Elasticity Imaging Using Short-Time Correlation

Transcription

1 82 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 Speckle Tracking Methods for Ultrasonic Elasticity Imaging Using Short-Time Correlation Mark A. Lubinski, Student Member, IEEE, Stanislav Y. Emelianov, Member, IEEE, and Matthew O Donnell, Fellow, IEEE Abstract-In ultrasound elasticity imaging, strain decorrelation is a major source of error in displacements estimated using correlation techniques. This error can be significantly decreased by reducing the correlation kernel. Additional gains in signal-to-noise ratio (SNR) are possible by filtering the correlation functions prior to displacement estimation. Tradeoffs between spatial resolution and estimate variance are discussed, and estimation in elasticity imaging is compared to traditional time-delay estimation. Simulations and experiments on gel-based phantoms are presented. The results demonstrate that high resolution, high SNR strain estimates can be computed using small correlation kernels (on the order of the autocorrelation width of the ultrasound signal) and correlation filtering. I. INTRODUCTION NE APPROACH to elasticity imaging is to apply a 0 static or quasi-static displacement at the body surface creating deformation within the tissue [l]-[5]. Induced internal displacements and corresponding strains must then be estimated. Correlation-based speckle tracking methods are commonly used in ultrasonic elasticity imaging to estimate tissue displacement [4]-[ll]. If the strain is estimated from the displacement derivative, any displacement noise will be amplified. Therefore, displacement measurements need to have small error. Induced strain in tissue compounds the problem, however, because it reduces signal coherence leading to increased error in displacements estimated using correlation [6]-[12]. By posing displacement estimation as a time-delay problem, some insight can be gained into why strain is such a large error source in correlation tracking. Time-delay estimation (TDE) using correlation has been studied extensively [13], and this body of research can be drawn upon. When the unknown time delay is constant, corresponding to the case of zero strain, the position of the maximum of the cross-correlation is the maximum likelihood estimator (MLE) when the correlation time-bandwidth product is large [14]. The MLE has the useful property of being able to asymptotically achieve the Cramer-Rao lower bound Manuscript received September 18, 1997; accepted July 7, This work was supported in part by NIH Grant #DK The authors thank ATL Ultrasound, Inc. for supplying the ultrasound scanner. The authors are with the Biomedical Engineering and Electrical Engineering and Computer Science Departments, University of Michigan, Ann Arbor, MI ( odonnelqeecs.umich.edu). (CRLB), the minimum error variance. When the delay is not constant, the nontrivial case for elasticity imaging, evaluating the performance of correlation delay estimation, becomes more difficult. One solution to the time varying delay problem is to find a new maximum likelihood estimator. When the delay is a linear function, corresponding to constant strain, one MLE is to time compand or stretch the signal prior to time delay estimation by correlation [15]. Some research in elasticity imaging has focused on companding to reduce strain decorrelation [S]-[lO], [16]. It should be noted that the MLE achieves the CRLB asymptotically in the limit that the observation time becomes infinite, but can approach the CRLB for large kernels. Thus, many MLE analyses require that the correlation tinie-bandwidth product is large, or equivalently, that the correlation kernel is much longer than the autocorrelation width of the signal. This assumption is required for many TDE applications in which the signal-to-noise ratio (SNR) is very low. However, ultrasound signals used for elasticity imaging gener- ally have high SNR, so that an increased correlation kernel is not needed to overcome noise in achieving low error variance. In elasticity imaging the tradeoff is rather one of precision versus spatial resolution. A larger correlation kernel will reduce error variance but will also reduce the spatial resolution of the imaging system. Therefore, the MLE may not be the optimal estimator for ultrasound elasticity imaging. To investigate the tradeoff between displacement (and strain) error variance and spatial resolution, in this paper we look at the effect of short correlation kernels, on the order of the autocorrelation width of the ultrasound pulse. The problem is modeled as a time-delay estimation, and results are demonstrated using simulations. All methods were tested using ultrasound data acquired from a set of gelatin phantoms. The results of these studies, detailed here, will show that a short correlation kernel along with filtering of the correlation functions produces a high strain SNR with high spatial resolution. 11. THEORY To gain insight into correlation processing for elasticity imaging, a one-dimensional (l-d) model is developed. Extension to the 2-D case is addressed later. The tradi IEEE

2 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 83 tional estimate of the cross-correlation function computed at time t = to between two signals rl(t) and rz(t) is: R(to, to+t/2 to + T ) = -.I' rt(t)7"2(t+7)4 T to-t/2 (1) where r is the correlation lag, T is the window or kernel length, and * is used to denote complex conjugation for complex signals. These signals, and all others discussed, are assumed to be functions of time t, but in pulse-echo ultrasound they simply are related to the corresponding spatial position y using the transformation y = tc/2, where c is the speed of sound in the medium, assumed here to be constant. Similarly, if the sound speed is constant, then the displacement of the medium, v(t), is simply related to the time delay, wt(t), between the signals ~ l(t) and ~a(t): -1 J I I I I I o Correlation Lag (psec) - Baseband - - Analvtic C w(t) = w.(t)-. 2 The time delay traditionally is estimated by the position of the maximum to + T ) = R(t0, to + T ) to+t/2 to+t/2 + J T?(t)Tl(t) dt + J r;(t + 7 )r2(t + T ) dt to-t/2 to-t/2 The cross-correlation coefficient is used for this estimate instead of the cross-correlation function itself [i.e., just the numerator of (3)] to minimize the effects of speckle fluctuations in r(t). If the correlation function is not normalized, then bright speckle spots will dominate the analysis. Because the ultrasound signal is at radio frequencies (RF), an alternative to peak detection is to use a complex representation of the signals and exploit phase information in the complex correlation. Time-delay estimation using complex signals is demonstrated in Fig. 1, where typical RF, complex baseband] and analytic cross-correlation coefficient functions are shown for a simple l-d deformation. Finding the position of the maximum of the RF correlation (shown as a dotted vertical line) in a sampled system requires peak interpolation. If the RF signal is sampled at reasonable rates, computationally intensive interpolation algorithms or biased estimators are generally required to find the peak [17]. The magnitude of the complex baseband correlation coefficient (or identically the analytic magnitude) matches the envelope of the RF correlation coefficient and will have the same maximum [18]. For complex baseband correlation, the phase (shown in the lower panel of Fig. 1) evaluated at peak lag is proportional to the displacement. Away from the peak lag, the baseband phase is a function of lag and of local frequency characteristics (as seen by the nonflat phase in Fig. 1). Consequently, either the position of the correlation peak must be estimated or the phase must be adjusted to properly estimate displacement [19],[20]. If (3) 6.8 i Correlation Lag (psec) Fig. 1. Typical RF, complex baseband, and analytic cross-correlation coefficient functions. analytic signals are used, the phase of the complex correlation function will cross zero at the peak lag [all, as shown in Fig. 1. Because the phase of the analytic correlation is well approximated by a straight line, the zero crossing can be accurately and quickly computed. The analytic phase crosses zero at multiple positions; therefore, the peak of the cross correlation must still be estimated to identify the desired zero crossing. However, the peak position estimate can be very coarse because it only needs to be within one-half wavelength of the true peak. To describe the characteristics of correlation in ultrasound elasticity imaging, the following signal model was used. Assume that measured ultrasound signals (RF A- lines) can be modeled with additive noise: where sl(t) is the noise-free ultrasound signal prior to deformation, s2(t) is the postdeformation signal, and n1 (t) and nz(t) are noise processes introduced in the imaging system. The random noise processes are assumed to be zero mcan, stationary, and uncorrelated to each other and to the noise free signals. For this I-D model, assume that all tissue motion induced by the applied deformation is strictly in the direction of the ultrasound beam. To further simplify the problem, the time delay, wt(t), corresponding to the tissue

