chapter 4. Experimental Methods 115 the summation in Equation 4.27 can be rewritten as L bi = Aik2 53 g(Tj\Ts, h) lj <7(rjlr<2> ki) Ax Ay (4.29) i=i L J '=I Nfc such equations for the full set of angular wave numbers can be placed together in the following matr ix equation: b = T l (4.30) which becomes the following when expanded fully: Axk\g\ Axk\g\ . . A^fgl h b2 A2k\~g\ A2k\~g22 . . A2klg2L h • = AxAy .  b N K 4 » r P n N K 4 x r P n N l i A »r P ANKKNk9I /LNKKNKg2 • • ANK^Nk9L h (4.31) The vector / is readily available since it is measured during a U C T experiment. The gf functions can be calculated to a first order approximation in the absence of refraction by tracing the straight-line paths that ultrasound waves travel through the tank and determining where the paths intersect tissue. Therefore, the matrix system in Equat ion 4.31 can be solved iteratively to determine the image stored in the vector I. Knowledge of / at any one angle provides only a subset of the final image. These subsets must each be solved for iteratively and then combined together to yield a final image. The advantage of this approach is two-fold. First , data can theoretically be corrected for fre-quency dependent attenuation. Second, the exact propagator is employed, which should theoretically result in a more accurate reconstruction in comparison to the method by Blackledge et al. Chapter 4. Experimental Methods 116 4.2.4 I T E R A T I V E S O L U T I O N The main difficulty with this method is that the matrix system is very large. For instance, there are 10000 unknowns in an image of 100x100 pixels. Techniques do exist to readily solve sparse systems with on the order of 10 6 unknowns, which is 100 times larger than the system at hand. However, the attenuated propagator matr ix is generally not sparse. A s an example, wi th 10 M H z sampling, a tissue radius of 80 mm, and the source and detector located 200 m m from the origin, the elements of A range from 1 to 0.009, wi th 99% of the values being greater than 0.01. The feasibility of this method has been evaluated in in i t ia l tests in this thesis. In doing so, two iterative techniques were implemented: the conjugate gradient method and the Jacobi iterative technique. The formulas for both are outlined in Appendix E . It is noted that in order to obtain convergence, the formulas were modified from their classical forms to include the application of a boundary condition that arises from the definitions of 7 P ( r ) and 7 K ( r ) and from the fact that the image extends past the tissue boundary. Under this condition, it was assumed that the image was sufficiently larger in width than the object such that the pixels in the first and last row and column corresponded to water. A s such, these pixels could be assigned image values of zero. Wi thout including the boundary condition in the solution, the iterative methods either converge on invalid solutions or do not converge at a l l . Results of the feasibility study are presented in Section 5.7. 4.3 N E C E S S A R Y N O I S E R E M O V A L In the prototype scanner, the detected ultrasound signals are digitized and noise is subsequently removed through digital signal processing (DSP) methods. Ultrasound signals are generally quite weak in amplitude because attenuation reduces the sound field amplitude by ~ 11 - 68% over the frequency range of 1-10 M H z for each centimeter that it travels through tissue [43, 51]. This becomes problematic when the amplitude of the Chapter 4. Experimental Methods 117 recorded data is on the order of the background noise in the ultrasound signals. Several sources of noise are apparent in any experiment. For instance, there is noise inherent in the speaker and microphone electronics and also due to vibrations in the surroundings. Noise also results from the electronic circuitry of the computer in which the A D C board is housed. A n additional source of unwanted signals is evident in backscatter imaging, in which minute signals from air bubbles in the water make up part of the background noise. This source of noise can be overcome, however, i f a degassing procedure is introduced into the system. Lastly, signals from tissue include not only the sound reflected or scattered from tumor boundaries and tissue interfaces, but speckle noise caused by sound scattered from many small structures wi thin the tissue that are too small to be resolved by the imaging system. A l l these sources of noise degrade the data that is input to the reconstruction algorithms. The rapid development of more powerful and economical computers has facilitated the use of real-time digital signal processing to improve the data before i t is input into a reconstruction algorithm. A s with al l quantitative imaging methods, the quality of the image is l imited by the quality of the reconstruction input data. Digi ta l filters can be used to remove the noise, but care must be taken not to destroy useful information. 4.3.1 A N A L Y S I S O F N O I S E IN T H E S I G N A L Several sources were used in the testing of the prototype scanner. However, upon averaging at least 200 different signals recorded at the same location using the same source, the noise in the resulting average signal has similar, although not identical, fea-tures regardless of the transducer used. As such, only two examples wi l l be provided in this section. Figures 4.4.A and 4.4.B illustrate typical ultrasound signals after they have been amplified by the S E A preamplifier and Panametrics receiver and digitized by the Gagescope 6012. In Figure A , ultrasound was backscattered from a 3 m m wide alu-minum rod that was insonified wi th the concave source. In Figure B , ultrasound from the Chapter 4. Experimental Methods 118 Concave Source Data 2.51 1 . • — 2 1.5 1 - 1 . 5 -- 2 _ 2 5 l 1 1 ' <— 0 50 100 150 200 Time (us) (A) Panametrics Source Pulse 0.51 . . 1 0.4 0.3 0.2 -0 .2 -0 .3 - 0 . 4 -- 0 . 5 ' •—' ' • ' - J 0 10 20 30 40 Time (us) (B) Figure 4.4: Figures A and B illustrate two examples of signal t ime profiles. In Figure A , ultrasound was backscattered from a 3 m m wide aluminum rod after insonification by the concave source. In Figure B , ultrasound from the Panametrics source was detected after transmission through water. Chapter 4. Experimental Methods 119 Panametrics source was detected after transmission through water. To reduce noise, 200 and 300 signals were averaged together in the backscatter and transmission examples, respectively. In addition, the source ringing that occurs in the first 9 ps was removed by substituting the voltage points in this region with those in the next 9 ps. A dc offset equal to the average signal amplitude has been subtracted from each example. It was assumed that the noise is generally constant in time throughout the detected ultrasound signal. In other words, the noise detected before the arrival time of the first scatter data were assumed to be similar to that which underlies the data. Hence, to create noise vectors for analysis, points were extracted from between 0-150 ps and 0-25 ps for the backscatter and transmission signals, respectively. The region of data for each example was estimated to lie from 168-192 ps and 29-33 ps for the backscatter and transmission signals, respectively. The data regions for the concave and Panametrics sources were found to have spectra that were band l imited by approximately 10 and 20 M H z , respectively. W i t h the data and noise vectors, signal-to-noise ratios (SNR's) were calculated. Several formulas for S N R are used throughout research and engineering, and a standard one is applied in this work, given by: MA S N R = 1 0 1 o g 1 0 ( ^ ) (4.32) where MD is the mean of the power in the data region, and MN is the mean of the power in the noise. The S N R in the unprocessed signals were 26 and 27 db for the backscatter and transmission examples, respectively. The corresponding Fourier spectra of the noise vectors are illustrated in Figure 4.5. Bo th noise spectra are relatively flat throughout the frequency range, wi th a few exceptions. There are high frequency spikes at ~ ± 2 0 M H z and ~ ± 2 5 M H z for the concave and Panametrics source, respectively, and both signals exhibit a very low frequency peak spread between ~ ± 0 . 2 M H z . The high frequency peaks are likely related to noise in the circuitry of the individual sources. Note that the spectrum of the backscatter signal includes significant components in the range of -6 and 6 M H z , which overlaps wi th the data spectrum. This was likely due to backscatter from Chapter 4. Experimental Methods 120 Noise Fourier Spectrum - Concave -10 o 10 Frequency (MHz) 1.2 £.0.8 O.0.6 E < 0.4 0.2 (A) Noise Fourier Spectrum - Panametrics -30 - 2 0 - 1 0 0 10 20 30 Frequency (MHz) (B) Figure 4.5: Figures A and B the Fourier spectra of noise in typical backscatter and transmission signals, respectively. The backscatter signal corresponds to the concave source, while the transmission signal corresponds to the Panametrics source. Chapter 4. Experimental Methods 121 air bubbles that were floating between the source and the aluminum rod. This can be overcome in the second generation system if a degassing procedure is introduced. Also of note are the higher Fourier amplitudes between 10 and 20 M H z in the noise spectrum of the transmission signal. The source of this has not been identified, although it is suggested that it may be either electrical noise or acoustic noise in the form of harmonics The unfiltered noise amplitude has been histogrammed for each example, and the results are presented in Figure 4.6. The sparse nature of the histograms is due to the digi tal sampling of the signal combined wi th the low level nature of the noise. The histograms indicate that the amplitudes in both noise vectors have a somewhat Gaussian distribution that is not quite zero-mean. 4.4 C O N V E N T I O N A L D E N O I S I N G T E C H N I Q U E S A l l the sources used in this thesis work are band limited. The concave and Panamet-rics sources in particular have upper frequency limits of approximately 10 and 12 M H z . Beyond these limits, the information contained in the signal does not contain valid data. A s such, to avoid reconstructing noise rather than data, conventional filtering techniques can be applied to denoise the data. The most common of these is the lowpass filter, which gradually zeros data beyond a certain frequency. In the prototype U C T scanner, a lowpass Butterworth filter was applied to the signals, wi th an amplitude response of In the above equation, n is the order of the filter and / 0 is the frequency at which the filter response is -3 db. Generally n was chosen to be 20 to facilitate a relatively sharp frequency cutoff. A 20th order lowpass filter with / 0 = 10 M H z for the concave source and / 0 = 12 M H z for the Panametrics source was applied to the respective noisy signals. The filtered [71]. G(f) 1 (4.33) Chapter 4. Experimental Methods 122 1500 Histogram of Noise Amplitude ; 1000 § 500 -07015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 Amplitude (Volts) (A) Histogram of Noise Amplitude 14001 1 1 1 — 1200 J2 1000 o a. I i-800 0) E 600 400 200 [ 0 1 -- 4 - 2 0 Amplitude (Volts) x 10"' (B) Figure 4.6: Figures A and B illustrate the amplitude histograms of noise in typical backscatter and transmission signals, respectively. The backscatter signal corresponds to the concave source, while the transmission signal corresponds to the Panametrics source. Chapter 4. Experimental Methods 123 Fourier data were then inverse transformed and the data and noise vectors were again each example. The amplitude histograms of the noise in the filtered signals are illustrated in Figures 4.7.A and 4.7.B. Evidently, the noise in the backscatter signal is s t i l l quite Gaussian, although it is not zero-mean. The noise in the filtered transmission signal no longer has Gaussian statistics. The problem lies in the low frequency noise components wi th / < 0.3 M H z , which is an effect that can be seen in some U C T data. To alleviate this problem the transmission signal was further filtered wi th a Butterworth highpass filter ( H P F ) , which has the following amplitude response: Again , n is the order of the filter and / 0 is the frequency at which the filter response is -3 db. n = 20 was chosen for the transmission signal to facilitate a relatively sharp frequency rise. U p o n inverse transforming and extracting the noise, the noise amplitudes were histogrammed, as illustrated in Figure 4.8.A. The concave source backscatter signal was also filtered wi th the H P F and the residual noise statistics are illustrated in Figure 4.8.B. The residual noise in both cases is now zero-mean Gaussian. These examples indicate that the noise characteristics vary slightly from source to source. Addi t iona l analyses indicated that the characteristics also vary slightly from experiment to experiment. Al though Butterworth filtering is a firm beginning in terms of noise removal, Section 5.5 w i l l illustrate that it does not yield data that are accurate enough for U C T image reconstruction. In order to further improve the data, two approaches have been studied in this thesis work. One approach involved wavelet denoising, which is discussed in the following Section. The other approach involved the application of a moving average filter, which again falls under the category of conventional filtering methods. The moving average filter is presently the most commonly used technique for removing noise that lies wi th in data band limits. This is a smoothing technique that has been in use for a number of decades. The filter is easy to implement and understand, and it is also optimal for extracted from the resulting signal. Lowpass filtering afforded a 3 db S N R increase in 1 (4.34) Chapter 4. Experimental Methods 124 Noise Histogram - After LPF 8001 1 1 1 . 700 -0 .015 -0.01 -0 .005 0 0.005 0.01 0.015 Amplitude (Volts) (A) Noise Histogram - After LPF 1201 . 1 1— Amplitude (Volts) (B) Figure 4.7: This figure illustrates the amplitude histograms of the noise in the backscatter (A) and transmission (B) signals after the application of a Butterworth L P F of order 20. Chapter 4. Experimental Methods 125 Noise Histogram - After L P F / H P F 801 1 1 < ' r-Ampl i tude (Volts) (A) N o i s e H is togram - After L P F / H P F 8001 1 1 1 1 r-Ampl i tude (Volts) (B) Figure 4.8: These figures illustrate the amplitude histograms of the noise in the Panamet-rics source transmission signal (A) and the concave source backscatter signal (B) after the application of a Butterworth H P F of order 20, in addition to a previous application of an order 20 Butterworth L P F . Chapter 4. Experimental Methods 126 reducing white noise while retaining a sharp step response. Hence, it is a useful filter for denoising time domain transient signals such as that found in pulsed U C T data [79]. Implementation of the moving average filter begins with choosing a window width that is an odd integer, ra. The width indicates the number of points that are averaged together in each operation. When the filter operates on a datum, the window is centered over the point and al l data in the window are averaged. The center datum is then replaced wi th the result. The window moves through the data unti l each point has been replaced wi th an average. Points near the beginning and end of the data vector are handled in the same manner, except that only 1 to (m-l) points can be included in the averages. A s the window size is increased, the smoothing or denoising effect is enhanced. Figure 4.9 illustrates the effect of the moving average filter. The top figure illustrates the unfiltered spectrum of the data region in the Panametrics source example from Section 4.3. The bottom figure illustrates the spectrum after it has been processed wi th a 5-point moving average filter. The advantage is that noise is indeed removed, but the disadvantage is that the features of the data are substantially changed. 4.5 W A V E L E T THEORY Wavelets are mathematical functions that are used as basis functions in wavelet the-ory, just as sinusoidal functions are the basis functions in Fourier theory. Figure 4.10 illus-trates four examples of commonly used wavelets, namely Symlet 6, Symlet 8, Daubechies 9 and Coiflet 5. Al though wavelets can represent functions of any variable, in this work they are used in the time domain. A s such, time wi l l generally be referred to in the fol-lowing discussion. Wavelets and Fourier basis functions are similar in that they are both localized in frequency. The main differences between the two types of basis functions are that wavelets are of compact support (localized in space) and they are scale-varying, whereas Fourier basis functions are not [34]. The support of a time domain function is defined as the interval in continuous time outside of which the function is zero, and com-Chapter 4. Experimental Methods 127 -20 -15 -10 - 5 0 5 10 15 20 Frequency (MHz) 41 i i i i i i r 3 --20 -15 -10 - 5 0 5 10 15 20 Frequency (MHz) Figure 4.9: These figures provide an example of moving average filtering. The top figure illustrates the unfiltered spectrum of the data region in the Panametrics source example from Section 4.3. The bottom figure illustrates the spectrum after it has been processed wi th a 5-point moving average filter. pact support means that this interval is finite. Basis functions that are not scale-varying, such as sines and cosines, are used to represent a particular function over its entire time interval. Scale-varying functions, such as wavelets, represent functions only over intervals of varying temporal width and location [34]. For instance, i f a function is defined over the domain from 0 to 1, it can be analyzed at different scales. A t the largest scale it can be analyzed over two successive intervals: (0, 1/2) and (1/2, 1). It can alternatively be viewed at a slightly finer scale over four successive intervals: (0, 1/4), (1/4, 1/2), (1/2, 3/4) and (3/4, 1). A n d so on. Wavelets of different scale are used to represent the function over the intervals of different scale. Chapter 4. Experimental Methods 128 Figure 4.10: This figure illustrates some well-known wavelet functions. A ) Symlet 6 B) Symlet 8 C) Daubechies 9 D) Coiflet 5. Chapter 4. Experimental Methods 129 The idea of scale-varying basis functions was formed in the 1930's but it was not unt i l after 1985 that wavelets came into being with the work of Stephane Mal la t and Ingrid Daubechies. Since then, wavelet analysis techniques have been widely developed in various fields that include astronomy, geophysics, acoustics, nuclear engineering, elec-tr ical engineering, neurophysiology, music, magnetic resonance imaging, optics, radar, and pure mathematics. The techniques are used in a variety of problems, from signal processing and the solving of partial differential equations to image compression and speech discrimination [34, 57]. Fourier analysis is generally better for data that are continuous in nature. Pulsed U C T data, however, are transient in nature. In other words, the data are not sinusoidal and the corresponding pulses exist for a short time only. It thus follows that pulsed U C T data may be well-suited to mathematical representation by wavelets, provided a suitable family of wavelet functions can be determined. 4.5.1 W A V E L E T AN A L Y S I S A N D T H E W A V E L E T T R A N S F O R M In wavelet analysis, a particular family of wavelets is chosen to represent the function at hand. The members of the family, which define an orthogonal basis, are generated through dilations and translations of the "mother wavelet," which is also known as the "analyzing wavelet" [18, 34]. The rescaling is performed by a scaling function, and for the discrete wavelet transform used in this thesis, the rescaling is always in powers of two. The translations always place wavelet family members of the same scale in an end to end fashion wi th no overlap. Wavelet analysis then processes the function at the different scales or resolutions, and forms a representation of the function that is the linear superposition of the scaled and translated family members. This is the wavelet transform, and it is similar to the F F T only in that it is a linear operation. The major difference between the two transforms is the scale-varying nature of the wavelet transform. A t larger scales, wavelets of larger support are used in the analysis, and gross features Chapter 4. Experimental Methods 130 are analyzed. A t small scales, wavelets of small support are used, and fine details are analyzed. A s such, the wavelet transform is a multi-resolution operation, and it is better at processing signals with sharp, transient features than Fourier analysis [34, 57]. In wavelet analysis, the short basis functions are of high frequency and small support (therefore wide bandwidth) and are used to isolate signal discontinuities. The low fre-quency long basis functions are of large support (therefore narrow bandwidth) and are used to perform detailed frequency analysis. In practice, this multi-resolution analysis is embodied in the operation of two filters. One is a smoothing filter (like a moving average filter), and the other picks out the details of the data. Together they are termed a quadrature mirror filter pair [34]. Wavelets are classified by their number of vanishing moments, n, and it is this number that is included in the wavelet name. A wavelet has n vanishing moments i f and only if its scaling function can generate polynomials of degree < n [57]. In addition, the Fourier Transform of the wavelet is n continuously differentiable if the wavelet has n vanishing moments [57]. 4.5.2 P R O C E S S O F W A V E L E T D E N O I S I N G One of the many applications of wavelet analysis is the removal of noise from signals, which is a relatively new practice that has already shown promise in the processing of electromyographic signals [77]. In computer simulations, there has been success in using wavelet analysis to detect acoustic signals in noise without a prior estimate of what the signals look like [15]. When the mother wavelet and by extension al l the basis functions in the corresponding wavelet family mimic the shape of the data, wavelet analysis techniques can be used to extract the data from the noise. A s already discussed, the wavelet transform essentially filters data in two complementary operations, one which smoothes data and looks at gross features and the another which brings out details [34]. The result is a set of coefficients at each scale or resolution that indicates how to Chapter 4. Experimental Methods 131 represent the data at that resolution as the linear superposition of the corresponding scale-dependent wavelets. This is similar to the coefficients that are calculated in the Fourier transform, which indicate how to represent the data as the linear superposition of the sinusoidal basis functions. Given that the mother wavelet has features in common wi th the data, only a few coefficients wi l l be significant and many can be reduced or removed by thresholding. The remaining coefficients can be used in an inverse wavelet transform to reconstruct the data. The result is a cleaner signal that s t i l l exhibits important features of the data. In particular, sharp features are not smoothed in choppy signals by wavelet denoising, which is not the case wi th conventional filtering methods. These wavelet denoising tech-niques have been developed by Donoho et al, and they are included in the M a t l a b ™ programming environment [24, 25, 60]. These methods are particularly efficient in the removal of Gaussian white noise [24, 25, 60]. The "wden" function in the M a t l a b ™ programming environment performs the de-noising of a I D signal, x, using wavelets [60]. This function incorporates work developed over the years by Donohoe and his associates. To use this function one must first specify whether "soft" or "hard" thresholding is to be performed. Hard thresholding is a crude technique that simply sets to zero al l the wavelet coefficients below a certain threshold. Soft thresholding, which is more mathematically complex, gradually reduces the wavelet coefficients towards zero i f they are below a certain value (this is also referred to as "shrinkage" in the literature). The "wden" function takes as input a threshold selection rule for denoising through the selective reduction of wavelet coefficients. The selection rules can be any of the following: • rigrsure - thresholding using Stein's Unbiased Risk Estimate • minimaxi - minimax thresholding Chapter 4. Experimental Methods 132 Stein's Unbiased Estimate of Risk, which is used in the "rigrsure" method, involves the computation of a quadratic loss function that attempts to estimate the risk involved in choosing a particular threshold value [3, 60]. T h e r i s k is that valid data are removed by the thresholding procedure. Risk estimates are computed for various thresholds, and the threshold that minimizes the risk is chosen. One threshold is computed for the entire wavelet transform data set, and the selection is thus based on the entire data set [3, 60]. " M i n i m a x i " threshold selection is based on the Min imax Principle. This type of filtering minimizes the "worst-case" data estimation error under the assumption that there is no prior knowledge of the noise statistics [60]. This is in contrast to the well known K a l m a n filter, which assumes that the noise properties are known and minimizes the "average" data estimation error. The "sqtwolog" method is a variant of the "minimaxi" method. It uses the thresholding in the latter method multiplied by J2 l o g 1 0 (length(x)) [60]. Final ly, the "heursure" method is simply an automated combination of "rigrsure" and "sqtwolog" that is heuristic, or based on tr ia l and error. If the S N R is very small, "rigrsure" thresholding wi l l compute a noisy signal. If this situation is detected and "heursure" is chosen, M a t l a b ™ wi l l automatically switch to "sqtwolog" thresholding A l l the threshold selection rules are based on the noise model that x(t) = f(t) + e(t), where e(t) and f(t) are the time dependent Gaussian white noise vector and signal vector, respectively (option "one" input to "wden") [60]. Non-Gaussian and/or non-nonwhite noise are handled by multiplicative rescaling of the wavelet transform output. If the noise model is white but not Gaussian, the noise characteristics and amplitude are based on a single estimation using the first level coefficients (option "sin" input to "wden"). If the noise is both non-white and non-Gaussian, an estimate of the noise is performed at each resolution level (option "min" input to "wden"). The wavelet denoising routine calls the function "wdec," which performs the I D wavelet decomposition of the data using the specific wavelet family chosen. The analysis [60]. Chapter 4. Experimental Methods 133 can proceed to a scale in which the viewing window includes a min imum of two data points, and recall that the scales are increased by factors of two. Thus, the wavelet decomposition can potentially proceed to L levels, where the length of the signal is 2L. Signals wi th lengths that are not powers of 2 are first padded wi th zeros. A t each level, denoising is performed based on the chosen parameters. The inverse transform is then performed at each level to yield several denoised representations of the original input. 4 .5 .3 T Y P E S OF WAVELETS USED There are literally a myriad of wavelet families that could be chosen for the denoising of data. Standard wavelets, such as the Daubechies or the Symlet, can be used, and one can also define wavelet families that are custom adapted to a given set of data. Each wavelet family corresponds to a new filter, and hence a thorough analysis of wavelet filtering techniques is beyond the scope of this thesis. Prel iminary work has been done, however, to gain some idea of the potential usefulness of wavelet analysis for denoising pulsed U C T data. In this preliminary study, mother wavelets from the Symlet and Daubechies groups were studied. Members within the corresponding families are both orthogonal and biorthogonal [60]. There are only a few Symlet wavelets, which can be of integer order n equal to 2 through 8, where n is the number of vanishing moments. There are many more Daubechies wavelets, which can have an order equal to any positive integer. Symlets have the most symmetry of al l the wavelets, while Daubechies have very l i t t le symmetry [60]. A smooth, continuous mother wavelet is required to ensure that the wavelet analysis w i l l not introduce discontinuities into the data [24, 25]. More smoothness is generally implied by more vanishing moments (higher order n) [57]. A s such, the study used the relatively smooth Symlet 8 and Daubechies 9 mother wavelets, which were illustrated in Figure 4.10. In addition, these wavelets exhibit features that mimic pulsed ultrasound data. Results of the wavelet denoising study, and a comparison wi th use of the moving Chapter 4. Experimental Methods 134 average filter, are presented in Section 5.5. 