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Out-of-plane motion estimation based on a Rician-Inverse Gaussian model of RF ultrasound signals: speckle.. Afsham, Narges 2012
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Title | Out-of-plane motion estimation based on a Rician-Inverse Gaussian model of RF ultrasound signals: speckle tracking without fully developed speckle. |
Creator |
Afsham, Narges Najafi, Mohammad Abolmaesumi, Purang Rohling, Robert N. |
Publisher | Society of Photo-Optical Instrumentation Engineers |
Date Created | 2012-03-05 |
Date Issued | 2012 |
Description | Fully developed speckle has been used previously to estimate the out-of-plane motion of ultrasound images. However, in real tissue the rarity of such patterns and the presence of coherency diminish both the precision and the accuracy of the out-of-plane motion estimation. In this paper, for the first time, we propose a simple mathematical derivation for out-of-plane motion estimation in which the coherent and non-coherent parts of the RF echo signal are separated. This method is based on the Rician-Inverse Gaussian stochastic model of the speckle formation process, which can be considered as a generalized form of the K-distribution with richer parameterization. The flexibility of the proposed method allows considering any patch of the RF echo signal for the purpose of displacement estimation. The experimental results on real tissue demonstrate the potential of the proposed method for accurate out-of-plane estimation. The underestimation of motion in ex vivo bovine tissue at 1 mm displacement is reduced to 15.5% compared to 37% for a base-line method. Copyright 2012 Society of Photo-Optical Instrumentation Engineers. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. |
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Conference Paper |
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Text |
Language | Eng |
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International Society for Optics and Photonics (SPIE) Publications (Electrical and Computer Engineering) |
Date Available | 2012-03-05 |
DOI | 10.14288/1.0107566 |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Citation | Afsham, Narges; Najafi, Mohammad; Abolmaesumi, Purang; Rohling, Robert N. Out-of-plane motion estimation based on a Rician-Inverse Gaussian model of RF ultrasound signals: speckle tracking without fully developed speckle. Medical Imaging 2012: Ultrasonic Imaging, Tomography, and Therapy, edited by Johan G. Bosch, Marvin M. Doyley Proceedings of SPIE, Volume 8320, 832017, 2012. http://dx.doi.org/10.1117/12.911710 |
Peer Review Status | Reviewed |
Scholarly Level | Faculty |
Copyright Holder | Rohling, Robert N. |
URI | http://hdl.handle.net/2429/41146 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/34910/items/1.0107566/source |
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Out-of-plane motion estimation based on a Rician-Inverse Gaussian model of RF ultrasound signals: speckle tracking without fully developed speckle N. Afsham, M. Najafi, P. Abolmaesumi, R. Rohling Dept. of Electrical and Computer Engineering, UBC, Vancouver, BC, Canada ABSTRACT Fully developed speckle has been used previously to estimate the out-of-plane motion of ultrasound images. However, in real tissue the rarity of such patterns and the presence of coherency diminish both the precision and the accuracy of the out-of-plane motion estimation. In this paper, for the first time, we propose a simple mathematical derivation for out-of- plane motion estimation in which the coherent and non-coherent parts of the RF echo signal are separated. This method is based on the Rician-Inverse Gaussian stochastic model of the speckle formation process, which can be considered as a generalized form of the K-distribution with richer parameterization. The flexibility of the proposed method allows considering any patch of the RF echo signal for the purpose of displacement estimation. The experimental results on real tissue demonstrate the potential of the proposed method for accurate out-of-plane estimation. The