3 84 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 displacement is approximated about the position of estimation to by a Taylor series expansion in which only first order terms are retained: where rto is the time delay at t = to, TO (= Tto ~ goto) is a constant time delay, EO is the instantaneous derivative of the varying time delay at t = to. If the sound speed is constant, then EO is also the instantaneous derivative of the displacement at t = to. If the mechanical properties of the tissue in this region are isotropic and uniform, and the deformation is small, then EO also will be the induced strain. Note in this formulation that strain is assumed to be constant over the correlation kernel. By examining the statistical properties of the correlation estimate using the ultrasound model described in Appendix A, the effects of the parameters used to calculate it can be seen. As shown in Appendix B, assuming sl(t) is wide sense stationary, the expected value of the crosscorrelation function estimate in (I) is given by. where R~I(T) is the autocorrelation function of sl(t), wrect is the window function: (7) and $(t, EO) is a function of the strain and the ultrasound pulse, p(t): where F- denotes the inverse Fourier transform. As shown in Appendix C, the function F(t,~o) describes the average strain-induced distortion of the ultrasound pulse produced by changes in interscatterer distances. If the strain is significant, then the ultrasound pulse is distorted by convolution with $(t, EO). As the strain decreases to zero, i.e., EO --7 0, the window term becomes: For zero strain, using the sifting property of convolution with delta functions, the cross correlation reduces as expected to a time-shifted version of the autocorrelation: E[fi(to,to +.)I = R11(? + TO). 01) Comparing (6) and (II), nonzero strain can be seen to time scale the autocorrelation, which broadens the correlation peak. The peak is further broadened by the low pass filtering characteristics of both the window and distortion functions in (6). In the limit of large electronic SNRs, the variance in the time-delay estimate is inversely proportional to the second moment of the signal cross spectrum (i.e., the Fourier transform of the cross-correlation function) [12], [22]. By the moments theorem of Fourier analysis, the second moment of the cross spectrum is proportional to tlie second derivative of the expected cross-correlation function evaluated at the lag corresponding to peak correlation. This means, as the correlation peak broadens, the second derivative at the peak decreases in magnitude, and the variance in the estimated time delay increases. To minimize the width of the correlation function near its peak value, and thus minimize time-delay error, the effects of strain must be reduced. One way to do this is to time compand, or temporally stretch, the deformed signal prior to Correlation. With nonzero strain, the correlation estimate of (1) is nonstationary because the delay betweeq sigqals varies with time (depth) In traditional time-delay estimation, the nonstationary effects of strain can be lessened or removed by time companding prior to cross correlation [15], [22]-[24]. This technique, sometimes called temporal stretching, also has been applied to elasticity imaging [8]- [lo], [16]. The postdeformation signal can be modified to include a companding term a to compensate for time compression, e.g., r3(t) = 7-2((1 - a)t), (12) and the cross-correlation function computed using the compensated signal. to+t/2 As shown in Appendix B, assuming sl(t) is wide-sense stationary, the expected value of this correlation function estimate is given by: Additionally, the distortion $(t,~g) also tends to a delta function: If the companding compensation is perfect, a = EO, the window term becomes:

4 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 85-0 and the expected value of the correlation function will be independent of the windowing function. However, strain effects have not been completely eliminated, which are more easily seen if R11(~) is described in terms of the ultrasound pulse. From Appendix B, if the positions of the scatterers in the initial distribution are assumed to be uncorrelated, then the expectation of the cross-correlation estimate and the companded cross-correlation estimate with perfect compensation become: *p* (-&) ' where K is a constant depending on the scattering strengths. Both these equations are functions of a shifted version of the ultrasound pulse and a scaled version of the pulse. The main difference is the windowing function in (16). If the companding is not perfect, the correlation function will not be independent of wrect(t). Thus, companding can be viewed as a technique that reduces the effect of the windowing function on the cross-correlation estimate. An alternative is simply to reduce the effect of the windowing function by reducing its length. This is a computationally simpler method to implement than companding, in which both the time-delay and the companding compensation must be searched, and the time-scaled signal recomputed for every independent value of a! [22]. A. Short Tame Correlation From (6) the effect of the windowing function can be reduced if T becomes smaller. In the same way that reducing the strain in (9) causes the window term to approach a delta function, reducing T also produces a near delta function. Obviously T cannot be reduced to zero because the cross-correlation function will broaden due to other terms. As noted earlier, the variance of the time-delay estimate is reduced if the width of the cross-correlation coefficient function [i.e., (3)] is kept as small as possible. As T decreases, the combined effects of the window and distortion terms reduce to a delta function. Consequently, the expected value of the cross-correlation function reduces to a shifted and scaled version of the autocorrelation function, in which the scaling is simply proportional to (1/1 - EO). As the window is reduced below the autocorrelation width of the ultrasound pulse, however, the normalization terms in (3) fluctuate wildly. These fluctuations can increase the variance in the estimated time delay greatly, and even produce peak hopping errors (see below). Thus, for a given pulse, p(t), there should be an optimum window length T such that the time delay variance is minimized. To explore the effect of window length on time-delay estimates, a simple 1-D deformation simulation was devel Time (psec) Fig. 2. Representative time-delay error for simple deformation simulation in constant 1% strain region, comparing large (dashed) and smaller (solid) kernels. oped. The simulation used a Gaussian pulse with center frequency of 5 MHz and fractional bandwidth of 50% at -3 db. The signals and scatterers followed the model described in (Al)-(A5) with 154 scatterers/" uniformly distributed in space (assuming a sound speed of 1,540 m/s) and scatterer strengths IC, uniformly distributed in the range [-1,1]. The random additive noise was modeled as Gaussian noise limited to the bandwidth of the ultrasound signals and scaled to produce a SNR of 40 db. The simulated signals were sampled at 20 MHz and were each 2,048 samples long. The simulated mechanical body consisted of two layers with different elastic moduli, where the first region experienced a (1.0%) constant strain and the second had a 2.0% constant strain induced. The time delay was estimated from the zero crossing of the phase of the analytic cross-correlation coefficient. Two simulations were run using different correlation kernels: the first with T = 1.35 psec and the second with T one-quarter the size (T = 0.35 psec). In Fig. 2, typical time-delay errors are plotted as a function of time for both cases over the 1% strain region. The reduced error variance for the smaller window correlation easily can be seen. The error was quantified by repeating the simulation 100 times using independent realizations both of random noise and of scatterer positions and strengths. The standard deviation of the time-delay error was computed for the 1 and 2% strain regions separately for comparison. In the 1% strain region, the standard deviation was 1.4 nsec for the longer window and 0.83 nsec for the shorter. As expected, the standard deviations in the 2% strain region were higher: 3.0 nsec and 1.5 nsec for the longer and shorter windows respectively. Note in Fig. 2 that the spatial resolutions of the two estimates differ. Thus, in exploring the tradeoffs between window size and variance, the resolution also must be considered. The smaller window gives increased spatial resolution and reduced variance. However, variance reduction is not without cost, as decreasing the window length also I --T

