REGULAR PAPER

Japanese Journal of Applied Physics 54, 07HF03 (2015)
http://dx.doi.org/10.7567/JJAP.54.07HF03

Compensation technique for the intrinsic error in ultrasound motion estimation
using a speckle tracking method
Hirofumi Taki1*, Makoto Yamakawa2, Tsuyoshi Shiina3, and Toru Sato4
1

Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
Advanced Biomedical Engineering Research Unit, Kyoto University, Kyoto 606-8501, Japan
3
Graduate School of Medicine, Kyoto University, Kyoto 606-8501, Japan
4
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
E-mail: taki@ecei.tohoku.ac.jp
2

Received November 27, 2014; accepted March 18, 2015; published online June 8, 2015
High-accuracy ultrasound motion estimation has become an essential technique in blood ﬂow imaging, elastography, and motion imaging of theheart wall. Speckle tracking has been one of the best motion estimators; however, conventional speckle-tracking methods neglect the effect of outof-plane motion and deformation. Our proposed method assumes that the cross-correlation between a reference signal and a comparison signal
depends on the spatio-temporal distance between the two signals. The proposed method uses the decrease in the cross-correlation value in areference frame to compensate for the intrinsic error caused by out-of-plane motion and deformation without a priori information. The root-meansquare error of the estimated lateral tissue motion velocity calculated by the proposed method ranged from 6.4 to 34% of that using a conventional
speckle-tracking method. This study demonstrates the high potential of the proposed method for improving the estimation of tissue motion using an
ultrasound speckle-tracking method in medical diagnosis. © 2015 The Japan Society of Applied Physics

1.

Introduction

Motion estimation has been employed in several ultrasoundtechniques such as blood ﬂow imaging,1–4) elastic imaging,5–8) phase aberration correction,9,10) and assessment ofcardiac function.8,11–15) Various motion estimation methods
based on non-Doppler techniques have been reported; they
are primarily classiﬁed into two tracking algorithms, speckle
tracking and elastic image registration. A speckle tracking
method calculates the similarity between the ultrasound dataof a reference window in one frame and those of comparison
windows in another frame. The shift between the reference
window and the best-match comparison window indicates thetissue movement at the reference-window position.16–19) In
contrast, an elastic image registration method deforms awhole image of one frame to adjust that of another frame.
After the deformation, the tissue movements at all positions
are estimated.7,20–24) Both of these methods are considered tobe accurate and reliable motion estimators. Heyde et al. havecompared these two algorithms for two-dimensional (2D)
myocardial motion estimation.25) Additionally, some researchers have reported other motion estimation methods
using a 2D measurement plane.26,27) However, several studies
have reported the diﬀerences between the results calculatedusing the 2D and 3D tissue motion estimators.28–32) Previous
studies have indicated that the diﬀerences might be caused by
the intrinsic error involved in the estimation of 3D tissue
motion in a 2D measurement plane.31,33)
The purpose of the present study is to investigate theintrinsic error involved in estimating 3D tissue motion in a
2D measurement plane using a conventional speckle-tracking
method and to propose a compensation method for the error
without a priori information. That is, our purpose is to
propose a method that compensates for the eﬀect of the outof-plane (elevational) velocity component on tissue velocity
estimation without the measurement of the out-of-plane
velocity component. Our previous study investigated adecrease in the cross-correlation as a result of introducing
several assumptions in a simulation study under a limited
condition.34) In our present study, we demonstrate thisdecrease in the estimation accuracy of tissue motion
calculated by a speckle tracking method and propose acompensation method that introduces one or two realistic
assumptions. Furthermore, we investigate the performance ofthe proposed compensation method in simulation andexperimental studies.
2.