4.6 INTERPOLATION M E T H O D S NECESSARY FOR U S E OF F F T The algorithm used to obtain images of minimally attenuating objects is a Fourier reconstruction method. Essentially, scatter data recorded at each angle about the tissue yield a subset of points in the Fourier space of the image. These points are situated on a radial grid in Fourier space, and must be interpolated to a Cartesian grid in order to do the inverse F F T to obtain the image. The interpolation of F F T data from the radial grid to a Cartesian grid is a somewhat difficult procedure that results in image reconstruction errors. A n idea of the effect of interpolation errors on image reconstruction was obtained through the following computer simulation. A sample image of 100 x 100 pixels was input to a F F T operation to obtain error-free Fourier data. The image was of a simple cylinder wi th a Gaussian cross-section. Errors were then added to the Fourier data, and the inverse F F T was performed on the substandard data. The locations of the radial and Cartesian grid points were assumed to be well known and free of error. It was assumed that the error in the interpolated data would increase in direct proportion wi th the decrease in the density of radial data upon which the computation is based. The sign of the error was generated randomly. The density of radial grid points can be calculated as follows, wi th reference to Figure 4.11. Here a small subset of points in the Fourier Transform of an image is shown, wi th data situated on a radial grid. There are = 4 source angles. The grid is thus defined by Nlfis spokes and by circles in Fourier space at discrete radii , r . The first and second circles are shown wi th radi i r\ and r 2 . The number of points in the ring defined by circles wi th radii T\ - S and r 2 + 5 is simply 2Ntpa, and the area of this r ing is « 7r(rf - T f ) . The density of points in the first ring is therefore Chapter 4. Experimental Methods 135 Figure 4.11: This schematic illustrates the radial grid point density in image Fourier space. B y extension, the density of points in the iVth r ing is IN Recall that NVt data points are superimposed at the origin of uw-space. Since a l l these data are used in the calculation of the center point, the density of data inside the Oth "ring" is thus 2N d = ^ (4.37) For a typical image wi th 200 NVt and r < 80 m m - 1 , the functions d and 1/d are plotted versus r in Figures 4.12.A and 4.12.B. The error is assumed to be proportional to 1/d and is thus linearly dependent upon radial distance from the origin in Fourier space. Chapter 4. Experimental Methods 136 Figure 4.12: Figures A and B illustrate the radial grid point density as a function of the wave number r, and the inverse of this function, respectively. Chapter 4. Experimental Methods 137 Figure 4.13 presents the results of a simulation in which errors were added to the Fourier data of an image. These errors were applied linearly wi th radial distance in Fourier space, wi th a minimum of 0% at the origin to a maximum of 20% at the largest distances. Figures 4.13.A and 4.13.B illustrate the original noise-free image and the corresponding Fourier Transform after adding noise, while Figures 4.13.C and 4.13.D illustrate the resulting noisy image wi th its corresponding residual image, respectively. Evidently, the interpolation errors have little effect on the image. In studying this problem, two general approaches were taken. The first involved the use of M a t l a b ™ interpolation methods that fall under the category of conventional numerical analysis. The second involved the use of a specialized interpolation routine known as the Unified Frequency-Domain Reconstruction ( U F R ) , which was developed by Kaveh and Soumekh[49]. These methods are explained and compared in the following sections. 4.6.1 M A T L A B ™ I N T E R P O L A T I O N The routine "griddata" in the M a t l a b ™ programming environment was evaluated as a means of interpolating Fourier data from the radial grid to the square grid necessary for the application of the inverse F F T . The points that "griddata" interpolates to usually lie on a square grid, which is where the routine gets its name. The underlying methods to choose from in "griddata" are: • linear - Triangle-based linear interpolation • cubic - Triangle-based cubic spline interpolation • nearest - Nearest neighbour interpolation The cubic spline interpolation is a common piecewise polynomial approximation. It produces a smooth surface wi th no discontinuities. In this method, the data are approx-imated by the fitting of cubic polynomials between each successive pair of nodes. This Chapter 4. Experimental Methods 138 Image - FT Data with No Error 46 48 SO 52 54 56 x (unitless) (A) Image - FT Data with 2 0 % Error 46 48 50 52 54 55 x (unitless) Fourier Data with 2 0 % Error tx (mm"1) (B) Residual Image - 2 0 % Error x (unitless) (D) Figure 4.13: Figures A illustrates the original noise-free image and its Fourier Transform after noise has been added. Figures B and C illustrate the resulting noisy image wi th its corresponding residual image, respectively. Chapter 4. Experimental Methods 139 method ensures that both the interpolating function as well as its first and second deriva-tives are continuously differentiable over the region of interest. The derivatives of the interpolant do not, however, agree wi th those of the original function, even at the data nodes. Linear interpolation assumes a piecewise linear function can be constructed be-tween any two neighbouring data points. Evidently, since polynomials are not involved, the method is simpler than the cubic spline interpolation. The interpolated surface is continuous, but it has discontinuities in the 1st derivative. The nearest neighbour inter-polation simply assigns to a grid point the value of the nearest node. Evidently, this is a rudimentary method wi th very little computation. The nearest neighbour produces a surface that is not smooth. 4.6.2 U N I F I E D F R E Q U E N C Y - D O M A I N I M A G E R E C O N S T R U C T I O N A second approach to the frequency domain interpolation problem is the Unified Frequency-Domain Reconstruction. This method interpolates data in the frequency do-main pursuant to the assumption that the image is of finite support (spatially l imited). The spatial l imitat ion is embodied in an indicator function, i, where %{x,y) = \ 1, (x,y) e imaged object 0, elsewhere The indicator function has a Fourier Transform denoted by I(u,v). If the object is a circle of radius r, then the F T of the indicator function is = (4.38) r y « -I- v where J x is the Bessel function of the first k ind of order 1. The U F R method begins wi th knowledge of the Fourier Transform, F(u,v), of an unknown 2D function, f(x,y), that is expressed in terms of Cartesian coordinates. / can be written as f(x,y) = i(x,y)f{x,y) (4.39) Chapter 4. Experimental Methods 140 Taking the Fourier Transform of both sides and applying the Convolution Theorem yields F(u,v) = FFT(i(x,y)f(x,y)) (4.40) = I(u,v) * F(u,v) The definition of convolution results in F{u,v) = jl I(u-u',v-v')F(u',v')du'dv' (4.41) Also , measured data for F{u', v') are only known at certain points on a non-Cartesian grid, which can be defined in terms of a transformation, T , as follows: u v T(x,r) Ti(x,r) T 2(x,r) (4.42) In this project, T(X,T) = COS(Y) sin(x) T sin(y) cos(x) 0 (4.43) du' du' dx dr dv' 8v' dx dr (4.44) for 0 < x < 27r and 0 < r < rmax. The transformation can be of any general form, however. The Jacobian for this particular transformation is therefore cos(x) -TCOS(X) sin(x) rcos(x) Changing variables form u' and v' to x a n d r in Equation 4.41 yields the following result: F{u,v) = j j I i u - T u v - ^ F ^ T ^ r d x d r (4.45) Equat ion 4.45 can be discretized, in keeping with the data that are available, to arrive at F M = Y,T,I^-Tuv-T2)F(T1,T2)rAXAr (4.46) allr allx Chapter 4. Experimental Methods 141 Equat ion 4.46 can now be used to calculate a Fourier Transform datum at any (u, V) pair based on the known values of data at the points defined by x a n d T. 4.6.3 SINC-BASED INTERPOLATION A sinc-based interpolation method for direct Fourier reconstruction has been devel-oped by Stark [82]. Al though this is considered to be an important addition to the field of Fourier-based reconstruction, Kaveh and Soumekh claim that the U F R approach , performs better than the sinc-based approach [49]. Furthermore, the approach is highly intensive computationally, making its incorporation into a viable imaging system very difficult [70]. 4.6.4 COMPARISON OF T H E INTERPOLATION M E T H O D S A Gaussian object characterized by an F F T wi th radially-situated points was inter-polated using the various methods described. Figure 4.14.A illustrates the profile along the it-axis in the original data. In this profile, there are 400 discrete radial points, and in the data to be interpolated, there were 200 such profiles at equally spaced angles. These data were interpolated to a Cartesian grid of size 50 x 50 points to yield a 2D function, finterp- The distance from the origin was calculated for each Cartesian point, and the corresponding Gaussian amplitude was calculated to determine the expected F F T value at each point, given by / . This was compared to the interpolated F F T value, and an error term in the form of a squared residual was calculated as follows: residual 2 = \finterp - f\2 (4.47) The error distributions for a l l three M A T L A B ™ interpolation methods as well as the U F R method are plotted in Figure 4.14.B. This plot shows the percentage of Cartesian grid points for which the residual 2 was less than some value e, where e is on the x-axis. Comparing the data for al l four methods, it is evident that the M A T L A B ™ methods Chapter 4. Experimental Methods 142 Radial F F T Profile 150 co V c^ -rt 100 "D in cr = 50 o (A) Error Plot For Interpolations / • • i 10 10" 10 (B) Figure 4.14: A Gaussian object defined by a radial grid was interpolated using the various methods. Figure A illustrates the profile along the w-axis i n the original data. In this profile, there are 400 discrete radial points, and in the data to be interpolated, there were 200 such profiles at equally spaced angles. Figure B plots the percentage of Cartesian grid points for which the residual 2 was less than some value e, where e is on the rc-axis. Shown are the results for the M A T L A B ™ interpolation methods and the U F R method. Chapter 4. Experimental Methods 143 are superior to the U F R method. The best method is the M A T L A B ™ cubic spline interpolation, for which only 4.0% of the points had a residual 2 > 10~ 6 . The second best method is the M A T L A B ™ linear interpolation, for which only 6.9% of the points had a residual 2 > I O - 5 . CHAPTER 5 RESULTS In the course of this research, extensive theoretical tests as well as a key tissue phan-tom experiment were performed to determine the capabilities of both the method by Blackledge et al (Direct Fourier Method), wi th and without the approximate attenuation correction, as well as the algebraic reconstruction technique wi th attenuated propagators. The theoretical tests were accomplished v ia computer simulations of scattering, while the experimental work involved the construction of a phantom and the subsequent collection and reconstruction of data from a waterbath setting. In analyzing the behavior of the reconstruction algorithms, point-spread-functions (PSF's) were investigated. This in -volved the simulated scattering of ultrasound from one or more infinitesimal points. The image reconstruction process in turn produced a Gaussian-shaped P S F , rather than an infinitesimal spike. The width of the P S F is of interest in evaluating the reconstruction algorithms. The P S F can be described by its full width at half its maximum ampli-tude (full-width-half-max, ie: F W H M ) . The F W H M , in turn, is a measure of the image resolution of the reconstruction algorithm. Various results are presented in the following sections, and key points are outlined here as follows: • In the algorithm derivations, the simplification {VSTj « ikhjSTj) has no discernible effect when imaging objects with «/p values in the range of human tissue. • The simplification ( V 5 F j . « ikhjSTj) (case of attenuation) has no discernible effect when imaging objects wi th n/p values in the range of human tissue. • Dispersion results in the incorrect calculation of scatter T O F ' s by up to 0.5 ps, as well as the incorrect determination of pulse shape. • The cylindrical source model describes line sources accurately to wi th in only 3-5%. 144 Chapter 5. Results 145 The Direct Fourier Method yielded the following results in computer simulations: • The simplification (STj « exp(ik(fij • r + \Tj\))/yJk\rj\) results in a miscalculation of scatter T O F ' s by up to 2 /xs. . • The T O F error due to both this assumption and the isochrone warping discussed in Section 2.3 results in distorted, halo-shaped P S F ' s wi th reduced amplitudes. • The T O F error worsens as \r\/a increases, where a is the distance from the origin to the source or detector, and r is the position vector of any scatter point. • Dispersion distorts P S F ' s into low amplitude halos. If the dependence of acoustic speed upon frequency can be determined, images can be corrected reasonably well. • Va l id jp and jK cross-sections can be generated only in l imited cases for which |r| < 0.05a and in which there is no attenuation. • For these l imited cases, the algorithm exhibits image resolution of ~ 0.07 m m • For these l imited cases, the algorithm reconstructed point amplitudes quantitatively to wi thin 97 ± 3% and 95 ± 4% of their actual values for 7 K and 7 P , respectively. • Beyond |r| < 0.05a, P S F ' s become distorted into low amplitude halos. • Images improve directly with the number of views when r 7^  the origin. • The approximate correction using dt and dw failed. Using the Direct Fourier Method, a key experiment was done in which an agar-graphite cyl indrical tissue phantom was imaged. The following results were obtained for the phantom, which had a negative 7 K and R and a diameter of 3 mm: • The resulting yK and R cross-sections were primarily negative. The F W H M values were 2.94 ± 0.06 m m and 1.94 ± 0.04 for 7« and R, respectively. • A valid 7 P image could not be calculated due to very narrow P S F for R. • P S F ' s were severely distorted by errors of only 1-2% in the measurement of a. The following results were determined v ia computer simulations of the A R T : • Va l id jp and 7 K cross-sections were generated wi th the A R T for relatively large values of |r| (tested up to |r| = 0.3a) and in the presence of strong attenuation. • For these cases, the A R T exhibits image resolution of ~ 0.05 mm. Chapter 5. Results 146 • For these cases, the A R T reconstructed point amplitudes quantitatively to wi th in 100 ± 14% and 99 ± 4% of their actual values for ^ K and yp, respectively. • Strong attenuation can be corrected to within 0.2%. • The A R T has difficulty converging for images of more than 20 x 20 pixels (~ 10 m m x 10 mm) due to ill-conditioning of the propagator matrix. Finally, this work lead to the following key results regarding noise filtering: • In simulation, wavelet denoised data required smoothing to remove residual chop-piness, resulting in an image improvement of only 10% over data that was only smoothed. • In experiment, wavelet denoising with the Daubechies 9 and Daubechies 20 wavelet families offered no improvement of data compared to lowpass filtering. Details of these and other results are presented in the sections to follow. 5.1 E F F E C T OF VARIOUS ASSUMPTIONS From time to time in the derivation of the image reconstruction algorithms, simplify-ing assumptions have been applied to render the mathematics tractable. In this section, the effect of three major assumptions wi l l be presented. 5.1.1 T H E ASSUMPTION THAT WSTj « ikhjSrj This assumption was applied in the derivation of the method by Blackledge et al. In Appendix B it was shown that V 5 r j . = n j { i f c - i ^ — } 5 R J . (5.1) after which the real term was dropped due to its small magnitude relative to the imaginary term. This procedure is cited in the work by Blackledge etal as well as Norton and Linzer [10, 66]. However, this implies that VSrj does not impart a phase shift when it operates on an ultrasound field. Hence, the assumption holds more meaning than i f the terms Chapter 5. Results 147 being compared in magnitude were either both real or both imaginary. In addition, VSTj is further operated on in the remainder of the reconstruction algorithm derivation. Hence, it is good judgement to verify that indeed the final effect of the dropped term remains negligible. Appendix C outlines the derivation of an expression for the scattered field, P s , when Equat ion 5.1 is used. To reiterate, the result is Pa(rd,rs,uj) = cx2A(uj)k2x (5.2) / S(k\r - r.|) ( 7 „ ( r ) + * 7 „ ( r ) ) S(k\r - rd\) d 2 r where $ = cos(0)(l + Q ' ° 1 4 2 8 ' m m - 1 - 0 .00714 2 mm- 2 ) (5.3) Recall that this expression is being compared to the one based on the approximate VSrj, in which $ is simply equal to cos(0). In order to determine the effect of using the approximate V i S r •, a computer simulation was performed in which a pulsed ultrasound field was scattered from a very small point wi th compressibility 7 K i and density 7 P i . The point was located at r = (0, 0), and it was assumed to be a square of width dx. The magnitude of dx was assumed to be determined by the Nyquist frequency in the data. A s discussed in Section 2.1, when a transient ultrasound field insonifies a region, the maximum spatial resolution that is possible from the data is \Nyq/2, which is one half the wavelength of the Nyquist frequency in the system being investigated. The grid on which the discrete scatter points lie in a computer simulation must then have a spacing, dx, of at most A J V ^ / 2 in order to closely approximate a "continuous" object relative to the wavelengths in the ultrasound field. Each individual scatter point then has a square element of area associated wi th it, wi th dimensions d x x d x , from which ultrasound is assumed to scatter. The study at hand considers only one such scatter point with a square element of area of width AJVJ , 9 / 2 . The small scatter point was assumed to be immersed in a nonattenuating fluid. Since for a l l intents and purposes the ultrasound in simulation traveled through only a uniform Chapter 5. Results 148 l iquid wi th a speed of sound equal to that of water, refraction was not an issue due to the lack of acoustic impedance interfaces. The grid spacing was thus equal to . The source and detector transducers were located a distance of a = 100 m m from the origin, wi th a relative angle of 9 between the beam direction and the detector angle. The coordinates of the source were fixed at r s = (0, —a). The insonifying field was simulated from a line source that produced a Ricker wavelet defined as in Equat ion 2.46 by Yr = ^ { u 2 - \ } e W ( - u 2 ) (5.4) In Equat ion 5.4, recall that time t ranges over both negative and positive values, wi th the simulation experiment starting at t = 0. Also, / 0 is the peak frequency of the wavelet. Figure 2.8 in Section 2.2.6 illustrated the time profile and the Fourier spectrum of a Ricker wavelet with OJ0 = 7.5 M H z , which has a corresponding pulse length of ~ 2.6 mm. Using the 2D approach, Equation 5.2 can be transformed to a summation over scatter points. Since only one point was considered in this computer simulation, the resulting expression for the detected scatter field, pa, is simply Ps{rd,rs,cu) = a2A{oj)k2 5(fc|r - r s | ) ( 7 k 1 + $ 7 p l ) S(fc|r - r d | ) d x d y (5.5) = a2A(oj)k2S(ka)(^Kl + ^ pl)S(ka)dx2 (5.6) Note that dy has been replaced with dx since the element of area is square. The above expression can be further improved by substituting the exact propagator g(r\rj,k) for the approximate one given by aS(k\r — Tj\). This results in ps(rd, rs, OJ) = A(u) k2 #(r|r s , k) (yKl + $7pi) 0(r|r r f, k) d x 2 (5.7) Recall ing from Section 2.2.1 that the 2D Green's function is ^(r | r j ,A;) = - ^ 0 1 ( A : | r - r J . | ) (5.8) Chapter 5. Results 149 yields the following expression for the exact scattered field from the small point: pes(rd,ra,u) = A(OJ) k2H^ka) ( 7 „ i + $ 7 p i ) H^ka) d x 2 (5.9) = ~ A(UJ) k2 Hl{ka) Hl{ka) d x 2 x { 7 k 1 + cos(0) 7 / 9 l + cos(0) ( ° - ° 1 4 2 8 t m m - 1 - ^ 0 . 0 0 7 1 4 2 m m - 2 ) 7 p l } In contrast, the reconstruction algorithm assumes that the point produces a scatter field given by the approximate expression Ps(rd, rs, OJ) = a2A(oj) k2 S(ka) S(ka) d x 2 ( 7 k 1 + cos (0) 7 p i ) (5.10) The fields described by Equations 5.9 and 5.10 were compared in the computer sim-ulation to determine the effect of the simplifying assumptions. The first point to note is that the approximate field is nearly identical to the exact field when 0 = 90°. This results because $ = 0 at this value of 9 and also because aS is an excellent approxima-tion to the exact Green's function propagator, as discussed in Section 2.2.6. A s such, compressibility image reconstruction based on detected side scatter can potentially give good results wi th the original method of Blackledge et al. The second point to note is that the error term in P s , which is e = cos(0) ( ° - 0 1 4 2 8 i m m - i _ _L 0 . 0 0 7 1 4 2 m m - 2 ) 7 , 1 (5.11) has the largest magnitude in both backscatter and transmission experiments, in which |cos(0)| = 1. Thus, backscatter imaging was investigated in this simulation as the worst case scenario regarding error in the model. Furthermore, the error term increases in magnitude directly wi th the value of | 7 p i | . A value of \jpi\ = 0.2 is considered relatively large for soft tissue, which has densities similar to that of water [10, 43]. A s such, 7 p l was kept less than or equal to this value. Lastly, within the case of backscatter imaging, the effect of the error term is evidently largest when e increases in magnitude relative to the ( 7 „i + c o s ( | ) 7 p i ) term. It was therefore expected that the error in the ultrasound field would grow infinitely large when 7 / t l is equal to 7 p i . Chapter 5. Results 150 The computer simulation investigated the error in ps for three values of 7 p l , equal to {0.05,0.1,0.2}, and —0.2 < jKi < 0.2. Since tissue is also similar to water in its compressibility characteristics, this range generally covers most situations likely to be found in practical imaging situations [10, 43]. For every (7^1, 7 p l ) pair, the Fourier data pes and ps were computed, together with the residual Fourier spectrum, p r = p e s - P s (5.12) A measure of the error in the Fourier data was obtained by computing the ratio of the sum of the power in pr relative to that in pes, given by: Q i = Jr^ 4 x 100% (5.13) The denominator of Q i goes to zero when 7 K l is equal to 7 P i . A s such, measures of error regarding the change in Fourier amplitude and phase were also calculated. The amplitude and phase of the residual field, pr, is simply the change in Fourier amplitude and phase that results from the application of the simplifying assumption that V 5 F j . « iknjSTj. Therefore, the amplitude of pT was summed over frequency to yield Q2 as follows: Q 2 = S|Pr| (5-14) A s well, the phase angle for each frequency was calculated by * » ™ { f ) = t a n ( r e a l M / ) ) } ( 5 1 5 ) and the absolute value of $phase was summed over frequency to yield Q 3 as follows: Q 3 = £ | $ p w | (5.16) The simulation work indicated that these measures of error depend only on ^r 1 and not on the specific values of the two parameters. Results for Q 1 ( Q 2 and Q 3 are illustrated in Figure 5.1, and Table 5.1 outlines some representative points in the Q i plot. Evidently, as expected the error in the model is worst and in fact increases towards infinity as jKi Chapter 5. Results 151 % Error in Sum ol Power 10 , 0f ' ' ' 10"5 I I I I I - 1 0 1 2 3 \i 1 1D1 ( u n i t l e s s ) ( B ) (C) Figure 5.1: Figure A illustrates Q i as a function of i n the case of no attenuation. Evidently, the Qx error increases towards infinity as 7 ^ approaches J p l . Figures B and C illustrate Q 2 and Q 3 , respectively, as a function of Chapter 5. Results 152 (uni t less ) Q i (%) (un i t less ) Q i (%) -1.0 6 . 7 x l 0 - 5 3 2 . 9 x l 0 - 4 -0.5 1 . 5 x l 0 - 4 2.5 4.5 x l O " 4 0.0 4.4.x 10" 4 2 8 . 7 x l 0 - 4 0.5 2 . 2 x l 0 " 3 1.5 3 . 0 x l 0 - 3 0.95 0.24 1.05 0.26 0.995 24 1.005 24 Table 5.1: The above data illustrate representative points in the graph of Q i versus ^ in the case of no attenuation. approaches 7 ^ . However, except for values of 2 2 1 in the range of 0.95-1.05, the total Q i error is s t i l l less than 0.26%. As such, image reconstruction based on the Fourier data should theoretically provide valid results up to this point. Fortunately, jK and jp for soft tissue generally have opposite sign, as illustrated wi th some representative values in Table 5.2. This results in a Q i error of less than 2 x l 0 ~ 4 % for Fourier data, according to Table 5.1. T i s s u e T y p e 7« 7p IP average soft tissue -0.1286 0.0566 -0.44 liver -0.1398 0.0566 -0.40 muscle -0.1875 0.0741 -0.38 fat 0.0810 -0.0504 -0.62 Table 5.2: The above data illustrate 7 p and 7^ values for some representative soft tissues. 5.1.2 T H E A S S U M P T I O N T H A T VSTj « ikhjSTj The computer simulation results presented in the previous section are valid for sit-uations in which attenuation can be ignored. However, this is not generally the case i n tissue imaging and as such the same analysis must be done for the attenuated propagator, STj, in order to test the effect on the model of the assumption that VSTj « ikhjSTj. This Chapter 5. Results 153 assumption was applied in the derivation of the A R T that is aimed at the reconstruction of jK and 7 P for attenuating objects. In Section 4.2.2, it was shown that VSrj « Srj{ikhj - T3} (5.17) where T 3 = (k2 1.4053 x 1 (T 5 m m + 1.2255 x I O - 2 k)(x + y) (5.18) = (Bk2 + Ck){x + y) The term in term in (x + y) was then dropped because it is 3 orders of magnitude less than the imaginary component in the above expression. Therefore, it was assumed that V 5 r . « ikhjSrj (5.19) A s in the case of no attenuation, this implies that VSTj does not result in a phase shift when it operates on an ultrasound field. Hence, this assumption also has more meaning than if the terms being compared were either both real or both imaginary. Like V 5 F j . , the expression for VSTj is further operated on during the remaining derivation of the reconstruction algorithm. Hence, it is necessary to verify that the final effect of the dropped term is in fact negligible. Appendix C outlines the derivation of an expression for the scattered field wi th at-tenuation included, P s , when Equation 5.17 is used. To reiterate, the result is Ps(rd,rs,u)) = a2A(uj)k2x (5.20) / S(k\v - r , | ) ( 7*(r) + $ 7 „ ( r ) ) S(k\r - rd\) d 2 r where $ = C 0 S ( ^ + - l ( 5 A ; 2 + CA;)2|x + y | 2 (5.21) - - (Bk2 + Ck) {cos((ps) + cos((pd) + sm((ps) + sin(<^d)} Chapter 5. Results 154 Recall that this expression is being compared to the equation for ps based on the approx-imate VSTj, in which $ simply equals cos(f9). In this computer simulation, it was again assumed that the same pulsed ultrasound field was scattered from a point identical in nature to that described in Section 5.1.1, wi th one exception. The point was assumed to be situated in a circular cross-section of tissue wi th radius equal to 0.2a and wi th its center at the origin. The coordinates of the source were again fixed at r s = (0, —a), where a = 100 mm. The tissue was assumed to be immersed in water, and refraction at the water/tissue interface was ignored. W i t h this geometrical setup, the scattered ultrasound detected in simulation for any view always traveled through a distance of dw and dt in water and tissue, respectively. In the identical manner as outlined in Section 5.1.1, the nearly exact scattered field in the presence of attenuation from this small point was derived. The only difference in the derivation is that attenuation factors precede the propagators, and $ has a different form. The result is as follows: pes{vd,Ts,uj) = -^A(oj)k2 ATTwATTt Hl0(ka) x (5.22) (7*i + *7pi) A T T ^ A T T t H'(ka) d x 2 ATTW and ATTt are defined as in Equations 2.64 and 2.66 to be A T T f ( d t , k) = l O " * 1 * * (5.23) ATTw(dw,k) = 1 0 - * ^ f e 2 where Xt = ~ ° M (5-24) X t 4 0 7 r M H z - m m V ' -0 .00022c 2 Xw 8 0 T T 2 M H Z 2 • m m In contrast, the reconstruction algorithm assumes that the point produces a scattered field given by the approximate expression Ps(rd,rs,uj) = a2A(u) k2 ATTWATTtS(ka) x (5.25) Chapter 5. Results 155 A T T w A T T t S(kd) d x 2 ( 7 k 1 + cos (0 ) 7 p l ) Evident ly $ is very different in the case of attenuation in that it is dependent upon k, ipa and (fd, in addition to 9. Analysis indicates that the imaginary part of $ , which accounts for the majority of the error in the model, is zero for all frequencies and for al l combinations of (ps and (pd in the case of backscatter imaging. Furthermore, given that \k\ < 42 m m - 1 for a typical imaging experiment, the real part of $ is always equal to cos(f?) to at least 4 significant figures. Hence, for backscatter imaging, pes is exactly equal to ps. The model was found to be most in error for transmission imaging, in which 9 = 0°. However, 9 = 90° was studied instead because it corresponds to compressibility imaging, which is a focus of this thesis. The value of the imaginary part of $ was calculated for k = 1 m m - 1 and 9 = 90° for the full range of (pd and <ps. This indicated that the error in the model was equal and opposite in magnitude for (<pd, <pa) — (0°, 90°) and (180°, 270°) due to the sinusoidal trigonometric term in Equation 5.21. Bo th sets of angles were studied. Aga in the error in ps was studied for 7 p l equal to {0.05, 0.1, 0.2} and —0.2 < 7 k 1 < 0.2 by computing the functions Q i , Q 2 and Q 3 defined in Equations 5.13 through 5.16. A s in the case of no attenuation, the simulation indicated that these quantities depend only on and not on the values of the individual parameters. However, the inclusion of attenuation shifts the Q i and Q 2 graphs when dt changes in magnitude relative to dw. Results for Q i , Q 2 and Q 3 are illustrated in Figures 5.2 and 5.3, and Table 5.3 outlines some representative Q i points plotted in Figure 5.2.A. The Q i error increases towards infinity as | ^ - | approaches zero, which is as expected. However, up to a value of | ^ | = 0.1, the Q i error is s t i l l less than 0.9%. A s such, 7 k imaging should theoretically provide valid results up to this point. Referring again to Table 5.2, it is evident that | ^ | is generally less than 0.35, which precludes a Q i error of less than 0.14% according to Figure 5.2. Note that Qi is a nearly symmetric function of | ^ | for dt = 0.3a, and that the graph deviates increasingly from symmetry about 0 wi th decreasing dt. Chapter 5. Results 156 % Error in Sum of Power Y k 1 / Y p 1 (unitless) (A) d=0.05a d=0.2a d=0.3a % Error in Sum of Power d=0.05a d=0.2a d=0.3a Y K 1 / Y p 1 (unitless) (B) Figure 5.2: The above figures illustrate Q i as a function of ^ for the case in which attenuation is considered. Figures A and B plot the results for ((pd, (p8) = (0°, 90°) and (180°, 270°), respectively. Evidently, the corresponding graphs for the two angle sets are mirror images, and the Q i error increases towards infinity as ^ approaches zero. Chapter 5. Results 157 Error in Fourier Amplitude i o - f . . _ 0 (B) Figure 5.3: Figures A and B illustrate Q 2 and Q 3 as a function of ^ for the case in which attenuation is considered. These figures represent the results for both (cpd, <pa) = (0°, 90°) and (180°, 270°), since the corresponding errors in each case are identical by virtue of the sum of absolute values. Furthermore, the phase error graphs are identical regardless of the value of dt-Chapter 5. Results 158 ^ (unitless) C*! (%), ^ = 0.2a Q i (%), ^ = 0.3a -0.0025 0.0025 -2.0 -1 -0.38 -0.15 -0.01 0.01 0.15 0.38 1 2 2 . 