5 86 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 increases the probability of finding a false peak (peak hopping) in the correlation search (i.e., when another peak becomes larger then the true one, the search hops to this false peak). Thus, there is a minimum window size at which false peak errors will dominate. In the simulation above, the search region was restricted to include only the true peak so that the variance due to jitter alone could be examined. When the simulation is repeated using a larger search region, a few false peaks are detected in the 2% strain region. B. Correlation Filtering One method to reduce the probability of peak hopping with small correlation windows and to reduce overall error is to spatially filter the correlation coefficient functions (3) prior to displacement estimation. This filtering should not be confused with filtering the estimates themselves. After a number of correlation coefficient functions have been computed at different positions, temporally (spatially) adjacent correlation coefficient functions are filtered along the time (space) direction. Note that this 1-D filtering is not performed in the correlation lag direction. In other words, each spatially adjacent correlation coefficient function is multiplied by a different weight (i.e., filter coefficient), and the resulting correlation coefficient functions are summed to produce the filtered correlation function. Also, note that this process is not the same as simply filtering the crosscorrelation function itself [i.e., just filtering the numerator of (311. Using the correlation coefficient function from (3), let the filtered correlation coefficient function used for timedelay estimation be defined as: if(t0, to + 7) = T2 (18) -Th/2 h(e)i(to + E, to + E + where J(to + [, to + E + T ) are correlation coefficient functions computed at adjacent positions, and h(z) is the filter function that is nonzero only for ltl 5 Th/2 and normalized such that : / h(t)dt = 1. (19) -Th/z The filtering operation is graphically depicted in Fig. 3. Naturally, filtering spatially adjacent correlation functions will reduce variance at a cost of reduced spatial resolution. Thus, the increased spatial resolution of the smaller correlation window can be used to reduce peak hopping and further lower error variance. The correlation filtering technique was tested with another 1-D simulation, where the discrete equivalent of (18) was used: Nh /b[n, = h[m]j[n + m, n + m m=-nh (20) Correlation Coefficient Filtered Correlation Functions Correlation Filter Coefficient Function Fig. 3. Graphical representation of correlation filtering. The filtering is performed in the time t direction (not the correlation lag T direction), so spatially adjacent correlation-coefficient functions are weighted and summed to produce the filtered correlation coefficient function l Time (psec) Fig. 4. Representative time-delay error for simple deformation simulation in constant 1% strain region, comparing large kernel (dashed) and smaller kernel with correlation filtering (solid). The simulation of Fig. 2 was modified to include filtering of the analytic correlation coefficient functions from the smaller window. The filter was a simple moving average, h[m] = 1/21, with Nh = 10 (Th = 1.05 psec). This filter was chosen to match the spatial resolution of the smaller window estimates to that of the larger window. Typical errors using the filtered correlation are shown in Fig. 4 for the same 1% strain region as Fig. 2, where the error from the longer correlation window is repeated for comparison. region was increased, no peak hopping occurred for filtered correlations. Error reduction clearly can be seen. Additionally, when the correlation search Quantitatively, for the filtered correlation time-delay estimates, the standard deviation of the time-delay error was 0.33 nsec in the 1% region and 0.57 nsec in the 2% region, yet no spatial resolution has been sacrificed compared to a larger correlation window. This is demonstrated in Fig. 5, where the strain is plotted near the step boundary. The strain was computed using a simple difference estimate: where the time difference t2 - tl was 0.65 psec. Fig. 5(a) shows typical strain estimates computed from a single re-

6 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 87...,....'. *E ' 10-14, I,,,,,,,,, o I True strain -- '.:, <- Time (pec) T= 1.35 pec (Ensemble Average) T= 0.35 pec, T, = 1-05 pec (Ensemble Average) Correlation Kernel Length (pec) (4 Fig. 6. Time-delay error standard deviation as a function of weighted correlation kernel length with correlation filter size adjusted to give nearly identical spatial resolution for all kernels. 1 0', r Time (psec) where KT is a constant: TI2 KT = / w(t)dt, (23) -T/2 and w(t) is still zero for It1 > T/2. Using a weighting function that smoothly decreases to zero at the tails, such as a Hanning window, high frequency noise passed by the rectangular window can be reduced. Similarly, the correlation filter in (18) can be a smooth function. C. Optimizing Correlation Kernel and Filter Lengths Fig. 5. Strain estimates at step boundary using large kernel (dashed) and smaller kernel with correlation filtering (solid). (a) Representative single realization and (b) ensemble average of 100 independent realizations. alization of scatterers and additive noise. The spatial resolution of the two techniques is more clearly visible in Fig. 5(b) where the strain estimates have been averaged over 100 independent realizations of scatterers and additive noise. Using the criterion established in [20] and [25], the spatial resolution of the two strain estimates is equivalent. An additional noise reduction method is to use a weighted correlation. In (l), a uniform, rectangular window is used. This equation can be rewritten more generally as: M To gain further insight into the optimal correlation kernel size, a simulation was performed varying the correlation window length. To compare the results, the spatial resolution of displacement estimates was kept constant by simultaneously filtering the correlation coefficient functions with varying filter lengths. Thus, as the length of the kernel was reduced, the filter size was increased. As before, the resolution was estimated using the criterion of [as], and the filter size was set so that the spatial resolution of the displacement was nearly identical (about 0.54 mm or 0.7 psec at c = 1540 m/s) for all kernels. The mechanical body was assumed to be homogeneous, rather than a layered material, and the simulation was repeated for four different strain values: 0.5%, l.0%, 2.0%, and 4.0%. The correlation window was modified so each kernel was weighted using a Hanning window, and the correlation filter was also Hanning shaped. Otherwise, the same parameters as those of Figs. 2 and 4 were used. The standard deviation of timedelay error was computed using 10 different 2,048 sample long A-lines, with independent realizations of speckle scatterers and noise used each time. The results are shown in Fig. 6, where the standard deviation is plotted versus window length. Note that the correlation window length along the abscissa is not the total window size T, but the fullwidth, half maximum (FWHM) of the autocorrelation of