Materials and methods

The above decrease in the cross-correlation between areference window and a comparison window has previously
been investigated in studies on atmospheric radar observations.35–37) In this section, we ﬁrst explain the intrinsic error
involved in a speckle tracking method that estimates 3Dtissue motion in a 2D measurement plane. Next, we verify
that the intrinsic error caused by the axial tissue velocity is
negligible. We then propose a compensation technique forthe intrinsic error involved in using a speckle tracking
method caused by out-of-plane motion. We account for theeﬀect of deformation on the cross-correlation. We deﬁne thecross-correlation coeﬃcient using ultrasound RF data, which
is one of the most established motion estimators. This meansthat we do not use the envelope cross-correlation. We thendescribe the simulation and experimental settings.
2.1 Intrinsic error of a speckle tracking method using a 2D
measurement plane

Generally, a speckle tracking method deﬁnes a reference
window in a frame and comparison windows in anotherframe. The ultrasound data of the reference window iscompared with those of comparison windows, and the tissue
motion at the reference-window position is estimated fromthe shift between the reference window and the best-matchcomparison window.38,39) Typically, the ultrasound data in asingle scan line are acquired in a transmit event. This meansthat ultrasound data in a frame are never acquired simultaneously, as emphasized in the current study, because thediﬀerence in measurement time between adjacent scan lines
in a frame is equal to the pulse-repetition time. Figure 1
shows the eﬀect of out-of-plane motion and deformationon the estimation of tissue motion using a speckle tracking

07HF03-1© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

Fig. 1. Forward shift in the measurement time of the best-matchcomparison window estimated using a speckle tracking method under thecondition of out-of-plane tissue motion and deformation, caused by thedecrease in the cross-correlation between the ultrasound data of a reference
window and that of a comparison window with the passage of time. The
measurement time τ = 0 is the acquisition time of a reference signal.

method. The existence of out-of-plane motion and deformation decreases the cross-correlation between the data of areference window and that of a comparison window with thepassage of time. Therefore, a larger decrease occurs at thecross-correlation value in a scan line measured at a latertime point. Because the eﬀect of out-of-plane motion anddeformation depends on the time passage from the acquisition time of a reference signal, the cross-correlation
decreases during the passage of time from the acquisition
time of a reference signal (Fig. 1). The conventional speckletracking method assumes that the cross-correlation curve hasa peak at the measurement time of τI, where τI is determinedby the lateral tissue motion. However, the decrease caused
by out-of-plane motion and deformation increases with thepassage of time. Therefore, this decrease will cause an
unexpected forward shift in the best-match time τM from theideal best-match time τI, where τI is measured when there isno out-of-plane motion and no deformation. Consequently,
the velocity estimated by a speckle tracking method in the
beam-scan direction vMx is smaller than the true tissue
velocity in the beam-scan direction vx.34) It should be notedthat ∣vMx∣ > ∣vx∣ when vx < 0.
2.2 Effect of the axial tissue velocity on a speckle tracking
methodA conventional speckle-tracking method searches for the
best-match comparison window in the measurement plane,
as shown in Fig. 2. The conventional method estimates thelateral and axial tissue motion velocities by using the lateraland axial distances (i.e., ξ and ξA) between the position of
the reference window and that of a comparison window,
respectively. This conventional method neglects the eﬀectof the out-of-plane distance because it only accounts forthe eﬀects of the lateral and axial distances on the crosscorrelation.
The out-of-plane distance between a reference window anda comparison window increases with the passage of time, as
shown in Fig. 3. In a case where out-of-plane motion exists,
there is no deformation within the window, and when aFig. 2. Schema of a conventional speckle-tracking method using a 2D
measurement plane. IS is the scan line interval. Each broken square denotes
the beam of each scan line.

Fig. 3. Schema for the comparison based on a speckle tracking method
using a 2D measurement plane. vx, vy, and vz are the lateral, out-of-plane(elevational), and axial tissue velocity components, respectively.

circular ultrasound beam spot is employed, the expectationof a cross-correlation value should depend on two parameters: the axial distance dA and horizontal distance dH between
the position of the reference window on the tissue at thecomparison time and the position of the comparison window.
In the present study, we deﬁne the beam spot in ameasurement range as the region where the echo intensityfrom a point scatterer in the region is larger than 25% of thepeak echo intensity from a point scatterer. The horizontal
distance is the distance perpendicular to the axial direction,
as shown in Fig. 3. The axial and horizontal distances are
expressed by Eqs. (1) and (2), respectively, as follows:



 c
nT cTPR ;ð1Þ
dA ¼ jvz   A j ¼   vz  22 
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dH ¼ ðvx   Þ2 þ ðvy Þ2 ;ð2Þwhere τ is the measurement time of a comparison window,
c is the sound velocity, TPR is the pulse repetition time, thesignal of the comparison window is acquired in the (nT + 1)th

