application of ultrasound-based velocity estimate statistics to strain-rate estimation

1464 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 50, no. 11, november 2003

Application of Ultrasound-Based VelocityEstimate Statistics to Strain-Rate Estimation

John I. Jackson and Lewis J. Thomas, Member, IEEE

Abstract—Quantification of the relative myocardial de-formation rate, or strain rate, is an emerging capability toaid a cardiologist in assessing myocardial function. Ultra-sound Doppler techniques can be used to compute tissuemotion relative to a transducer. The myocardial strain ratecan be computed as the localized spatial derivative of thetissue velocity. Such a strain-rate estimate is typically nu-merically noisy. We present the relevant speckle statisticsto faciliate the computation of the strain rate based on aweighted least squares regression, with statistically appro-priate weights.

I. Introduction

Strain rate, or relative deformation rate, is a measure ofthe relative motion of a region of an object in compari-

son to the motion of neighboring regions. Thus, strain rateis an appropriate tool for examining the localized effectsof myocardial fiber contraction and elongation that takeplace within a moving, beating heart. Strain rate is thetemporal derivative of strain, which is the relative spatialdeformation of an object. Alternatively and equivalently,strain rate can be considered, and computed as, the spatialderivative of an object’s velocity.

The application of strain to cardiology was proposedby Mirsky and Parmley [1]. Both the strain and the strainrate are quantifiable measures of the myocardial motion.As such, it has been proposed that they may be usefulfor diagnosing conditions such as ischemia and coronaryartery disease [2].

Strain rate has been measured clinically using magneticresonance imaging (MRI) [3]. Diagnostic ultrasound alsocan be used to measure strain rates, using two differentbut related techniques. Both techniques require at leasttwo acquistions of ultrasound data, separated by a shortamount of time. The cross-correlation technique, presentedby Kanai et al. [4], uses statistical cross correlation to tracktissue motion, and the strain rate is the difference in mo-tion between two regions divided by the distance betweenthe regions, and divided by the time between the ultra-sound data acquisitions. The other technique uses the tis-sue velocity computed from Doppler methods, then com-putes the spatial derivative, or gradient, of the velocity.This is generally known as the myocardial velocity gradi-ent (MVG) method, and it has been described by several

Manuscript received March 8, 2002; accepted March 12, 2003.The authors are with Siemens Medical Solutions USA, Inc.,

Ultrasound Division, Mountain View, CA 94039-7393 (e-mail:[email protected]).

Fig. 1. Illustration of strain rate.

authors [5], [6]. In this paper we consider strain rate es-timation that is based on the latter technique, in whichsimple autocorrelation is used to derive the velocity esti-mates [7].

Several articles have been published recently which de-scribe how ultrasound can be used to noninvasively com-pute myocardial strain and strain rates. D’hooge et al. [8]review the different types of strain that can exist withinthree-dimensional anisotropic tissue, describing both nor-mal strain and shear strain. They also review the differenceand relationship between the Lagrangian strain, which isbased on changes from a nominal initial object shape orlength, and natural strain, which is based on changes tothe current object shape or length.

It is the temporal derivative of the natural strain, thenatural strain rate, that is most conveniently computedfor moving myocardium with an ultrasound system. Thisis the strain rate used by Heimdal et al. [9], and this is thestrain rate that will be used throughout this paper.

The one-dimensional tissue strain rate is defined by:

ε̇ =dε

dt≈ v(d + ∆d) − v(d)

∆d, (1)

where d is the spatial position where the first velocity ismeasured, and ∆d is the spatial distance between velocitysamples. Strain rate is the spatial derivative of the tissuevelocity, and it is illustrated in Fig. 1. The tissue veloc-ity typically is measured in the direction of an ultrasoundline, and it is most convenient if the anatomy is alignedcorrespondingly. The spatial derivative then can be takenappropriately along the same direction. Angle correction,for measuring strain rates that are not along an ultrasoundline, has been considered by others [10].