underestimation of motion in ex vivo bovine tissue at 1 mm displacement is reduced to 15.5% compared to 37% for a base-line method. Keywords: Sensorless freehand ultrasound, speckle, fully developed speckle, out-of-plane motion 1. INTRODUCTION The deviation of a coherent field phase front from its original form, after confronting a random medium, results in a granular noise-like pattern in ultrasound images referred to as speckle1. Speckle is an informative signal rather than being random noise and it can be used to reveal information about the imaging system or the random medium2. On the other hand, the speckle formation process can be considered as the summation of small phase changes of the incident coherent signal produced by randomly distributed scatterers3. Such a process intrinsically suggests it can be modeled as a stochastic process. By modeling the speckle formation process, it is possible to obtain information about the echo signal amplitude and intensity distributions. Sensorless freehand ultrasound aims to eliminate the need of a position sensor mounted on the transducer with speckle tracking. The main challenge is the out-of-plane motion estimation. Three main categories have been introduced in the literature for the purpose of speckle tracking: regression-based, correlation-based, and learning-based methods. The regression-based method4 determines the best affine linear estimation of a circularly Gaussian distributed Radio Frequency (RF) signal correspondent to a Fully Developed Speckle (FDS) pattern and is closely related to the correlation of speckle patches. The majority of the remaining work on ultrasound speckle tracking is correlation-based. The second order statistics and low-order moments of the envelope ultrasound echo signals has been used to estimate the out-of-plane motion for more than two decades5. Many of the proposed correlation-based methods focus on the improvement of FDS detection by introducing different FDS detection methods, such as optimal low-order moments6, novel meshing7, and the use of the K-S test as a non-parametric goodness of fit8. Other efforts intend to increase the motion estimation accuracy by adapting the correlation curve and compensating the loss of coherency by using additional information such as correlation in the axial and lateral directions8, beam steering9, developing a heuristic method to consider the coherent part of the image10, using Maximum Likelihood Estimators (MLE) for motion estimation11, or incorporating the information of several noisy measurements in a probabilistic framework12. Recently a new learning based method of out-of-plane motion estimation on imagery of real tissue has been introduced13. They adapt the scale factor of the nominal correlation curve based on training data. In this paper, we have developed a novel correlation-based method that incorporates coherency in the derivation of correlation function and gives a general form of previously proposed correlation-based methods. For any patch of the image, the parameters of the Rician-Inverse Gaussian (RiIG) model are estimated from the RF signal, and from these Cum Laude Poster Award Medical Imaging 2012: Ultrasonic Imaging, Tomography, and Therapy, edited by Johan G. Bosch, Marvin M. Doyley, Proc. of SPIE Vol. 8320, 832017 · © 2012 SPIE CCC code: 1605-7422/12/$18 · doi: 10.1117/12.911710 Proc. of SPIE Vol. 8320 832017-1 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms parameters, the correlation coefficient is computed. Since we propose an analytical closed-form formula for the correlation, it is possible to estimate the out-of-plane motion. The remaining parameter needed in this formulation is the elevation width of the ultrasound Point Spread Function (PSF) in its elevation direction which can be found from a prior calibration process. 