07HF03-2© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

transmit event, and vx, vy, and vz are the lateral, out-of-plane(elevational) and axial tissue velocity components, respectively. A reference window is measured at τ = 0 in the ﬁrst
transmit event, and the lateral position of the reference
window is ξ = 0. The conventional method neglects theterm (vyτ)2 in Eq. (2). We introduce an assumption that thecross-correlation depends on the spatial distance dAH, which
satisﬁes the following equation:
d2AH ¼ d2A þ d2H ;ð3Þ

where α is a positive value. The eﬀect of the axial distance on
the cross-correlation may depend on half the pulse width andthat of the horizontal distance may depend on the beam-spot
width. Because the beam-spot width is larger than half thepulse width in most cases, α may be larger than 1. When ξ
is a constant (i.e., in each scan line), dAH is minimized atτ = τmin and τmin is expressed by
cnT cTPRvx  þ  vz
22nT cTPRmin ¼;
ð4Þ

2 ﬃ
c  2vzc22
 vz
vx þ v y þ 
2because typically ∣c=2−vz∣ ≫ ∣vx∣, ∣vy∣ and nTcTPR ≫ ξ.
Equation (4) indicates that dA should be 0 when dAH is
minimized subject to a constant ξ. That is, the axial positionof the best-match comparison window in each scan line
should be the same as that of the reference window at thecomparison time. Therefore, the cross-correlation depends on
the horizontal distance dH when the cross-correlation value ofthe axial-best-match position in each scan line is used.
The approximation in Eq. (4) is valid because the variationin the measurement time in the axial direction is muchsmaller than that in the lateral direction. The diﬀerence in themeasurement time between two adjacent scan lines with thesame depth is equal to the pulse-repetition time TPR. When
TPR = 0.12 ms and the scan line interval is 0.36 mm, thevariation in the measurement time in the lateral direction is0.33 ms=mm. In contrast, the variation in the measurement
time in the axial direction is 1.33 µs=mm because it isdetermined by the sound velocity c. When the tissue velocity
is less than 2 m=s, the tissue movement caused by a time
passage of 1.33 µs is less than 2 µm. Therefore, the variationin the measurement time in the axial direction is muchsmaller than that in the lateral direction, indicating thevalidity of the approximation in Eq. (4). Consequently, theunexpected position shift of the best-match comparison
window caused by the axial tissue movement is negligible.
2.3 Compensation method for the intrinsic error involvedin a conventional speckle-tracking method in a case with
out-of-plane motion

First, we describe the proposed compensation method in thecase of using a circular ultrasound beam, that is, the lateralbeam width is equal to the elevational beam width. We then
expand the proposed method to the case of using an ellipticalultrasound beam spot, that is, the lateral beam width isdiﬀerent from the elevational beam width. In this study, we
assume that ultrasound beams of all scan lines have the same
beam-spot shape.
Equation (4) indicates that the axial position of the bestmatch comparison window in a scan line should be the same

Fig. 4. Locations of cross-correlation values in the ξ–τ plane, where thevalues are acquired in a single measurement plane. ξ and τ are the distance inthe x direction and the diﬀerence in measurement time between a reference
window and a comparison window, respectively. Small dots denotemeasured axial-best-match cross-correlation values. Point Q is the bestmatch position in a comparison frame. Point R exists in a reference frame,
where it has the same correlation value as Q.

as that of the reference-window position at the comparison
time, namely dA = 0. Therefore, this is the key point of the
proposed method; we introduce a simple and realistic
assumption that the axial-best-match cross-correlation value
depends on the horizontal distance between the position of
the reference window at the comparison time and that of thecomparison window dH. First, the proposed method selects
the best-match position in the line segment of a comparison
frame, point Q, as shown in Fig. 4. The line segment of acomparison frame lC is determined by the lateral distance ξand measurement time τ at the peak cross-correlation position
along each scan line (ξ and τ at the axial best-match
positions) in a comparison window. Most conventional
speckle-tracking algorithms estimate the lateral tissue motion
by using the position of Q. Next, the proposed method
searches for the position of point R in a reference frame,
where R has the same correlation value as Q. Because thecross-correlation value at a point on the line l0c decreases
monotonically as the distance between the origin and thepoint increases, we can determine point R using interpolation
or extrapolation. The axial-best-match cross-correlation value
should decrease with the increase in the horizontal distancedH, and thus a contour of the cross-correlation value describes
an ellipse in the ξ–τ plane. The concept of the “crosscorrelation contour” is the same as the concept of contour
lines on a map, where the cross-correlation coeﬃcients are
identical on each contour. Because points Q and R have thesame correlation coeﬃcients, there is an ellipse contour thatpasses through them, as shown by the solid curve in Fig. 5.
Q represents the peak value of the cross-correlation in acomparison frame. Therefore, the ellipse contour must be
tangential to the line segment of the comparison frame lC at Q
because the intersection of the ellipse and lC at Q means that
Q does not have the peak cross-correlation value on lC. We
can determine the ellipse contour by using the two pieces
of information: it passes through R and it is tangential to lC at
Q. The proposed method estimates the lateral tissue motion
by using the ellipse contour. As shown in Figs. 4 and 5, no