7 x l 0 - 3 1 .2x l0~ 2 8 . 3 x l 0 - 2 5 - l x l O " 1 1 . 2 x l 0 2 2 . 0 x l 0 3 2 . 0 x l 0 3 1 . 2 x l 0 2 5.5X10"1 8.7X10"2 1 . 2 x l 0 ~ 2 3 . 5 x l 0 " 3 4 . 6 x l 0 - 3 1 . 9 x l 0 - 2 1 . 4 x l 0 _ 1 8 . 9 x l 0 ~ 2 2 . 0 x l 0 2 3 . 2 x l 0 3 3 . 2 x l 0 3 2 . 0 x l 0 2 9 - O x l O " 1 1 . 4 x l 0 _ 1 2 . 0 x l 0 ~ 2 5 . 5 x l 0 " 3 Table 5.3: The above data illustrate representative points in the graph of Q i versus ^ wi th attenuation considered (Figure 5.2.A. 5.1.3 T H E E F F E C T OF U S E OF AN APPROXIMATE STj Recal l that in Section 2.2.1 the original function STj used in the approximation to the Green's function propagator was substituted wi th a simpler expression, given by This resulted from the application of the far field assumption stated in expression 2.21. It was noted that this approximation results in a loss of phase information and ult imately is responsible for the relatively poor performance of the reconstruction algorithm. To obtain an idea of the magnitude of this effect, a computer simulation was performed in which the source and detector were situated at (0, -a) and (-a, 0), respectively, wi th a = 60 mm. The Ricker wavelet used in the simulations described in Section 5.1.1 was scattered from several points situated with |r| = 10 mm. Propagation of the pulsed ultrasound was calculated using both the exact and approximate Srj. The pulses corresponding to the exact and approximate STj are shown in Figures 5.4.B through 5.4.D wi th solid and dotted lines, respectively. Evidently, the pulse shape is largely preserved, but the pulse exp(ik(hj • r + |rj|)) A; 21 12 (5.26) Chapter 5. Results 159 G e o m e t r y Fo r S imu la t ion x 1 0 - 6 C o m p a r i s o n for Po int 1 8 3 5 8 4 8 4 . 5 8 5 8 5 . 5 7 0 . 5 7 1 7 1 . 5 7 2 T i m e (us) T i m e (us) (C) (D) Figure 5.4: Figure A illustrates the experimental geometry in the computer simulation used to test the effect of the approximate Srr Scatter propagated from points 1, 3 and 5, al l wi th r = 10 mm, are illustrated assuming use of both the exact 5 r . (solid line) and the approximate STj (dotted line). Chapter 5. Results 160 Error in T O F Versus Distance 0.5i . 1 > 1 Distance From Origin (mm) Figure 5.5: This figure plots the error in T O F for scatter points that lie along the diagonal in the 4th quadrant of the Cartesian grid. Evidently, the error increases non-linearly as the distance from the origin increases. height is reduced in error in some cases. Also , the time-of-flight is always overestimated wi th use of the approximate STj. The maximum error in the T O F is lps for the simulation shown. Computer simulations indicate that the error in the T O F is always on the order of 0-2 ps, being 0 ps for points at the origin and worsening as the value of | r | / a increases. To illustrate this, Figure 5.5 plots the T O F error given r s = (0, - 100 ) , rd = ( -100 ,0) , and scatter points at r = (d, — d) mm, where d equals integers between 0 and 8. Evidently, the error in the time-of-flight increases non-linearly as the distance from the origin increases. Chapter 5. Results 161 Furthermore, the error becomes as large as 7 ps when d is equal to 30 mm. A s such, if smaller objects can be imaged or the source and detector can be placed further from the object, the reconstruction of 7 K and jp w i l l improve. However, this is generally not possible. The source must be close enough to the object such that the ultrasound field intensity is sufficient to generate detectable scatter upon insonification of the object. In addition, the detector sensitivity drops off at large distances and as such must be kept wi thin 10-15 cm of the object. Hence, the T O F error is an effect that cannot be removed by geometry considerations alone. Section 2.3 discussed the image reconstruction process that occurs wi th the analysis of data from each view. Figure 5.6.A sketches a closeup of an example ell iptical isochrone that is "warped" into a line by the image reconstruction routine. A s is illustrated, a point on the original isochrone is thus misplaced further away from the source and detector in this process. This in turn corresponds to an overestimation of the T O F of the scatter from the point shown. A s such, the "warping" of the isochrones is due at least in part to i the use of the approximate STj, which incorrectly overestimates time-of-flight. The error in T O F based on purely geometrical considerations is plotted in Figure 5.6.B. Here, the distance from the origin is measured along the incorrect linear isochrone passing through the origin wi th rs = (0 , -100) m m and rd = (-100,0) mm. The error in the T O F is calculated from the linear displacement that was experienced during the "warping" procedure by a point situated at the corresponding distance from the origin along the linear isochrone. Evidently, the T O F errors are significant and moreover are larger than the errors due to the use of the approximate STj, which accounts only in part for the "warping". This warping is a significant drawback of the reconstruction algorithm by Blackledge et al. Chapter 5. Results 162 . A — • — j E r r o r in Object ^ ^ • J Placement Portion of Perpendicular Bisector of Line Between Source and Detector (A) Error in TOF Versus Distance 12, , . , , 0 5 10 15 20 25 30 x (mm) (B) Figure 5.6: Figure A illustrates the misplacement of an example point when an isochrone is "warped" from an ellipse to a line. Figure B illustrates the error i n T O F that wi l l be associated with points along a linear isochrone through the origin as a result of the "warping" process. Chapter 5. Results 163 5.2 G E N E R A L N O T E S R E G A R D I N G T H E C O M P U T E R S I M U L A T I O N S 5.2.1 S I M U L A T I O N O F O B J E C T A N D S C A T T E R Both the method by Blackledge et al and the algebraic reconstruction technique were extensively tested through computer simulation. A s described in Section 5.1, ultrasound was scattered in simulation from objects made of one or more discrete points on a Carte-sian grid. Each point had associated with it a square element of area wi th a wid th of dx. Recall from Section 5.1 that dx can be at most Ajv O T /2 , where \Nyq is the wavelength in water (20°C) corresponding to the Nyquist frequency of the data that are being sim-ulated. The object was immersed in water, and a two dimensional model of scattering was used. For purposes of simplicity, refraction was not considered, and attenuation was incorporated in only a few simulations as noted. The source and detector were situated at a distance of | r s | = a and | r d | = a from the origin. The time dependence of the pulsed ultrasound field was described by the Ricker wavelet, given in Equat ion 5.4, wi th uo equal to 20 M H z , unless otherwise noted. A s such, the pulse length was ~ 1.0 mm. Returning to the assumption that $ cos(0) and summing scatter contributions from all N discrete points in the object situated at locations r^, respectively, yields from Equat ion 5.7 the following expression for the non-attenuated scattered field N 1 P s { r d , T s , u ) = — A ( w ) x (5.27) i=i 1 0 k2 Hl(k\Ti - r , | ) (yK(Ti) + cos(6)7p{Ti)) H^k\n - rd\) d x 2 The field was calculated according to Equation 5.27 for each view about the object. ps was then generally input to the reconstruction algorithm without any further processing. However, in a few simulations as noted in the following sections, the field was further operated on to impart attenuation or dispersion effects. Attenuation was studied for both the method by Blackledge et al and the A R T . However, since only preliminary studies were performed wi th the A R T , the study of dispersion was l imited to the former method only. Chapter 5. Results 164 5.2.2 POINTS REGARDING ANALYSIS In analyzing the behavior of the reconstruction algorithms, P S F ' s were investigated by scattering ultrasound in simulation from an isolated grid point at some known location. The point has an associated square element of area wi th particular values of 7 K and jp and dimensions of d x x d x . A s already indicated, image reconstruction spreads this t iny point out into a generally Gaussian shape called the P S F . The F W H M of the P S F is a measure of the image resolution of the reconstruction algorithm. A s discussed in Section 5.1.1, the resolution should be on the order of \Nyq/2. Consequently, in water the F W H M of the point-spread-function given fNyq equal to 10 M H z should be approximately 0.074 mm; the expected F W H M ' s for other values of fNyq scale inversely wi th frequency. Also of interest is the amplitude of a point-spread-function for a given reconstruction relative to the actual value of the point in the original image. The results in the sections to follow illustrate that the reconstructions must always be normalized in order to reproduce a jK or 7 P cross-section that exhibits amplitudes similar to the original object. Furthermore, the normalization factors for yK, jp and R are not necessarily equal. The "percentage ratio matrix" and the "power sum residual" are two other measures of performance that were used in the following sections. Often 7 K and 7 P cross-sections were produced for objects composed of an M x N grid of points, wi th the purpose of determining the quantitative capabilities of the imaging algorithm. Following normalization of the reconstructed image, / , the amplitudes of the M x N corresponding point-spread-functions were placed in an array, A , and compared point-by-point wi th the amplitudes, A 0 , in the original object, IQ. This comparison is herein referred to as the "percentage ratio matrix" of / relative to I0, and its elements are defined by: PRii = % = 1 , M , j = 1 , N (5.28) When two different reconstructions, I i and I2, of the same object must be compared, the similarity between the two images can be measured in terms of a power sum residual. Chapter 5. Results 165 Firs t a residual image, 7 r , is computed by Ir = h - / 2 (5-29) The ratio of the sum of the power in Ir relative to that in Ix is then computed by Q = x 100% (5.30) E Ii 2 In this thesis, Q is termed the "power sum residual." It w i l l become evident from the following results that the reconstructed image, / , is not an exact scaled version of the original object, I0, for either the method by Blackledge et al or the A R T . A s such, it is necessary to choose a physical location, ( x j y , yN), at which the corresponding yK or 7 P amplitudes in both I and IQ are extracted for comparison to create a normalization factor, £. (£JV,2/AT) was chosen to be the origin. The motivating factor was that this point experiences the most accurate reconstruction v ia the method by Blackledge et al as a result of the various approximations based on the far field assumption, such as that discussed in Section 5.1.3. As such, the reconstructed image was normalized by. a factor of e J ( ° ' ° ) r i m 5 = 4 p i ( 5 ' 3 1 ) This convention for choosing £ was retained in tests of the algebraic reconstruction tech-nique for lack of a better rule and to facilitate the comparison of the two methods. 5.2.3 DISCUSSION O F A P P L I E D F I L T E R I N G Reference is made to lowpass filtering throughout the following sections. Firs t of a l l , this always refers to Butterworth filtering. Secondly, when the simulation was to test the algebraic reconstruction technique, the source and scatter data Fourier spectra are each filtered and an image is iteratively solved for. This is not the case, however, when the simulation was to test the method by Blackledge et al. Recal l from Section 2.2.9 that lowpass filters applied to the source and data spectra are rendered ineffective by Chapter 5. Results 166 mathematical operations in this reconstruction algorithm. Instead, the lowpass filtering must be performed along each ray of data in the Fourier domain of the image as described in Section 2.2.9. A lowpass filter with a particular cutoff, / 0 , that is designed for the frequency dependent scatter data must be transformed into an equivalent L P F for the r-dependent radial data, where r = \A/ 2 + v2 is the position along the ray in the Fourier domain of the image. This equivalent filter has a cutoff of r 0 = s in ( - ) (5.32) Co I where 6 is the angle between the incident beam direction and the detector position vector, Td, and Co is the speed of sound in water at 20°C. 9 is equal to 90° and 180° for jK and reflectivity imaging, respectively. Therefore, the cutoff value of r for ryK imaging, given by (T 0 ) 7 k , is smaller by a factor of when compared to the cutoff for reflectivity imaging, given by (TQ)R. Often the Fourier data are not only filtered, but they may additionally be truncated at some frequency, fa, that is greater than or equal to the band l imit frequency, / 2 , of the source. B y extension, then, the corresponding r truncation limits, (TT)1K and (TT)R, also differ by a factor of A n increase in the maximum value of T means there is an increase in the maximum spatial frequencies, umax and vmax, of the data. According to the discussion in Section 2.2.7, this in turn decreases both the image width and pixel spacing. In order to be able to subtract R from ^ K to obtain jp, it is best to reconstruct the two images such that they have the same coordinates, thus circumventing the need to interpolate R at new (x, y) coordinates. (TT)1K and (TT)R must therefore be made equal, and care must be taken to ensure that the corresponding yK and R truncation frequencies, [fa)lK and (fa)R, do not lie in a region of valid data. The following sections present results for simulations of the method by Blackledge et al in which the ultrasound source was a Ricker wavelet wi th OJ0 equal to either 20 M H z or 6 M H z . The source with OJ0 = 20 M H z has a Fourier spectrum amplitude that drops to less than 0.7% beyond / = 9.0 M H z and only 0.0006% of the source power spectrum lies Chapter 5. Results 167 beyond this frequency. A s such, a L P F wi th / 0 = 9.0 M H z was often applied in this thesis as the optimal filter for these data. After filtering, only 0.0002% of the power spectrum lies beyond 9.0 M H z , so the upper band l imit , / 2 , is considered to be this frequency. The expected image resolution is equal to A / 2 / 2 , which is 0.08 m m in water at 20°C. For good measure, when data truncation was applied to these data, fr was set at 11.3 M H z or higher. To experiment with lower frequency reconstructions, sometimes a much more l imi t ing lowpass filter with / 0 = 2.6 M H z was applied to the simulated data for the source wi th OJ0 — 20 M H z . After filtering, only 0.0001% of the power spectrum lies beyond 3.5 M H z , so the upper band l imit , / 2 , was set to this frequency. The corresponding expected image resolution in water at 20°C is 0.21 mm. For good measure, when data truncation was applied to these data, fT was set 4.0 M H z or higher. The source wi th u>0 — 6 M H z has a Fourier spectrum amplitude that drops to less than 0.7% beyond / = 2.7 M H z , and only 0.0008% of the source power spectrum lies beyond this frequency. A s such, a L P F with / 0 = 2.7 M H z was often applied in this thesis as the optimal filter for these data. After filtering, only 0.0009% of the power spectrum lies beyond 2.7 M H z , so the upper band l imit , / 2 , was considered to be this frequency. The expected image resolution was then A / 2 / 2 , and in water at 20°C this is 0.27 mm. For good measure, when data truncation was applied to these data, fa was set at 3.4 M H z or higher. 5.3 T E S T S O F T H E M E T H O D B Y B L A C K L E D G E et al IN C O M P U T E R S I M U L A T I O N Various simulations were performed to test the method by Blackledge et al. In the following sections, generally only image profiles and percentage ratio matrices are pre-sented since these are more illustrative of key features than the corresponding images. The full-width-half-maxima of the point-spread-functions were measured along both the x and y axes, and these values were always found to be equal. In a l l simulations 7 K was obtained from side scatter imaging, and jp was obtained by reconstructing the reflectiv-Chapter 5. Results 168 ity through backscatter imaging and subtracting this from 7 K . For the sake of brevity, results for either jK or jp imaging, but generally not both, are presented since the same patterns were seen in this type of reconstruction. Also, the results d id not depend upon the actual values of jK and jp. This was expected by virtue of the linear nature of the Fourier transform. The simulation of scatter from discrete elements of area means that the amplitude of ps is inversely dependent upon the sampling frequency, fs. A n image based on ps w i l l also be inversely dependent upon fs for two reasons: first, there are no factors in the reconstruction algorithm to cancel this dependency, and second, the I F F T is a linear transformation. A s an example, data wi th / s i = 60 M H z have a corresponding element of area equal to M = {\^Y (5.33) 1 Jsl = 6.08 x 1 0 _ 4 m m 2 Data wi th fs2 = 30 M H z has a corresponding element of area equal to A, = {\^}2 (5.34) * Js2 = 2.43 x l O " 3 m m 2 which is also equal to A2 = A1x # ) 2 (5.35) Js2 The two corresponding images wi l l also exhibit this ratio of amplitudes due to the linear nature of the I F F T . A s such, two images with different sampling frequencies must be normalized to some common sampling frequency in order to be properly compared. This approach has been made note of in the following sections where it was applied. In the various tests of the method by Blackledge et al, it was observed that under certain circumstances the reconstructed image, / , was approximately a scaled version of the original object, I0. For instance, this was the case when the following conditions held: Chapter 5. Results 169 • Scatter point locations were close to the origin. • Ricker wavelet had UJQ equal to 20 M H z . • Sampling frequency was greater than 70 M H z . • No truncation of Fourier data. • Lowpass filter wi th / 0 = 2.6 M H z was applied. • The number of views, Nv, was > 50. A s discussed in Section 5.2, the expected image resolution is 0.21 m m given the value of u)0 and the L P F choice. A n example of reconstructions that satisfy the conditions outlined above is illustrated in Figures 5.7.A and 5.7.B, in which the sampling rate was 75 M H z and Nv was 50. The scattering object in simulation consisted of 9 isolated points spaced 1.5 m m apart on a square grid. The 7« and 7 P values of the points in the scattering object were equal' to IK = 7«0 1 2 3 1 2 3 1 2 3 7p = IpQ 1 2 3 1 2 3 1 2 3 (5.36) where 7 K n and 7 K n were equal to 0.04 and 0.0909, respectively. The normalization factor, £, was found to be dependent upon the previously mentioned parameters as well as the overall number of points in the Cartesian grid in Fourier space. The percentage ratio matrices comparing the normalized reconstruction, 7, to the original object I0 for jK and 7 P imaging, respectively, where PR-f. -98.1 96.9 92.0 99.4 100.0 95.9 97.9 96.6 91.8 91.7 97.2 90.7 100.7 100.0 96.1 91.7 97.2 90.7 (5.37) There is evidently excellent agreement between the expected and reconstructed values. The average over al l entries in PRyK and PRlp were 97 ± 3% and 95 ± 4%, respectively, Chapter 5. Results 170 Figure 5.7: Figures A and B illustrate example reconstructions in which the resulting images are nearly scale copies of the original image. Chapter 5. Results 171 -0.05 h •0.1' • ' 1 A - 4 -2 0 2 4 x (mm) Figure 5.8: This figure illustrates the profile of 7 P along the x-axis. indicating a relatively quantitative reconstruction algorithm for this object. Note, how-ever, that some points i n the reconstructions are not cyl indrical ly symmetric, part icularly in the 7 P image, due to the effect of scatterer location discussed in Section 5.3.2. Figure 5.8 illustrates the profile of the reconstructed 7 P along the x-axis. The features of this profile are typical of al l other profiles along the x and y axes in the jK and jp images. Note that the value of the background pixels oscillates in the range of -0.011 to 0.008 and -0.03 to 0.02 for the 7* and jp images, respectively. The average F W H M of a l l the points i n the 7^ and jp images is 0.244 ± 0.004 m m and 0.192 ± 0.002 m m , respectively. Since the image resolution is expected to be 0.21 m m , the F W H M for jp is Chapter 5. Results 172 excellent in that it falls below this value, and the F W H M for 7 r e varies from theory by - 16%. 5.3.1 T H E E F F E C T O F F I L T E R I N G A N D D A T A T R U N C A T I O N A s discussed in Sections 2.2.8 and 2.2.7, the interpolation to a square grid necessitates a reduction of the data Nyquist frequency in order to increase the pixel size of the resulting image and ensure the reconstruction is practically computable. This can be accomplished either through sampling data at a higher frequency and subsequently truncating the data in frequency space, or by reducing the sampling frequency at the outset. These options in combination wi th different lowpass filter cutoffs were studied in simulation for jK reconstruction. Together with use of a simulated Ricker wavelet source wi th u>0 equal to 20 M H z , the following combinations of data processing approaches were investigated: • C a s e 1 - A p p l y L P F with / 0 = 2.6 M H z to data sampled at / = 20 M H z , wi th no need to truncate data. • C a s e 2 - A p p l y L P F with / 0 = 9.0 M H z to data sampled at / = 20 M H z , wi th no need to truncate data. • C a s e 3 - A p p l y L P F with /„ = 2.6 M H z to data sampled at / > 20 M H z and truncate data at 11.3 M H z . • C a s e 4 - A p p l y L P F with / 0 = 9.0 M H z to data sampled at / > 20 M H z and truncate data at 11.3 M H z . To test the relative performance of these options, the grid used in the simulations in Section 5.3, again wi th jK = 0.04, was reconstructed subject to the above itemized circumstances. For Cases 1 and 2, the data were sampled at 20 M H z , whereas the sampling frequency was 75 M H z for the other two tests. There was always 100 views and a Ricker wavelet wi th u0 equal to 20 M H z was used. Upon normalization relative to the center point, the percentage ratio matrices of I relative to IQ for Cases 1 through 4 were Chapter 5. Results 173 the following: PR% 93.3 96.9 93.9 96.0 100.0 97.2 93.3 96.9 93.9 76.6 80.8 78.2 80.8 100.0 80.8 76.6 80.8 78.2 PR2 PR* = 42.4 59.7 43.9 55.9 100.0 61.0 42.4 59.6 43.9 61.9 82.3 63.1 81.4 100.0 82.0 61.9 82.3 63.1 (5.38) Evidently, the application of a lowpass filter with / 0 = 2.6 M H z wi th either truncated (Case 3) or non-truncated data (Case 1) yield the best results. However, the application of a L P F wi th / 0 = 2.6 M H z to data that have been sampled at 20 M H z (thus no need to truncate data) is best and affords a percentage ratio improvement of 16-22% throughout the image relative to Case 3 wi th a higher sampling frequency and truncation of data. Figures 5.9. A and 5.9.B illustrate the reconstructed jK profiles along the x-axis for Case 1 and Case 3. Note that, for comparison purposes, the Case 1 image has been normalized to a sampling frequency of 75 M H z . Recall that this normalization was discussed in Section 5.3, and that it involves mult iplying the image by a factor of ( f | ) 2 -Analysis of a l l the profiles indicates that the F W H M values for the peaks in Case 3 are not consistent, ranging from 0.206 mm at the center to 0.278 m m elsewhere. A s discussed in Section 5.3, this is in comparison to the expected value of 0.21 m m given that a lowpass filter wi th / 0 = 2.6 M H z was applied. The F W H M values of al l peaks in Case 1 had an average value of 0.271 ± 0.005 mm, which is off from the expected value by ~ 29%. 5.3.2 T H E E F F E C T O F S C A T T E R E R L O C A T I O N Given a particular source and detector distance equal to \TJ\, computer simulations indicate that the reconstructed P S F for both jK and 7 P imaging exhibits more error in Chapter 5. Results 174 Y - Case 1 0.08| 1 1 0.07-0.06-x (mm) (A) Y - Case 3 0.081 . . 0.07 0.06 | -0.01 r x (mm) (B) Figure 5.9: Figures A and B illustrate the reconstructed 7 r e profiles along the x-axis for Cases 1 and 3, respectively. Recall that the Fourier data for Case 3 has been truncated at a frequency of 3.4 M H z , while that for Case 1 has not been truncated and hence spans the range of 0-20 M H z . Chapter 5. Results 175 terms of amplitude and lack of Gaussian shape as the location of the scatterer moves further from the origin. Also, for a given scatterer located at r , the error in the P S F increases as M- increases. This is not surprising due to the approximations that were applied in the derivation of the reconstruction algorithm by Blackledge et al. Figure 5.10.A illustrates the yK reconstruction for a point located at (5,5) mm, while Figures 5.10.B and 5.10.C illustrate the jK reconstruction and corresponding x-axis profile for a point located at (0,0) mm, both with ^yK equal to 0.04. In both reconstructions, there were 100 views, the sampling frequency was 67.5 M H z , Fourier data were lowpass filtered with / 0 = 9 M H z , and data were truncated beyond 11.3 M H z to boost the image pixel size. The point at the origin has a uniform Gaussian-like P S F wi th an amplitude of ~ 4.25 x 1 0 - 3 , while that at (5,5) m m has a hollow oval-shaped P S F with a maximum amplitude of only 5% that of the center point. The expected image resolution is 0.08 m m . The F W H M of the origin point is ~0.07 mm, which falls below the expected value. The ring of the point at (5,5) m m has large dimensions of approximately 0.4 m m x 0.8 mm. The halo effect in the shape of the P S F is expected from the T O F errors discussed in Section 5.1.3. For each view, scatter is being reconstructed further away from the source and detector than it should be, and the effect is the creation of a ring upon reconstruction of all views. The ring has a much lower amplitude than the point reconstruction should have in part because sound energy is being spread out over a larger area, as opposed to being concentrated into a point. However, most of the energy is being spread out along rays throughout the image that can be seen emanating from the scatterer location. This halo effect is a significant result that seriously l imits the capabilities of this reconstruction algorithm. It wi l l become evident, however, in the following section that this effect can be alleviated by the collection of data at a large number of views. Chapter 5. Results 176 Figure 5.10: This figure illustrates the effect of scatterer location on the reconstruction of yK points. Figure A illustrates the result for a point located at (5,5) mm, while Figures B and C present the image and z-axis profile results for a point at (0,0) m m . Chapter 5. Results 177 3.5 4 4.5 5 5.5 6 6.5 x (mm) Figure 5.11: This figure illustrates the effect of the number of views on the reconstruction of 7 r e for a point located at (5,5) m m given 400 views of data. This is in comparison with Figure 5.10.A, which illustrates the same reconstruction with 100 views of data. 5 . 3 . 