2. METHODS 2.1 Speckle formation process As mentioned, the speckle formation process can be modeled as a stochastic process. It can be stated as the summation of ܰ complex phasors as below 2: ܣ ൌ |ܽ| ே ୀଵ ݁ ൌ ܺ ݆ ܻ. (1) ܣ is the complex RF signal and ܰ is the number of scatterers in a resolution cell. The number of scatterers in a resolution cell for a randomly distributed medium follows a Poission process 14. If the number of scatterers is large enough, the amplitude of the radio-frequency signal, ܴ ൌ ඥܺଶ ܻଶ, has a Rayleigh distribution. In this case in-phase (I) and quadrature (Q) components of the RF signal, ܺ and ܻ, are zero-mean Gaussian distributed. The Rayleigh distribution is correspondent to FDS. If the variances of I and Q components of the RF signal are Γ distributed, the RF echo signal is K distributed14. In a general form, ܺ and ܻ are compound random variables as follows: ܺ ൌ ߤ௫ ߚ௫ܼ √ܼ ௫ܰ, (2-1) ܻ ൌ ߤ௬ ߚ௬ܼ √ܼ ௬ܰ. (2-2) ௫ܰ and ௬ܰ are normally distributed with a covariance matrix of identity. If Z is Γ distributed, with ߤ௫ ൌ ߤ௬ ൌ 0, ܴ has a Rician distribution. If ߚ௫ ൌ ߚ௬ ൌ 0, ܴ becomes homodyned-K distributed15. There is no explicit analytical expression for homodyned-K Probability Density Function (PDF) and its second order statistics; hence, it is more difficult to use for the application of the out-of-plane motion estimation. Considering the Inverse Gaussian (IG) distribution for ܼ and zero ߤ௫ and ߤ௬, the resulting distribution is the RiIG 16. The physical interpretation of RiIG is that the complex RF signal is a combination of two independent Brownian motion with drifts ߚ௫ and ߚ௬ and the Inverse Gaussian (IG) first passage time. The drift part models the presence of coherency in the echo signal. In this statistical process, ܼ models the speckle-free part of the signal and ௫ܰ , ௬ܰ correspond to the FDS part. A previous study on RiIG shows that it outperforms K, and Nakagami distributions in modeling ultrasound echo signal14. Moreover, it has some interesting characteristics that make it suitable for the application of out-of-plane motion estimation. It is possible to estimate the parameters of RiIG distribution fairly well, even from a few samples14. Moreover, its posterior distribution formula is available in a closed form that makes it possible to estimate the coherent part, ܼ, in a Maximum A-Posterior (MAP) manner16. Finally, the model allows for separation of the coherent and non- coherent parts of the correlation function, as will be described next. 2.2 Motion estimation based on second order statistics For a linear rectangular array, the point spread function (PSF) of the returned echo amplitude can be simplified as17: ݃ሺݔ, ݕ, ݖሻ ൌ ݃ሺݔ, ݖሻ_݃ሺݕሻ௨௧__ ൌ ܭ݃ሺݔ, ݖሻ. ݏ݅݊ܿଶሺ ݕ ߪ௬ሻ (3) where y indicates the out-of-plane or elevation direction. ݏ݅݊ܿሺݕሻ equals ௦ሺగ௬ሻగ௬ and ߪ௬ ൌ ఒబ௭బ . ܦ௬ refers to transducer dimension in the elevation direction. ݖ is the distance along the beam axis and ߣ is the ultrasound wavelength in its central frequency. ܭ represents a constant factor. Proc. of SPIE Vol. 8320 832017-2 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms For a random process ܣ and spatial point spread function of an imaging system ݃ሺݔ, ݕ, ݖሻ, if the displacement is only in elevation direction, based on the linear system theory, the autocorrelation function of the backscattered complex process can be written as2: ܴሺ∆ݕሻ ൌ ܴሺ∆ݕሻ כ ݃ሺെ∆ݕሻ כ ݃ כሺ∆ݕሻ, (4) where כ indicates the convolution operator and ݃ כ is the complex conjugate of ݃. In the case of a rectangular array ݃ሺݕሻ can be approximated by ݏ݅݊ܿଶ function and it gives: ܴሺ∆ݕሻ ൌ ܭଶܴ ሺ∆ݕሻ כ ݏ݅݊ܿଶሺ∆௬ఙሻ כ ݏ݅݊ܿ ଶሺ∆௬ఙሻ. (5) The very simple approximation of the convolution of two ݏ݅݊ܿଶ functions is a Gaussian one. However, such an approximation may be one of the sources of the out-of-plane motion underestimation based of Gaussian correlation curves reported in previous papers. Since the convolution of two ݏ݅݊ܿଶ functions can be determined by term-wise Fourier transform2, it gives: ܴሺ∆ݕሻ ൌ ܴሺ∆ݕሻ כ మ ሺగ∆௬ሻమ ሺ1 െ ݏ݅݊ܿ ൬ ଶ∆௬ ఙ ൰ሻ. (6) Under Rayleigh conditions, where the microstructure of the imaging sample is uncorrelated as a result of randomly scattered fine particles in the imaging sample, and due to the averaging over uniformly distributed phases we have 2: ܴR౯ౢሺ∆ݕሻ ൌ 2ߪௗଶ ߜሺ∆ݕሻ. (7) ߪௗଶ is the variance of the diffuse part. In this case, the autocorrelation