07HF03-3© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

window and the comparison window τ and the other is thehorizontal distance dH. The conventional speckle-tracking
methods neglect the former parameter and the horizontal
distance in the y direction.
In this case, we introduce an additional assumption thatthe axial-best-match cross-correlation value depends on thespatio-temporal distance given by the following equation:
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
d0H ¼ d2H þ 2 ;ð6Þ
where β is a positive number. When an elliptical ultrasoundbeam is employed, a contour of the cross-correlation value
depicts the ellipse given by
Fig. 5. Axial-best-match cross-correlation map in the ξ–τ plane. ξ and τ
are the distance in the x direction and the diﬀerence in measurement time
between a reference window and a comparison window, respectively. ξM andτM are the measured distance and time diﬀerence using a conventional
speckle-tracking method, respectively. ξI and τI are the measured distanceand time diﬀerence using the proposed compensation technique, respectively.

a priori information is required regarding the tissue motion
for the proposed compensation method.
The cross-correlation value on an ellipse contour isconstant, and thus the horizontal distance dH is also constanton a contour. Because the derivative of Eq. (2) with respectto ξ becomes vxτ − ξ = 0, the maximal point of an ellipse
contour, point S in Fig. 5, is located on the line vxτ − ξ = 0.
Therefore, the proposed compensation technique estimates
the lateral tissue motion velocity vx by localizing the positionof S, where the localization of S requires only the derivativeof the acquired function of the ellipse contour with respect
to ξ. This compensation method is termed full correlationcompensation (FCC). The gap between points Q and S
should be the intrinsic error in motion estimation in the lateral
direction based on a speckle tracking method using a 2D
measurement plane.
When an elliptical ultrasound beam spot is employed,
Eq. (2) is modiﬁed toqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dE ¼ ðvx   Þ2 þ ðBR vy Þ2 ;ð5Þ
where BR is the ratio of the beam-spot size in the x direction tothat in the y direction. This equation is based on the modiﬁedassumption that the axial-best-match cross-correlation value
depends on the square measure of the overlap between the
beam spot of the reference window at the comparison time
and that of the comparison window. Equation (5) indicatesthat a contour of the cross-correlation value also depicts anellipse in the ξ–τ plane. Its maximal point of τ in terms of ξ
also indicates the true velocity vx because the derivative ofEq. (5) with respect to ξ also becomes vxτ − ξ = 0. Therefore,
in the case where an elliptical ultrasound beam spot is
employed, the proposed compensation method can be usedwith no additional modiﬁcation.
2.4 Compensation method for the intrinsic error involvedin a speckle tracking method in the case with both out-ofplane motion and deformation

When tissue deformation occurs, the axial-best-match crosscorrelation value should depend on two parameters: one isthe diﬀerence in measurement time between a referenceðvx   Þ2 þ ðB2R v2y þ Þ2 ¼ D;

ð7Þwhere D ¼ d02
H is a positive constant. Equation (7) indicatesthat a contour of the cross-correlation value also depicts anellipse in the ξ–τ plane, where its maximal point of τ interms of ξ also indicates the true velocity vx. Therefore, theproposed compensation method can be used with noadditional modiﬁcation.
2.5 Estimation of tissue motion using cross-correlation
coefﬁcients