7 88 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY % 2.Ph % - 0.5% Correlation Filter Length (psec) (a) can be compared to the resolution for the rectangular case. For each strain, time-delay error decreases with increasing correlation filter size until it reaches a plateau where the time delay error is nearly constant. Additionally, the strain was computed using (2l), where for each correlation filter the time difference was changed so it equaled the half-width at half maximum of the autocorrelation of the filter. Thus, as the spatial resolution of the displacement decreased, the resolution of the strain estimate was decreased accordingly. In the plateau region of Fig. 7(a), where the time-delay error variance, a:, is nearly constant, the decrease in strain resolution should produce an identical decrease in tlie strain-error variance, 0,". Using (2l), and a simple propagation of error analysis, the strain variance is: O ~ ' ~ l l * t l ~ l ~ t I l l Correlation Filter Length (psec) Fig. 7. Using fixed Hanning weighted correlation (2' = 0.35 psec). (a) Time-delay error standard deviation, and (b) strain signal-tonoise ratio as a function of correlation filter length. the Hanning window so that weighted correlation can be compared to a rectangular window. As expected, the timedelay error increases with increasing strain at any kernel length, but the minimum of each curve gives nearly the same value for the desired window size. This FWHM size was about 0.2 psec (7' = 0.35 psec), approximately the full width at half maximum of the pulse envelope and halfwidth at half maximum of the RF autocorrelation function. To fully benefit from the reduced variance of the smaller correlation kernel, a minimum correlation filter size is required. Another simulation in which the kernel window was fixed and the correlation filter length was varied illustrates this point. The kernel window was a 0.35 psec Hanning window, producing the minimum-time delay error as shown in Fig. 6, and all other simulation parameters were identical. The standard deviation of time-delay error is plotted as a function of the Hanning filter length in Fig. 7(a). Once again the filter length shown is the FWHM of the autocorrelation of the Hanning filter so the results assuming that the estimates at these positions are independent and that the time-delay estimator is unbiased. This strain error reduction produces a linear increase in the strain signal to noise ratio (SNR,): E[&] SNR, -. ff, The tradeoff between spatial resolution and strain SNR is shown in Fig. 7(b), where the SNR, from the simulation used to produce Fig. 7(a) is plotted along the same axis for the same four strain values. The results for 4 and 2% strain are nearly identical so the lines overlap. The slight waviness is due to the discrete size of the difference estima,te of the strain. For larger filter sizes, the linear relationship between spatial resolution and SNR is clear. In the plots above, only four values of strain were shown. In a real elastic body, a continuum of strain values will be present. Therefore, the strain SNR was computed for a number of strain values repeating the simulation used to produce Fig. 7. These simulations also used a 0.35 psec Hanning correlation kernel but kept the Hanning correlation filter fixed at a 1.55 psec. Additionally, the SNR, was computed for the case of a rectangular correlation kernel of length 1.35 psec with no filtering. The length of tlie Hanning filter was chosen so the spatial resolution of the displacement and strain for these two cases were nearly identical. The strain was computed for both kernel sizes using (2l), where the time difference was 0.65 pee. The results are shown in Fig. 8(a) and the increased strain SNR with the smaller, filtered correlation kernel is significant. In fact, these simulation results are comparable to, or greater than, the theoretical maximum SNR, (in which the Cramer-Rao lower bound of error variance is achieved) predicted by Varghese and Ophir [9], [26] for temporal stretching using the same spatial resolution and system parameters. The improvement in strain SNR seen in Fig. 8(a) is most pronounced where error due to strain decorrelation

8 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 89 is much larger than that due to additive noise. The smaller kernel size results in increased additive noise error, but it is more than offset by the decrease in error due to correlation filtering and reduction of strain effects within the kernel. This tradeoff between additive noise and strain decorrelation error is further demonstrated in Fig. 8(b). The strain SNR was recomputed with the same simulation, only changing the additive noise power. The 40 db SNR results from Fig. 8(a) are repeated in Fig. 8(b) for comparison purposes, along with results at 30 db and 20 db SNR. At low strain values, the gain from small kernels with filtering is not as great because the pre- and postdeformation signals are highly correlated. Thus, displacement error due to additive noise may be comparable to that from strain decorrelation, especially at low SNR values. At higher strains, strain decorrelation is the dominant error source, and smaller correlation kernels are more effective. I - Filtered small kernel Large kernel 1, I s 0.; 0 3 Strain (a)... D. Extension to Two-Dimensional Estimation The 1-D methods described earlier can be extended to estimate motion in both the axial (along the ultrasound beam) and lateral (perpendicular to the beam) directions. The correlation function will become 2-D as will the weighted kernel, correlation filter, and search region. The computed position of the correlation peak can be used as a preliminary motion estimate, and the axial displacement can be refined using the zero crossing of the 2-D analytic correlation phase. Unfortunately, lateral displacements cannot be improved using phase zero crossings as there is no lateral carrier frequency. There is a local instantaneous frequency in this direction (near DC) and the phase of the 2-D analytic correlation will pass though zero at the lateral lag corresponding to the lateral displacement. However, because the phase is linear and the slope of the phase is near zero, small changes in slope become larger errors in the lateral position of the zero crossing. More importantly, near the correlation peak, the 2-D analytic phase becomes zero not only at a point, as in the 1-D case, but along a line corresponding to each 1-D peak in the axial direction. Thus, in the 2-D case, an estimate of the lateral displacement is computed from the peak of the correlation function, and the zero crossing of the 2-D analytic correlation at this lateral lag is used to estimate the axial displacement. The statistical theory and simulation results above suggest that a small weighted correlation kernel and larger correlation filter can be used to reduce displacement and strain variance while retaining the same spatial resolution as that from a large correlation kernel. The abilities of complex correlation tracking methods and their extension to 2-D were experimentally tested using gelatin based phantoms. a cn 10- s -._ E : I ;I,' < _--- Filteredsmall kernel Large kernel.. 04' I t Strain (b) Fig. 8. Strain signal-to-noise ratio (SNR): (a) using Hanning weighted kernel (T = 0.35 psec) with Hanning correlation filter (Th = 1.55 psec) (solid) and unweighted, unfiltered kernel (T = 1.35 psec) (dashed); and (b) repeated simulation results using additive SNR values of 40 db (solid), 30 db (dotted) and 20 db (dashed) EXPERIMENTAL METHODS To test short window correlation methods for speckle tracking, experiments were performed on gelatin-based phantoms designed to quantitatively simulate the mechanical and ultrasonic properties of soft tissue. For the results described here, two rectangular phantoms were used, both constructed using 4.2% by weight gelatin (Knox Gelatine, Inc., Englewood Cliffs, NJ) and 0.5% by weight microspheres (Amberlite Strongly Acidic Cation Exchanger, Sigma Chemical Co., St. Louis, MO) with a diameter of 75 to 150 pm acting as ultrasonic scattering centers. Both were 145 mm long by 100 mm wide. A 117-mm tall homogeneous phantom was first constructed. A second 80-mm tall gel was fabricated with two cylindrical holes extending the length of the phantom. The 16-mm and 13-mm diameter holes were backfilled with 8.4% by weight gelatin to simulate circular hard inclusions in the imaging plane. These inclusions had a shear modulus about 2.5 times