As the spatial derivative of the velocity estimate, thestrain-rate estimate is very susceptible to even small varia-tions or errors in the velocity. Consider the velocities mea-sured from an interventricular septum, shown in Fig. 2.By using only the velocities of adjacent sample points(∆d = 0.11 mm), the estimated strain rate shown in Fig. 3is obtained. It is clear that the error in the strain-rate es-

0885–3010/$10.00 c© 2003 IEEE

jackson and thomas: velocity estimate statistics for strain-rate estimation 1465

Fig. 2. Example of the velocity estimate along an ultrasound line.

Fig. 3. Strain-rate estimate obtained from adjacent velocity esti-mates.

timate shown in Fig. 3 is too large to derive much, if any,useful information. The error can be reduced by increasing∆d, at the cost of decreased spatial resolution. An exam-ple with ∆d = 10 mm is shown in Fig. 4. Alternatively,the estimate can be low-pass filtered to reduce the localvariations. Fig. 5 shows the strain rate obtained using a∆d of 5 mm, followed by a spatial 5 mm rect, or uniform,low-pass filter. Using this method, each strain-rate sam-ple is a function of all of the velocities that occur within arange of 10 mm. A further improved strain rate estimate isgiven by the slope of the best fit regression line, computedfor the velocities along each ∆d range of data, in whichthe best-fit line is computed using the conventional leastsquares regression. The result is shown in Fig. 6. The ob-jective of this paper is to improve on this least squares re-gression estimate by introducing an appropriate weighting

Fig. 4. Strain-rate estimate using velocity estimates separated by10 mm.

Fig. 5. Strain-rate estimate using velocity estimates separated by5 mm and low-pass filtered over 5 mm.

factor based on the statistics of the errors in the velocityestimates.

The errors in the strain-rate estimates are the directresult of errors in the velocity estimate. The velocity es-timate, v̂, is equal to the true velocity of the tissue at apoint in space, plus errors (noise) introduced from varioussources:

v̂ = v + nq + ns + nt + nc. (2)

The noise sources include quantization of the signal phase,nq; random phase associated with the moving speckle, ns;thermal and quantization noise in the receiver and beam-former, nt; and acoustic clutter, nc, or signal from acoustictargets other than the desired tissue.

In this paper, we will consider the statistics of nq andns, assuming these terms to be the most significant sources


Fig. 6. Strain-rate estimate using least squares estimation, each gen-erated from 10 mm of velocity data.

of error when the signal to thermal-noise ratio (SNR) ishigh and acoustic clutter is minimal. We also will assumethat the velocity estimate is obtained from the differencein phase between two acoustic signals, which correspond tothe same location in space and are obtained at two pointsin time, separated by a pulse repetition interval (PRI). Wethen will use the resulting statistics to derive an improvedestimate of the strain rate, using an appropriately weightedleast squares regression.

II. Methods

Consider the process of estimating the tissue velocity.The ultrasound signal is of the form:

x(t) = A(t) cos[2πfct + φ(t)]. (3)

The signal is quadrature demodulated, as shown in Fig. 7,to obtain:

y(t) = A(t) exp[−iφ(t)]. (4)

The signal from one location in space is measured twice,separated in time by the PRI. The first-lag autocorrelationfor the two samples is simply:

R1 = y(0)y∗(1), (5)

and the velocity estimate is given by:

v̂ =c · arg(R1)4πfc · PRI

, (6)

where c is the speed of sound in the medium, fc is thecenter frequency of the ultrasound signal, and PRI is thetime interval between the data samples. The arg is typi-cally computed over a range of −π to π, but the range canbe adjusted to any 2π interval.

Fig. 7. Obtaining the complex ultrasound signal for velocity estima-tion.