The cross-correlation coeﬃcient is one of the mostestablished motion estimators.38,39) Therefore, we refer tothe motion estimator using the cross-correlation coeﬃcientsas the conventional speckle-tracking method. In this study,
we applied the proposed compensation technique, FCC,
to a speckle tracking method using the cross-correlationcoeﬃcient in simulation and experimental studies. The crosscorrelation coeﬃcient between two successive frames at theaxial-best-match comparison window is deﬁned as follows:CC ðxR ; zR ; ÞzR XþWz =2fðxR ; zÞgðxR þ ; z þ uÞ

z¼zR Wz =2¼ max vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð8Þ
u u zR þWz =2
zR X
þWz =2u Xtf 2 ðxR ; zÞ
g2 ðxR þ ; z þ uÞ
z¼zR Wz =2z¼zR Wz =2

where xR and zR are the x and z coordinates of the referencewindow position, u is the search range in the axial direction,i.e., in the z direction, Wz is the window size in the axial
direction, and fðx; zÞ and gðx; zÞ are the ultrasound RF data of
the reference and comparison frames, respectively. The crosscorrelation coeﬃcient between scan lines in the reference
frame at the axial-best-match comparison window is deﬁned
as follows:CR ðxR ; zR ; ÞzR XþWz =2fðxR ; zÞfðxR þ ; z þ uÞ

z¼zR Wz =2¼ max vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð9Þ
u u zR þWz =2
zR X
þWz =2u Xtf 2 ðxR ; zÞ
f 2 ðxR þ ; z þ uÞ
z¼zR Wz =2z¼zR Wz =2

2.6

Simulation setup
We investigated the performance of the proposed compensation method in a simulation study based on a 3D ray-tracing
algorithm. In the simulation study, we assumed that the07HF03-4© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

position of the kth scatterer in the tissue at the nth transmit
event can be expressed by the following equations:PSk ðÞ ¼ PSk ð0Þ þ ðv þ vRk Þ; ¼ ðn  1ÞTPR ;

ð10Þ
ð11Þ

where v and vRk are the translational components of the tissue
motion velocity and random motion velocity, respectively.
We investigated the performance of the proposed method
under the conditions of −1 ≤ vx ≤ 1 m=s, 0 ≤ vy ≤ 1.5 m=s,
and vz = 0 m=s, where vx, vy, and vz are the x, y, and z
components of v, respectively. In this setting, the out-ofplane motion was simulated by vy. The norm of the random
motion vRk follows a Gaussian distribution, where thedirection of vRk is random (i.e., its z component is not zero),vy is the out-of-plane motion, and vRk causes deformation.
In the case with deformation, we assumed that the norm of
the random motion velocity follows a Gaussian distribution
with an average value of zero and a standard deviation of
0.05 m=s.
We calculated the ultrasound data at the nth transmit event,i.e., at the measurement time of τ, using the following
formulae:X2fn ðzÞ ¼
2dH sðz  jPSk ðÞ  ðxn ; 0; 0ÞjÞ;
ð12Þ
k

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃspSny ðÞ 2
pSkx ðÞ  xn 2
dH ¼
þ
;
By
Bx

ð13Þ

where the center frequency of s(z) in the time domain is
5 MHz and its −3 dB fractional bandwidth is 60%, the soundvelocity c is 1500 m=s, and the sampling frequency is
30 MHz. xn is the x coordinate of the scan line at the nthtransmit event, pSkx(τ) and pSky(τ) are the x and y coordinatesof the position vector PSk(τ), and Bx and By are the −6 dB
beam-spot widths in the x and y directions, respectively. The
center of the tissue with scatterers was situated at a depth of
40 mm. The tissue size was 30, 20, and 15 mm in the x, y, andz directions, respectively. Scatterers were distributed randomly in the tissue and their density was ﬁve scatterers per
cube of the wavelength at the center frequency, as originallydescribed in a previous study.40) Therefore, a total of
1.67 × 106 scatterers was used. We employed a 2D spatial
median ﬁlter with a kernel size of 3 × 3 in the tissue velocity
estimation, which has been commonly used to eliminate
outliers.26,41) The grid size of the measurement region was
1.08 and 1.0 mm in the x and y directions, respectively.
Consequently, by using the 2D spatial median ﬁlter, eachsample was selected from nine estimated velocities in theregion size of 3.24 and 3.0 mm in the x and y directions,
respectively. This size is suﬃciently larger than the speckle
size.
To suppress the contribution of out-of-plane motion anddeformation, the diﬀerence in measurement time between
the reference window and a comparison window should beshortened. We thus employed the setting that each frame
consists of three scan lines and the successive frame of
the reference frame was used for the comparison frame. We
acquired the ultrasound data of ﬁve frames to calculate theexpectation of the cross-correlation coeﬃcient. The pulserepetition time TPR was 0.12 ms and the window size in theaxial direction Wz was 1.0 mm. The scan line interval IS was