of the output process only reveals the information of imaging PSF (݃ሺݕሻ ሻ rather than the information of medium and it can be used to estimate the out-of-plane motion (∆ݕሻ. In the case of RiIG process, autocorrelation function can be written as: ܴሺ∆ݕሻ ൌ ൏ ሺߚܼଵ ඥܼଵ ௫ܰଵ ݆ሺߚܼଵ ඥܼଵ ௬ܰଵሻሻሺߚܼଶ ඥܼଶ ௫ܰଶ ݆ሺߚܼଶ ඥܼଶ ௬ܰଶሻሻכ ൌ ൏ ඥܼଵܼଶ ൏ ௫ܰଵ ௫ܰଶ ൏ ඥܼଵܼଶ ൏ ௬ܰଵ ௬ܰଶ ߚଶ ൏ ܼଵܼଶ . (8) We assumed that ߚ and ߚ are the same for the closely positioned frames and ߚ ൌ ඥߚଶ ߚଶ. 1 and 2 subscripts represent the process at the positions ݕଵ and ݕଶ respectively. Considering ܼଵ and ܼଶ fully correlated for two adjacent frames, without loss of generalization, it can be assumed ܼଶ ൌ ܼ݇ଵ or ܼଵ ൌ ܼ݇ଶ such that 0 ݇ 1. Considering the fact that ܼ models the variance of the first passage time of the process, it is plausible to suppose that ܼ is independent of the normally distributed part. Under these circumstances: ܴሺ∆ݕሻ ൌ √݇ ൏ ௫ܰଵ ௫ܰଶ ൏ ܼ √݇ ൏ ௬ܰଵ ௬ܰଶ ൏ ܼ ݇ߚଶ ൏ ܼଶ . (9) Since the Rayleigh condition goes for the normal part of RiIG distribution and the variances of these normal distributions are unit, we can replace ൏ ௫ܰଵ ௫ܰଶ by ߜሺ∆ݕሻ (see 2 for more details on the derivation of Rayleigh and K correlation functions). ܴሺ∆ݕሻ ൌ 2√݇ ൏ ܼ ߜሺ∆ݕሻ ݇ߚଶ ൏ ܼଶ . (10) In this new correlation function, the variance of the diffuse part is replaced by 2√݇ ൏ ܼ (compare with Eq. 7) and ݇ߚଶ ൏ ܼଶ represents the effect of coherent part. Substituting the moment of ܼ from14 in Eq. 9 and apply the result in Eq. 5, we may write: ܴሺ∆ݕሻ ൌ ܴሺ∆ݕሻ כ మ ሺగ∆௬ሻమ ሺ1 െ ݏ݅݊ܿ ൬ ଶ∆௬ ఙ ൰ሻ. (11) After some simple arithmetic: Proc. of SPIE Vol. 8320 832017-3 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms ܴሺ∆ݕሻ ൌ ܭଶ݇ߚଶ ൬ߜߛ൰ ଶ ൬1 1ߜߛ൰ ߪ௬ ଶ ܭଶ ൬ߜߛ൰ 2√݇ ሺߨ∆ݕሻଶ ൭1 െ ݏ݅݊ܿ ቆ 2∆ݕ ߪ௬ ቇ൱ ൌ ܴ ܴሺ∆ݕሻ. (12) ܴ is the coherent part of the correlation and ܴ is the Rayleigh part which out-of-plane displacement can be estimated from. To eliminate the scaling factor K, it is feasible to use correlation coefficient instead: ߩ ൌ ܥሺ∆ݕሻܥሺ0ሻ ൌ ܴሺ∆ݕሻ െ ߤሺݕଵሻ כ ݃ሺݕሻ ൈ ߤሺݕଶሻ כ ݃ሺݕሻ ߪଶ , (13) where ߤሺݕሻ is mean of the process at position ݕ and ߪଶ is the variance of the process. Assuming ܣ is a stationary process to its second order, and using the explicit terms for RiIG moments14, from Eq.12 and Eq. 13 the correlation coefficient is given by: ߩ ൌ ݇ߚଶߪ௬ଶ 2ߛଶ√݇ሺ 1 െ ݏ݅݊ܿ ൬2∆ݕߪ௬ ൰ ሺߨ∆ݕሻଶ ሻ ߚଶߪ௬ଶ 43ߪ௬ଶ ߛ ଶ . (14) The maximum value of function ሺ ଵି௦൬మ∆ ൰ ሺగ∆௬ሻమ ሻ equals ଶ ଷఙమ and 0 ݇ 1, so ߩ is less than 1 for all the values of ∆ݕ, as expected. ߩ is directly calculated from the ultrasound RF signal. All the parameters in the right side of Eq. 14, including ݇, ߚ, and ߛ, can be estimated from data based on Expectation Maximization algorithm (EM) explained in14,16. ߪ௬ is the variance of the imaging system PSF at different depths, which can be known from the system manufacturer or can be calibrated from a speckle phantom. The second approach has been followed in this work. The mathematical expression for MAP estimation of the coherent part (ܼ) of the RF signal can be written as: ̂ݖ ൌ arg max |ோሺݖ|ݎሻ. (15) Since the posterior distribution of ܼ, |ோሺݖ|ݎሻ, is given in a closed form16, ̂ݖ is explicitly determined. For the details and discussion on the parameter estimation of RiIG model see14. 3. EXPERIMENTAL SETUP The image acquisition system consists of a 10 MHz 2D linear probe (SonixGPS, Ultrasonix Inc., Richmond, BC, Canada). The elevation movement of the phantom is created by means of a linear motor stage (T-LSR150B, Zaber Technologies Inc., Vancouver, BC, Canada) with equal steps of 0.0635 mm. The ultrasound probe is fixed during the experiments and the phantom is placed on top of the motor stage. We considered the frames to be parallel with no in- plane motion. Firstly in a calibration process, a phantom with a large number of randomly distributed scatterers is scanned. At four different axial depths of the image (0.57, 0.95, 1.33, 1.75 cm) a patch of 100 RF sample× 25 pixel is considered. The PSF elevation width of the ultrasound transducer ( yσ ) is estimated at each depth from the selected patch to minimize the difference of true displacement and