A. Quantization Noise

Consider first the effect of quantizing the phase,arg(R1), to N levels. For example, the Acuson SequoiaTM

ultrasound system (Siemens Medical Solutions USA, Inc.,Ultrasound Division, Mountain View, CA) quantizes phaseto 6 bits, or 26 = 64 levels. The error introducedby this phase quantization is uniformly distributed over(−π/N, π/N) radians. The standard deviation of a uni-form distribution is the width of the distribution dividedby

√12, thus:

σq =2π

N· 1√

12(7)

=π

N√

3. (8)

Using (6) with c = 1540 m/s to convert from radians tovelocity in [m/s], and it follows that the quantization errorstandard deviation, scaled in m/s, is given by:

σq =1540

4πfc · PRI· π

N√

3(9)

=222

N · fc · PRI. (10)

Eq. (10) shows that the quantization noise will be re-duced for increased PRIs or increased frequency. For mo-tion of a given velocity and for a given frequency, a prac-tical upper bound on the PRI is limited by the Nyquistsampling criteria, which will introduce velocity uncertain-ties based on 2π phase uncertainties.

B. Speckle Noise

Consider now the significantly more interesting problemof determining the statistics of correlated speckle. Specif-ically, consider the signal from tissue to be fully devel-oped, coherent speckle, which has been generated by alarge number of point scatterers located at random loca-tions over the distance of at least the size of the systempoint-spread function, or system impulse response. Sucha signal is considered to be a complex Gaussian random


variable with independent and equally distributed real andimaginary components. Let the signal from a location inspace d, measured at time t, denoted sd(t), be character-ized as a complex random variable, x = xI + ixQ, from aprobability density function (pdf) given by:

f(x) = fx(xI) · fx(xQ) (11)

=1

2πσ2 exp

[−1

2

(x2

I + x2Q

σ2

)], (12)

where σ is the standard deviation of both the real andimaginary components of the random variable.

Assume that, a time τ later, the tissue has moved adistance δ with an average velocity v. Let vr be the velocityin the direction of the ultrasound beam. The signal thatwill now be returned from the same location in space issd(t + τ), which is the signal from the tissue that hadbeen at location d−δ, which has moved and “accumulatedphase” based on its change in distance from the transducer(as measured along the direction of the ultrasound beam):

sd(t + τ) = sd−δ(t)eikvr ·τ , (13)

or

sd(t + τ) = [sd(t) + n(δ)] eikvr ·τ , (14)

where k equals 2π divided by the wavelength of the centerfrequency of the signal.

The term n(δ) reflects the change in the speckle signalbetween two locations separated by a distance δ. Becausethe velocity is computed based on the difference in phasebetween sd(t + τ) and sd(t), any phase difference betweenthese two points that is due only to speckle statistics willresult in an error in the velocity estimate. Assume that thetwo locations have equal statistical properties (equal val-ues of σ), that their real components are correlated withcorrelation coefficient ρ, and that their imaginary com-ponents also are correlated with the same correlation co-efficient. The correlation coefficient is a function of thethree-dimensional system point-spread function, evaluatedat the three-dimensional location δ. For reference, considertwo real random variables, x and y, each normally dis-tributed with mean 0, variance σ2, and correlated withcorrelation coefficient ρ. Their joint pdf is given by:

f(x, y) =1

2πσ2√

1 − ρ2exp

{−1

2

[x2 − 2ρxy + y2

σ2 (1 − ρ2)

]}.(15)

Now consider the situation in which x, ρ, and σ are known,but y is not. In this case, f(y|x) has a normal distributionwith mean ρx and variance σ2(1 − ρ2):

f(y|x) =1√

2π σ√

1 − ρ2exp

{−1

2

[(y − ρx)2

σ2 (1 − ρ2)

]}.(16)