3 T H E E F F E C T O F T H E N U M B E R O F V I E W S Section 2.3 described how each view at which data are recorded yields information along one direction through the image space. A s the number of views increases, the object thus becomes more well-defined and the reconstructed image looks better. This effect can be seen in Figure 5.11 in which a point at (5, 5) m m with 7 r e equal to 0.04 is reconstructed with 400 views of data. The sampling frequency was 67.5 M H z , Fourier data were lowpass filtered at 9.0 M H z , and data in Fourier space were truncated at 11.3 M H z to boost the pixel size. Comparing this reconstruction to that in Figure 5.10.A, Chapter 5. Results 178 which was based on only 100 views of data, the effect of increasing Nv is dramatic. The halo becomes more tightly bound and cylindrically symmetric, and the amplitude of the P S F increases by a factor of 2.5. However, the maximum 7 K amplitude is s t i l l only 12% of that for the point at (0,0) mm, due to the effect of scatterer location discussed in Section 5.3.2. The change in maximum pixel amplitude in the previous example appears to result due to the misplaced location of reconstructed sound energy discussed in the previous section. This effect is not seen for a point at the origin, for which sound energy is not misplaced in the reconstructed image. The number of views was also set to 400, and jK was again reconstructed for a point located at (0,0) m m with 7^ equal to 0.04. The result was nearly identical to that for 100 views, shown in Figures 5.10.B and 5.10.C, and as such is not illustrated here. The power sum residual comparing the two images was negligible at 0.001%. Given that sound energy is being reconstructed in the proper location, i t makes intuitive sense that the image amplitude does not depend upon the number of views. A n increase in Nv simply defines the Fourier data of the image along additional radial lines. This does not increase the intensity in the Fourier domain of the image, but simply makes the space better defined. Due to the linear nature of the Fourier Transform, then, the intensity in the image does not increase either. • 5.3.4 T H E E F F E C T O F C A R T E S I A N G R I D SIZE IN F O U R I E R S P A C E The method by Blackledge et al has at its heart an interpolation from a radial grid to a Cartesian grid in the Fourier domain of the image, after which the data are operated on by the inverse F F T to yield an image. The pixel size in the x and y dimensions in the reconstructed image is inversely dependent upon the maximum values of u and v in the radial grid in Fourier space. The pixel size is thus obviously constant regardless of the fineness of the Cartesian Fourier grid. However, as more points are added to the Cartesian grid along the u and v dimensions, more points are added to the reconstructed Chapter 5. Results 179 image along the x and y dimensions, respectively. A s discussed in Section 2.2.7, the spacing of points in the Fourier domain should technically be not less than the largest distance between any two points on the radial grid. This is termed the optimal grid spacing. However, achieving the opt imal grid spacing leads to some data requirements that are difficult to satisfy at the development stage. For instance, in order to reconstruct a square image of only 75 m m with an insonifying source that has a cutoff frequency of 5 M H z , there must be 6400 views of data. The corresponding optimal Cartesian Fourier grid has 720 x 720 points. Interpolating from a radial grid wi th 6400 rays to a Cartesian grid of this size is out of the question at the development stage however. The M a t l a b ™ interpolation is highly intensive in terms of memory and C P U time when performed inside the M a t l a b ™ workspace. For instance, the processing required for a typical simulation with 100 views of data and a 800 x 800 grid can take well over 24 hours to complete due to the interpolation. In a future version of the prototype system, the interpolation processing could likely be embedded in a dedicated electronics unit that would be orders of magnitude faster. Due to these limitations, in this thesis work the spacing in the Cartesian Fourier space grid has generally been selected to be less than the optimal value. This choice has led to quite satisfactory results. To illustrate, Figures 5.12.A and 5.12.B present the results of a 7 K image reconstruction for a point at the origin wi th jK equal to 0.04 in which the number of pixels was optimal at 100 x 100 (A) and in which it was increased to 400 x 400 (B) . In both reconstructions, the sampling frequency was 67.5 M H z , the number of views was 50, and data in Fourier space were not truncated. Evidently, there is excellent agreement between the two images, and it is thus not expected that the Cartesian grid size is an important factor in the image reconstruction. Upon interpolating the 400 x 400 image to a 100 x 100 grid, the power sum residual comparing the two images was negligible at 0.06%. Chapter 5. Results i -100x100 Grid -1 -0.5 0 0.5 1 x (mm) (A) y _ 400x400 Grid 11I , . . 1 -1 -0.5 0 0.5 1 x (mm) Figure 5.12: Figures A and B illustrate the reconstruction of jK wi th Cartesian grid in Fourier space of 100 x 100 and 400 x 400, respectively. Chapter 5. Results 181 5.3.5 T H E E F F E C T O F S A M P L E R A T E The image normalization factor, £, was found to be constant wi th frequency for 7 K , R, and jp imaging. In testing this, al l three types of reconstructions were performed for an isolated scatter point located at the origin. In each simulation, 7 r e and yp were equal to 0.04 and 0.0909, respectively. A Ricker wavelet with UJ0 equal to 20 M H z was used wi th a subsequent 9.0 M H z lowpass filter. Da ta were truncated in Fourier space at 11.3 M H z to boost the pixel size. 100 views of data were simulated for each reconstruction, and simulations were performed at several frequencies between 30 and 75 M H z . In order to properly compare the images with different sampling frequencies, they were all normalized to a sampling frequency of 75 M H z . Recall that this normalization was discussed in Section 5.3, and that it involves mult iplying each image by a factor of ) 2 , where / s l is the data sampling frequency corresponding to the image under consideration. Figure 5.13.A illustrates £ as a function of frequency for the non-normalized 7 K , jp, and R images. The average values of £ are as follows: 6^  = 7.87 ±0 .01 (5.39) £7„ = 15.78 ± 0 . 0 4 £ R = 12.29 ± 0 . 0 3 Evidently, upon proper normalization, £ is constant for every type of imaging. 5.3.6 T H E E F F E C T O F D ISPERSION A s discussed in Section 1.6, the speed of sound in water and tissue is dependent upon frequency. Al though the change in acoustic velocity is less than 0.5% over 1-20 M H z , analysis indicates that this results in significant changes in the shape of the pulse and the corresponding Fourier spectrum [43]. Figures 5.14.A and 5.14.B illustrate the effect of dispersion on the shape of the pulse scattered from a point at the origin, when the source and detector are located at a distance of 60 m m and 200 m m from the origin, Chapter 5. Results 182 C, V s F r e q u e n c y 100 80 8 60 .2? "E 5 40 20 30 40 50 60 70 80 F r e q u e n c y (MHz) (A) C, V s F r e q u e n c y 20 r 15 - 1 0 5 20 40 60 80 F r e q u e n c y (MHz ) (B) Figure 5.13: Figure A plots the frequency dependence of £ for the non-normalized jK R and l p images, while Figure B illustrates the same results for images which have been normalized to a common sampling frequency of 75 M H z . - - - R Chapter 5. Results 183 (A) .-6 Scattered Signal i , i i 1 1 1 269 269.5 270 270.5 271 271.5 272 Time (\is) (B) Figure 5.14: Figures A and B illustrate the effect of 1% dispersion over the frequency range of 1-20 M H z on the shape of a pulse scattered from a point at the origin. In A and B the source and detector were situated at a distance of 60 m m and 200 m m from the origin, respectively. The solid and dashed lines illustrate the pulses in the absence and presence of dispersion, respectively. Chapter 5. Results 184 respectively. Evidently, the effect worsens as the propagation distance increases. If dispersion is not corrected for in the reconstruction algorithm, it has the effect of creating dramatic halos and interference patterns due to the misplacement of ultrasound energy during the image reconstruction process. Figure 5.15 illustrates this effect. Figure A shows the P S F in the absence of dispersion for a point located at (3, 3) m m wi th jK equal to 0.04. Da ta were sampled at 67.5 M H z , and a low Nv = 50 was chosen, which explains the poor shape of the P S F . However, since the halo effect is likely unavoidable at larger distances wi th this algorithm even for Nv on the order of a few hundred, useful information can be gained by starting with a P S F of this shape. Figure 5.15.B illustrates the effect in the presence of 1% dispersion over 1-20 M H z , wi th an f2 dependence. This dependence was chosen for educational purposes only, on the premise that more complicated functions of frequency may be more difficult to correct for. L i t t l e has been done to date towards the experimental characterization of ultrasound dispersion in water and tissue. The halo and interference pattern effect is extreme and completely destroys the P S F . Sound energy that was in the P S F is spread along curved rays throughout the image. This is due to the destruction of valid phase information in the modelling of the scattered ultrasound field. Evidently then, even this small percentage dispersion can not be neglected in ultrasound computed tomography. Figure 5.15.C illustrates the 7« image based on the same dispersive data, although this time the dependence upon frequency was included in the tomography algorithm and thus corrected for. This approach restored the point-spread-function to a reasonably good extent, wi th very litt le change in the shape of the function. The minimum and maximum values in the image did change in going from the case of no dispersion to that of dispersion correction, wi th min ima of -0.017 and -0.022, respectively, and maxima of 0.053 and 0.055, respectively. A s such, the percentage changes in the reconstructed jK minimum, maximum and range were 26%, 4% and 9%, respectively. It thus appears that U C T data can be corrected to yield useful reconstructions i f the functional dependence of acoustic velocity upon frequency Chapter 5. Results 185 Y k - No Dispersion x (mm) (A) y - Dispersion yK - Corrected x (mm) x (mm) (B) (C) Figure 5.15: Figures A and B illustrate the 7 K reconstructions for the scattering object given the absence and presence of dispersion, respectively, while Figure C illustrates the same reconstruction based on corrected data. Chapter 5. Results 186 is known. The power sum residual comparing the two images is 22%. • 5.3.7 R E C O N S T R U C T I O N O F A S Q U A R E B L O C K Thus far reconstructions of 7 r e and 7 P have been shown only for points that are infinitesimal with respect to the wavelengths in the insonifying ultrasound pulse. This section presents results for the reconstruction of jK and 7 P for a square block in order to further examine the capabilities and limitations of the algorithm by Blackledge et al, subject to its underlying assumptions. The block had a width of 1.85 mm, a uniform jK equal to 0.04, and a uniform 7 P equal to 0.0909. Insonification was simulated wi th a Ricker wavelet that had OJ0 equal to 6 M H z . A L P F was applied wi th a cutoff frequency of 2.6 M H z , and Fourier data were truncated at 3.4 M H z to boost the pixel size. These images were based on 100 views of data sampled at 30 M H z . This sampling frequency would preclude a point spacing of 0.05 m m in the scatter simulation procedure, but to reduce the computation time, this was increased to 0.074 mm. Figures 5.16.A and 5.16.B illustrate the reconstructed 7 K image and its corresponding z-axis profile, while Figures 5.17.A and 5.17.B illustrate the same for 7 p . Note that the profiles have been normalized and are plotted wi th the normalized profiles of the original object. The square block is reasonably well defined in al l dimensions in both the 7 K and 7 P images. Aside from a higher amplitude edge, the pixels on the top of the block are relatively uniform. Those within a radius of 0.9 m m have an average amplitude of 1.4 ± 0.1 in the fK image, while the corresponding pixels in the 7 P image have an average amplitude of 3.1 ± 0.2. Beyond the object past a radius of 1.3 mm, 7 K oscillates wi th an average value of -0 .03 ± 0.04, while 7 P oscillates wi th an average value of -0 .06 ± 0.06. The F W H M measured at. heights in the range of 0.5-0.55 on the normalized 7« profile is 1.85 ± 0.01 mm, which is the width of the original object. Similarly, the F W H M measured at heights in the range of 0.5-0.53 on the normalized 7 P profile is again 1.85 ± 0.01 m m . Note that the maximum amplitudes in the reconstructions are quite high relative to previous examples Chapter 5. Results 187 Figure 5.16: Figures A and B illustrate the resulting jK image and x-axis profile, respec-tively, of a square block. Note that the profile has been normalized and is plotted with the normalized profile of the original object. Chapter 5. Results 188 Figure 5.17: Figures A and B illustrate the resulting jp image and x-axis profile, respec-tively, of a square block. Note that the profile has been normalized and is plotted wi th the normalized profile of the original object. Chapter 5. Results 189 due to the relatively large size of the element of area associated wi th each scatter point. If the images were normalized to a sampling frequency of 75 M H z , the maximum pixel amplitudes for 7 k and 7 p would become 0.11 and 0.25, respectively, which is on the order of previous grid results. 5.4 E F F E C T O F A T T E N U A T I O N A N D T H E A P P R O X I M A T E C O R R E C T I O N Figure 5.18 illustrates the effect of attenuation on the reconstruction of 7 r e for a single point located at (0, 0) m m in a cylindrical tissue. The point was characterized by 7 k equal to 0.04, and it was centered inside a tissue with a radius of 20 mm. The source and detector were located at a distance of 100 m m from the origin. The Ricker wavelet had coo equal to 20 M H z , and the sampling frequency was 67.5 M H z . The data were processed wi th a Butterworth L P F of order 20 wi th a cutoff of 9 M H z , and there were 80 views. Da ta were truncated at 11.3 M H z in order to boost the pixel size. As such, the image resolution was expected to be 0.08 mm. Figure 5.18 plots the 7 k profile along the x-axis both in the absence and presence of attenuation in tissue and water. The profile along the y-sxis was identical. As expected, attenuation reduced the amplitude of the point-spread-function significantly by 32%. The F W H M was 0.078 ± 0.002 m m and 0.098 ± 0.002 m m in the absence and presence of attenuation, respectively. Also , note that the ratio of the F W H M height to width is ~8:1 in the absence of attenuation and ~2:1 in the presence of attenuation. This pattern can be explained by the "beam hardening" that occurs due to attenuation. Water and tissue attenuate ultrasound in a stronger manner as frequency increases, shifting the insonifying pulse spectrum such that there is relatively more energy in the lower frequency range. This in turn translates into a "rounding" of the point-spread-function. In analyzing the effect of attenuation, real world objects must be modeled. The reconstruction algorithm must be tested in simulation on tissues made of scatterers that experience widely different effects, depending upon the source and detector locations. A s Chapter 5. Results 190 Y X Profiles 'K 0.4I 1 -0.1 1 ' 1 -0.5 0 0.5 x (mm) Figure 5.18: This figure illustrates the effect of attenuation in tissue and water on the reconstruction of 7r e for a single scatter point located at (0, 0) mm. Shown are the x-axis profiles for jK in both the absence (solid line) and presence (dotted line) of attenuation. such, tissues wi th widths on the order of 60-80 m m must be reconstructed. This could potentially be accomplished i f the Nyquist frequency is reduced to ~ 2 M H z through the truncation of data in Fourier space, and i f additionally the Cartesian grid has on the order of 900x900 points. However, the number of views would necessarily be on the order of several hundred to ensure that the P S F energy for points wi th large | r | is not spread out along rays throughout the image. The computing required for this type of reconstruction is presently not available in the prototype scanner. In addition, points wi th large |r | would exhibit a halo effect, and the influence of attenuation on P S F shape Chapter 5. Results 191 would be difficult to discern. To facilitate testing for the effect of attenuation as well as the capabilities of the approximate attenuation correction described in Section 4.1, an upward scaling in at-tenuation magnitude was applied to the scattering system. In this manner, distances in the image can be kept to a minimum for ease of reconstruction, and the expected point-spread-functions are Gaussian-like in shape for ease of analysis. A s such, the magnitude of attenuation in tissue was scaled up by a factor of 1000; attenuation in water was not scaled. A s in Section 4.1, a test was done to determine the opt imal form of the ap-proximate attenuation correction for this situation. Errors in the scatter amplitudes for small tissue sizes on the order of 1 m m were found to be minimized wi th an approximate attenuation correction coefficient given by 1 /C, where ( J — ^Q-(1000xt)(0.45dt)fc x I Q - X W (Gdu,) k2 (5 40) Xt a n d Xw a r e defined according to Equations 2.64 and 2.66 in Section 2.4, and xt has a multiplier of 1000 due to the upward scaling of attenuation in tissue. The dt and dw multipliers have the same order of magnitude as those for the case of realistic attenuation in tissue. Figure 5.19 illustrates the reconstructed jK image in the absence of attenuation for a grid of 9 closely spaced but isolated points of equal amplitude. The number of views was 50, the sampling frequency was 60, and there was no truncation of data i n Fourier space. The data were processed wi th a Butterworth L P F of order 20 wi th a cutoff of 9 M H z . Note that the point-spread-functions are al l well-defined wi th F W H M ' s of ~ 0.07 mm, whereas the expected image resolution is 0.08 m m for data wi th a band l imit frequency of ~ 9 M H z . Figure 5.20.A illustrates the same grid of points reconstructed in the presence of the scaled attenuation. The effect is dramatic, wi th al l but 4 outside points disap-pearing from the image. In addition, the remaining points no longer have cylindrical ly symmetric P S F ' s , but the F W H M values of these P S F ' s changed very li t t le. The am-plitudes of these remaining P S F ' s were reduced to one half their original value. -Figure Chapter 5. Results 192 Y - 50 Views 1 v -1 - 0 . 5 0 0.5 1 x (mm) Figure 5.19: This figure illustrates the original grid of points for which 7* was recon-structed in the absence of attenuation. 5.20.B illustrates the same grid of points reconstructed wi th data corrected according to Equation 5.40. Unfortunately the missing points did not reappear. Instead, a distinctive interference pattern appeared, and the remaining points were severely overcompensated to over 40 times their original value. Their F W H M values were also reduced. Evidently, then, it appears as though the approximate attenuation correction can not be applied to ultrasound computed tomography. Chapter 5. Results 193 Y - With At tenuat ion -1 -0.5 0 0.5 1 x (mm) (A) Y - Corrected -1 -0.5 o 0.5 1 x (mm) (B) Figure 5.20: Figure A illustrates the effect of scaled attenuation in tissue and realistic attenuation in water on the reconstruction of 7 k for the grid of points in Figure 5.19. Figure B illustrates the image for the grid based on scatter data that have been corrected for attenuation in tissue and water using the approximate attenuation coefficient defined in Equation 5.40. Chapter 5. Results 194 5.5 C O M P A R I S O N O F F O U R I E R A N D W A V E L E T D E N O I S I N G M E T H O D S Prel iminary tests of the moving average filter and several wavelet denoising methods have been done in simulation. Recall that the nature of the noise in pulsed U C T data was analyzed in Section 4.3.1. The unprocessed signal was shown to have pr imari ly zero-mean Gaussian white noise, which lends itself to removal v ia wavelet denoising techniques. It was also shown that data which have been lowpass filtered beyond the source spectrum is often also zero-mean Gaussian and white for the most part. In cases where a low frequency band clutters the signal, subsequent high pass filtering was shown to render a signal wi th zero-mean Gaussian white noise. A s such, wavelet processing can be applied at any stage of signal processing. In order to test both Fourier based and wavelet based denoising techniques, a data signal was simulated that represented scatter from an infinitesimal point located at (0, 0) mm. The insonifying Ricker wavelet had OJO equal to 20 M H z , and the sampling frequency was always 75 M H z . There were 100 views of data. Using the "randn" function in the M a t l a b ™ programming environment, Gaussian white noise was added to the data such that the S N R was equal to 35 db. A different noise vector was added to each view so that noise was not correlated. Figure 5.21 illustrates a closeup of the pulse region in the noiseless signal and a noisy signal from one angle, respectively. As discussed in Section 4.5.3, this study adopted the relatively smooth Symlet 8 and Daubechies 9 mother wavelets, which were illustrated in Figure 4.10. Note that the Ricker wavelet was not used as a mother wavelet because this would have involved the custom creation of a wavelet family, which was beyond the scope of this thesis. Soft thresholding was always used, and results presented here are for the "huersure" threshold selection rule, which yielded slightly better results than the other methods. A l l three noise rescaling models (options "one", "sin" and "min") were tested, and there was no discernible difference in the results. The denoised signal for a given mother wavelet was identical for resolution levels 1, 2 and 3. The denoised pulse result for level 4, Chapter 5. Results 195 time (us) time (us) Figure 5.21: This figure zooms i n on the region around the data pulse i n the noiseless and noisy signals, respectively. however, had an amplitude that was only about 25% that i n the original data. Results for resolution level 1 are presented in this thesis. The denoised signal using the Symlet 8 wavelet family is illustrated in Figure 5.22. Evidently, the result is somewhat choppy even though smooth mother wavelets were chosen. Also shown is the wavelet denoised signal after it has been smoothed with a 5 point moving average filter. The result is very similar to the noiseless data. The following subsections w i l l illustrate when this smoothing operation is necessary. The difference between the denoised signals for the Daubechies 9 and Symlet 8 families is very subtle, and in fact there is often a zero residual between the results of the two approaches for a given view. A s such, the Daubechies 9 Chapter 5. Results 196 27.5 time (us) time (us) Figure 5.22: Figures A and B illustrate the denoised signal given filtering wi th the Symlet 8 and Daubechies 9 families, respectively, both with (right) and without (left) subsequent smoothing. denoised signal is not shown. 5.5.1 R E C O N S T R U C T I O N W I T H N O I S E - F R E E A N D N O I S Y D A T A Figure 5.23 illustrates the x-axis profiles for 7 r e reconstructions using both noise-free and noisy data. Figure A presents the noise-free case, while Figures B through C illustrate noisy reconstructions given lowpass filter cutoff frequencies of 9.0, 6.6 and 2.6 M H z , respectively. Recall that a L P F wi th /o = 9.0 M H z is nominal for the Ricker wavelet used in this simulation. Figure B indicates that nominal lowpass filtering is not Chapter 5. Results 197 Noiseless Image Noisy Image ( C ) (D) Figure 5.23: Figure A illustrates the x-axis profile of the jK reconstruction in this s im-ulation for noise-free data. Figures B through D present the x-axis profiles for noisy data wi th a S N R = 35 db, subject to lowpass filtering wi th f0 = 9.0, 6.6 and 2.6 M H z , respectively. Chapter 5. Results 198 feasible wi th noisy data. The reconstruction based on data wi th a slightly lower cutoff of 6.6 M H z was able to recover the shape and amplitude of the main peak, although the background is st i l l very noisy. Fi l ter ing wi th a very low cutoff of 2.6 M H z broadened the peak as expected and increased the peak amplitude. It also produced a background amplitude that increases with distance from the origin, when in fact it should be relatively flat. Evident ly then, even data that look reasonably good with a relatively low S N R of 35 db can not produce useful images regardless of the Butterworth filtering that is applied. 5.5.2 P R E L I M I N A R Y RESULTS OF NOISE F I L T E R I N G N A R R O W E R B A N D W I D T H D A T A Prel iminary tests have been done to determine the effect of noise filtering on narrower bandwidth noisy U C T data. Da ta were denoised and then processed wi th a L P F wi th /o = 2.6 M H z , rather than the nominal 9.0 M H z . 7 K image reconstruction was compared for data filtered wi th a 5 point moving average filter and the Daubechies 9 wavelet family. Fi l ter ing wi th the Symlet 8 wavelet has not been presented because the result overlaps wi th that of the Daubechies 9 family at this graph magnification. The images were normalized to the range 0-1, as was the image produced wi th noiseless data processed wi th the same L P F . The normalized x-axis profiles are illustrated in Figure 5.24. The result for the moving average filter is for al l intents and purposes identical to that wi th no denoising at a l l , shown in Figure 5.23.D, wi th a corresponding power sum residual between the two images of only 0.005%. Thus the moving average filter does not afford any quality improvement. Image power sum residuals for various filtering tests (compared to images based on noiseless data) are tabulated in Table 5.4. Evidently, the Daubechies 9 wavelet family does an excellent job of denoising this low frequency data, even without prior smoothing. This indicates that the choppy features of the denoised data correspond to Fourier data above 2.6 M H z . Chapter 5. Results 199 Compar ison Along X - A x i s x (mm) Figure 5.24: This figure compares the x-axis profiles for 7« reconstructions based on noise-free data as well as data filtered wi th the Daubechies 9 wavelet family and a 5 point moving average filter. In a l l cases a lowpass filter was applied wi th / 0 = 2.6 M H z , and the resulting images were normalized to the range of 0-1. W I D E R B A N D W I D T H D A T A If the reconstruction of wavelet filtered data in the higher frequency range in desired, data smoothing is necessary due to the choppy features that result from the filtering process. This is illustrated in Figures 5 . 2 5 . A - B and 5 .25 .C-D, in which lowpass filters were applied wi th / 0 equal to 6.6 and 9.0 M H z , respectively. Figures A and C present the non-normalized 7 r e x-axis profile for the non-smoothed data. Figures B and D compare the normalized profiles based on noise-free data and those which have been Symlet 8 Chapter 5. Results 200 S y m l e t 8 C o m p a r i s o n (B) C o m p a r i s o n 0 1 x (mm) (C) (D) Figure 5.25: These figures illustrate the effect on 7 K reconstruction that is afforded by smoothing Symlet 8 filtered data. Figures A and C illustrate the x-axis profile for non-smoothed data that have been lowpass filtered wi th /o = 6.6 and 9.0 M H z , respectively. Figures B and D compare the normalized profiles based on noise-free data (solid line) and those which have been wavelet filtered and smoothed wi th a moving average filter (dashed line). A g a i n , a L P F has been applied, wi th / 0 = 6.6 and 9.0 M H z i n Figures B and D , respectively. Chapter 5. Results 201 F i l t e r A p p l i e d I m a g e P o w e r S u m R e s i d u a l (%) None - Noisy 22 5 P t MoveAve 22 Symlet 8 3.7 Symlet 8 4- 5 P t MoveAve 3.8 Daubechies 9 0.9 Daubechies 9 + 5 P t MoveAve 0.8 Table 5.4: The above data are the image power sum residuals for different filtering methods, together wi th the application of a L P F wi th / 0 = 2.6 M H z . filtered as well as smoothed with a 5 point moving average filter. Evidently, the wavelet denoised data must be smoothed before the image reconstruction process. Similar results were noted in the case of Daubechies 9 wavelet filtering. Several tests have been done wi th wider bandwidth data to compare the efficiency of the following three noise filtering approaches: • Symlet 8 wavelet filtering followed by 5 point moving average filter (herein "Sym8"). • Daubechies 9 wavelet filtering followed by 5 point moving average filter (herein "Daub9"). • 5 point moving average filtering alone (herein "MoveAve") . These approaches were tested in combination wi th subsequent lowpass filtering during image reconstruction, with / 0 equal to both 6.6 and 9.0 M H z . The preliminary results are illustrated in Figure 5.26. Tables 5.5 and 5.6 present image power sum residuals for the various approaches, subject to lowpass filtering wi th / 0 equal to 6.6 and 9.0 M H z , respectively. Evidently, filtering wi th the Daubechies 9 wavelet family, followed by 5 point moving average filtering, results in the most accurate reconstruction of noisy data in this study. However, it appears as though the smoothing operation is essentially the l imi t ing factor, since the Daub9 method affords absolute and relative decreases in the power sum residual of only 0.1% and ~ 10%, respectively, in comparison to the MoveAve approach. Chapter 5. Results 202 (A) Compar ison Along X - A x i s Noiseless Moving Ave 1 0.9 0.8 ~ 0 . 