0.36 mm. The search range u ranged from −1 to 1 mm. Whenthe pulse-repetition time TPR is 0.12 ms and the tissue
movement in the lateral direction vx is 1.0 m=s, a typical
speckle-tracking method requires a scan line interval IS oflarger than 0.24 mm to catch up with the reference window inthe successive comparison frame. Because the scan line
interval of the data acquired by a commercial diagnostic
ultrasound (US) device was 0.12 mm, we deﬁned the scanline interval IS as 0.36 mm, which is three times the scan line
interval acquired by the commercial US device used in theexperimental study.
2.7

Experimental setup
Experiments were conducted using a Hitachi EUB-8500 US
device with a 5 MHz linear array to acquire raw in-phase andquadrature (IQ) data. The scan line interval in the x direction
was 0.12 mm. IQ data were converted to radio frequency(RF) data using the RF data oversampling technique.42) The
ultrasound beam-spot width at a depth of 40 mm was 0.6 and1.9 mm in the x and y directions, respectively. We prepared aModel 049 elasticity phantom (Computerized Imaging
Reference Systems). In the experimental study, we also
employed a setting where each frame consists of three scanlines and the successive frame of the reference frame wasused for the comparison frame. The parallel movement of
the probe in the elevational direction simulated the tissue
movement in the elevational direction. The tissue movementin the lateral direction was simulated by the selection ofsuitable scan lines. Because the setting intervals of the tissue
motion velocity in the x and y directions were 0.5 and0.25 m=s, the x and y components of the tissue motion in apulse-repetition time were multiples of 0.06 and 0.03 mm,
respectively. We thus acquired the ultrasound data of several
measurement planes, where the measurement plane intervalin the y direction was 0.03 mm. Because the scan line intervalof the linear array probe was 0.12 mm, we constructed theultrasound data with a 0.06 mm scan line interval in the x
direction using linear interpolation. The locations of theselected scan lines were calculated from the location of thescan line after the compensation for the tissue movement at
the measurement time. In the experimental study, we also
used a 2D spatial median ﬁlter with a kernel size of 3 × 3.
We used 10 samples to investigate the performance of theconventional method and the FCC technique, with the
exception of the 2D-image study.
3.

Results

3.1 Investigation of the FCC technique in a simulation
study with out-of-plane motion

Figure 6 shows the tissue motion velocity in the beam-scandirection at a depth of 4 cm estimated using a conventional
speckle-tracking method and the FCC technique in asimulation study based on a 3D ray-tracing algorithm. No
random motion existed and the ultrasound beam-spot width
was 0.6 and 1.9 mm in the x and y directions, respectively.
Each average value was calculated using 10 samples. The
tissue motion velocity estimated using the conventional
speckle-tracking method decreased as the velocity of out-ofplane motion increased. This result is consistent with themethod described in Sect. 2.1. The root-mean-square errors(RMSEs) of the averaged velocities estimated using the

07HF03-5© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

(a)

(b)
Fig. 6. Tissue motion velocity in the beam-scan direction at a depth of
4 cm estimated using the conventional speckle-tracking method and the
proposed FCC compensation technique in the simulation study, where thetissue velocity in the z direction is 0 m=s. The true velocities in the x direction
are (a) 1.0 and 0.5 m=s and (b) −0.5 and −1.0 m=s. Ten samples areaveraged. The tissue motion in the y direction, i.e., the out-of-plane motion,
ranges from 0 to 1.5 m=s and no deformation occurs. The ultrasound beamspot width is 0.6 and 1.9 mm in the x and y directions, respectively.