1. The Problem: Changes in Phase from Changes inSpeckle: The tissue velocity will be estimated from twosamples of a complex ultrasound signal, denoted sd(t) andsd(t + τ). The signal sd(t + τ), as shown in (14), is differ-ent from the first signal because it is from a different, butcorrelated, target and because of phase accumulation re-sulting from tissue motion along the direction of the ultra-sound beam. After both signals have been measured, thequestion to be answered is: “How much uncertainty is therein the phase because the signals come from slightly differ-ent, but correlated, locations in a field of speckle?” Putanother way, the problem is to determine the pdf for thedifference in phase between sd(t) and sd(t)+n(δ). For nota-tional convenience, let sd(t) = xI + ixQ = rx exp(iθx) andlet sd(t) + n(δ) = yI + iyQ = ry exp(iθy). Let θ = θy − θx.Mathematically, the problem is to find f(θ|rx, ry).

We will assume that the values of σ and ρ are knowna priori. In practice, they can be estimated from the localdata.

2. The Solution: The Correlated Speckle Phase Distribu-tion: We start by applying (16) to the real and imaginaryrandom variables (xI , xQ, yI , and yQ) and concluding:

f(yI , yQ|xI , xQ) =1

2πσ2 (1 − ρ2)·

exp{

−12

[(yI − ρxI)2 + (yQ − ρxQ)2

σ2 (1 − ρ2)

]}. (17)

Using yI = ry cos(θy), yQ = ry sin(θy), and the coordi-nate transform Jacobian of 1/ry, (17) can be rewritten as:

f(ry, θy|xI , xQ) =ry

2πσ2 (1 − ρ2)·

exp

{−1

2

[r2y −2ryρ(xI cos θy +xQ sin θy)+ρ2

(x2

I +x2Q

)σ2 (1 − ρ2)

]}.

(18)

We are interested only in the phase of sd(t) + n(δ) rel-ative to the phase of sd(t). Therefore, it is convenient touse θ = θy − θx. And (18) then can be rewritten as:

f(ry, θ|xI , xQ) = f(ry , θ|rx) (19)

=ry

2πσ2 (1 − ρ2)·

exp

{−1

2

[r2y − 2ρrxry cos θ + ρ2r2

x

σ2 (1 − ρ2)

]}. (20)

Now recall that:

f(θ|rx, ry) =f(ry , θ|rx)f(ry |rx)

. (21)


Fig. 8. The correlated speckle phase distribution.

The denominator of (21) is computed by integrating (20)over θ:

f(ry|rx) =ry

2πσ2 (1 − ρ2)exp

{−1

2

[r2y + ρ2r2

x

σ2 (1 − ρ2)

]}·

∫ π

−π

exp[

ρrxry cos θ

σ2 (1 − ρ2)

]dθ. (22)

The integral is simplified by recognizing the integraldefinition of I0, the zero order modified Bessel function ofthe first kind, defined by:

I0(α) =1π

∫ π

0eα cos θdθ, (23)

and by defining:

α =ρrxry

σ2 (1 − ρ2). (24)

It is sometimes convenient to compute α in terms ofthe mean of the (Rayleigh distributed) magnitude of thespeckle, µ, rather than the standard deviation, σ, of thereal part (or of the imaginary part) of the complex signal.Because µ = σ

√π/2, an alternate definition for α is:

α =πρrxry

2µ2 (1 − ρ2). (25)

Using (23), the integral term in (22) is equal to 2πI0(α),and it follows that:

f(θ|rx, ry) =1

2πI0(α)eα cos θ. (26)

We refer to (26) as the relative phase distribution ofcorrelated bivariate-Gaussian random variables, or, moresimply, the correlated speckle phase distribution. The dis-tribution is shown in Fig. 8 for several values of α. Thevariance of the distribution is shown in Fig. 9. The asymp-

Fig. 9. Variance of the correlated speckle phase distribution.

totic variance values of π2/3 and 1/α also are included inFig. 9. For values of α less than 0.5, the distribution isclose to uniform, in which case essentially no reliable in-formation can be inferred about the phase. For values ofα greater than 2, the distribution is similar to a Gaussiandistribution with a standard deviation of 1/

√α. This is

readily derived from using a two-term Taylor series expan-sion for the cosine term in the exponent of the distribution.For values of α greater than 10, the distribution is visuallyvirtually indistinguishable from a Gaussian. Thus,

f(θ|rx, ry) ≈ 12π

, for α < 0.5, (27)

and

f(θ|rx, ry) ≈√

α

2πe− 1

2 αθ2, for α > 2.