7 W tn Q> = 0.6 c ^ T^o.5 fo.4 E ™0.3 0.2 0.1 0 (B) Compar ison Along X - A x i s Noisele: S y m S D a u b 9 -1 0 1 x (mm) (C) (D) Figure 5.26: These figures compare the MoveAve, Sym8 and Daub9 noise filtering ap-proaches. Normalized 7« x-axis profiles are shown. Figures A and B were based on data lowpass filtered wi th f0 = 6.6 M H z , while data corresponding to Figures C and D were lowpass filtered wi th / 0 = 9.0 M H z . Chapter 5. Results 203 F i l t e r A p p l i e d I m a g e P o w e r S u m R e s i d u a l (%) None - Noisy 113 MoveAve 0.8 Sym8 1.2 Daub9 0.7 Table 5.5: The above data present the image power sum residuals for the different filtering methods, together wi th the application of a L P F wi th / 0 = 6.6 M H z . F i l t e r A p p l i e d I m a g e P o w e r S u m R e s i d u a l (%) None - Noisy 795 MoveAve 1.1 Sym8 1.1 Daub9 1.0 Table 5.6: This table presents image power sum residuals for the various filtering tech-niques, together wi th the application of a L P F wi th / 0 = 9.0 M H z . PRELIMINARY CONCLUSIONS REGARDING NOISE FILTERING The noise filtering results presented in this thesis hint that wavelet denoising alone has the potential to be superior to the conventional smoothing of pulsed ultrasound data. This was evident in the processing and reconstruction of narrower bandwidth data. The low frequency cutoff was able to remove the higher frequency choppy features introduced into the data by the wavelet denoising process. B y extension, i f a smooth mother wavelet can be custom-made to mimic the data, it is likely that the denoised signal would not exhibit high frequency choppy features. These data would not require subsequent smoothing with the moving average filter. In likelihood then, the custom wavelet filtered data would produce superior images, given that the behavior in the lower frequency region can be extrapolated to the higher frequencies. Chapter 5. Results 204 5.6 EXPERIMENTAL TESTS OF THE METHOD BY B L A C K L E D G E et al 5.6.1 TEMPERATURE DEPENDENT SPEED OF SOUND Al though the acoustic velocity, cn, in water was set at 1.48 mm/pis in the simulations, the true speed of sound in the experimental setting varies wi th water temperature, T . A s such, the following experimentally determined equation by Greenspan and Tschiegg was adopted to calculate cn [38]: c 0 = 1.402736 mm/ps (5.41) A = 5.03358 T - 0.0579506 T 2 + 3.31636 x 10~ 4 T3 -1.45262 x I O - 6 T 4 + 3.0449 x 10" 9 T 5 „ = O + A ^ S L (5.42) In the above equation T is measured in degrees Celsius, and the resulting cn has an error of ± 0 . 0 0 0 1 mm/ps [38]. Figure 5.27 illustrates the resulting acoustic velocity as a function of water temperature. The method by Greenspan and Tschiegg provides an estimate of the speed of sound in the absence of dispersion at a pressure of 1 atmosphere, which is equal to 101.3 kPa . A t the time of the following experiments, the air pressure at the lab location was 101.0 kPa , and hence this formula was considered accurate for this work. Furthermore, for the purpose of these experiments, the acoustic velocity in the fresh water in the water tank was assumed to be approximately the same as that in distilled water, described by Equation 5.41. 5.6.2 INTRODUCING A GENERAL SOURCE Recall that the reconstruction method by Blackledge et al calls for a pulsed line source wi th a field that is dependent upon frequency and space according to the following equation: p 0 ( r , w) = A(u)g(i\is, u) (5.43) Chapter 5. Results 205 1.53 Speed of Sound in Distilled Water 10 15 20 25 30 Temperature (Degrees Celsius) Figure 5.27: This figure illustrates the speed of sound in disti l led water versus water temperature. The pressure is assumed to be 1 atmosphere and dispersion is not included. where g(r\Ta,uj) is the 2D Green's function propagator and A(uf) is the amplitude spec-t rum of the source. A s such, the desired source has zero phase at the source location (or alternatively at time zero), and its field spreads out cyl indrical ly from r s . A s dis-cussed in Section 3.1, attempts were made to approximate this source field through the use of different transducers ranging from a line source for non-destructive evaluation to a custom-made concave source. However, the numerous difficulties described in the Apparatus Chapter l imited the sources available for imaging experiments to only the Panametrics V326 /5 .0MHz/0 .375" . Use of this source introduced wrinkles into the ex-perimental work in two aspects. First , the Panametrics field is beam-shaped rather than cylindrical . However, this feature simply l imited the experimental work to the imaging Chapter 5. Results 206 of only small objects whose cross-section was entirely insonified by the laterally narrow ultrasound field. The second and most important wrinkle introduced by use of the Pana-metrics source is that the field does not have zero phase at the face of the transducer. This was determined by detecting the field at a particular location, r ,^ and backprop-agating it to r s . Equation 5.43 models propagation, which mathematically is the act of operating on a source field by the Green's function. Backpropagation is simply the inverse operation of dividing a frequency dependent field in a point-by-point fashion by the frequency dependent Green's function. The backpropagated field at rs for the signal detected at is then described by Pbp(rs, = p0(rd, u))/g(rd\rs, u) (5.44) where p0 is the detected source field. The pulsed field of the Panametrics transducer was recorded at a distance of 76.3 m m from the transducer face. This distance was measured accurately by using the Panametrics source to both insonify the hydrophone and measure the strong signal backscattered from the hydrophone. Knowledge of | the time-of-flight of the backscatter signal, together with the calculation of the acoustic speed in wafer (27 °C for this particular study) allowed for the accurate determination of the propagation distance. 500 signals were averaged in this experiment to reduce the noise in the data. The field was backpropagated to the source, and the result was not simply equal to the amplitude spectrum of the source, A(u). Rather, the backpropagated field was equal to an ini t ia l field, Pinu, wi th non-zero phase. It was then hypothesized that the correct model of the field from the Panametrics source is Po(r,u) =pinit(u})g(r\Ts,uj) (5.45) Pinit(v) is thus substituted for A(u) in the reconstruction algorithm by Blackledge et al, and the functions F l and F2 , defined in Equat ion 2.27, become F l = apinit{u)-^ (5.46) [2ika c o s ( | ) ) 2 Chapter 5. Results 207 ™ 2 / \ i exp(2zfca) a The advantage of using the Panametrics beam source is that both backscatter and sidescatter signals are not contaminated by the background field that reaches the de-tector without first interacting with the scattering object. Thus according to Equat ion 2.26 the scattered field can simply be written as ps(rd,rs,u) tt F2ip0((pa,k) (5.47) 2 exp(2ika) tt a Pinit{oj)k ibd(<ps,k) a This model was tested by comparing the results of Green's function operation on source fields measured at different distances from the Panametrics transducer. The key in this study is that the fields, PA and pB, measured at two Points A and B , respectively, can both be backpropagated to the source location to yield new complex Fourier spectra, {pbp)A and (pbp)B, which should be similar. Furthermore, the re-propagation of (PbP)A to Point B should produce a new field, PA-+src->B, that is similar to pB measured at Point B . The backpropagation and re-propagation of PA is described by the following mathematics: PA^src^B{rB,U) = —r— -g{TB\Ts,uj) (5.48) g{rs\rA,oj) Note that the field cannot simply be propagated from Point A to B due to the spatial dependency that is embodied in the Green's function. Operat ing on P A wi th ^(rB|r^) would in effect model the scattering of ultrasound from Point A and the propagation of this secondary cylindrical source of sound from Point A to B . This is quite different than the situation at hand in which the source pulse simply travels from TS past Point A to Point B . Three pairs of points were investigated with distances equal to (33.1, 38.6) mm, (38.6, 54.1) mm, and (76.3, 88.1) mm. Again , the distances from the Panametrics trans-ducer were accurately determined through detection of a backscatter signal from the hydrophone, and 500 signals were averaged to reduce noise in the data. {pbp)A and Chapter 5. Results 208 (Pbp)B were compared in a power sum residual calculation according to Equat ion 5.30, as were PA^STC-^B and PB- Recall for example that the power sum residual of PA^STC-^B with respect to pB is = Z \ P A - + ^ B - P B ? X 1 0 Q % ( 5 4 Q ) Given a non-dispersive speed of sound in water, the power sum residuals for a l l pairs of points were ini t ial ly very large in the range of 40-160% for the comparison of (pbp)A and {Pbp)B and in the range of 30-123% for the comparison of PA->STC-*B and PB- Dispersion was then added in the following ad hoc manner in an attempt to improve the source model. A simple linear model of frequency dependent acoustic velocity was introduced in which the fractional range of dispersion over 0-20 M H z was chosen to be some value, fi. Thus, the corresponding range over frequencies from 0 M H z to f^yq in the imaging experiment was calculated to be range d j s p , given by: = i , x ^ (5.50) Recal l from Section 5.3.6 that dispersion is on the order of 1% over the range of 1-20 M H z . As such, fi was varied from 0-0.01. The range of acoustic speed was then calculated to be range c = c 0 x vangedisp (5.51) where cn is the acoustic speed in the absence of dispersion. The min imum acoustic speed over the frequency range was not assumed to be c 0 . Rather, it was calculated to be an ini t ia l value equal to cinit, given by Cinit = c 0 - / 2 x range c (5.52) in which / 2 was allowed to vary between -0.75 and 0.75. Finally, the acoustic speed as a function of frequency, / , was calculated by A B S ( / ) _ _ v c(f) = Cinit + range, x (5.53) JNyq Chapter 5. Results 209 Note that this model does not necessarily mimic physical reality. Rather, it was intro-duced for the sole purpose of reducing the difference between (pbP)A and (pbP)B, as well as the difference between PA-^STC^B and PB, thereby leading to a workable model of Panametrics source field propagation. Upon calculating the frequency dependent speed of sound, PA^STC->B and pB were recalculated based on the new corresponding frequency dependent wave numbers, k(f). The best case scenario for the point sets equal to (33.1, 38.6) m m and (38.6, 54.1) m m was found to be fx = 0.01 and f2 = 0.5, which resulted in power sum residuals in the range of 16-20% for the comparison of (pbp)A and (pbP)B and in the range of 14-15% for the comparison of PA->STC->B and ps- For the point set equal to (76.3, 88.1) mm, the optimal value of fi was smaller at 0.005. This resulted in a power sum residual of only 1% for the comparison of (Pbp)A and {pbP)B and 2.5% for the comparison of PA^STC-^B and pB. The corresponding dispersive speed of sound is plotted in Figure 5.28.A. Figure 5.28.B compares the pulses corresponding to {pbp)A and (PbP)B and those corresponding to PA-^STC^B and P B , respectively, where A = 76.3 m m and B = 88.1 mm. There evidently is excellent agreement. B y virtue of the above experiment, the 2D Green's function was adopted as the propagator for the Panametrics source over the range of 30 to 60 m m subject to use of the dispersive speed of sound for which / i = 0.01 and / 2 = 0.5. F ina l ly a source spectrum was determined as follows for use in the reconstruction algorithm. The reflectivity experiment (backscatter imaging) was performed with the Panametrics source/detector situated at Abs = 31.4 mm. The source field for this reconstruction was determined by comparing the fields for Point Set 1, with A = 33.1 m m and B - 38.6 mm. The fields measured at Points A and B were both backpropagated to the source location to yield new fields, {Pbp)A and (pbp)B, respectively. The corresponding source spectrum, (pinit)bs was chosen to be the average spectrum given by (JWH. = » + (5.54) The power sum residuals comparing [pinit)bs to {pbP)A and (pbP)B were relatively low at Chapter 5. Results 210 Dispersive Speed of Sound 1.511 1 1 — 1 Frequency (MHz) (A) co 4 1 » 2 T3 i o Q. <-2 -4 - 6 5 8 Comparison of Pulses i 1 A . ~ j f Measured Propagated 5 8 . 5 5 9 5 9 . 5 x (mm) 6 0 (B) Figure 5.28: Figure A illustrates the dispersive speed of sound that yielded the least difference between PA-+ src->B and PB, where A = 76.3 m m and B = 88.1 m m . Figure B compares the time dependent pulse corresponding to PA^STC-+B wi th the pulse measured at Point B . Chapter 5. Results 211 3.3% and 2.4%, respectively. The jK experiment (sidescatter imaging) was performed wi th the Panametrics source and the hydrophone situated at Ass = 51A mm. Similarly, (Pinit) ss for this reconstruction was determined by comparing the fields for Point Set 2, wi th A = 38.6 m m and B = 54.1 mm. These fields were backpropagated to the source location to yield (pbp)A and (ptpJB, and Pinit was again chosen to be the average of these fields. The power sum residuals comparing (pinit)ss to (Pbp)A and (pbp)B were again low at 2.6% and 3.6%, respectively. A s further verification of this exercise, Figures 5.29.A and 5.29.B compare the real and imaginary parts of (pinit)bs and (pinit) ss for Point Sets 1 and 2. Evidently, there is excellent agreement even though the distances of the points vary widely. O f final note in this section is the fact that the source amplitude falls to 0.012% of its maximum frequency response at 12 M H z . Furthermore, only 0.02% of the power spectrum lies outside of the range from -12 to 12 M H z . As such, a low pass filter was applied to the corresponding experimental data with a cutoff frequency of / 0 = 12 M H z . 5.6.3 E X P E R I M E N T A L W O R K W I T H T I S S U E P H A N T O M S C H A R A C T E R I Z I N G T H E G R A P H I T E / G E L A T I N P H A N T O M W i t h the Panametrics source introduced into the reconstruction algorithm, the next step in the experimental work was the characterization of the graphite-gelatin tissue equivalent material ( G G - T E M ) described in Appendix D . This material was used due to its non-toxic nature. However, it is difficult to obtain consistent speed of sound results wi th different batches of this material [14]. Furthermore, the speed of sound of the material was not listed in the literature supplied with the formula obtained from another group. A s such, the G G - T E M was characterized with the apparatus illustrated in Figure 5.30. The apparatus was made of a centrifuge tube with a diameter of \ inch. A 3 m m wide rod was placed through the tube, parallel to the r im of the tube. It was necessary to reduce the depth of the G G - T E M in the centrifuge tube to approximately 9 m m in order Chapter 5. Results 212 Comparison of Real Spectra 4000 3000 2000 > T3 CL <-1000 -2000 -3000 -4000 L l 1 S e t 1 S e t 2 -20 -10 0 10 20 Frequency (MHz) -20 (A) Comparison of Imag Spectra -10 o 10 Frequency (MHz) (B) Figure 5.29: Figures A and B compare the real and imaginary parts, respectively, of (Pinitjbs and (pinit)ss for Point Set 1 = (33.1, 38.6) m m and Point Set 2 = (38.6, 54.1) m m . Chapter 5. Results 213 3 mm rod Plastic tube spacer Transducer Active Element \ T E M tube Figure 5.30: This figure illustrates the apparatus that was used to measure the speed of sound in the tissue equivalent material that the phantom was composed of. to obtain a strong signal from the rod that was easily distinguished from the background of backscatter signals returning from the attenuating tissue equivalent material. It was also necessary to ensure that the tissue equivalent material was not touching the active element of the transducer, since this caused significant ringing of the transducer. This ringing was overcome by attaching an 11 m m deep spacer made of the top of a centrifuge tube of the same size to the tube containing the G G - T E M . The two tubes were attached wi th waterproof duct tape. Placing the spacer against the transducer d id not have this effect because the tube was wider than the active element. The speed of sound in the G G - T E M was determined in the following manner. Firs t the empty apparatus was immersed in distilled water and insonified wi th the Panametrics Chapter 5. Results 214 source, which was placed against the centrifuge tube. The T O F of the backscatter signal from the rod was recorded. The temperature of the distilled water was 21°C, and hence the speed of sound in water was c 0 = 1.4857 ± 0.0001 mm///s (dispersion was ignored in this experiment). From the T O F of 26.75 ± 0.02 /xs, the distance from the transducer face to the rod was calculated to be 19.87 ± 0.02 mm. The backscatter experiment was then repeated wi th the centrifuge tube filled wi th G G - T E M , and the plastic spacer again attached. The time-of-flight was 26.40 ± 0 . 0 2 ps. Accounting for the length of the spacer, the speed of sound in the tissue equivalent material was calculated to be CTEM = 1-530 ± 0.002 mm/ps (5.55) The density of the G G - T E M was also measured by comparing the mass of a centrifuge tube full of distilled water at 28°C to that of the same tube full of the tissue equivalent material. Figure 5.31 illustrates the density of water as a function of temperature ac-cording to 71st Edi t ion of The Handbook of Chemistry and Physics [55]. A t 28°C, the density of water is 0.99624 g / m l . Accounting for the mass of the tube, the ratio of the mass of a tube of distilled water at 28°C to that of a tube of G G - T E M was 0.91924, leading to a G G - T E M density of PTEM equal to 1.084 ± 0.008 g / m l . From the speed of sound and density of the tissue equivalent material, the compressibility, KTEM, can be calculated according to Equation 1.7: KTEM = -5 = (0-394 ± 0.003) x 10~ 3 _ g _ (5.56) (^EMPTEM ps2mm2 W i t h the G G - T E M characterized, jK and 7 p can be calculated for any experiment in-volving the material. EXPERIMENTAL S E T U P A string phantom was made using the "candle method" described in Appendix D , in which string wi th a weight on the end is dipped into the graphite-gelatin tissue equivalent material several times, allowing 5 minutes in between for each layer to solidify. In this Chapter 5. Results 215 Density of Water Vs Temperature 1.002] 1 1 • r— Temperature f C) Figure 5.31: This plot illustrates the density of water as a function of temperature. manner, a phantom with diameter equal to 3 . 0 ± 0 . 3 m m was constructed. The large error in the diameter was due to the slightly undulating cylindrical surface of the phantom. The phantom was suspended i n the middle of the tank wi th a plumb bob tied to its bot tom to facilitate the centering of the phantom in the middle of the tank. For the reflectivity experiment, the source/detector transducer was situated at (—31.41 ± 0.03, 0) m m . For the 7 K experiment, | r 8 | and |rd| were l imited to a min imum of 50 m m due to the construction of the arm that holds the hydrophone. The source and detector positions were (0, -51 .35 ± 0.05) m m and (-51.41 ± 0.05, 0) mm, and the value of a = | r s | = | r d | i n the reconstruction algorithm was chosen to be the average of the distances. T H E D A T A A N D A S S O C I A T E D S I G N A L P R O C E S S I N G 500 signals digitized at 60 M H z were averaged together to reduce the noise in the signal. Al though it was desirable to collect data at many angles, it was found that the skin on the surface of the phantom did not prevent it from absorbing water and increasing Chapter 5. Results 216 in diameter over time. A s such, data were recorded at only one angle due to the several minutes required for the collection of a reasonable number of views by manually moving the apparatus (recall the internal corrosion of the traction drive). This view was copied for a l l other views, which was a valid approach for the cylindrically symmetric phantom. In fact, this approach has been published in the literature due to similar problems wi th graphite-gelatin tissue phantoms [32]. Figures 5.32.A and 5.32.B illustrate the data for backscatter and sidescatter, respectively. The signal-to-noise ratios for the backscatter and sidescatter signals in the absence of any filtering are 34 db and 10 db, respectively. The noise in the transmission data is identically a sawtooth of ± 0 . 5 m V , and there is a possibility that this is a digitization error of the Gagescope A D C board. However, given that the board is a 12 bit module, and that the voltage range was set at ± 1 0 0 m V , the digitization error was expected to be on the order of 0.05 m V . It is thus more likely that this sawtooth noise is an effect of either the hydrophone/preamp or the Panametrics receiver/amplifier. Wavelet filtering was tested on these data. Initially the Daubechies 9 was adopted as the mother wavelet, in keeping wi th the best case simulation results presented in Section 5.5.2. The "huersure" threshold selection rule was tested wi th "min" noise rescaling model for the most general approach in the event that the noise was not identically white and Gaussian. It is suggested in the Wavelet Toolbox Guide that level 5 is generally considered to be a good choice of denoised signal in wavelet filtering, so this was the basis of the following analysis [60]. Fi l ter ing the backscatter data wi th the Daubechies 9 family increased the S N R from 34 db to 39 db, but afforded very little improvement in the data. The power sum residual of the Fourier spectrum for the wavelet filtered data compared to that for the unfiltered data was only 0.6%. Furthermore, it was possible to obtain the same improvement by simply lowpass filtering the data wi th / 0 = 12 M H z . Upon applying the L P F to both the wavelet filtered data and the original data, there was no discernible difference between the two corresponding Fourier spectra, wi th a negligible Chapter 5. Results 217 Figure 5.32: Figures A and B illustrate backscatter and sidescatter data for a graphite-gelatin tissue equivalent phantom. Chapter 5. Results 218 power sum residual of 0.006%. It was noted that the data had features in common wi th the Daubechies 20 mother wavelet, illustrated in Figure 5.33.A. Fi l ter ing the backscatter data wi th this family produced identical results when compared to filtering wi th the Daubechies 9 family. Fi l ter ing of the sidescatter data produced slightly different results. The power sum residual of the Fourier spectrum for the Daub20 filtered sidescatter signal relative to the original signal was 63%. Figure 5.33.B illustrates the filtered signal, wavelet decomposed at level 5. Figure 5.33.C illustrates a closeup of a noisy region of the signal, comparing the filtered and non-filtered versions. Evidently, there is a dramatic improvement in the noise level, and in fact the S N R was increased to 28 db, up from 10 db. However, it was again possible to obtain the same improvement by simply lowpass filtering the data wi th a cutoff frequency of / 0 = 12 M H z . Upon applying the L P F to both the wavelet filtered data and the original data, there was no discernible difference between the two corresponding Fourier spectra, with a negligible power sum residual of 0.06%. The same results were obtained when the sidescatter data were wavelet filtered wi th the Daubechies 9 family. Given the results of these preliminary tests on experimental data, wavelet filtering was not applied in the reconstructions to follow. R E S U L T S F O R 7 K A N D R E F L E C T I V I T Y The compressibility imaging was performed in water wi th a temperature of 25°C, wi th a speed of sound and density of approximately 1.497 mm// i s and 0.99705 g / m l , respectively. 7 r e is therefore -0.12 given the value of KTEM calculated in Section 5.6.3. The reflectivity experiment was performed in water wi th a temperature of 28°. The speed of sound and density of water for this temperature are 1.5047 mm/ps and 0.99624 g / m l . Given PTEM determined in Section 5.6.3, 7 p is 0.09. The expected value of reflectivity is therefore R = 7 k — 7 p = —0.21.' In calculating the data, ipo((ps,k), in the 2D Fourier domain of the R or 7 r e cross-Chapter 5. Results 219 Daubechies 20 Mother Wavelet 20 30 40 Time (unitless) Filtered Signal - Daub20 (A) 68 70 Time (us) 2 1.5 1 > 0 5 CD N 8 o >-0.5 -1.5 Zoom of Signal - Daub20 79.2 Orig D a u b 2 0 79.4 79.6 Time (us) 79.8 (B) (C) Figure 5.33: Figure A illustrates the Daubechies 20 mother wavelet. Figures B illustrates the sidescatter data for the phantom after it has been denoised wi th the Daubechies 20 family. Figure C compares a noisy region both before and after denoising. Note that the denoised line is identically equal to the x-axis. Chapter 5. Results 220 section during image reconstruction, there is a deconvolution of the form ^ k ) ~ P s ( r d , r s , o ; ) (5.57) a*pinu(u>)ke-?^ Since both P s and Pinu are characterized by noise, the deconvolution is subject to error and results in ipg data characterized by spikes. Even after lowpass filtering the data, the inherent noise was emphasized by the deconvolution and resulted in images wi th significant circle and streak artifacts that essentially destroyed the features of the image. The following smoothing approach was thus adopted in an ini t ia l attempt to overcome this problem. For a real valued image, only the real part of the corresponding 2D Fourier data has any effect on the resulting image upon inverse transforming. This is due to the existence of the following symmetry: F T ( - « , - « ) = FT+(u,v) (5.58) where u and v are the coordinates in Fourier space. The imaginary components of the 2D Fourier data therefore cancel out. A s such, the real part of ipg for the backscatter data along each radial line in Fourier space was smoothed twice with a moving average filter wi th a window size equal to 5. The real part of ipg for the sidescatter data were similarly processed, but these data required 3 passes of the filter due to the relatively poor signal-to-noise ratio of 28 after lowpass filtering. In both experiments, the imaginary part of ipg was left as is. Last ly it should be noted that the backscatter data were normalized by the amplitude spectrum of (pinit) ba in order to remove any dependence upon the transducer response. Recall that the response of the hydrophone is relatively flat and as such this is not an issue wi th sidescatter data. Upon smoothing the data, promising reflectivity and 7 K images were obtained for the G G - T E M phantom. Figures 5.34.A and 5.34.B illustrate the reflectivity image and corresponding x-axis profile, while Figures 5.34.C and 5.34.D present the resulting jK image and x-axis profile. These images were based on 100 views of data. K e y points are as follows. First , the images indicate a negative R and 7*, as expected. Second, the Chapter 5. Results 221 Reflectivity Reflectivity x (mm) x (mm) (C) (D) Figure 5.34: This composite presents imaging results for a 3 m m wide G G - T E M phan-tom based on 100 views of data. Figures A and B illustrate the reflectivity image and its corresponding x-axis profile, while Figures C and D present the 7 r e image and its corresponding x-axis profile. Chapter 5. Results 222 full-width-half-max values are reasonable. Measured from zero amplitude, the F W H M of the rr-axis profile of jK is 2.94 ± 0.06 mm, which is in excellent agreement wi th the phantom width of 3.0 ± 0 . 