estimated ones from a sequence of 40 frames with equal steps of 0.0635 mm displacements. Then, the same experiment is performed on real tissue. At each depth, similar to the depths selected in the calibration phase, a random window of 100 RF sample× 25 pixel is selected. Here for the sake of simplicity we used non overlapping windows, but it is possible to apply this method on any arbitrary window at the selected depth. The parameters of the RiIG model are estimated as explained previously. Since all of the parameters used in Eq. 14 are now determined, the out-of-plane motion is computable. Note that the value of yσ for each depth is different and comes from the calibration process. Proc. of SPIE Vol. 8320 832017-4 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms Fig. 1 shows parameters: RF sample× samples of t parameters. Z of IG model. 1 indicates th Fig. 1 To evaluate implemented patches are r the proposed of our imple method. Fig.2 (a) depth) One of the m presence of novel correla RiIG model provides the samples of e β , γ andδ . T 25 pixel patch he phantom is is determined It is observed at as the numb (a) Estimated PD the proposed and a Gaussia are within the method occur mentation of (a) Bovine ultraso (b) Displacem ain concernin coherency cau tion-based me of speckle for possibility of stimated distr he EM algorit es at the same compared to from MAP e that estimated er of samples F in comparis method, the n curve has be real tissue and s at greater de a base-line m und image (re ent estimation 5. D g issues in FD ses underestim thod of out-o mation proces dividing the c 4. ibutions of R hm has been u center positio the one whic stimation and PDFs follow increases, the on with real PD base-line m en fitted on th no FDS patch pths in compa ethod at 1mm (b) d star indicate using base-lin using th ISCUSSIO S-based free ation of out- f-plane motion s which is a g orrelation of tw RESULTS and Z from a sed to estimat n. Figure 1 su h is generated it is compared the real ones f closer estimati (b) F from (a) 20 ethod of low e correlation c is found at th rison with the displacemen s FDS patch an e FDS method e proposed me N AND CO hand ultrasoun of-plane motio estimation f eneral represe o patches int phantom imag e these param mmarizes the r from the clo to the PDF re airly well, eve on to the real d × 20 (b) 50× -order mome urve of the ph e third depth. FDS-based m t is 37% com d blue x show for three patch thod NCLUSIO d is the rarity n as the disp or non-FDS p ntation for ul o two terms. T e. RiIG is a eters from 20× esult. The PD sed-form form sulted from cl n for small nu ata is acquire 50 and (C) 100 nts for FDS antom data. As Underestimat ethod. The av pared to 15.5 (c) s randomly se es (c) Displac N of FDS patch lacement incr atches. The m trasound echo he one that sh distribution wi 20, 50× 50, F of the RF am ula and the e osed-form form mbers of samp d. (c) × 100 patche detection18 h Fig. 2(a) show ion of displace erage underes % with the p lected patch at ement estimat es in real tiss eases. We pro ethod is based intensity. Thi ows the cohe th three and 100 plitude stimated ulation les. Fig. s as been s, FDS ment in timation roposed each ion ue. The posed a on the s model rency of Proc. of SPIE Vol. 8320 832017-5 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms the patches and the one that comes from the randomly located scatterers. In this frame work it is possible to use almost all patches of the images regardless of being FDS and take advantage of plenty estimations and make the final out-of- plane motion estimation more robust. Here we just evaluated the out-of-plane motion estimation for the frames with fixed elevation displacement. It is possible to determine the complete out-of-plane transform between two frames based on the elevation distance of tree different corresponding points on the frames10. Experiments on a tissue sample suggest that considering the coherent part significantly improves both the precision and the accuracy of the out-of-plane motion estimation. Our results indicate by choosing a suitable frame