(28)

The correlated speckle phase distribution is intuitivelyconsistent. If the two Doppler signals are totally uncorre-lated, then ρ is 0, and thus α is 0, and the distributionis uniform. This is consistent with the expectation thatthe relative phase between the two Doppler signals will betotally random, and thus no information can be inferredabout the velocity of the moving speckle. If the Dopplersignals are well correlated (ρ close to 1) and both signalsare relatively strong (rx/σ and ry/σ are large), then α islarge and there is relatively little error in the phase intro-duced by the statistics of the moving speckle. However, ifeither of the signals is small, corresponding to being neara “null” in the speckle, then α will be small and the phasewill be close to uniformly random.

3. The Correlated Speckle Product Distribution: Tounderstand the practical implications of the correlatedspeckle phase distribution, it is important to know whatvalues of α are typical [see (24) or (25)]. In this section, wederive the probability density function of α. We begin by


finding the distribution of rxry . Recall that rx is Rayleighdistributed:

f(rx) =rx

σ2 exp[−1

2

(rx

σ

)2]

, (29)

and the distribution of ry given rx is given by (see Pa-poulis [11], equation 6-52):

f(ry |rx) =ry

σ2 (1 − ρ2)I0

[ρrxry

σ2 (1 − ρ2)

]·

exp

{−1

2

[r2y + ρ2r2

x

σ2 (1 − ρ2)

]}.

(30)

Because f(rx, ry) = f(rx)f(ry|rx) and the distributionof z = rxry is:

fz(z) =∫ ∞

0

1ry

· frx, ry

(z

ry, ry

)dry , (31)

it follows, after a little algebra, that the distribution of zis given by:

fz(z) =z

σ4 (1 − ρ2)I0

[ρz

σ2 (1 − ρ2)

]·

∫ ∞

0exp

{−1

2

[z2

r2yσ2 (1 − ρ2)

+r2y

σ2 (1 − ρ2)

]}dry

ry.(32)

The integral is conveniently found in Gradshteyn andRyzhik [12], equation 3.471.9, using a substitution ofx = r2

y. The integral incorporates the zero order modi-fied Bessel function of the second kind, K0, which can bedefined by:

K0(x) =∫ ∞

0

cosxt√t2 + 1

dt (x > 0). (33)

Thus,

fz(z) =

z

σ4 (1 − ρ2)I0

[ρz

σ2 (1 − ρ2)

]K0

[z

σ2 (1 − ρ2)

]. (34)

The final step is to convert from z to α, where α = βz [see(24)], recalling that:

fα(α) =1

|β| fz

(α

β

). (35)

From (34) and (35) it follows that:

fα(α) =α

(1 − ρ2

)ρ2 I0(α)K0

(α

ρ

). (36)

We refer to (36) as the distribution of the scaled productof the magnitude of correlated bivariate-Gaussian randomvariables, or, more simply, the correlated speckle productdistribution. This distribution is plotted in Figs. 10 and11, using different scales.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

α

ρ = 0.80

0.85

0.90

0.95

0.98

Fig. 10. The correlated speckle product distribution.

Fig. 11. The correlated speckle product distribution (note the loglogscale).

Fig. 12. The cumulative density function (CDF) of the correlatedspeckle product distribution.


The values of α for which the cumulative density func-tion (CDF), Fα(α) =

∫ α

0 fα(τ)dτ , is equal to 0.05, 0.25,0.50, 0.75, and 0.95 are shown in Fig. 12. The graph inFig. 12 can be used to infer, for example, that for ρ = 0.9,50% of the values of α will be less than 6, and hence 50%will be greater than 6.