9 90 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 larger than the surrounding material, as estimated from independent measurements of the elastic modulus using a gold standard deformation system [27], [28]. The concentration of scatterers in the inclusions was the same as that of the background. A data capture system for elasticity imaging was built using a clinical ultrasound scanner (Advanced Technologies Laboratory, Seattle WA) attached to a custom integrated circuit board. Digital RF data were collected at real time rates and stored on disk for later processing. For the results shown below, a 3%" linear array transducer was used, and RF data were filtered and converted to complex baseband data sampled axially at 20 MHz and laterally at 3.3 A-lines/". The mean power spectrum was approximately Gaussian with a center frequency of 4.7 MHz and a 35% fractional bandwidth at -3 db. The center frequency and bandwidth are slightly less than those assumed in the simulation. Displacements and strains were induced in the phantoms using controlled surface deformation. The ultrasound transducer was placed vertically into a custom holder, whose bottom was a 75-mm by 130-mm rectangular Plexiglas plate. This pad had a beveled hole in the center so the transducer face was at the same level as the bottom of the block. The plate increased the area over which the deformation was applied so the imaging plane more closely approximated a plane-strain state. The probe holder was attached to an unidirectional positioning slider (Velmex, Bloomfield, NY) used to control vertical motion. The correlation was computed using complex baseband signals with a kernel size of 0.8 mm (lateral) by 0.3 mm (axial, 0.39 psec), weighted to approximately equal the point spread function of the system. The kernel size was measured by the FWHM of its autocorrelation. The correlation coefficients were filtered using a 1.5 mm (lateral) by 1.0 mm (axial, 1.3 psec FWHM) separable Hanning filter. Lateral displacements were estimated by fitting the peak of the magnitude of the baseband correlation coefficient to a 2-D second order polynomial and finding the position of its maximum. After converting the baseband correlation about the peak to its equivalent analytic values, the axial displacement estimate was computed using the zero crossing of the correlation evaluated at the lateral lag determined from the previous peak estimation. Strain estimates were computed from axial displacements using a simple l-d difference filter along the axial direction for correlation windows separated by 0.77 mm (1.0 psec). IV. EXPERIMENTAL RESULTS In Fig. 9(c), a B-scan of the homogeneous phantom is shown prior to deformation, displayed over a 32-mm by 106-mm area. The transducer, which was at the top of the image, was positioned in the center of the phantom. For both phantoms, the deformation was applied at the top surface by vertically moving the transducer (and surrounding plate) downward. Thus, all displacements are with respect to the face of the transducer. Additionally, all Fig. 9. Homogeneous phantom results: (a) lateral displacement, (b) axial displacement, (e) B-scan, and (d) normal axial strain.

10 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 91 displacements and strains shown below were computed in the reference frame of the initial transducer position. For the homogenous phantom, a single 2.0 mm displacement was applied at the surface with a preloaded height of 108 mm, resulting in an average applied strain of about 1.9%. Estimated lateral displacements for the homogeneous phantom are presented in Fig. 9(a) over the same area as the B-scan, as are all results for the homogeneous phantom. This image is displayed on a linear gray scale in which full black represents 0.5 mm motion to the left, and full white represents 0.5 mm motion to the right. In Fig. 9(b) the axial displacements also are shown using a linear gray scale in which black represents no motion and white represents 2.0 mm downward motion. The resulting normal, axial strain image is displayed using a linear gray scale in Fig. 9(d) in which zero strain is black and white depicts 3.8% strain. Away from the upper and lower boundaries, the image is mid gray as expected. The strain SNR was estimated to be 8.8 using the sample mean and standard deviation in the middle of the phantom in which the strain was nearly constant. The spatial resolution was estimated by measuring the FWHM of the strain autocovariance in the same region. The axial resolution was 0.8 mm and the lateral resolution was 2.0 mm. The average applied strain in the inclusion phantom also was , with a preload height of 79 mm and a single 1.5 mm surface displacement. In Fig. lo(c), the B-scan for this phantom is shown over a 32-mm by 77-mm area using the same pixel size as that for the homogeneous phantom. Thus the widths of the images are the same, and the difference in image height reflects the difference in phantom height. The transducer was positioned directly over one of the two inclusions, and the bottom of this inclusion can be seen due to slight scatterer settling during phantom construction. In Fig. lo(a) the lateral displacement image is shown. Again a linear gray scale is used where black is 0.25 mm motion to the left and white 0.75 mm to the right. The asymmetric lateral motion is due to the second inclusion, which was just to the left of the image and 40.5 mm from the top of the phantom. The estimated axial displacement is shown in Fig. 10(b) on a linear gray scale over the range 0 to 2.0 mm downward motion (black to white). The corresponding axial strain image, shown in Fig. 10(d), is much more detailed. This image is displayed using the same gray scale as that used for the homogenous phantom (0 to 3.8% strain). The inclusion is clearly visible with a high definition of the boundary. The edge of the second inclusion is just visible midway up on the left side of the image. The strain SNR was estimated to be the same as the homogenous phantom (8.8) in the background material and lower (7.8) in the inclusion, where the strain was reduced. The spatial resolution was measured to be the same as that in the homogeneous phantom. V. DISCUSSION Fig. 10. Inclusion phantom results: (a) lateral displacement, (b) axial displacement, (c) B-scan, anti (d) normal axial strain. Simulations and phantom experiments show that the MLE assumption of large correlation time-bandwidth

