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Out-of-plane motion estimation based on a Rician-Inverse Gaussian model of RF ultrasound signals: speckle… Afsham, Narges; Najafi, Mohammad; Abolmaesumi, Purang; Rohling, Robert N. 2012

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Cum Laude Poster Award  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-ofplane 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  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 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: | |  .  (1)  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, , 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 and are compound random variables as follows: general form,  and  √  ,  (2-1)  √  .  (2-2)  are normally distributed with a covariance matrix of identity.  If Z is Γ distributed, with 0, has a Rician distribution. If 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 and , 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 and and the Inverse Gaussian (IG) first passage time. The drift part models the presence of coherency in the drifts echo signal. In this statistical process, models the speckle-free part of the signal and , 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 noncoherent 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: , ,  ,  _  _  where y indicates the out-of-plane or elevation direction.  ,  _  .  equals  dimension in the elevation direction. is the distance along the beam axis and central frequency. represents a constant factor.  (3) and  .  refers to transducer  is the ultrasound wavelength in its  Proc. of SPIE Vol. 8320 832017-2 Downloaded from SPIE Digital Library on 05 Mar 2012 to Terms of Use: http://spiedl.org/terms  For a random process and spatial point spread function of an imaging system , , , 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: ∆  ∆  ∆ ,  (4)  is the complex conjugate of .  where indicates the convolution operator and In the case of a rectangular array  ∆  function and it gives:  can be approximated by ∆  ∆  ∆  ∆  .  (5)  The very simple approximation of the convolution of two 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 functions can be determined by term-wise Fourier transform2, it gives: ∆  ∆  ∆  1  ∆  .  (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: ∆  R  2  ∆ .  (7)  is the variance of the diffuse part. In this case, the autocorrelation of the output process only reveals the information of imaging PSF ( rather than the information of medium and it can be used to estimate the out-of-plane motion (∆ . In the case of RiIG process, autocorrelation function can be written as: ∆  (8)  . and are the same for the closely positioned frames and We assumed that represent the process at the positions and respectively.  . 1 and 2 subscripts  and fully correlated for two adjacent frames, without loss of generalization, it can be assumed Considering or 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: √  ∆  √  .  (9)  Since the Rayleigh condition goes for the normal part of RiIG distribution and the variances of these normal by ∆ (see 2 for more details on the derivation of Rayleigh and K distributions are unit, we can replace correlation functions). ∆  2√  ∆  In this new correlation function, the variance of the diffuse part is replaced by 2√ represents the effect of coherent part. Substituting the moment of  (10)  .  (compare with Eq. 7) and  from14 in Eq. 9 and apply the result in Eq. 5, we may write: ∆  ∆  ∆  1  ∆  .  After some simple arithmetic:  Proc. of SPIE Vol. 8320 832017-3 Downloaded from SPIE Digital Library on 05 Mar 2012 to Terms of Use: http://spiedl.org/terms  (11)  ∆  1  1  2√ ∆  2∆  1  ∆ .  (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: ∆  ∆ 0  ,  (13)  is mean of the process at position and is the variance of the process. Assuming is a stationary where 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: 2∆  1 2  √  ∆ 4 3  (14)  .  ∆  The maximum value of function  equals  ∆  and 0  1, so  is less than 1 for all the values of ∆ , as  expected. is directly calculated from the ultrasound RF signal. All the parameters in the right side of Eq. 14, including , , and is the variance of , can be estimated from data based on Expectation Maximization algorithm (EM) explained in14,16. 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: ̂  arg max  |  | .  (15)  Since the posterior distribution of , | | , is given in a closed form16, ̂ 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 inplane 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 ( σ y ) 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 σ y 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 Terms of Use: http://spiedl.org/terms  4. RESULTS Fig. 1 showss samples of estimated e distributions of R and Z from a phantom imagge. RiIG is a distribution wiith three parameters: β , γ and δ . The T EM algoritthm has been used u to estimatte these param meters from 20 × 20, 50 × 50, and 100 hes at the samee center positioon. Figure 1 suummarizes the result. r The PD DF of the RF am mplitude RF sample × 25 pixel patch samples of the t phantom iss compared to the one whicch is generatedd from the closed-form form mula and the estimated parameters. Z is determined d from MAP estimation e and it is comparedd to the PDF reesulted from closed-form form mulation of IG model.. It is observed d that estimatedd PDFs follow the real ones fairly f well, eveen for small numbers of sampples. Fig. 1 indicates thhat as the numb ber of samples increases, the closer estimatiion to the real data d is acquired.  (c) (a) (b) Fig. 1 Estimated PD DF in comparison with real PD DF from (a) 200 × 20 (b) 50 × 50 and (C) 1000 × 100 patchees To evaluate the proposed d method, thee base-line method m of low w-order momeents for FDS detection18 has h been implementedd and a Gaussiaan curve has beeen fitted on thhe correlation curve c of the phhantom data. Ass Fig. 2(a) show ws, FDS patches are rare r within the real tissue andd no FDS patchh is found at thhe third depth.. Underestimattion of displaceement in the proposedd method occurrs at greater deepths in compaarison with thee FDS-based method. m The avverage underesstimation of our impleementation of a base-line method m at 1mm m displacemennt is 37% com mpared to 15.55% with the proposed p method.  (b) (c)) (a) Fig.2 (a) Bovine ultraso ound image (reed star indicates FDS patch annd blue x show ws randomly seelected patch att each depth) (b) Displacem ment estimation using base-linne FDS methodd for three patchhes (c) Displaccement estimattion using the proposed meethod  5.  D DISCUSSIO ON AND CO ONCLUSIO ON  One of the main m concernin ng issues in FD DS-based freehand ultrasounnd is the rarityy of FDS patchhes in real tisssue. The presence of coherency cau uses underestim mation of out--of-plane motioon as the dispplacement incrreases. We prooposed a novel correlaation-based meethod of out-oof-plane motionn estimation for f non-FDS patches. p The method m is basedd on the RiIG model of speckle forrmation processs which is a general g represeentation for ulltrasound echoo intensity. Thiis model provides the possibility of dividing the correlation of tw wo patches intto two terms. The T one that shhows the coheerency of  Proc. of SPIE Vol. 8320 832017-5 Downloaded from SPIE Digital Library on 05 Mar 2012 to 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-ofplane 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] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]  Abbott, J.G., and Thurstone, F.L., “Acoustic speckle: Theory and experimental analysis,” Ultrasonic Imaging 1(4), 303-324 (1979). 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). Middleton, D., [An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue], 1st ed., Wiley-IEEE Press (1996). 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