C. Weighted Least Squares Strain Rate Estimation

The motivation to compute the statistics of the quanti-zation noise and of the change in phase from slightly decor-related speckle was to determine the variance in the errorof the velocity estimate, so that a statistically sound esti-mate of the strain rate could be computed. Using the errorvariances derived in the previous sections, it is straightfor-ward to estimate the strain rate using a weighted leastsquares regression. To minimize the expected regressionerror, the weights are set equal to the inverse of the es-timated variance of the error of each velocity estimate.Thus, the regression equation to be solved is:

Wy = WAx, (37)

where W is a diagonal matrix with element wii inverselyproportional to the variance of the error of the ith mea-surement; y is an N × 1 vector of velocity estimates; A isan N × 2 matrix, with 1’s in the first column and the po-sition of the velocity estimate in the second column; andx is a 2×1 vector, whose second element is the strain rateto be estimated.

To determine the value of wii, consider the variance ofthe various components of the error. For example, using64 quantization levels, the standard deviation of the phasequantization error is π/(64

√3) = 0.028 radians (or 1.6

degrees). Now consider the relative phase of very well cor-related speckle, with, for example, ρ = 0.99. For ρ = 0.99,the median value of α is a little more than 50 (see Fig. 12).The variance of the error in the phase with α = 50 is1/50 or 0.02 radians2, so the standard deviation of thephase error is

√0.02 = 0.14 radians or 8.1 degrees. Thus,

even for well correlated speckle, the phase error from theslight speckle decorrelation typically will dominate overthe quantization error. If we also assume a high signal tothermal noise ratio, and that acoustic clutter is minimal,then the speckle decorrelation can be considered the dom-inant component of the velocity error.

Now, although the exact variance of the correlatedspeckle phase distribution is a complicated function of α(see Fig. 9), the results of Section B, 3 indicate for ρ > 0.9,most of the α’s will be greater than 5, and thus will have anerror variance very close to 1/α. This is particularly con-venient because it implies that an appropriate value forthe weighting term wii would be proportional to α. Thespecification of wii can be further simplified by assumingthat the cross correlation of the signals and the mean sig-nal level are relatively constant within the analysis region,∆d. If these assumptions are reasonable, then we assume

Fig. 13. The velocity estimate and the scaled product of the ampli-tudes of the signals used to generate that velocity estimate.

Fig. 14. Strain-rate estimated using unweighted and weighted regres-sions.

that ρ/[σ2(1 − ρ2)] is constant for each region of each ul-trasound frame in which the strain rate is to be computed.With this assumption, it follows that:

σ2s ∝ 1

rxry, (38)

and thus a reasonable weighting for each velocity estimateis simply the product of the amplitudes of the two originalsignals:

wii = rxry. (39)

III. Results

The effect of the amplitude of the ultrasound signals onthe accuracy of the velocity estimate is clearly shown in


Fig. 15. Strain-rate image derived using an unweighted regression.

Fig. 16. Strain-rate image derived using a weighted regression.


Fig. 13. Note how the discontinuities and apparent errorsin the velocity estimates all occur where the product of thesignal amplitudes is relatively small. Conversely, in regionsin which the signal amplitudes are relatively large, the ve-locity estimate is relatively well behaved. By incorporatingthe signal-amplitude information [using (39)] into the re-gression (37), we obtain the strain-rate estimate shown inFig. 14. For comparison, Fig. 14 also shows the strain-rateestimate obtained using an unweighted least squares re-gression. As before, 10 mm of velocity data were used togenerate each sample of the strain-rate estimate.