11 92 IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS, AND FREQUENCY CONTROL. VOL. 4fi. NO 1. JA~~JARY 1999 products may not be necessary for ultrasound elasticity imaging. Using small correlation kernels, on the order of the autocorrelation width of the ultrasound signal, and correlation filtering, high SNR strain estimates can be computed with high spatial resolution. System parameters used in the 1-D simulations were chosen to match those of other publications so results could be directly compared. Unfortunately, this means that the strain SNR measured in the phantoms cannot be directly compared to simulations due in part to the lower center frequency and smaller bandwidth of the transducer used in the experiments. These effects can be accounted for in 1-D simulations, but the experimental results will still exhibit lower SNR, due to decorrelation in the lateral direction. Just as axial strain reduces coherence due to the scatterer motion, so too do lateral and shear strains, which are ignored in the 1-D simulation. In addition, the correlation window must be larger laterally than axially due to the reduced spatial resolution in the lateral direction, increasing the effect. Nevertheless, the measurements confirm the conclusion of the simulations that small correlation kernels with correlation filtering can minimize the effects of strain decorrelat ion. The strain SNR can be improved by increasing the correlation filter size, or by spatially filtering the estimates prior to strain computation, at a cost of spatial resolution. As discussed earlier, if the correlation filter is large enough, the tradeoff between resolution and SNR, becomes linear. Thus, in previously published results, [6], the strain SNR was high& (30 compared to 8.8), but at a lower resolution (greater than 3 mm by 3 mm versus 2 mm by 0.8 mm). One of the main advantages of this technique over strain compensation by time companding is its simplicity. Because it ib very simple, real-time implementation in hardware has been explored. Temporal stretching, however, is a computationally intensive algorithm because companding compensation must be searched and the time scaled signal recomputed for every value of Q tried [22]. These computations are in addition to correlation calculations. The performance of the small kernel, filtered-correlation estimator may be improved using estimates of the strain to compand the correlation function prior to filtering, as in the deskewed short-time correlator by Betz [22]. However, companding adds additional complexity and, from explorations using 1-D simulations, the improvement is slight, even when the strain is constant and known exactly. Additionally, if nonoverlapping windows are used, as by Betz [22], the resolution of the estimate will be reduced. The displacement and strain estimator presented here is not intended to be the final step in producing an elasticity image. The algorithm is designed to produce estimates suitable for elasticity reconstruction [29]-[31]. The resulting Young's modulus image removes many of the strain artifacts caused by boundary conditions, such as the low strain regions seen in the lower corners of Fig. 10(d) [31]. In general, the complete strain tensor is required for elasticity reconstruction, so 2-D tracking provides the necessary lateral displacements in addition to maximizing the corre- lation. Lateral estimates can be improved greatly prior to reconstruction using incompressibility proccssing [32], [33]. Additionally, multiple displacement arid strain estimates can be accumulated to larger average strains, improving the strain SNR [6]. This also will allow multiple displacement estimates computed at different average strain values to be combined so low arid high strain regions in the image can be processed at different average strains to optimize strain SNR throughout the image [34]. APPENDIX A To describe the characteristics of correlation processing in ultrasound elasticity imaging, the following signal model was used. The measured ultrasound signals were modeled as additive noise processes described by (4), repeated here: where sl(t) is the noise free ultrasound signal prior to deformation, sz(t) is the postdeformation signal, and nl(t) and nz(t) are noise processes introduced in the imaging system. The random noise processes are assumed to be zero mean, stationary, and uncorrelated to each other and to the noise free signals. To model ultrasound backscatter. consider the case where the tissue consists of randomly positioned microscopic scatterers so a 1-D convolution can be used: 2 (A-2) where * denotes the convolution operator, p(t) is the ultrasound pulse, dl(t) is the initial scatterer distribution with scatterer strengths IC, each at depth corresponding to time ~,,l, and b(t) denotes a Dirac delta function. The displaced signal is similarly defined: sz(t) = P(t) * dz(t) 2 (A-3) For this 1-D model, assume that all tissue motion induced by the applied deformation is strictly in the direction of the ultrasound beam. To further simplify the problem, the time delay, ut(t), Corresponding to the tissue displacement is approximated about the position of estimation to by a Taylor series expansion, where only first order terms are retained: where Ttto is the displacement at t = to. TO (= TtTtu +E&) is a constant displacement, EO is the instantaneous derivative

12 LlIBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 93 of the displacement at t = to (assumed to be constant over the correlation interval), T is the integration time of the correlator, and T~~~ is the maximum size of the search region over which the correlation is computed. If the mechanical properties of the tissue in this region are &(to, to + T ) = [; isotropic and uniform. and the deformation is small, then 0 will be the induced strain. The scatterers in this region will be displaced by the applicd deformation, such that their new positions 7,,2, will be: The cross-correlation function for elasticity can be expressed in terms of more traditional time delay estimation: to+t/2 t+t+to - J' si(t)sl ( 1 - Eo ) dt] to-t/2 * $(T, 0) + [noise terms], (A-11) where the terms involving noise processes can be similarly described. 72,2 = T2,l - ~T(TZ,l), (A-5) and using (A3), the postdeforniation signal will be: 2 (A-6) Note that a positive strain will result in a time compression of the scatterer distribution; therefore, time compresses the postdeformation signal. Conversely, a negative strain produces a. timc dilation. This notation is arbitrary and is used here to be consistent with the standard definitions of strain. To relate the elasticity delay estimation problem to traditional time-delay estimation, consider the signal $(t, EO), defined in (8). As shown in Appendix C, the postdeformation signal can thcn be expressed as: Thus, sa(t) can be described as a time delayed and time scaled version of the initial signal sl(t) convolved with a distortion function that dcperids on the ultrasound pulse and the time-delay rate. By defining the window function wrect: APPENDIX B Because the time-delay rate 0 is generally nonzero, the time delay is not constant, and the correlation estimate depends on the evaluation time. Thus, the expectation of the nonstationary cross-correlation function must be obtained by computing the ensemble average rather than a time average. Following Remley [35], let RJ (to, to + T ) denote the jth realization of the ensemble. Using (A9) the expectation is then: E[@(to, to + T)] = 1 E [ T[W'&T - to)t;*(-t)l * 4. (B-1) Substituting (4) and ('28) into (Bl) yields: si (-) 1 - Eo *F(T,&")], where the assumption of no signal-noise or noise-noise correlations forces the signal-noise and noise-rioise terms to zero. Writing the correlation as an integral and placing the expectation inside the integral yields: E[&(to, 1 T to + T)] = -$(T, Eo) * the cross-correlation function in (1) can be described as a convolution: 1 &I. to + T ) = -[W,,,t(-T - to)t;(-t)] * T2(T). T (~-9) To evaluate this expectation, assume that the ultrasound signal is stationary in the statistical sense. If the microscopic scatterers in the region around to are randomly positioned for each realization with the same average scattering strengths, the ultrasound signal will exhibit fully developed speckle characteristics and be stationary in the wide sense [36]. Using the assuniption of stationarity, the expec- This descriptiorl makes it easier to see that $(t, Eo) can lje separat,ed from the time-de1a-y estimation. Combining tation in (B3) simply be the autocorrelation of s1 (t):