distance (around 0.6 mm in this experiment) the underestimation over a long distance can be considerably reduced. More experiments are needed to demonstrate ability on tracking in vivo. 6. ACKNOWLEGMENTS This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canadian Institutes of Health Research (CIHR). REFERENCES [1] Abbott, J.G., and Thurstone, F.L., “Acoustic speckle: Theory and experimental analysis,” Ultrasonic Imaging 1(4), 303-324 (1979). [2] Wagner, R.F., Smith, S.W., Sandrik, J.M., and Lopez, H., “Statistics of speckle in ultrasound B-scans,” IEEE transactions on sonics and ultrasonics 30(3), 156-163 (1983). [3] Middleton, D., [An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue], 1st ed., Wiley-IEEE Press (1996). [4] Prager, R.W., Gee, A.H., Treece, G.M., Cash, C.J.C., and Berman, L.H., “Sensorless freehand 3-D ultrasound using regression of the echo intensity,” Ultrasound in Medicine & Biology 29(3), 437-446 (2003). [5] Chen, J.-F., Fowlkes, J.B., Carson, P.L., and Rubin, J.M., “Determination of scan-plane motion using speckle decorrelation: Theoretical considerations and initial test,” International Journal of Imaging Systems and Technology 8(1), 38-44 (1997). [6] Martí, R., Martí, J., Freixenet, J., Zwiggelaar, R., Vilanova, J.C., and Barceló, J., “Optimally discriminant moments for speckle detection in real B-scan images,” Ultrasonics 48(3), 169-181 (2008). [7] Rivaz, H., Boctor, E., and Fichtinger, G., “A robust meshing and calibration approach for sensorless freehand 3D ultrasound,” in Medical Imaging 2007: Ultrasonic Imaging and Signal Processing 6513, 651318-8 (2007). [8] Hassenpflug, P., Prager, R.W., Treece, G.M., and Gee, A.H., “Speckle classification for sensorless freehand 3-D ultrasound,” Ultrasound in Medicine & Biology 31(11), 1499-1508 (2005). [9] Rivaz, H., Zellars, R., Hager, G., Fichtinger, G., and Boctor, E., “Beam Steering Approach for Speckle Characterization and Out-of-Plane Motion Estimation in Real Tissue,” in IEEE Ultrasonics Symposium, 2007, 781-784 (2007). [10] Gee, A.H., James Housden, R., Hassenpflug, P., Treece, G.M., and Prager, R.W., “Sensorless freehand 3D ultrasound in real tissue: Speckle decorrelation without fully developed speckle,” Medical Image Analysis 10(2), 137-149 (2006). [11] Rivaz, H., Kang, H.J., Stolka, P.J., Zellars, R., Wacker, F., Hager, G., and Boctor, E., “Novel reconstruction and feature exploitation techniques for sensorless freehand 3D ultrasound,” in Medical Imaging 2010 SPIE Proceeding, 76291-76299 (2010). [12] Laporte, C., and Arbel, T., “Combinatorial and probabilistic fusion of noisy correlation measurements for untracked freehand 3-D ultrasound,” IEEE Transactions on Medical Imaging 27(7), 984-994 (2008). [13] Laporte, C., and Arbel, T., “Learning to estimate out-of-plane motion in ultrasound imagery of real tissue,” Medical Image Analysis 15(2), 202-213 (2011). [14] Eltoft, T., “Modeling the amplitude statistics of ultrasonic images,” IEEE Transactions on Medical Imaging 25(2), 229-240 (2006). [15] Eltoft, T., “A new approach to modeling signal amplitude statistics by the K distributions,” in Proceedings of the 7th Nordic, 62-65 (2006). Proc. of SPIE Vol. 8320 832017-6 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms [16] Eltoft, T., “The Rician inverse Gaussian distribution: a new model for non-Rayleigh signal amplitude statistics,” IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society 14(11), 1722- 1735 (2005). [17] Chen, J.-F., Fowlkes, J.B., Carson, P.L., and Rubin, J.M., “Determination of scan-plane motion using speckle decorrelation: Theoretical considerations and initial test,” International Journal of Imaging Systems and Technology 8(1), 38-44 (1997). [18] Prager, R.W., Gee, A.H., Treece, G.M., and Berman, L.H., “Analysis of speckle in ultrasound images using fractional order statistics and the homodyned k-distribution,” Ultrasonics 40(1), 133-137 (2002). Proc. of SPIE Vol. 8320 832017-7 Downloaded from SPIE Digital Library on 05 Mar 2012 to 137.82.117.28. Terms of Use: http://spiedl.org/terms
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