conventional speckle-tracking method were 0.082 and0.056 m=s when the x components of the tissue motion
velocity vx were 1.0 and 0.5 m=s, respectively. The y
components of the tissue motion velocity, namely thevelocity of out-of-plane motion, ranged from 0 to 1.5 m=s.
We calculated the RMSE vERR by the following formula:vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u X
u 1 vyM
vERR ¼ t
ð14Þ
jvEx ðvy Þ  vTx j2 ;Ny v ¼0
ywhere Ny is the number of variations in the out-of-planemotion velocity vy, vyM is the maximum value of vy, vEx(vy) isthe average estimated lateral velocity after 2D spatial medianﬁltering at the out-of-plane motion of vy, and vTy is the true
lateral velocity. The RMSE indicates the estimation error of
the lateral tissue velocity compared with the true lateralvelocity.
The RMSEs using the FCC technique were 0.007 and0.017 m=s when vx = 1.0 and 0.5 m=s, respectively. The
averages of all the tissue velocities in the range of 0 ≤vy ≤ 1.5 m=s estimated using the speckle tracking algorithm
were 0.939 (0.030) and 0.465 (0.033) m=s when vx = 1.0
and 0.5 m=s, respectively. Standard deviations are given inparentheses. Because 10 samples existed in each out-of-planemotion, 70 samples were used for the stochastic investigation. In contrast, the averages using the FCC techniquewere 1.000 (0.028) and 0.512 (0.034) m=s. In both cases ofvx = 1.0 and 0.5 m=s, the estimated velocities using theconventional method and the FCC technique are statisticallysigniﬁcant (P < 10−10; Student’s t-test). When vx = 1.0 and
0.5 m=s, in the range of 0 ≤ vy ≤ 1.5 m=s, the employment of
the FCC technique suppressed the RMSE of the estimatedvelocity to 8.7 and 30%, respectively, of that using theconventional speckle-tracking method. The FCC techniquealso succeeded in preventing overestimation of the tissue
motion velocity when no out-of-plane velocity existed, i.e.,vy = 0 m=s. In addition, the FCC technique compensatedfor the large eﬀect of the out-of-plane velocity at vy = 1.5m=s, and it suppressed the estimation error at ðvx ; vy Þ ¼
ð1:0 m=s; 1:5 m=sÞ and (0.5 m=s, 1.5 m=s) to 11 and 2.6%,
respectively, of that using the conventional speckle-tracking
method.
The poor performance of the conventional speckletracking method and the FCC technique in the case ofvx < 0 m=s was caused by the wide beam-spot interval on thetissue. Because the scan line interval was set to 0.36 mm andthe pulse-repetition time was set to 0.12 ms in the current
study, the tissue moved by −0.06 mm in the beam-scandirection at vx = −0.5 m=s during a single pulse-repetition
time. Therefore, the beam-spot intervals on the tissue in the x
direction were 0.24, 0.30, 0.42, and 0.48 mm at vx = 1.0, 0.5,
−0.5, and −1.0 m=s, respectively. Because we employed abeam-spot width of 0.6 mm, a beam-spot interval of 0.42 mm
or wider should suppress the accuracy in calculating the peakcross-correlation coeﬃcient at the successive frame. The
FCC technique uses the peak value of the cross-correlationcoeﬃcient in the successive frame to compensate for the error
in the estimation of motion, and thus the low accuracy incalculating the peak value decreases the performance of the
FCC technique.
Figure 7 shows the estimated velocity in the beam-scandirection when no random motion existed and the ultrasoundbeam-spot width was 0.6 mm in both the x and y directions.
The conventional speckle-tracking method used in the current
study considerably underestimated the velocity in the beamscan direction as the out-of-plane motion increased, as
compared with the results obtained using the beam-spot sizesof 0.6 and 1.9 mm in the x and y directions (Fig. 6). The large
underestimation of velocity should be caused by the narrow
beam-spot width in the y direction, which emphasizes theeﬀect of out-of-plane motion on the estimation of the velocity
in the beam-scan direction, as shown in Eq. (5). When the ycomponent of the tissue motion velocity ranged from 0 to
1.0 m=s, the RMSEs of the averaged velocities estimatedusing the conventional speckle-tracking method were 0.276,
0.198, 0.141, and 0.181 m=s when the true tissue motion
velocity in the beam-scan direction was 1.0, 0.5, −0.5, and−1.0 m=s, respectively. In contrast, the RMSEs using the
FCC technique were 0.027, 0.013, 0.029, and 0.077 m=s.
The averages of all the tissue velocities in the range of0 ≤ vy ≤ 1.0 m=s estimated using the speckle tracking method
were 0.810 (0.095), 0.364 (0.057), −0.590 (0.052), and−1.154 (0.083) m=s when vx = 1.0, 0.5, −0.5, and −1.0 m=s,respectively. In contrast, the averages using the FCC
technique were 0.990 (0.084), 0.507 (0.050), −0.472