13 94 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 Combing the previous equations with the translation of the time axis t + (t + to) yields: Following the same steps and again assuming that sl(t) is wide sense stationary and w(t) is even: E[& (to, to + T)] = +(T, 1 Eo) * E[I&(tO,tO+T)] = 11-a15((1-a)T,Eo)* T 1-a 1-CY i 1 To + Eoto (EO - a)t Wrect ( Eo P- T a ) * Eo t ). (B-11) The integration can be expressed as a convolution using the translation t If the companding compensation is perfect, a = 0, the ---f t/~o, window term becomes: 1 E[@ (to, to + T)] = -$(T, Eo) * - (1 -a). T aieo* Wrect (r) + 6(7), 00 (Eo- a)t Eo - 01) (B-12) 1 - Eo Eo - Eo (B-6) and the expected value of the correlation function will be: R1l + (r + 1 To - Eo ) ' Assuming that w(t) is even so that w(t) = w(-t), and using the definition of TO from (5) yields: Hence, the mean correlation function is independent of the position of evaluation to, assuming that TO and EO are constant throughout the signal. The expectation also is independent of the windowing function. However, strain effects have not been completely eliminated. This becomes more clear if the autocorrelation of sl(t) is described in (to, to + = EO) * -Wrect terms of the ultrasound pulse. If the positions of the scatterers in the initial ultrasound +Tto, p8) scatter distribution dl(t) are assumed to be uncorrelated, R1l (G) then R11(~) becomes: COT (:)* This result differs in form from that of Remley [35] due to the addition of the 5(7,EO) term and because this result is written in the reference frame of the original signal and that of Remley [35] is expressed with respect to the autocorrelation of the shifted signal. From the relationship between the pre- and postdeformation signals described in (CS), the relationship between the two autocorrelation functions can be shown to be: Rll(7) = KdT) *p*(--7), (B-14) where K is a constant depending on the scattering strengths. Substituting (B14) into (B8) and using the definition of F(t, q) from (C1) yields the expectation of the cross-correlation function: E[g(h, to + T)] so the expected value of the correlation function can be expressed in either geometry. The expectation of the time companded correlation function also can be computed. Combining (4), (la), (13), and (C8) in a similar manner as above and again assuming no signal-noise or noise-noise correlations yields: Similar substitution into (B13) returns the expectation of the companded cross-correlation estimator with perfect compensation: E[AA(to,to +.)I = U@((l- Q)T,EO) * T A)] 1 -Eo dt. (B-10) Comparison of (B15) and (B16) shows that both are functions of a shifted version of the ultrasound pulse and a scaled version of the pulse. The main difference is the window function. Thus, companding can be viewed as a technique that reduces the effect of the windowing function on the cross-correlation estimator.

14 LUBINSKI et al.: SPECKLE TRACKING USING SHORT-TIME CORRELATION 95 APPENDIX C To relate displacement and strain estimation in elasticity imaging to classic time-delay estimation, it is useful to define the following function of the ultrasound pulse, p(t): where F-l denotes the inverse Fourier transform. This function follows the analysis of Alam and Ophir [lo]. Consider the postdeformation signal: and its Fourier transform: This frequency domain signal can be rewritten as: To relate this signal to the predeformation one, observe that the Fourier transform of the initial signal is simply: By such that: the postdeformation signal can be expressed as a function of the initial signal in the frequency domain as: and in the time domain as: The signal $(t, EO) can be considered a distortion function produced by strain-induced changes in interscatterer distances. If the strain is significant, the ultrasound pulse is distorted by convolution with 5(t, EO). REFERENCES M. Bertrand, J. Meunier, M. Doucet, and G. Ferland, Ultrasonic biomechanical strain gauge based on speckle tracking, Proc. IEEE Ultrason. Symp., 1989, pp H. Ponnekanti, J. Ophir, and I. Cespedes, Axial stress distributions compressors in elastography: An analytical model, Ultrasound Med. Biol., vol. 18, no. 8, pp , S. Y. Yemelyanov, A. R. Skovoroda, M. A. Lubinski, B. M. Shapo, and M. O Donnell, Ultrasound elasticity imaging using Fourier based speckle tracking algorithm, Proc. IEEE Ultrason. Symp., 1992, pp J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi, and X. Li, Elastography: A quantitative method for imaging the elasticity of biological tissues, Ultrason. Imagtny, vol. 13, pp , A. R. Skovoroda, S. Y. E:melianov, M. A. Lubinski, A. P. Sarvazyan, and M. O Donnell, Theoretical analysis and verification of ultrasound displacement and strain imaging, IEEE Trans. Ultrason., Ferroelect., fieq. Contr., vol. 41, pp , May M. O Donnell, A. R. Skovoroda, B. M. Shapo, and S. Y. Emelianov, Internal displacement and strain imaging using ultrasonic speckle tracking, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 41, pp , May I. Cespedes, M. Insana, and J. Ophir, Theoretical bounds on strain estimation in elastography, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 42, no. 5, pp , M. Bilgen and M. F. Insana, Deformation models and correlation analysis in elastography, J. Acoust. Soc. Amer., vol. 99, no. 5, pp , T. Varghese and J. Ophir, Performance optimization in elastography: Multicompression with temporal stretching, Ultrason. Imaozno. vol. 18. DD [lo] S. k. ilam and J Ophir, Reduction of signal decorrelation from mechanical compression of tissues by temporal stretching: Applications to elastography, Ultrasound Med. Biol., vol. 23, no. 1, pp , [ll] T. J. Hall, M. Bilgen, M. F. Insana, and P. Chaturvedi, Phantoms for elastography, Proc. IEEE Ultrason. Symp., 1996, pp [12] W. E. Walker and G. E. Trahey, A fundamental limit on delay estimation using partially correlated speckle signals, IEEE Duns. Ultrason., Ferroelect., Frey. Contr., vol. 42, pp , Mar [13] G. C. Carter, Coherence and Time Delay Estimation. Piscataway, NJ: IEEE Press, [14] C. H. Knapp and G. C. Carter, The generalized correlation method for estimation of time delay, IEEE?Fans. Acoust., Speech, Signal Processing, vol. ASSP-24, pp , Aug [15] -, Estimation of time delay in the presence of source or receiver motion, J. Acoust. Soc. Amer., vol. 61, no. 6, pp , [16] I. Cespedes and J. Ophir, Reduction of image noise in elastography, Ultrason. Imagin.9, vol. 15, pp , [17] I. Cespedes, Y. Huang, J. Ophir, and S. Spratt, Methods for estimation of subsample time delays of digitized echo signals, Ultrason. Imaging, vol. 17, pp , [18] R. C. Cabot, A note 011 the application of Hilbert transform to time delay estimation, IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, pp , [19] N. A. Cohn, S. Y. Emelianov, M. A. Lubinski, and M. O Donnell, Displacement and strain imaging for an elasticity microscope, Proc. IEEE Ultrason. Symp., 1996, pp [20] -, An elasticity microscope. Part I: Methods, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, pp , Nov [21] T. Loupas, J. E. Powers, and R. W. Gill, An axial velocity es- timator for ultrasound blood flow imaging, based on a full evaluation of the Doppler equation by means of a two-dimensional autocorrelation approach, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 42, pp , July [22] J. W. Betz, Comparison of the deskewed short-time correlator and the maximum likelihood correlator, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp , Apr J. C. Hassab, B. W. Guimond, and S. C. Nardone, Estimation of location and motion parameters of a moving source observed