3 mm. Measured from zero amplitude, the F W H M of the z-axis profile of R is very small at 1.94 ± 0.04 mm, which varies by 33% and 35% from the low and central values of the object diameter. As such, the agreement is not good, although the result is reasonable given the error in the source model, the necessary smoothing, and the application of the Born approximation in the development of the reconstruction algorithm. B o t h R and 7„ have primarily negative profiles, as expected. The minimum values of R and 7 K where the phantom is situated are Rmin = —16.5 and {jK)min = —14.6, respectively. Normalization factors, £, are therefore required to obtain the actual values of 7 K and R, as in the simulation tests. £R and £ 7 k are furthermore not the same, which was again seen in simulation. Due to the large shortfall in the F W H M of the R image, a valid cross-section for 7 P could not be calculated and is therefore not presented here. It should be noted that the distance, a from the origin to the source or detector is a very important parameter in the reconstruction of experimental data. If this value is accurately known, the expected image wi l l result, while errors in a can cause dramatic effects on the image. A s an example, Figure 5.35 illustrates the effect on yK when the parameter a is too large by 1% and 2%. A n error of 1% leaves (pi^mm as is, but reduces the F W H M (measured from zero) to 2.00 ± 0.04 mm. A n error of only 2% completely destroys the image. Note that an error in a was not the cause of the smaller F W H M for the reflectivity image presented in this section, a was varied to test this and 2.0 m m was the largest F W H M that was possible in the reconstruction of the data. F I N A L R E M A R K S R E G A R D I N G T H E S O U R C E A s discussed in Section 3.1, it was possible to detect a reasonable signal 3-5 cm directly in front of the line source. These data were used to determine i f in fact the source model is able to properly describe the field from a broadband pulsed line source, Chapter 5. Results 223 - 1 0 0 10 x (mm) Figure 5.35: This figure illustrates the x-axis profile of the 7 r e image that results when a is increased in error by 1% and by 2%. as expected. A s wi th the Panametrics transducer, the pulsed field was measured at two points situated A = 46.6 m m and B = 59.4 m m from the line source. The fields PA and PB were backpropagated to the source location to yield (PbP)A and (PbP)B, which were compared. (Pbp)A was then re-propagated to Point B to yield a new field PA^src-+B, which was compared to the measured field pB. Again it was necessary to add dispersion according to Equations 5.50 through 5.52 in order to reduce the power sum residuals, and values of fi and / 2 equal to 0.003 and -0.6, respectively, were found to be opt imal . The resulting power sum residuals were 21% for the comparison of (Pbp)A and (»(, p )B and 23% for the comparison of PA ->• src->s and pB- The results of this exercise are therefore slightly worse than for the study of the Panametrics source. Figure 5.36.A compares the pulses Chapter 5. Results 224 150 100 | 50 •o 0 Q. E -50 < Comparison of Sources -100 -150 1 0 2 4 Time (us) B P from A B P from B (A) Comparison of Sources 40 42 Time (us) M e a s u r e d P r o p a g a t e d (B) Figure 5.36: Figure A illustrates the dispersive speed of sound that yielded the least difference between pA-ysrc->B and pB, where A = 46.6 m m and B = 59.4 m m . Figure B compares the time dependent pulse corresponding to PA-tsrc-*B wi th the pulse measured at Point B . Chapter 5. Results 225 corresponding to (pbP)A and {Pbp)B, while Figure 5.36.B compares those corresponding to PA->STC->B and PB- Al though there is reasonable agreement, the power sum residuals indicate that the source model does not do a better job of describing the line source as compared to the Panametrics source. This is unfortunate given that the source terms in the reconstruction algorithm are aimed at a description of line sources. 5.7 T E S T S O F T H E A R T IN C O M P U T E R S I M U L A T I O N Recall from Section 4.2.3 that images, / , equal to 7 K or and 7 P can be obtained through the application of an algebraic reconstruction technique to the data. This algorithm solves the following system of equations: b = T l (5.59) where Ti:j = Atki g(rj\rs, ki) Ij g(rj\rd, k{) AxAy (5.60) and g) = g{rj\Ys,ki)g{rj\rd,ki) (5.61) h = ps(rd,rs,ki) Ai = A(cji) = A(ki) In testing the A R T , scatter data were simulated for 1 or more discrete points making up an object. These Fourier data were calculated for each view about the object as in Section 5.3 v ia Equat ion 5.27. Again , the width of the element of area associated wi th each scatter point was determined by the Nyquist frequency of the data. Evident ly then, there is an element of area associated with both the left hand side and right hand side of Equat ion 5.59. As in the testing of the method by Blackledge et al, these elements of area, denoted by ax and ct2, are essentially factors that scale the resulting images. A n increase in a\ together wi th al l other terms being kept constant w i l l scale / upward. In Chapter 5. Results 226 contrast, an increase in a2 alone wi l l scale I downward. Care must therefore be taken when comparing different images. In the following tests, ay and a2 were consistently made equal. This had the effect of cancelling these terms and allowing for the direct comparison of different images. The full development and testing of an algorithm involving iterative techniques is generally regarded as an involved project in its own right. As such, the scope of this thesis allowed the inclusion of only a few investigations that were aimed at determining the potential capabilities and limitations of the method. The results of these preliminary tests w i l l serve as a foundation for future work in this research project. Since the a im of this work was to evaluate the potential of the pulsed U C T methods at hand to provide quantitative images of tissue, the following key points were investigated in the A R T approach: • Image resolution. • Ab i l i t y to correct for frequency dependent attenuation. • A b i l i t y to produce properly scaled images. • Ab i l i t y to reconstruct images for which is not <C 1, where Tj is the position vector of either the source or detector. In addition, since the A R T involves iterative techniques, an over-riding factor is the ability of the reconstruction algorithm to converge on the correct image, I. In a l l the simulations performed, the convergence criterion was that the error was less than 1 x 1 0 - 7 . In doing simulations to study these key points, it became clear that computing power, the speed of convergence, and the ability of the algorithm to converge were three major obstacles. For instance, the A R T performed poorly in reconstructing larger images on the order of 30 x 30 or 20 x 20 pixels because the error was reduced very litt le in each iterative step. As such, convergence was almost nonexistent. Given that the scope of this thesis could not include an in-depth study into convergence methods and issues, these preliminary tests were necessarily l imited to images wi th fewer than 20 x 20 pixels. This Chapter 5. Results 227 guideline was advantageous in that it l imited the size of the propagator array, A , and therefore reduced the required computing time and memory. To further reduce the C P U time and memory requirements, the incident field Fourier spectrum was l imited to less than 4000 discrete frequencies. A s wi th the implementation of the method by Blackledge et al, the number of fre-quencies in the Fourier data were further reduced by truncating the data set beyond a reasonable frequency. In the examples presented in the following sections, insonification was simulated with a Ricker wavelet that had u>0 equal to 20 M H z . Recall from Section 5.3.1 that a lowpass filter wi th / 0 equal to 9.0 M H z can be applied to the corresponding scatter data without loss of valid information, and that the Fourier data can be subse-quently truncated anywhere beyond 11.3 M H z . Therefore, the U C T data simulated in this section were processed with a lowpass Butterworth filter of order 20 wi th / 0 = 9.0 M H z , and data for both the 7 K and R reconstructions were truncated at ( / r ) 7 r e and (/T)K equal to 14 M H z . Since the same (x, y) image grid is set up for ^ K and R imaging before data processing begins, the truncation frequency can be the same for both image recon-structions. Recall that wi th the method by Blackledge etal, setting ( /T) 7 k equal to (}T)R resulted in different image sizes and pixel coordinates. The above truncation frequency was expected to have no effect on image reconstruction. For good measure though, this assumption was tested in simulation, and the power sum residual, given by Equat ion 5.30, was computed for the image based on truncated data relative to that based on the full original data set. The power sum residual was zero for al l tests. 5.7.1 C H O I C E O F I T E R A T I V E M E T H O D Four methods were considered for the iterative solution of the linear system, y = A x , in the A R T . These were the Conjugate Gradient (CG) method, the Jacobi Iterative (JI) technique, the Gauss-Seidel (GS) method, and the Successive Under-Relaxation (SUR) technique, a l l of which are outlined in Appendix E . In this appendix it was indicated that Chapter 5. Results 228 the JI and G S methods require that A be diagonally dominant and that every diagonal element of A be nonzero. Given the nature of the propagator matrix, the condition of diagonal dominance is impossible to satisfy. In fact, the sum of the absolute value of al l non-diagonal elements along a row in a typical propagator matrix is generally 2 orders of magnitude larger than the absolute value of the diagonal term. Therefore, the JI and G S methods can not be used. The S U R and C G methods were tested to determine if they are able to converge upon an image solution in the A R T reconstruction problem. Several simulations were done for a simple point scatterer located at the origin, with the source and detector located 100 m m from the origin. Values of OJ ranging from 1 x 1 0 - 4 to 0.9 were tested, but convergence with the S U R method was not possible. In fact, for values of a; less than 1 x 1 0 - 3 , the method would diverge both immediately and rapidly. In contrast, the C G method converged in 98 iterations. Thus, the C G method is the only approach of the four that was available for use within the scope of this project. 5.7.2 A T T E N U A T I O N C O R R E C T I O N A N D C H O I C E O F X ° This section presents the results of 7 r e and 7 P reconstruction for a 3 x 3 grid of isolated points. W i t h this grid, the ability of the A R T to reconstruct images in the presence of attenuation was studied. In addition, the effect of the choice of the ini t ia l image vector, x°, was also investigated. The grid was identical to that used in Section 5.3, wi th the exception that the x and y spacing between points was 0.26 m m instead of 1.5 mm. In other words, points in the nth column had 7* and 7 P equal to 0.04n and 0.0909n, respectively. Da ta for 100 views were sampled at 45 M H z . This test of attenuation correction is comparable to the study of the approximate correction, which was done in Section 5.4. For instance, the spacing of the points in the latter study was 0.20 m m in both the x and y dimensions, and the same Butterworth filter was used. Data were not truncated, though, so in effect a more complete data set Chapter 5. Results 229 was used in Section 5.4. A s in the case of the method by Blackledge et al, it was not possible to perform an A R T reconstruction of a grid of points with large x and y spacing due to computing power and memory restrictions. As well, convergence issues that were beyond the scope of this thesis l imited the physical extent in the reconstructed images. A s such, attenuation in tissue was scaled up by a factor of 1000 as in the testing of the approximate attenuation correction in Section 5.4 in order to facilitate the study of attenuation effects over smaller distances. Attenuation in water was left as is. Since the maximum distance between a scatter point and the origin was only 0.28 mm, the distance from the origin to the source or detector was set to a correspondingly small value of 2 mm. A n in i t ia l image vector, x ° , must be selected to begin the iterative process. This was found to have an effect on the solution that the A R T converged upon. Prel iminary tests involving both the grid at hand as well as other scattering objects indicated that the matr ix system involving propagators in the absence of attenuation does not have only one solution, but rather has several similar solutions that the A R T converges upon given different ini t ia l estimates, x° , of the image. In contrast, when attenuation was considered in the system and a correction was applied, the image solution can vary widely given different ini t ia l estimates, x ° , indicating a generally poorly behaved system. Two approaches to selecting x° were studied. In the first, x° was set equal to [0.2 ... 0.2] for each view, regardless of whether 7 K or jp was being imaged. It was found, however, that the reconstruction of 7 K data for each detector angle converged upon a view wi th generally positive pixel amplitudes (since 7 r e was equal to equal to 0.04) wi th a maximum value of that was in the range of 0.3-0.35. Similarly, the reconstruction of reflectivity data for each detector angle converged upon a view with generally negative pixel amplitudes (since R was equal to -0.509, to yield a 7 P equal to 0.0909) wi th a minimum value that varied between -0.9 and -1 . Thus, in a second reconstruction a custom x° was determined for each view. For jK imaging, data for the i t h view were reconstructed, and the maximum Chapter 5. Results 230 pixel value, rf, was calculated. x° for the reconstruction of the next view was then chosen to be [rf...rf]. Da ta for the first view was ini t ial ly processed with x° = [0.1...0.1]. The maximum pixel value, rj, was determined, x° was set equal to x° = [r?...r/], and data for the first view were reprocessed. The case of 7 P imaging followed identically, except that instead determining the maximum pixel value for each view, the minimum pixel value was calculated and used as a basis for x ° . Figures 5.37.A and 5.37.B illustrate the 7 K and jp image solutions in the absence of attenuation given the custom varied x ° , while Figures 5.38.A and 5.38.B illustrate the same given a constant x ° . Evidently, the choice of x° can have a large effect on the maximum pixel amplitude in the absence of attenuation. Table 5.7 presents the maxi-I m a g e T y p e x° C h o i c e M i n (un i t less ) M a x (un i t l e ss ) 7* Constant 0 3.6 7K Custom Varied 0 43.7 7P Constant 0 17.7 7P Custom Varied 0 78.2 Table 5.7: The above data compare the maxima and minima for jK and 7 P imaging of a grid of points in the absence of attenuation, given a x° that is either constant or custom varied. mum and minimum pixel values for the different images in the absence of attenuation. Al though these images are not identical, they are in fact nearly scale versions of one another. To verify this, the images were normalized to one and the power sum residual was computed for the custom varied x° image versus the constant x° image. The power sum residuals for the 7 r e and 7 P images were 0.1% and 2.0%, respectively. A s such, there is li t t le difference in the absence of attenuation between the two methods of choosing x° after the images are normalized. The most noticeable effect of the custom varied x° method, however, is that it pro-duces very inferior images in the presence of attenuation. Figures 5.37.C and 5.37.D Chapter 5. Results 231 Figure 5.37: Figures A and B illustrate the 7K and jp image solutions for a 3 x 3 grid of isolated scatter points in the absence of attenuation given the custom varied x ° . Figures C and D illustrate the same in the presence of attenuation. Chapter 5. Results 232 -0.2 0 x (mm) -0.2 o x (mm) (B) y - Comparison (C) (D) Figure 5.38: Figures A and B illustrate the 7 K and 7 P image solutions for a 3 x 3 grid of isolated scatter points in the absence of attenuation, wi th a constant x° equal to [0.1 ... 0.1]. Figure C and D compare the x-axis profiles for 7 / c and 7 p , respectively, i n the presence (dashed line) and absence (solid line) of attenuation. Note that the dashed and solid lines in Figure C overlap identically. Chapter 5. Results 233 illustrate jK and 7 P for the grid in the presence of attenuation given the custom varied x ° . Evidently, the method does not have the ability to correct for attenuation in this particular example. The reconstruction of a square block in Section 5.7.4, however, w i l l illustrate an example in which the custom varied x ° worked quite well. These studies serve to illustrate that iterative methods do not always behave in an intuitive manner, and that their use requires an extensive understanding of convergence issues. Returning to the method involving the constant x ° equal to [0.2 ... 0.2], these pre-l iminary results indicate that the A R T is able to reconstruct images well in the presence of attenuation given a x ° that promotes convergence to valid images. Figures 5.38.C and 5 . 3 8 . D compare the x-axis profiles with (solid line) and without (dashed line) attenua-tion for the 7 r e and 7 P images, respectively. Note that the corresponding images have not been normalized or scaled in any way. The agreement is so good that the lines are nearly superimposed. In fact, the power sum residual is 0.004% and 0.009% for 7« and 7 P imaging, respectively. A n important final note regarding the quantitative imaging potential of the A R T is that not al l the pixels corresponding to water in Figure 5.38 are equal to zero as they should be. In addition, the heights of the peaks do not exhibit the same ratio along each row nor do they exhibit the 1:2:3 ratio exactly. The percentage ratio matrices of the normalized images relative to the actual scatter objects were 89 92 103 98 93 100 112 100 132 104 100 105 89 92 103 98 93 100 Evident ly the jK result is problematic in that the points on the positive x-axis has an error of 32%. However, although the A R T is not fully quantitative, it behaves comparably wi th the method by Blackledge et al. The average over a l l entries for the percentage ratio matrices presented in Expression 5.37 (best case scenarios for Blackledge method) were 97 ± 3% and 95 ± 4% for ^ K and 7 p , respectively. In comparison, the averages for this Chapter 5. Results 234 example are 100 ± 14% and 99 ± 4% for 7 K and 7 P imaging, respectively. Thus, result for 7« is reasonable, while that for 7 P is better than the result for the method by Blackledge et al. 5.7.3 R E C O N S T R U C T I O N O F P O I N T S A s in the tests of the method by Blackledge etal, imaging experiments were simulated for single isolated points to determine the image resolution capabilities of the A R T . In addition, point reconstructions were able to test the ability of the A R T to image objects wi th location r such that ^ is not <C 1, where is the position vector of either the source or detector. The images were based on 40 views, and the iterative process for each view was begun with a constant x° equal to [0.2 ... 0.2]. In the first test, reconstructions were done for a point at (0,0) m m in a non-attenuating background fluid wi th an acoustic velocity equal to that of water. The point was char-acterized by 7 K and 7 P equal to 0.04 and 0.0909, respectively. The source and detector were located at a distance of 100 m m from the origin. The data were sampled at both 30 M H z and 45 M H z for comparison. Figures 5.39.A and 5.39.C illustrate the reconstructed P S F ' s for jK and 7 P , respectively, given data sampled at 45 M H z . Figures 5.39.B and 5.39.D illustrate the P S F profiles along radial lines at the angles [0°, 22.5°, 45°, 67.5°]. A profile for 90° is not shown since it is identical to that for 0°. Also, note that the profiles for 22.5° and 67.5° overlap. The F W H M of the jK P S F measured at 0.113 along both the x and y axes is 0.053 ± 0.001 mm. Note that this parameter was found to be equal along both the x and y axes, as was the case for al l other cases presented in these sections. Since the cutoff frequency in the data is effectively 9 M H z after the application of the low pass filter, the expected image resolution is approximately 0.08 mm. A s such, the A R T is provides excellent results in this respect. The F W H M for the 7 P P S F measured at a height of 0.285 along both the x and y axes is 0.049 ± 0.001 mm. Note that the 7 K reconstruction looks noticeably wider than the 7 P reconstruction. Chapter 5. Results 235 Figure 5.39: This figure illustrates the reconstructed P S F ' s for a point located at (0,0) m m with jK and 7 P equal to 0.04 and 0.0909, respectively. Chapter 5. Results 236 I m a g e S a m p l i n g F r e q u e n c y ( M H z ) M a x i m u m (uni t less ) t (un i t less ) B a c k g r o u n d (un i t l e ss ) 7* 30 0.224 5.62 ~ 0.04 7« 45 0.225 5.61 ~ 0.04 1/> 30 0.570 6.27 ~ 0.09 7P 45 0.569 6.25 ~ 0.09 Table 5.8: The above data illustrate the maximum pixel values, rough background values and the image normalization factors, £, for the various reconstructions of a point at (0,0) mm. This is simply an artifact of the M a t l a b ™ plotting routine and the image coarseness that results from the necessarily l imited number of pixels. In reality, the reconstructions are cylindrically symmetric. This is illustrated in Figures 5.39.B and 5.39.D, which plot the P S F profiles along radial lines at various angles for the 7 K and 7 P images, respectively. The average F W H M for the 7 K image is 0.051 ± 0.001 mm, while that for the jp image is 0.048 ± 0.001 mm. The F W H M along the x and y axes given data sampled at 30 M H z was again 0.053 ± 0.001 m m for jK and 0.049 ± 0.001 mm for 7 P . The average F W H M over radial lines at angles of [0°, 22.5°, 45°, 67.5°] was 0.052 ± 0.001 m m for jK and 0.047 ± 0.002 m m for 7 P . It should also be noted that as expected the P S F height scaled directly wi th the actual value of the corresponding 7 K or 7 P . The P S F heights were nearly identical for the same image type wi th data sampled at 30 M H z and 45 M H z , and the nonzero background values were the same. Table 5.8 outlines the maximum pixel values, normalization factors (£) required to reproduce the original object, and the rough background values of the images. The second set of simulations tested the ability of the A R T to reconstruct images for a point for which M- was not much less than 1. Recall from Section 5.3.2 that the method by Blackledge et al had difficulty with this and produced halos rather than Gaussian P S F ' s as a result of the simplifying assumptions used in the derivation of the Chapter 5. Results 237 image reconstruction algorithm. Experimental parameters were identical to those in the previously reported simulation, except that the source and detector were a distance of 2 m m from the origin, the point was located at (-0.7,0) mm, and there were 100 views of data. Figures 5.40.A and 5.40.B illustrate the reconstructed P S F ' s for jK and 7 P , respectively. For comparison, Figure 5.40.D presents the 7„ P S F for image reconstruction using the method by Blackledge et al, given the same simulation conditions. Evidently, the A R T produces a far superior well-defined Gaussian P S F based on far less data, as compared to the halo-shaped P S F in Figure 5.40.D. Figure 5.40.C illustrates the normalized x-axis profiles for the two A R T produce images. Evidently, the 7 P profile is much neater in terms of shape and background than the 7 K profile. Note that the F W H M values are wider than for the previously shown point at the origin. However, the point-spread-function F W H M for jK and 7 P measured along both the x and y axes is 0.051 ± 0.001 m m and 0.048 ± 0.001 mm, respectively, which agree within error wi th the values for the point at the origin given 45 M H z data sampling. The average F W H M over radial lines at angles of [0°, 22.5°, 45°, 67.5°] was 0.051 ± 0.002 m m for jK and 0.046 ± 0.002 m m for 7 P . Not only are the F W H M values comparable, but the heights of the point-spread-functions for the off center point are very close to those for the point at the origin. The maximum pixel values in the jK and 7 P images for the off center point are 0.227 and 0.571, respectively, and these differ by only 0.9% and 0.4% from the values for the point at the origin. Given that ^ — 0.35 for this example, the algebraic reconstruction technique behaves very well and in a far superior manner when compared to the method by Blackledge et al. Recall from Section 5.3.2 that, for the latter method, the P S F for a point at (5,5) m m for which ^ = 0.05 had a halo shape and an amplitude that was reduced by 92% when compared to a point at the origin. Chapter 5. Results 238 - 0 .78 -0 .76 -0 .74 -0 .72 - 0 . 7 -0 .68 -0 .66 -0 .64 x (mm) E E C - 0 x (mm) 0.251 0 21 °' 1 5 l 0.11 o . o s l o i -0.051 -0.11 - 0 . 1 5 1 - 0 . 2 | -0.251 (B) -1 - 0 . 9 - 0 . 8 - 0 . 7 - 0 . 6 - 0 . 5 x (mm) (C) (D) Figure 5.40: Figures A and B illustrate the reconstructed P S F ' s for a point located at (-0.7,0) m m wi th jK and 7 P equal to 0.04 and 0.0909, respectively. Figure D presents the 7 r e P S F for the same point reconstructed under the same conditions using the method by Blackledge et al. Figure C presents the normalized x-axis profiles of the images produced v ia the A R T . Chapter 5. Results 239 I m a g e A t t e n u a t i o n P i x e l A v e r a g e O v e r D i s c (un i t l ess ) 7* N 2.01 ± 0 . 0 3 7« Y 1.96 ± 0 . 0 3 7 P N 6.7 ± 0 . 2 7 P Y 6.5 ± 0 . 1 Table 5.9: The above data present the average pixel value over a disc of radius 0.19 m m for the 7„ and ^yrho of the square block using the A R T . 5.7.4 R E C O N S T R U C T I O N O F A S Q U A R E B L O C K This section provides an idea of the ability of the A R T to reconstruct objects that have a plateau region. Results are presented for a square block of width 0.38 m m wi th 7 K and 7 P equal to 0.04 and 0.0909, respectively. The data were sampled at 45 M H z . Attenuation in tissue was again scaled up by a factor of 1000, and attenuation in water was left as is. The source and detector were again placed at a distance of 2 m m from the origin, xo was varied for each angle of data based on the maximum value (for jK) or minimum value (for R) of the reconstruction for the previous view. Figures 5.41.A and 5.41.B illustrate the reconstructed 7 K and 7 P images in the absence of attenuation, while Figures 5.41.C and 5.41.D compare the x-axis profiles for the original image and the reconstruction in both the presence and absence of attenuation for 7 K and 7 P , respectively. Note that the image figures are not normalized, whereas the x-axis profiles a l l correspond to normalized images. The images reconstructions in the presence of attenuation were very similar in shape to their counterparts in the absence of attenuation. W i t h normalization, the corresponding power sum residuals for jK and 7 P imaging were 0.15% and 0.06%, respectively. It should be noted that the A R T produced reconstructions with a definite plateau region, and Table 5.9 presents the average pixel value over a disc of radius 0.19 m m centered on the origin. Evidently, determining a custom x ° for each view worked well in this example, particularly in the case of 7 P reconstructions. Chapter 5. Results 240 x (mm) x (mm) (C) (D) Figure 5.41: Figures A and B illustrate the 7 k and 7 p reconstructions, respectively, in the absence of attenuation for a square block of width 0.38 m m and with 7 k and 7 p equal to 0.04 and 0.0909, respectively. Figures C and D compare the normalized x-axis profiles for the original image (solid line), and the reconstructed 7 k and 7 p in the presence (dotted line) and absence (dashed line) of attenuation. C H A P T E R 6 CONCLUSIONS A N D F U T U R E W O R K 6.1 G E N E R A L R E M A R K S This thesis entailed significant development and testing of a prototype scanner for pulsed ultrasound computed tomography that is aimed at reconstructing 2D cross-sections of spatially varying 7^ and 7 p . In particular, compressibility beyond the palpable region can be investigated only through ultrasound imaging, and this information may be highly useful for tissue characterization. A t the beginning of this work, it was hoped that a sys-tem could be developed that would lay the groundwork for a future scanner for use in the early detection of breast cancer. To this end, a direct Fourier method introduced by Blackledge et al was incorporated into the system, as well as an algebraic reconstruc-t ion technique involving attenuated propagators that was developed in this thesis. B o t h Fourier and wavelet based digital signal processing methods were also incorporated to remove noise from the data. A proof of concept study has been completed that has i l luminated the capabilities and limitations of this first generation U C T scanner. Fur-thermore, a strong foundation has been laid for continued work in pulsed U C T , which is a new area of research in the Cancer Imaging Department of the B C Cancer Research Centre. It was hypothesized that this system would be able to reconstruct quantitative images wi th resolution on the order of 1 mm, thereby providing a basis upon which to bui ld a second generation system for use in the early detection of breast cancer. The major focus of this thesis was therefore the testing of the scanner in simulation. This provided a controlled environment in which to observe algorithm behavior under the best conditions. 