The effect of the statistical weighting also is seen bycomparing Figs. 15 and 16, which show the strain ratethroughout the cardiac cycle for the interventricular sep-tum of a healthy, 7-year-old male. The data was obtainedusing an Acuson Sequoia C256�echocardiography systemwith a 3V2c transducer, using an apical four-chamberview. The harmonic signal returned at 3.5 MHz was ac-quired using custom hardware and was processed off-line.The PRI was 1600 µs. As in the previous examples, eachstrain-rate estimate was based on 10 mm of velocity infor-mation. Using a smaller range would improve the responseto locally varying changes in strain rate, but it also wouldincrease the error throughout the image. Fig. 15 uses astandard linear regression, without any weightings. Fig. 16used identical input signal, plus the inclusion of the weight-ing term. The increased quality of the resulting image isreadily apparent. In particular, note the improved homo-geneity of the strain-rate estimates over time. For example,note the significant decrease in the number of green “is-lands” in the midst of the contracting yellow regions andthe expanding blue regions.

The reduction in estimation error in Fig. 16 relative toFig. 15 also can be quantified. Although some regionalvariations in strain rate are expected, one approach tomeasuring the accuracy of the strain-rate estimates is tosubtract the mean of the strain-rate estimate along theentire line of interest from the estimates at each locationand compare the standard deviation of the residuals. Usingthis technique, the residuals in the unweighted regressionhave a standard deviation of 0.5366s−1, and the residualsof the weighted regression have a standard deviation of0.4074s−1, which is a reduction of 24%. Alternatively, wecan subtract the mean of the strain rate estimate, followedby a subtraction of the strain-rate estimates from consec-utive frames of ultrasound data. This will highlight ran-dom frame-to-frame variations in the strain-rate estimate.Using this method, the residuals from the unweightedregression have a standard deviation of 0.6075s−1, ver-sus 0.4280s−1 for the weighted regression, or a reductionof 30%.

IV. Conclusions

We have presented the statistics associated with aphase-based velocity estimate obtained from two ultra-sound measurements, and we have shown how these statis-tics can be appropriately incorporated into an estimate of

strain rate. The resulting strain-rate estimate is relativelyimmune to errors in the velocity estimate that are concur-rent with nulls in the speckle in the ultrasound signal.

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John I. Jackson was born in Salt Lake City,UT, on October 24, 1962. He received his B.A.degree in mathematics from the University ofUtah, Salt Lake City, UT, in 1985, and hisM.S. and Ph.D. degrees in electrical engineer-ing from Stanford University, Stanford, CA,in 1986 and 1992, respectively.

Following his graduate work developingdata acquisition pulse sequences and imagereconstruction methods for magnetic reso-nance angiography, Dr. Jackson joined Acu-son Corporation (now Siemens Medical Solu-

tions USA, Inc., Ultrasound Division, Mountain View, CA) in 1991.While with Acuson and Siemens, he has worked to improve the di-agnostic imaging performance of several echocardiography systems.He currently works within the Ultrasound Innovations Group. Hehas authored or coauthered 17 technical papers and holds 19 U.S.patents.


Lewis (Tom) Thomas (M’97) received hisPh.D. degree in physics from WashingtonUniversity in St. Louis, MO, where he in-vestigated tissue characterization under JimMiller.

He held several positions at the GE Re-search Center in Schenectady, NY, where hisresearch covered both nondestructive evalu-ation (NDE) and diagnostic imaging withultrasound as well as high-resolution x-rayimaging for NDE. Since 1999 Dr. Thomas hasfocused exclusively on diagnostic ultrasound

imaging while at the Ultrasound Division of Siemens Medical Solu-tions, Mountain View, CA, where he is now the Director of Ultra-sound Innovation.

His work has included coded excitation schemes, adaptive flowimaging algorithms, and novel beamforming techniques. All of hiswork is motivated by applying information theory and statisticalanalysis to optimize signal-processing algorithms. He is an inventoron 23 issued patents and an author on 11 archival publications.

application of ultrasound-based velocity estimate statistics to strain-rate estimation

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