15 96 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 46, NO. 1, JANUARY 1999 from a linear array, J. Acoust. Soc. Amer., vol. 70, no. 4, pp , [%4] W. B. Adams, J. P. Kuhn, and W. P. Whyland, Correlation compensation requirements for passive time delay estimation with moving source or receivers, IEEE Trans. Acoust., Speech, Szgnal Processing, vol. ASSP-28, no. 2, pp , [25] N. A. Cohn, S. Y. Emelianov, and M. O Donnell, An elasticity microscope. Part 11: Experimental results, IEEE Trans. UZtro.son., Ferroelect., Freq. Contr., vol. 44, pp , Nov [26] T. Varghese and J. Ophir, A theoretical framework for performance characterization of elastography: The strain filter, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, pp , Jan [XI R. Erkamp, P. A. Wiggins, A. R. Skovoroda, S. Y. Emelianov, and M. O Donnell, Gold standard system for reconstructive elasticity imaging, Ultrason. 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O Donnell, Nonlinear estimation of the lateral displacement using tissue incompressibility, IEEE Trans. Ultrason., Ferroelect., Prey. Contr.. vol. 35, pp , [34] M. O Donnell, S. Y. Emelianov, A. R. Skovoroda, M. A. Lubinski, W. F. Weitzel, and R. C. Wiggins, Quantitative elasticity imaging. Proc. IEEE Ultra.son. Symp., 1993, pp [35] W. R. Remley, Correlation of signals having linear delay, J. Acoust. Soc. Amer., vol. 35, no. 1, pp , [36] R. F. Wagner, S. W. Smith. J. hl. Sandrik, and H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Sonics UZtrasm., vol. 30, pp hlay Mark A. Lubinski (S 94) received the B.S. degree in electrical engineering (with biomedical engineering option) in 1990 from Carnegie Mellon University and the M.S. in bioengineering and electrical engineering (systems) in 1993 and 1994, respectively, from the University of Michigan, Ann Arbor. Before beginning his graduate work, he worked as a computer engineer in the Department of Neurophysiology at the Children s Hospital of Pittsburgh. While at the University of Michigan he was a National Science Foundation Graduate Fellow in Bioengineering and a GAANN Fellow. He is currently a graduate student research assistant at t,he University of Michigan, working in the Biomedical Ultrasonics Laboratory, pursuing a Ph.D. in biomedical engineering, and researching ultrasonic elasticity imaging. He is a member of Tau Beta Pi and Eta Kappa Nu. His research interests include signal processing, medical imaging, and motion estimation. Stanislav Emelianov (M 94) was born in May He received the B.S. and M.S. degrees in physics in 1986 and 1989, respectively, froni Moscow State University, and the Institute of Matheniatical Problems of Biology of the Russian Academy of Sciences, Russia. In 1989, he joined the Institute of Mathematical Problems of Biology, where he was engaged in both mathematical modeling of soft tissue biomechanics and experimental studies of noninvasive methods in medical diagnostics based on tissue elasticity variations. Following his graduate work, he moved to the University of Michigan, Ann Arbor, as a post-doctoral fellow in the Electrical Engineering and Computer Science Department working on applications of imaging systems for medical diagnosis and nondestructive testing. Dr. Emelianov is currently a research scientist in the Biomedical Ultrasonics Laboratory at the University of Michigan and involved primarily in the theoretical and practical aspects of ultrasound elasticity imaging. He is the author of several scientific papers. His research interests are in the areas of tissue biomechanics, medical imaging systems, and nondestructive material testing. Matthew O Donnell (M 79-SM 84-F 93) received the B.S. and Ph.D. degrees in physics from the University of Notre Dame, Notre Dame, IN, 1972 and 1976, respectively. Following his graduate work, he moved to Washington University in St. Louis as a postdoctoral fellow in the Physics Department working on applications of ultrasonics to medicine and non-destructive testing. He subsequently held a joint appointment as a Senior Research Associate in the Physics Department and a Research Instructor of Medicine in the Department of Medicine at Washington university. In 1980 he moved to General Electric Corporate Research and Development in Schenectady, New York, where he continued to work on medical electronics, including NMR and ultrasound imaging systems. During the academic year, he was a visiting fellow in the Department of Electrical Engineering at Yale University investigating automated image analysis systems. Most recently, he has worked on the application of advanced VLSI circuits to medical imaging systems, including catheter arrays. In a bold move during 1990, Dr. O Donnell moved to the Elect,rical Engineering & Computer Science Department at the University of Michigan in Ann Arbor, Michigan, where he is currently a Professor. Dr. O Donnell is a member of Sigma Xi, the American Physical Society, and is a fellow of both the IEEE and the AIMBE. He has authored or coauthored over 100 archival publications, including 2 receiving best paper awards, numerous presentations at national meetings, and 40 patents.