241 Chapter 6. Conclusions and Future Work 242 The reconstruction algorithm by Blackledge et al has been thoroughly examined, and key preliminary studies of the A R T have been conducted. In addition, the method by Blackledge et al has been tested in a tissue phantom experiment. Contrary to the hypothesis, the method by Blackledge et al works well in only very l imited cases. For instance, there is excellent image resolution of ~ 0.07 m m for points in the absence of attenuation for which |r| < 0.05|r-j|, where Tj is the distance from either the source or detector to the origin. Grids of such points were reconstructed in a quantitative fashion to wi thin 5% of their actual values of 7 K and 7 p . However, this method has far more limitations than capabilities due primarily to the lack of an available attenuation correction, an inherent T O F error, and lack of a model that can very accurately describe real-world transducers. These factors currently prevent the method from reconstructing quantitative images wi th useful resolution in most simulated and experimental settings. The A R T performed dramatically better. It consistently exhibited image resolution of ~ 0.05 m m for points wi th |r| equal to at least 0.3|r-j| in both the absence and presence of attenuation. This method also reconstructed grids of such points in a quantitative manner to within 1% of their actual values. Most important, in simulation the A R T was able to properly correct for attenuation to within an error of less than 0.2%. However, the algebraic reconstruction technique has difficulty converging on an image solution for 7 K and 7 P cross-sections of more than 20 x 20 pixels, currently rendering it impractical. These theoretical results do indicate, though, that the method holds promise. The non-periodic nature of the pulsed U C T data motivated the study of wavelet denoising methods, wi th appropriate wavelet families used in both the simulation en-vironment and the experimental setting. Typica l theoretical data required smoothing wi th a moving average filter after wavelet filtering in order to remove residual choppy features, and these data resulted in an image that had only 9% less error (relative to noiseless data) as compared to data processed solely with the moving average filter. In the experimental setting, wavelet analysis performed no better than lowpass Butterworth Chapter 6. Conclusions and Future Work 243 filtering wi th respect to noise removal. The effect of dispersion on image reconstruction was studied for the method by Black-ledge et al and shown to have a dramatic effect on algorithm performance. A typical value of 1% dispersion over 1-20 M H z had the effect of distorting Gaussian point-spread-functions into halo-shaped functions with amplitudes dramatically reduced by a factor of ~ 250. This was due to the destruction of valid phase information in the modelling of the scattered ultrasound field. A correction based on knowledge of the dispersive speed of sound was able to restore the P S F to within reasonable limits, wi th percentage errors of 26%, 4% and 9% in the minimum, maximum and range, respectively, of the corrected 7«, cross-section. Al though dispersion was not studied for the A R T , it w i l l indeed have a similar effect on the image reconstruction process since field phase information is equally important in this method. A s such, precise knowledge of the dispersive nature of the speed of ultrasound in tissue and water is required in order to develop pulsed ultrasound computed tomography into a viable imaging technique. This thesis has shed light on several areas that are in need of further research in order to develop a potentially viable imaging system. These areas involve the application of better digital signal processing techniques for noise removal, the modification of the scanner wi th respect to the stepper motor apparatus and the source, and the improvement of the reconstruction algorithms in terms of attenuation correction and isochrone warping in the method by Blackledge et al, and the problem of convergence in the algebraic reconstruction technique. These future directions are discussed in the following sections. 6.2 K E Y F U T U R E W O R K 6.2.1 A P P A R A T U S I M P R O V E M E N T S O f primary importance to the success of the method by Blackledge et al is the in-corporation of a pulsed cylindrical source whose field is identically described by the 2D Green's function propagator. This source model is necessary for the mathematics to Chapter 6. Conclusions and Future Work 244 reduce to a tractable method. Both the amplitude and Fourier phase characteristics of the field as a function of space and frequency must be modelled by the Green's function, and ideally the source would have zero phase at its face as well. It was expected that the Panametrics beam source would not necessarily be modelled identically by the 2D Green's function, but the line source theoretically should have been. However, it was found that the model varied from reality to a greater extent with the line source than wi th the Panametrics source. A s such, it may not be possible to obtain a source that identically fits the model. The scientific literature currently does not offer an answer due to the lack of experimental studies in pulsed ultrasound computed tomography. It may ultimately be necessary to have a transducer custom-made to fit the model as closely as possible. Due to the 3D nature of ultrasound propagation, it may be that even a custom made source wi l l not be described by the 2D Green's function to within an acceptable error. A n alternative may be to commission the surveying of the field from a suitable source at a continuum of points in space. From these data, a frequency dependent function of the source phase and amplitude characteristics could be built up throughout the imaging space. This would, of course, be an expensive option. Furthermore, this would require development of the algebraic reconstruction technique, since it is the only one of the two methods in which a general propagator can be incorporated. Essentially, a specialized propagator would be used to model the source field, and the 2D Green's function would be used ini t ial ly to model scatter propagation. A 3D propagator could eventually be included for scatter. A discussion regarding development of the A R T is included in Section 6.2.4. The stepper motor apparatus must also be redesigned and rebuilt. It is suggested by this author that a gear mechanism be installed, with a separate gear controlling the movement of each of the source and detector transducer arms. This design w i l l avoid the use of a bushing mechanism that is susceptible to corrosion. Al though it w i l l not Chapter 6. Conclusions and Future Work 245 be possible to attain the maximum possible theoretical resolution of A / 2 , where A is the maximum frequency in the source spectrum, the apparatus wi l l work well and afford resolutions on the order of 1 mm, which is sufficient for the early detection of breast cancer [69]. 6.2.2 I M P R O V E D D I G I T A L SIGNAL P R O C E S S I N G B o t h the theoretical and experimental tests of wavelet denoising suggest that this method may be l imited in its ability to reduce noise in U C T data. In simulation, the moving average filter was equally effective for removing noise, and in experimental tests, wavelet filtering fared no better than lowpass Butterworth filtering. It is true that the success of wavelet denoising often depends on the creation of a custom wavelet family that mimics the ultrasound data features. However, the Daubechies 20 mother wavelet illustrated in Figure 5.33 indeed mimics the general features of the experimental data, yet wavelet denoising was no more effective than lowpass filtering. It is suggested that some effort be directed towards defining and testing a custom wavelet family based on the data features in order to obtain closure regarding this area of research. 6.2.3 I M P R O V E M E N T O F M E T H O D B Y B L A C K L E D G E et al I N C O R P O R A T I O N O F E L L I P T I C A L P R O J E C T I O N S Section 5.1.3 illustrated that the elliptical data isochrones are "warped" into lines due to the simplifying assumption that jjrjr = ^ << 1, which is applied in the development of the reconstruction algorithm. In particular, this assumption leads to an approximate STj function in the 2D propagator that overestimates the T O F of ultrasound from scatter points not located at the origin. The overestimation error increases wi th distance from the origin. The simplifying assumptions are necessary for tractable mathematics leading to the tomographic algorithm. The T O F errors had the effect of creating low amplitude halo-shaped point-spread-functions, rather than full height Gaussian P S F ' s , even for Chapter 6. Conclusions and Future Work 246 points wi th ^ as small as 0.05. This effect is a significant result that seriously limits the capabilities of this reconstruction algorithm. Section 2.3 illustrated that the isochrone "warping" manifests itself in each view through the reconstruction of a partial image of consecutive linear isochrones that are parallel to the line joining the source and detector. The intensity of each isochrone is directly proportional to the intensity of the point in the data signal wi th the T O F corresponding to the isochrone. It was shown that in the absence of time-of-flight error, these curves would be elliptical rather than linear. A s such, a spatial correction must be developed that can be applied to each view in order to redistribute the image intensity into the proper pixels. The same correction can be applied to each view as it wi l l depend only on the relative position of the source and detector, which defines the T O F information. The correction would follow from the geometrical description of the isochrones discussed in Section 2.3. Upon correcting the partial image for each view angle, the images can be summed to obtain an improved cross-section of yK or reflectivity. I N C O R P O R A T I O N O F A N A R R O W - B E A M S O U R C E A n important extension to the method by Blackledge et al would be the incorporation of a narrow beam pulsed field to replace the cylindrical field. The creation of a field wi th a width on the order of 1 m m is quite feasible wi th conventional phased array technology. This source would facilitate the application of a spatial mask to the partial image reconstructed for each view angle. The mask would be defined by the insonification region. Information would thus be dramatically more localized in each view, which should result in an improved cross-section of jK or reflectivity upon the summation of the part ial images. The narrow beam source would also facilitate a frequency dependent attenuation correction. In the data region of the signal detected at each angle, the time samples could be corrected as follows. A group of 64 points, herein termed a signal portion, Chapter 6. Conclusions and Future Work 247 can be extracted wi th the time sample in question at the center. The T O F values in tissue of the samples in the signal portion differ by a maximum of ~ 1.1 fis, and the corresponding attenuation in tissue varies over the small range of ~ 0.01 — 0.6 db (0.007-0.07% of field amplitude). The average T O F in tissue can be calculated for the time samples in the signal portion, and this can be used to determine a frequency dependent correction for the F F T of the signal portion. Given the small range of T O F values, this should not result in significant error over the points. The least amount of error w i l l be associated wi th the point at the center of the signal portion. A correction can similarly be applied for attenuation in water. Upon correcting the F F T , the data can be inverse transformed to produce a corrected signal portion in the time domain. The time point at the middle can now be extracted to replace its corresponding point in the original signal. A first order image can be reconstructed with data that have been corrected by assuming an average attenuation coefficient throughout the tissue. The original data can then be corrected again using the attenuation characteristics determined by the first order image. A second and likely better image can be reconstructed using these data. This process can be repeated unti l there is little change in the image. Again , the amplitude and Fourier phase characteristics of the field as a function of space and frequency must be modelled by the 2D propagator in order to incorporate a narrow beam source into the reconstruction algorithm by Blackledge et al. Combined wi th a spatial mask corresponding to the insonification region, this source model may be feasible. If, on the other hand, a specialized source propagator must be determined through an analysis of the field surveyed at a continuum of points in the insonification region, the narrow beam source can only be incorporated into the A R T . Development of the method using a narrow beam field can be tested in a prototype using a specialized narrow beam source kindly provided by M r . Jerry Posakony. This pulsed source has a field wi th F W H M values of 3-5 m m over the distance of 3-7 cm from the transducer. A s such, it may be sufficient for use in a proof of concept study. Issues involved in the Chapter 6. Conclusions and Future Work 248 development of the algebraic reconstruction technique are discussed in the next Section. 6.2.4 I M P R O V E M E N T O F T H E A L G E B R A I C R E C O N S T R U C T I O N T E C H N I Q U E The A R T must be substantially improved so that it can reconstruct images wi th dimensions greater than 20 x 20 pixels. A n analysis of the condition number of the attenuated propagator matrix, T , indicates that it is ill-conditioned. The condition number of the coefficient matrix A of a linear system, y = A x , is a nonnegative number that estimates the amount by which small errors in either y or A can change the solution, x. A small condition number suggests that the iterative solution of the system wi l l not be sensitive to errors, while a large condition number indicates that small data or arithmetic errors may result in enormous errors in x. The condition number for a matr ix A is usually defined by Condi t ion(A) = ||A|| • | |A _ 1 | | (6.1) A condition number of infinity indicates that A is not invertible. For the various recon-structions attempted, the attenuated propagator matrix had a condition number that varied from a value on the order of 10 2 0 to a value of infinity. This indicates that the systems were very ill-conditioned, which explains the difficulty in obtaining convergence. A s such, in order to develop a viable reconstruction algorithm, research must be done to determine i f the matrix system can be pre-conditioned using existing algorithms in mathematics. It is suggested that the focus of this research be the method of singular value decom-position ( S V D ) . This is a least squares approximation method that can be used to solve for x in the equation y = A x , when A is singular and the system is ill-conditioned. In brief, when A is a matr ix of dimension M x N, where M > N, this method decomposes A into the following: A = U • W • V T (6.2) U and V are both column-orthogonal matrices, wi th dimensions M x N and N x N, Chapter 6. Conclusions and Future Work 249 respectively. The matrix W is a diagonal matrix wi th elements Wj greater than or equal to zero (the singular values) [74]. S V D then works with this decomposition of A to solve for x v ia a least squares approximation algorithm. A n alternative and potentially better solution may also be possible through the application of thresholding to the W matrix. This is the standard process of zeroing small wfs, which are responsible numerically for the ill-conditioning of A . The condition of the matrix system can also likely be improved i f a narrow beam pulsed ultrasound field could be incorporated into the A R T . The propagation matrix would thus become sparse due to the narrow insonification region. The matr ix system may then reduce to a form that can be solved in a fast and feasible manner using iterative techniques. W i t h the introduction of this source, the attenuation correction described in Section 6.2.3 would not be necessary since the propagators are attenuated. However, the concept of reconstructing successive images wi th an increasingly better characterization of attenuation should also be applied here. The first image can be obtained assuming an average attenuation. A second image can then be reconstructed using propagators that embody the attenuation information of the first image. This process can be repeated unti l there is little change in the image. 6.3 A D D I T I O N A L F U T U R E W O R K 6.3.1 T E S T S W I T H B E T T E R P H A N T O M S Extensive expansion of the experimental work is required to fully test the prototype U C T scanner. Upon improvement of the reconstruction algorithms, a C I R S ™ 3D u l -trasound calibration phantom should be acquired to test the system resolution. This phantom mimics average human tissue with inclusions similar to tumors wi th various sizes and attenuation coefficients. Addi t ional tissue equivalent phantoms can also be constructed wi th inclusions to mimic inhomogeneities in tissue. A method should be devised for coating the phantoms wi th a substance that is relatively impervious to water. Chapter 6. Conclusions and Future Work 250 6.3.2 S T A C K I N G O F P L A N A R V I E W S T O C R E A T E 3 D I M A G E S The present U C T prototype uses algorithms that include a 2D model of sound propa-gation to provide 2D planar cross-sections of 7 K and reflectivity through the object. This is in keeping wi th the majority of U C T reconstruction algorithms [21, 32, 47, 53, 80]. The methods could be extended to combine consecutive, stacked planar images into a 3D composite image, properly accounting for overlapping pixels in neighbouring planes. Visual izat ion software must also be written that can change the image orientation and scan through the different planes. The U C T scanner must also be modified to include the automated and accurate vertical movement of the transducers. 6.4 F I N A L R E M A R K S In closing, extensive new results have been established regarding the potential viabil i ty of pulsed ultrasound computed tomography using both a direct Fourier method and an algebraic reconstruction technique. It has become evident in the course of this work that pulsed U C T is an involved and multifaceted problem that requires extensive research and development. W i t h continued effort in the key areas discussed in the preceding sections, it is possible that pulsed U C T could become a viable imaging technique. 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Waves in Focal Regions. Bris tol : Hilger (1986). BIBLIOGRAPHY 257 [82] H . Stark, J . W . Woods, I. Paul , R . Hingorani, "Direct Fourier Reconstruction in Computer Tomography," I E E E Transactions on Acoustics, Speech and Signal Processing, ASSP-29, No. 2 (1981). [83] N . A . Thacker and T . F . Cootes. Vis ion Through Optimization. Leeds: Computer Based Learning Uni t , University of Leeds, Copyright 1994. [84] J . J . W i l d , D . Neal, "The Use of High Frequency Ulstrasonic Waves for Detecting Changes of Texture in L iv ing Tissue," Lancet, 1, p. 655 (1951). [85] J . J . W i l d , J . M . Reid , "Further Pi lo t Echographic studies on the Histologic Structure of the L i v i n g Intact Breast," American Journal of Pathology, 28, p. 839 (1952). [86] J . J . W i l d , J . M . Reid, "Echographic Visualization of Lesions of the L i v i n g Intact Human Breast," Cancer Reveiws, 14, p. 277 (1954). [87] G . Wojcik, J . Mou ld , F . L i z z i , N . Abboud, M . Ostromogilsky, D . Vaughan, "Non-linear Model ing of Therapeutic Ultrasound," 1995 I E E E Ultrasonics Symposium Proceedings, p. 1617 (1995). [88] T . Yokoyama, K . Nagai, K . Mizutan i , "Imaging Properties of Diffraction Tomog-raphy by Compound-Scanning Transmitter-Receiver Pai r Transducers," Japanese Journal of Appl ied Physics, 32, p. 2510 (1993). [89] From the website "www.ynLt.u-tokyo.ac. jp/publicat ions/papers95/IEEE95/chung/ nodel6.html." [90] From the website "pipeline.keck.hawaii.edu:3636/realpublic/observing/interfacvinst /ktelgde-l .html." [91] From the website "pcim.com/articles/1997/art0007/ell l .htm." [92] From the University of Vi rg in ia Health System website "www.med.virginia.edu/ medicine/cl inical/pathology/educ/innes/text/wcd/myelo.html." A P P E N D I X A D E T A I L E D D E R I V A T I O N O F M E T H O D B Y B L A C K L E D G E et al The reconstruction method of Blackledge et al was the starting point for the study involved in this thesis. This method is aimed at the reconstruction of images of com-pressibility and density for minimally attenuating objects. W i t h this approach, the object being imaged is modelled as a nonhomogeneous fluid immersed in a uniform background fluid. The following pages outline the derivation in detail with references to three works: the paper by Blackledge et al [10], the paper by Norton and Linzer [66], and Section 6.2 of the book by Morse and Ingard [62]. In working through the math, minor errors were found in the work of Morse and Ingard, while several errors were found in the paper by Blackledge et al. To aid the reader in following the derivations, errors in the references have been noted. A.l T H E W A V E E Q U A T I O N The physics of the sound/fluid interaction is described by the Chernov Equation, which is a wave equation with the following form [10, 62, 66]: V - ( ^ V p ( r , * ) ) = « ( r ) ^ p ( r , 0 ( A . l ) Note that this equation is expressed incorrectly in the paper by Blackledge et al, wi th a negative sign on the right hand side [62, 66]. In Equation A . l , p(r,t) is the ultrasound pressure field at any time, t, during the measurement process and at any location, r, in the measurement region. Since the 3D problem is difficult to solve, this model considers 258 Appendix A. Detailed Derivation of Method by Blackledge et al 259 2D pressure fields only. Thus, r = xx + yy, where x and y are unit vectors in the xy-plane. p(r) and n{r) are the spatially-varying density and compressibility in the image region. The term K0^p(v,t)-yVMr,t) (A.2) is added to both sides of Equation A . l [10, 62]. K0 and po are the compressibility and density of the background fluid, which is typically water. Note that the step cited here in Reference [62] is incorrect in that the time derivative includes a factor of p and thus has improper units. Let t ing the time derivative terms on the left and right hand side be T l and T 2 respectively for simpler notation, the wave equation can be reduced as follows: V - ( - V p ) + T l - - V 2 p + — V 2 p = T 2 (A.3) P Po Po { V - ( - V p ) - - V 2 p } + { T l + - V 2 p } = T 2 P Po Po The expression inside the first pair of curly brackets can be reduced further as follows: V - ( - V p ) - — V 2 p = V - ( - V p ) - V - ( — V p ) (A.4) P Po P Po = v . { ( i - i - ) V p } P Po PPo PO P = - - V - ( T p V p ) Po where a 7P(r) function has now been defined as 7 p ( r ) = ( A . 5 ) Substituting Equation A . 4 into Equation A . 3 yields - - V - ( 7 p V p ) + T l + - V 2 p = T 2 (A.6) Po Po V 2 p + p 0 T l = p 0 T 2 + V - ( 7 p V p ) Appendix A. Detailed Derivation of Method by Blackledge et al 260 It is now necessary to reduce the time derivative terms in Equation A . 6 . For simpler notation, V terms on the left and right hand side wi l l be referred to as D I and D2, respectively, and ^p(r, t) w i l l be written as p. Equation A .6 can then be rewritten as D l + p 0 « o P = poK(r)p + p0K0p + D2 (A.7) D I + po (KO - K(T))P = PQKQP + D2 Noting that the speed of sound in the background fluid is c 0 = ^/KOPO, the above equation can be further reduced: D l + - ^ - j ( K o - K ( r ) ) p = p0KQp+ D2 (A.8) m _ J L ^ i W p = P o K o P + m CQ KQ D l - 3 7«(r)p = ^ P + D2 Co where a 7 / t(r) function has now been defined as 7 . W = (A .9) KQ Note that the 7«(r) definition differs from that of 7K(r) in that the background parameter, rather than the tissue parameter, is the denominator. Substituting the V terms back into Equation A .8 results in the final wave equation V 2p(r,*) - ^P(r,t) = ^ ^ p ( r , t ) + V • (7pVp(r,t)) (A.10) In the paper by Blackledge et al, this equation is quoted incorrectly wi th a factor of -1 in the time derivative on the right hand side. Equation A . 10 is a time dependent wave equation wi th a forcing term that is a function of 7«(r) and 7P(r). The forcing term gives rise to scatter due to inhomogeneities. For completeness, it is noted that i f the tissue had been modelled as a medium that is solid, linear, isotropic and viscoelastic, the forcing function would take on a more complicated form involving tensor notation [47]. Interested readers are referred to the work by Iwata and Nagata done some years ago [44]. Appendix A. Detailed Derivation of Method by Blackledge et al 261 A . 2 FOURIER D O M A I N AND B O R N APPROXIMATION The solution to Equation A . 10 has time dependence embodied in an exp(iwt) term. Therefore 1 82 (hn)2 4 T L P M ) = = -k2P(r,t) (A.ll) CQ Ot CQ where k = UJ/CQ is the wave number in the background fluid. Substituting this expression for the p terms into Equation A . 10 and applying the Fourier Transform to both sides yields V 2 p ( r , UJ) + k2p(r, UJ) = -A; 2 7 ) t ( r )p( r , UJ) + V • ( 7 p ( r )Vp ( r , UJ)) (A.12) Note that the expression quoted in the chapter by Morse and Ingard is incorrect in that the V • (7 p(r)Vp(r, UJ)) term includes an extra negative sign. The solution to Equat ion A.12 at any detector location wi th position vector r = r<* is given by p(rd,uj) = p0(rd,uj) + k2 g(r\rd, k)-yK(T)p(r, UJ) d 2 r (A.13) - f p(r | r d , fc)V-(7p(r)Vp(r ,w))d 2 r Note that the signs preceding the two integrals in the paper by Blackledge et al are both opposite to what they should be. In Equation A.13, P0(TS,UJ) is the incident ultrasound as a function of UJ at any detector position r^, and ^(rlr^, k) is the 2D Green's function that describes 2D wave propagation in the background fluid between any two locations described by r and r^. It is common knowledge that g(r\rd, k) is the solution to the following equation (V2 + k2)g(r\rd,k) = -82(r-TD) (A.14) which is given by ^(r|r d,A;) = - ^ 0 1 ( f c | r - r d | ) (A.15) where HQ is the Hankel function of the first kind [81]. The Green's function is symmetric wi th respect to the interchange of r and vd. Note that the solution quoted in the reference Appendix A. Detailed Derivation of Method by Blackledge et al 262 by Blackledge et al is incorrect by a factor of -1, which results in the wrong pulse shape upon propagation of the incident field. Equat ion A.13 is nonlinear in that the total field p(rd,u)) is not only being solved for, but it is also found within the integral. Thus, it is not possible to determine an exact analytical solution to this equation without linearizing it. In order to do this, the commonly used Born approximation is applied, in which the interacting sound field is assumed to be approximately equal to the incident field. In other words, the tissue is assumed to be a weak scatterer of sound. The application of the Born approach yields Note again that the signs preceding the two integrals in the paper by Blackledge et al are both opposite to what they should be, which is the error carried over from Equat ion A . 3 INTRODUCING T H E P U L S E D L I N E SOURCE The 2D-based image reconstruction method assumes that the interrogating pulse derives from a line source, which is generally considered to be a field that is easily obtainable in the lab [10, 32]. The incident field at any given location, r, due to a source wi th position vector, r a , is written (A.16) A.13. Po(r,w) = A(u)g(r\Ts,k) (A.17) and Equation A.17 can now be expressed as p(rd,rs,uj) = A(u)g(rd\rs, k) + (A.18) Appendix A. Detailed Derivation of Method by Blackledge et al 263 Note again that the signs preceding the two integrals in the paper by Blackledge et al are the opposite of what they should be due to the error propagated through the derivation from Equation A.13. The incident pulse is furthermore assumed to be band l imited wi th values of UJ from tti to Q2 only. The source and detector are also assumed to be significantly far from any point of ultrasound scatter, located at r in the image space such that the following relation holds for r^ - equal to rs or r^: | * | | r - r j | » l (A.19) for every wave number that satisfies the expression - < |*| < — (A.20) c 0 c 0 These assumptions allow the exact Green's Function to be approximated by a simpler function given by 0(r|r,-,fc) « aS{k\T-rA) (A.21) e x p ( « f c | r - r j | ) ( f c | r - r j | ) 2 .exp(3i7r/4) a = % i— 2(2TT)2 Note that the value of a quoted in the paper by Blackledge et al is incorrect by a factor of i, resulting in the wrong phase information. B y extension, Vg ~ V S 1 , which is derived in Appendix B to be VS = ikhjS (A.22) l r - r j l To simplify notation in the derivation to follow, STj w i l l be used to denote S(k\r — Tj\). Substituting the approximate expressions for g and V# into Equation A . 18 yields p(vd,vs,u) tt aA(u)(S(k\rd-T$\) + a2A(oj)k2 f STd7«(r) ST, d 2 r - (A.23) a2A(u) f S r d V • ( 7 p ( r ) V S r J d 2 r JSRJ Appendix A. Detailed Derivation of Method by Blackledge et al 264 Note again that the signs preceding the two integrals in the paper by Blackledge et al are both opposite to what they should be, since this error was carried over from Equat ion A.13 . A . 4 DERIVATION OF E Q U A T I O N 2.16 Several steps are involved in deriving Equation 2.16 in Section 2.2.1, given by p(rd,Ts,u) * aA(oj)S(k\Td-rs\) + a2A(oj)k2I (A.24) 1 = fsMr)+cos(ehP(v))Srad2v from Equation A.23 in the preceding section. Most of the steps involve reducing the integral / 5 ( f c | r - r d | ) V - ( 7 p ( r ) V 5 ( A ; | r - r s | ) ) d 2 r (A.25) to the expression k2 I Srd(nd-ns)jp(r)STsd2T (A.26) which wi l l now be shown. B y the Chain Rule, V • 0 S r , 7 p ( r ) V S r . ) = V 5 r - . ( 7 p ( r ) V 5 r . ) + 5 r d V . ( 7 p ( r ) V 5 r . ) (A.27) Therefore, the desired integral can be written as / S r d V - ( 7 p ( r ) V S r J d 2 r = / V • (STdlp(r)VSrJ d 2 r (A.28) " / V S r d . ( 7 p ( r ) V S r J d 2 r B y the Divergence Theorem, / V • { S r - 7 „ ( r ) V S r j } - f { 5 r d 7 p ( r ) V 5 r . } • <fl (A.29) where C is a contour enclosing the extent of 9t.2 (the 2D image space), and dl is the infinitesimal vector line element that is always tangential to C. However, because the Appendix A. Detailed Derivation of Method by Blackledge et al 265 contour C lies inside the background fluid, 7 P = 0 everywhere on the contour and the integral in Equat ion A.29 is therefore zero. A s such, Equation A.28 has now been reduced to / S r d V - ( 7 p ( r ) V S r J d 2 r = - / VSTd • ( 7„(r)VS r .) d 2 r (A.30) Recalling again that VSTj — ik hj STj results in the expression fsTdV • ( 7 p ( r ) V 5 r J d 2 r = - f (ik)2hdSTd • h s l p S r e d 2 r (A.31) J 5 R 2 J S R 2 = k2f 5 r d ( n d - n s ) 7 p ( r ) 5 r s d 2 r which is the desired result in Equation A.26. Lastly, an expression for f i s • iLj must be derived. In order to do this, an imaging geometry must be defined. Figure A . l illustrates the experimental U C T setup in which a source transducer wi th position vector r s insonifies an object located about the origin wi th a pulse that has an amplitude spectrum A(t). The scattered field is detected by a second transducer wi th position vector r .^ Bo th the source and detector are situated at a distance a from the origin, and at angles of <ps and ipd, respectively. Note that <ps is always larger than <pd. The unit vectors n s and hd can then be written as hs = cos(</?s)x + s in(^ s)y (A.32) hd = cos(<£>d)x + sin(v?d)y 9 describes the angle between the pulse propagation direction and the detector angle, and it is defined in terms of the other angles by 6 = <pd-<ps + n; (A-33) W i t h these definitions, n s • hd = cos(^ d ) cos(<£ s) + sin(<pd) s in(^ s ) (A.34) Appendix A. Detailed Derivation of Method by Blackledge et al 266 Figure A . l : This schematic illustrates the U C T experiment geometry upon which a reconstruction algorithm is developed. Vectors r i and are two examples of r , which is the location vector of scatter points wi thin the image. Appendix A. Detailed Derivation of Method by Blackledge et al 267 A trigonometric identity states that cos(£i ± t2) = cos(ii) cos(£ 2) T s in(i i ) s in( i 2 ) (A.35) and thus n s • fid = cos((pd — (ps) = cos(# — TT) = — cos(9) (A.36) Note that the paper by Blackledge et al quotes a value for this expression that is incorrect by a factor of -1. Substituting this expression for the dot product into Equat ion A.31 results in / 5 r d V • ( 7 p ( r ) V 5 r J d 2 r = -k2 f SId cos(0) 7 p ( r ) 5 r . d 2 r (A.37) Replacing the second integral in Equation A.23 with this expression yields the following equation for the total field at detector position rd p(rd,rs,Lj) = aAioj^Siklrd-TsD + ^ A ^ k 2 x (A.38) f STd ( 7 K ( r ) + c o s ( % p ( r ) ) 5 r s d 2 r B y extension, the scattered field is given by 2A(u) k2 / Srd ( 7 K ( r ) + cos(6») 7p(r)) STs d 2 r (A.39) A . 5 SIMPLIFICATION OF E Q U A T I O N A . 3 8 The following steps involve the reduction of Equation A.38 to yield Equat ion 2.23 in Section 2.2. Recall that Equation 2.23 expresses the total field at any location as p(rd,rs,uj) « J = X = aA(u){S(k\Td-rs\) + aJ) exp(*fc(|r«j| + |r, |)) (A.40) k [ v d 2 r JM2 exp(ikhd • r ) ( 7 „ ( r ) + cos(0)7 p(r)) exp(ikhs • r) Appendix A. Detailed Derivation of Method by Blackledge et al 268 Substituting the full expression for S into Equation A.38 leads to p (r d , r s ,u ; ) » aA{u)(S(k\rd - rs\) + a2A(uj)k2 x (A.41) r exp(i* | r - r d |) exp(ifc|r - r s |) 2 / — hnin + cos l f lWJr) ) —• ^ - d r to ( * | r - r „ | ) i W K ) K n p K " ( * | r - r . | ) i Comparing Equations A.40 and A.41 , it is evident that the key step in the derivation is to show that exp(iA:|r - vd\) exp(ik\r - r,|) _ ^ ^ |r — r d | 2 |r — r s | 2 exp(ifc(|r d| + \rs\)) exp^-fcfirf _ r^ exp(^ns. r) ( | rd | | r s | ) 2 In order that the exponential terms on the left hand side of Equation A.42 reduce to those on the right hand side, it must be shown that \T - Tj\ = hj • r + (A.43) where Tj is equal to r s or r^. |r — Tj\ can be rewritten as Ir - r - l 2 | r - r . - | = 7 J (A.44) (r - r,) • (r - r^ ) (r ~ rj) • r _ (r - r,-) • Tj \r-Tj\ \r-Tj\ = T 1 - T 2 Recalling that (A.45) it is evident that T l is simply the first term of Equation A.43. Thus, a l l that remains is to show that T 2 reduces to - | r j l - Analytically, T 2 can be reduced mathematically by deriving a different expression for (r — r^ ) as follows: v-Vj = TjrJ1{r-TS)=rj(Tj1T-rJ1Tj) (A.46) Appendix A. Detailed Derivation of Method by Blackledge et al 269 The inverse of r^ - is COS(T7)X + - sin(r;)y (A.47) Tj a a where rj is the source or detector angle. Substituting this expression for r^ - into Equat ion A.46 yields r — r, = r,- (— cos(ri) + — sin(r?) — 1) J a a The physics model now assumes that the object is being insonified in the far field of the source, so that |r |/ |rj | is always <C 1. This means that - « 1 (A.48) a ^ « 1 a and therefore r - Tj = Tj x (S - 1) = -TJ (A.49) where S « 0. Substituting the above expression into Equation A.44 allows T 2 to be expressed as T 2 « - ^ - ^ r (A.50) I - T j l = - k i l and an alternate expression for |r — r j^ has thus been shown to be r,- = 3\ ~ | r - r I 1 0 I1  l3\ = hj-r+\Tj\ + | r , | (A.51) Substituting this expression into S and noting that, according to Equat ion A.49 , 1 1 (A.52) Appendix A. Detailed Derivation of Method by Blackledge et al 270 yields the following result S(*|r-r,|) = T ' < * | , - , ; l ) (A.53) [klr-Tjlp expjikjhj T + |r,-|)) K21 r 12 Note that this approximation results in a loss of phase information and ultimately is responsible for the relatively poor performance of the reconstruction algorithm. This w i l l be discussed in more detail in Section 5.1.3 Equat ion A.38 finally reduces to P(rd,Ts,oj)ttaA(u)S(k\Td-rs\) + a2A(oj)kT [ * d 2 r (A.54) where T = QMiH\rd\ + \rs\)) ( A 5 5 ) ( | r d | N ) 2 X = exp( i fcn d - r ) (7 K ( r ) + cos(^)7 P ( r ) )exp( iA;n s - r ) Thus, Equat ion A.40 is derived. A . 6 S U B S T I T U T I O N F O R r s A N D rd The following wi l l illustrate the final step in the derivation of the reconstruction algorithm by Blackledge et al, which involves the substitution of r d = ahd = ax cos(ipd) + ay sin((pd) (A.56) r s = ahs = ax cos(</?s) + ay sin(y>s) into Equat ion A.54. It wi l l be shown that this process generates the final expression for the total ultrasound field measured at any point r , which is given by: . t . . . exp(2ifcacos(|)) akexp(2ika) , . . . . , . __. pe{T,u>) tt aA{u){-^- 7 ^ 4 ^ + LMVs,k)} (A.57) ( 2 « a c o s ( | ) ) 2 a 1>9(<Ps,k) = I exp{-i{ux + vy))(jK(r) + c o s ( % p ( r ) ) dxdy Appendix A. Detailed Derivation of Method by Blackledge et al 271 where u and v are defined by 9 9 u = - 2 f c s i n ( - + ^ ) s i n ( - ) (A.58) Zi Z v = 2fccos(^ + <p s)sin(^) Note that the signs quoted for both u and v in the paper by Blackledge et al are opposite to what they should be. This mathematical reduction depends on simplifying the term \rd — r s|, the square of which can be expressed as follows: \rd - r. |2 = | a {cos(ipd) - cos(p,)}x + a {s in(^ d ) - sin(p,)}y | 2 (A.59) = a2 {cos(^ d ) - cos(v3 s)} 2 + a2{sin(ipd) - sin(<^)} 2} = a2 {cos2(<pd) + sm2((pd) + cos2((ps) + sin 2(v? s)} -a2 {2 cos(<pd) cos((ps) - 2 sm((pd) sin(<ps)} = 2 a 2 { l - cos((pd) cos((ps) - sm(ipd) sin(ips)} Also , cos(#) must be expressed in terms of <pd and ips through use of trigonometric identities as follows cos(0) = cos(y>d — ips + 7r) (A.60) = cos(v?d - ips) cos(7r) - sm(ipd - <ps) sin(vr) = - COs(lfd - ips) - - cos(yd) cos(^ s ) - s in(^ d ) sin(<ps) Substituting this expression for cos(f9) into Equation A.59 yields | r d - r s | 2 = 2 a 2 { l + cos(0)} (A.61) According to a trigonometric identity l + cos(i) = 2 c o s 2 ( ^ ) (A.62) Z Appendix A. Detailed Derivation of Method by Blackledge et al 272 Thus, | r „ - r , | = { 2 a 2 { 2 c o s 2 ( ^ ) } } * (A.63) = 4a cos(x) This expression for the norm can now be substituted into Equation A.54, together wi th the relation \rd\ = a = | r s | , to yield ( \ s exp(2ifca cos(f)) 2 exp(2ifca) . p ( r d , r s , a ; ) « a i 4 ( w ) - ^ + a2 A{u) k — ^ '-^ (A.64) (2«fca cos ( | ) )2 a fa = / exp(zfcnd • r ) ( 7 K ( r ) + cos(0)7 p(r)) exp(zfcn s • r ) d : 2 r The final step involves reducing the ipe term by showing that exp(ifc(n<j • r + n s • r ) ) = exp(—i{ux + vy)) (A.65) where u and v are defined in Equation A.58. Expanding the exponent argument in Equat ion A.65 as follows k{nd • r + hs • r } = k (cos(<pd)x + sin(y>d)y) • (arx 4- yy) + (A.66) k (cos(^s)x + sin(<ps)y) • (xx + yy) = kx {cos(ipd) + cos(<ps)) + ky (sin(<pd) + s in(^ s ) ) results in alternate expressions for u and v given by u = -k (cos(<pd) + cos(ips)) (A.67) v = -k (sm((pd) + s in(^ s )) Substituting 6 — ipd - cps + 7r and applying the following trigonometric identities s in 2 (^ ) = i ( l - c o B ( 0 ) (A.68) sin(t) = 2 s in (^)cos( - ) sin (ti + t2) = sin (ti) cos (t2) + cos (ti) sin (t2) Appendix A. Detailed Derivation of Method by Blackledge et al 273 allows the term (cos(^ s) + cos(y?d)) to be further reduced to arrive at (cos(<pfl) + cos(<£>d)) = cos(ips) - cos(<pd + TT) (A.69) = C0S(<£>S) - COs(0 + ips) = cos(<ps) — {cos(#) cos(y>s) — sin(0) sin(<ps)} = cos(^ s ) (1 — cos(0)) + sin(0) sin(</?s) 8 9 8 = 2 cos(<£>s) s i n 2 ( - ) + 2 s in( - ) cos(-) sin((ps) 8 8 6 = 2 S " 1 ^ ) ( c o s ( ^ ) s in( - ) + s i n ( ^ ) cos(-)} 8 8 = 2 s i n ( - ) s i n ( - + Similarly, wi th the application of the following trigonometric identity cos( the term (sin(y?s) + sm(<pd)) can be reduced to yield ti ± t2) = cos(ti) cos(£ 2) T sin(*i) sin(t 2) (A.70) (sin(yj s) 4- sm((pd)) = sin(<ps) - sm((pd + ir) (A.71) = sin(<ps) - sin(0 + <ps) = sin((ps) — {sin(0) cos(<ps) + cos(0) sin(yj s)} = sin((^ s) (1 — cos(0)) — sin(0) cos(<ps) 8 8 8 = 2 sm(ips) s i n 2 ( - ) - 2 s in( - ) cos(-) cos(p,) 8 8 8 = - 2 s in( - ) {cos(v? s) cos(-) - sinfa,) s in ( - )} 8 8 = - 2 s in( - ) cos( - + <ps) Substituting these results into Equation A.67 yields 9 8 u = - 2 f c s i n ( - ) s i n ( - + c/>,)) (A.72) 8 8 v = 2k s in( - ) cos( - + (ps) Appendix A. Detailed Derivation of Method by Blackledge et al 274 Thus, the final equation for the total ultrasound field has been proven to be p(rD,Ts,oj) tt F1 + F2 xip0((p9,k) (A.73) Fl = aA(u) exp(2ika cos(f)) i i (2ika cos(|))5 ™ 9 . / x , exr)(2ika) F2 = a>2A(u)k—^ '-i>e(<Ps,k) = / exp(-i(ux + vy))(^yK(r) + cos(0)7 P(r)) d x d y 9 9 u = -2k s i n ( - ) s i n ( - + <pa)) 9 9 v - 2k s in ( - ) cos( - + ips) The above equation provides the basis for the reconstruction algorithm by Blackledge et al described in detail in Section 2.2.2. A closing note regards a quick dimensional analysis of both the intermediate and final solutions to the wave equation, given by Equations A.54 and A.73, respectively. Looking at the first term of Equation A.54, the dimensions are the same as that of A(u>), since both a and S are dimensionless. A(CJ) is the Fourier Transform of the incident pulse amplitude as a function of time, which is given by some function A(t). The units of A(t) w i l l generally be volts since this is what is measured by the detector transducer. Voltage is not involved in the Fourier Transform integral, so the final units of A(u) are V s . The units of the incident field are the same. The units of the second term in Equat ion A.54 are X is unitless, so the dimensions of the 2D spatial integral are m 2 , and the dimensions of the final expression are again V s , which is consistent with the first term. A n analysis of Equat ion A.73 yields the same results. The first term has dimensions V t ( A 7 5 ) (2ika cos ( | ) )2 Appendix A. Detailed Derivation of Method by Blackledge et al 275 while the second term has dimensions [a2 A { o j ) k <»p(2»fca) M ( p ^ k)] s V s ( A 7 6 ) tj)Q is the 2D spatial integral of a unitless function, so its units are m 2 , and the final dimensions of the second term are again V s as expected. APPENDIX B DERIVATION OF X7S In order to develop the reconstruction algorithms included in this thesis, knowledge of VS'(A;|r — Tj\) is required, where aS(k\r — r,-|) is an approximation to the 2D Green's function propagator that describes ultrasound propagation in the far field of the source. In this appendix, VS , ( /c | r — r^ -1) is derived for the situation without attenuation due to tissue and water included. This exercise results in a correction to the value quoted in Blackledge et al [10]. To simplify notation in the following derivation, Srj w i l l denote S(k\r — Tj\). It is also shown in this appendix that V5(A; | r — Tj\) and V5'(A;|rj — r|) are equal. B . l D E R I V A T I O N O F V5 ' ( /c | r — r^ -1) The starting point of this calculation is the expression for the propagator term that neglects attenuation effects, which is given by Sr. = e X p ( ^ | r - r f l ) ( B . l ) (k\r-ij\)2 Lett ing f(x,y) = |r — r^ -1 for simplicity, Equation B . l can be rewritten = exp{ikf(x,y)) , g ^ T j (*/(x,y))* From this one can derive VSV, > where V 5 r . = V{exp{ikf(x,y))k-? f{x,y)-$} (B.3) = k~* f(x,y)~* V{exp(ikf(x,y))} + k~% exp(ikf{x,y))V{f(x,y)~%} 276 Appendix B. Derivation of V 5 277 Equat ion B.3 V exp(ikf(x, y)) can be expanded as V exp(ikf(x, y)) = ik exp(ikf(x, y))Vf(xt y) (BA) Equat ion B.4 and Vf(x, y) can be expanded as follows: V / ( s , y ) = V{\(x-rx)x+(y-ry)y\} (B.5) = V { { ( s - r x ) 2 - r ( y - r v ) 2 } * } = \ w j ) { 2 { x - r x ) ± + 2 { y - r v m l r - r i l ( r - r , ) From this result, the form of Vf(x, y) 2 can be derived: V / (ar ,y)-* = -\f{x, y)"2 V / (x , y) (B.6) 1 nj-2f(x,v) Combining Equations B.6 and B.4, the result is (fc | r -r j - | )a 2 l r - r j l where Equat ion B .7 differs from the result quoted in the paper by Blackledge et al in that the factor of \ is not found within the curly brackets [10]. Recalling from Section A . 3 that the following assumption is applied: |r - r j l » 1 (B.9) Appendix B. Derivation of VS 278 or equivalently that k > , r (B.10) k - r j l 1 1 2 |r - r,-is assumed to hold in the derivation of the image reconstruction algorithms used herein, Equat ion B.7 reduces to V 5 r . = ik&jSrj ( B . l l ) This final result agrees wi th that quoted within the paper [10]. The reduction of Equat ion B.7 by dropping the real term is commonly found in the literature [10, 66]. However, it is important to note that this assumption implies that V 5 r j . does not impart a phase shift when operating on an ultrasound field. This effect of this assumption is studied in detail in Section 5.1.1. In support of this study, Appendix C shows the derivation of the scattered field, p s ( r d , r s , OJ), for the case in which the full expression for VSTj is used. B . 2 D E R I V A T I O N O F VS(k\rj - r |) In the derivation of the reconstruction algorithms, the propagator from the source to the scatter point, 5(A;|r 0 — r | ) , is equal to the propagator from the scatter point to the detector, S(k\r — r s | ) , by the Principle of Reciprocity. It w i l l now be shown that V5(A;|r_,- - r | ) and V<S'(A;|r - rj |) are also equal. Equations B . l through B.4 hold, wi th the exception that now f(x,y) — \TJ - r | . The expression for Vf(x,y) can be expanded as follows: Vf(x,y) = V{{(rx-x)2 + (ry-y)2}12} = \ j ^ ) { - 2 { r x - x ) ± - 2 { r y - y m (x-rx) (y~ry) — i i x i , i y Appendix B. Derivation ofVS 279 Thus, the result is the same as for VS'(A;|r — r^j), and it therefore follows that V5(fc | r i - r | ) = V5 (* |r-r i | ) (B.12) A s such, the Principle of Reciprocity holds for the gradient of the propagator as expected. APPENDIX C IMPROVED DERIVATION OF PS Recal l that in Sections B and 4.2.2, the gradient of G(k\r — r , | ) (the nonconstant portion of the propagator) was determined, where G — S in the case i f of no attenuation, and G = S when attenuation is considered. In both cases, VG(k\r — rj |) takes the form: VG(k\v - Tj\) = G(k\r - TA) (ik&i - A r . ) ( C . l ) In the above, j is equal to either s or 0 for propagation to the detector from any point or from the source to any point, respectively. A n important assumption in both the original method by Blackledge et al and the A R T developed in this thesis is that A r j , is negligible. This appendix expands upon the derivation of the expression [ G(k\r - r d | ) V • (lp(v)VG(k\r - r s | ) ) d 2 r (C.2) which is found in the development of both reconstruction algorithms used in this thesis. Specifically, the full expression for VG(A; | r — r^ -1) is retained, rather than applying the assumption that VG(A; | r - TA) tt ikhj G(k\r - TA) (C.3) To simplify notation in the following derivation, GTj w i l l denote G(k\r — Tj\)). The first steps in the derivation are identical to those outlined in Equations A.23 through A.30, which involve only vector calculus, with the exception that GTj and V G r j are substituted for STj and VSTj, respectively. These steps result in the expression / C 7 r d V • ( 7 P ( r ) V G r J d 2 r = _ / V G r d • ( 7 p ( r ) V G J d 2 r (C.4) 280 Appendix C. Improved Derivation of Ps 281 Recall ing again that V G r j . = GTj (ikhj — Ar.) results in the expression / GrdV • ( 7 p ( r ) V G r J d 2 r = - / {ik)2hdGId • h s l p G I s d 2 r (C.5) = -k2 G,dlPGTa{ikrid - Ar<J) • (ikhs - A r J d 2 r Recall ing from Equation A.36 that hs • hd = - cos(6>) (C.6) finally results in f G(k\r-rd\)V-Mr)VG(k\r-rs\))d2v (C.7) - -k2 f G(k\r - rd\) cpd, 9) 7 p ( r ) G(A:|r - r , | ) d 2 r ./SR2 where $ = cos(0) - l- (Ard • n s + A r s • hd) + ^ A r d • A r s (C.8) In the case of both attenuated and nonattenuated propagators, $ simply equals cos(0) upon the application of Assumption C.3. Therefore, the only change from the case in which A r j is ignored is that 7 p ( r ) is multiplied by a factor of $ rather than simply cos(0). A s per Equations A.39 and 4.25, the new expression for the scattered field is ps(vd,rs,u) = a2A(oj)k2x (C.9) / G(k\r - vd\) ( 7 # t ( r ) + $ 7 p ( r ) ) G(k\r - r , | ) d 2 r C . l NONATTENUATED PROPAGATORS REVISITED For the nonattenuated 2D Green's function propagator, aS(k\r - r^ -1), recall from Equat ion B.8 that a - 1 n ' (CIO) 2 r - rj Typica l ly |r - Tj\ w i l l be kept to greater than 0.8|rj | , where \TJ\ is the distance of the source or detector from the origin. Furthermore, | r j | w i l l typically be on the order of Appendix C. Improved Derivation of Ps 282 100-200 mm. A worst case scenario might be characterized by the outermost regions of tissue having |r — r^| « 0.7|r-j| and |rj | = 100 mm. This leads to A r . 0.00714 m m - 1 ( C . l l ) Substituting this expression into Equation C.7 and recalling that n s -Hd = — cos(0) yields $ = c o s m + 0-01428imm^ _ J _ 0.007142 m m - 2 ) (C.12) and thus according to Equation C.9, ps(rd,rs,u}) = a2A(u)k2x (C.13) / 5(*|r - r s |) ( 7 «(r ) + $ 7p(r)) S(k\r - rd\) d 2 r C . 2 A T T E N U A T E D P R O P A G A T O R S R E V I S I T E D For the attenuated 2D Green's function propagator, aS(k\v — r^]), recall from Equa-tion 4.23 that A r . = (A;2 1.4053 x 10" 5 m m + 1.2255 x 1 0 - 2 jfe)(x + y) (C.14) = (Bk2 + Ck)(x + y) This expression can be substituted into Equation C.7, and the following relations regard-ing the unit vectors, hs and n<f, can be applied: hd = xcos(> d ) + ysm(ipd) (C.15) n s = x c o s ( < £ s ) + y s i n ( v ? s ) Recall that (ps and <pd are the angles of the source and detector, respectively. The result for $ is thus $ = cos(0) + ^ ( £ f c 2 + Cfc) 2 | x + y | 2 (C.16) - - (Bk2 + Ck) {cos(ips) + cos(<£d) + sin(y>s) + sin(v>d)} k Appendix C. Improved Derivation of Ps and according to Equation 4.25, the new expression for the scattered field ps(rd,rs,u) = a2A(u)k2x f S(k\r - r.\) ( 7 « ( r ) + $7pW) S(k\r - rd\) d : APPENDIX D PHANTOM CONSTRUCTION The following formula for a tissue equivalent material was provided by M r . Authur Worthington of the Ultrasound Imaging Group at the Ontario Cancer Institute. It uses powdered graphite to achieve the desired attenuation, which mimics that in soft tissue. The formula is meant to produce a phantom with a speed of sound of 1.54 mm/us at 22°C and an attenuation coefficient of -1.1 M H Z b c m . This attenuation is achieved through use of fine powdered graphite with a granule size rating of at most 325 mesh. Note that mesh sizes are the standard method of measuring graphite granules, and this rating is simply the size of the holes in the finest wire mesh that can pass al l the graphite granules in a sample. The attenuation coefficient can be changed by varying the coarseness of graphite. Materials such as metal or wooden rods and bal l bearings can be placed in the phantom as it hardens to test the imaging of non-uniform objects. This formula is for a cylindrical phantom that is 10 cm deep and 2 cm in radius. The volume is therefore 125 c m 3 , and this is approximately equal to the volume of water that is used to create the phantom. Hence, the amount of water used is 125.7 c m 3 , wi th a weight, w of 125.7 g. Given that W is the total weight of the phantom, the formula requires that w = 0 7 4 1 ( m ) Thus, the total weight of the phantom is 169.6 g. The following formula completes the phantom: • 5% gelatin powder by weight, 8.5 g • 0.9% N a C l by weight, 1.5 g • 20% graphite powder (325 mesh) by weight, 33.9 g 284 Appendix D. Phantom Construction 285 The instructions for constructing the phantom are as follows: • Degas water by boiling. • A d d N a C l and gelatin. Stir well and wait unti l solution clarifies. • A d d graphite a bit at a time and stir well. • Let cool to about 40-50°C. • Pour into mold. • Rotate mold periodically as phantom solidifies to keep graphite from settling. A n alternate method of constructing cylindrically symmetric phantoms that works very well is the "candle method." Essentially, a "wick" made of string is successively dipped into the phantom mixture, allowing 2-3 minutes in between for quick drying. A weight is tied to the end of the wick so that it can be easily plunged into the mixture. In this manner, cylindrical phantoms of length 6-10 cm and wi th diameters of ~2-6 m m can be easily constructed. The phantom construction can also be varied by changing the mixture to one wi th a different speed of sound and/or attenuation coefficient after a certain diameter is reached. APPENDIX E ITERATIVE METHODS In applying the reconstruction algorithm that includes attenuated propagators, one must solve a system of linear algebraic equations, given by y = A x , for x . This has been done iteratively, and several methods were studied in this effort: the conjugate gradient (CG) method, the Jacobi Iterative (JI) method, the Gauss-Seidel (GS) method, and the Successive Under-Relaxation (SUR) method. The algorithms for each are described below, and results of their use are presented in Section 5.7.1. E . l C O N J U G A T E G R A D I E N T M E T H O D The conjugate gradient method solves the matrix system by minimization of a resid-ual error function. It is furthermore a Kry lov Subspace solver, which means the following. The iterated x is an evolving function of both the ini t ia l guess x 0 as well as the residual vector and the matrix A . Functions of this sort belong to Kry lov Subspace. The conju-gate gradient method finds an x by systematically searching through K r y l o v Subspace for the x that minimizes the residual. The minimization of the residual produces a sequence of coefficients, and vectors, pW, upon which the iterated x is built . The C G method minimizes the residual of the matrix equation along a set of conjugate directions [39, 83]. This is defined as a set of directions such that a line minimizat ion along any direction in the set does not interfere wi th any other previous minimizations that were done along any other directions in the set. In other words, having performed a line minimization along a direction u, a new direction, v must be chosen for the next minimizat ion step such that minimizing along v w i l l not "spoil" the minimizat ion along u. 286 Appendix E. Iterative Methods 287 Minimiza t ion along conjugate directions is an efficient approach to the iterative solution of a linear system. The following illustrates the algorithm for the conjugate gradient method, when an array equation y = A x ( E . l ) must be solved for x given knowledge of A and y . The matrix A can be square or overdetermined. The algorithm begins by calculating the ini t ia l residual, r^°\ and a measure of error, e^°\ given by .(0) ,(o) A x ' 0 ' - y TN r2 ^ 3 = 1 3 x 100% (E.2) TN ii2 2^=1 Vj where N is the length of x , or equivalently the number of pixels in the image. Also, the first conjugate vector, p(°\ is calculated by p(0) = _ A ' R 0 > ) (E.3) Whi le is greater than a very small l imit , rj, which in practice was chosen to be 1x10 7 , the following set of commands is executed [39]: = l A ' r ^ 1 ) ! „«. = x ( - i ) + i W p ( « - i ) = |A'rW| | A P (gW)2 («W)2 (E.4) jv^)2 (s(0)2 In practice when using this algorithm in the reconstruction of attenuating objects as described in Section 4.2, it was necessary to apply a simple boundary condition in order to achieve convergence. Under this condition, it was assumed that the image was sufficiently larger in width than the object such that the pixels in the outermost border corresponded Appendix E. Iterative Methods 288 to water and hence had values of zero. Hence, the boundary condition manifests itself as a vector mask of l ' s and O's. This mask is multiplied point-by-point wi th the vector x during each iteration in order to apply the boundary conditions at each step. The error given by Equat ion E.2 is recalculated, and when it becomes less than n, the algorithm is assumed to have converged upon a good solution, x , that satisfies Equat ion E . l . E.2 JACOBI ITERATIVE T E C H N I Q U E The Jacobi iterative method is a technique to solve Equation E . l iteratively for x when A is square. The technique converts the system y = A x into an equivalent system of the form x = T x + c, where T is some n x n matrix and c is some vector [13]. Upon selecting an ini t ia l guess, xo, at the solution, the vector x is iterated by computing x « = T x ( i - i ) + c (E.5) To convert y = A x into the form x = T x + c, one solves each successive equation in the system for successive elements of x . As an example, equations 1 and 2 in the system are rearranged to solve for x\ and x2, respectively. The procedure is quite simple as follows. The A;th equation can be written as n Vk = ^AkjXj (E.6) n = Akj Xj + Akk xk 3=1, j^k which can be rearranged to yield j=l,frk A k k A k k = { t =£±xj + 0.zk} + f -j=l, &k A k k A k k Appendix E. Iterative Methods 289 Combining the equations for al l elements, Xk, of x yields a matrix equation of the form x = T x + c, where ^ 0, k = j and <* = f~ ( E - 8 ) A s such, the elements Xk of the iterated solution can finally be calculated as follows: j=l,jjLk k k k k A s in the case of the C G method, a mask vector of l ' s and 0's to distinguish pixels in tissues from those in water can be multiplied point-by-point wi th x in order to apply boundary conditions during each iteration. A t the end of each iteration, a residual vec-tor rW = A x ' * ' — y is determined, and from this the error given by Equat ion E.2 is calculated. The iteration continues unti l the error becomes less than some l imit rj. E v i -dently, this algorithm requires that al l diagonal elements of A be nonzero. Furthermore, the algorithm has difficulty converging and often can not converge unless the matr ix is diagonally dominant. In a diagonally dominant matrix, \Akk\>=^2\Akj\{oTJ = l,2,...,k-l,k + l,...,n (E.10) 3 Again , a boundary condition was applied in which the pixels in the outermost border were assumed to correspond to water and hence have values of zero. E.3 G A U S S - S E I D E L I T E R A T I V E T E C H N I Q U E The Gauss-Seidel is an improved version of the Jacobi Iterative method. The JI method successively calculates the k elements of the current ith. iterated solution using only the previous solution, x^%~l\ Since the new solution is assumed to be a better Appendix E. Iterative Methods 290 approximation to the desired x , it would seem useful to compute the A;th element of x^ with the (A;-l) newly calculated elements of x^ together with elements (k+1) through n of the previous solution, x^_1K The Gauss-Seidel algorithm makes use of this variation and thus calculates the elements xk of the iterated solution as follows [13]: v f^e-l A _(») v^ n A „(«-l) , „. x(i) _ L,j=i AkjXj - 2^j=fc+i AkjXj + Vk ^E n ^ Akk Again , a l l diagonal elements of A must be nonzero, and the method has difficulty converg-ing when A is not diagonally dominant. A s per the previous two methods, a boundary condition was applied in which the pixels in the outermost border were set to zero. E.4 S U R ITERATIVE TECHNIQUE This method is a modification of the Gauss-Seidel iterative algorithm that results in convergence of some systems that would otherwise not converge wi th the G S method. The S U R method calculates the elements xk of the iterated solution by the following equation [13]: *if> = ( i - U) xt1] + OJ ~ A^xt ~ A^4~1] + Vk { E 1 2 ) Akk OJ is some positive fraction that is less than 1, and the proper selection of this parameter can facilitate the solution of systems that fail to converge wi th the G S method. Unfortu-nately, though, OJ must be chosen in what is best described as a "hit and miss" manner [13]. Due to the division by Akk, al l diagonal elements of A must be nonzero. Note that if OJ is chosen to be some positive value greater than 1, the method is referred to as the Successive Over-Relaxation algorithm, which is used to accelerate the solution of systems that do in fact converge wi th the Gauss-Seidel method [13]. A boundary condition was again applied in which the pixels in the outermost border were set to zero. 

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