07HF03-6© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

Fig. 8. Tissue motion velocity in the beam-scan direction at a depth of
4 cm estimated using the conventional speckle-tracking method and the FCC
technique in the simulation study, where the tissue velocity in the z direction
is 0.5 m=s. The true velocities in the x direction are 1.0 and 0.5 m=s. The
tissue motion velocity in the y direction ranges from 0 to 1.5 m=s and no
deformation occurs. The ultrasound beam-spot width is 0.6 and 1.9 mm in thex and y directions, respectively.

(a)

(b)
Fig. 7. Tissue motion velocity in the beam-scan direction at a depth of
4 cm estimated using the conventional speckle-tracking method and the FCC
technique in the simulation study, where the tissue velocity in the z direction
is 0 m=s. The true velocities in the x direction are (a) 1.0 and 0.5 m=s and(b) −0.5 and −1.0 m=s. The tissue motion velocity in the y direction ranges
from 0 to 1.5 m=s and no deformation occurs. The ultrasound beam-spot
width is 0.6 mm in both the x and y directions.

(0.065), and −1.072 (0.062) m=s. In all cases, the estimatedvelocities obtained using the conventional method and the
FCC technique were statistically signiﬁcant (P < 10−7).
When vx = 1.0 and 0.5 m=s, the employment of the FCC
technique not only suppressed the RMSE of the estimatedvelocity to 9.7 and 6.4%, respectively, of that calculatedusing the conventional method, but also decreased thestandard deviation of the estimated velocity. This result
indicates that the employment of a narrow slice thickness
causes a large estimation error using the conventional
speckle-tracking method and that the FCC techniqueeﬀectively compensates for the large estimation error.
When vx = −0.5 and −1.0 m=s, the beam-spot intervals
on the tissue were 0.42 and 0.48 mm, respectively. The wide
beam-spot intervals caused the decrease in the performanceof both the speckle tracking method and the FCC technique,as shown in Fig. 6(b). However, in the case of using the
0.6 mm beam spot, the FCC technique succeeded insuppressing the RMSE of the estimated velocity to less thanhalf of that using the conventional speckle-tracking method,
as shown in Fig. 7(b). The improvement using the FCC
technique might have originated from the large underestimation of the tissue velocity using the speckle tracking

method. This result indicates that when out-of-plane motion
causes a large estimation error, the FCC technique may
suppress the error even under the severe condition that thebeam-spot interval on the tissue is wider than half of thebeam-spot width.
Figure 8 shows the tissue motion velocity in the beamscan direction at a depth of 4 cm estimated using theconventional speckle-tracking method and the FCC technique, where the tissue velocity in the z direction, vz, is
0.5 m=s. No random motion existed and the ultrasound beamspot width was 0.6 and 1.9 mm in the x and y directions,
respectively. The tissue motion velocity shown in Fig. 8 isvery close to that when the tissue velocity in the z direction
is 0 m=s shown in Fig. 6(a). In both cases using the conventional method and the FCC technique, the estimatedlateral velocities in the cases of vz = 0 and 0.5 m=s are
statistically insigniﬁcant (P > 0.25). This result indicates thevalidity of our supposition that the cross-correlation depends
on the horizontal distance dH. This result supports the ﬁndingthat the estimation error of the lateral tissue velocity causedby the axial tissue movement is negligible.
3.2 Investigation of the FCC technique in a simulation
study with both out-of-plane motion and deformation

Figure 9 shows the estimated tissue motion velocity in the
beam-scan direction at a depth of 4 cm, where random motion
existed and the ultrasound beam-spot width was 0.6 and1.9 mm in the x and y directions, respectively. Each averagevelocity was calculated using 10 samples. The RMSEs of theaveraged velocities estimated using the conventional speckletracking method were 0.159, 0.119, 0.077, and 0.150 m=swhen vx = 1.0, 0.5, −0.5, and −1.0 m=s, respectively. The y
components of the tissue motion velocity ranged from 0 to
1.5 m=s. In contrast, the RMSEs of the averaged velocities
estimated using the FCC technique were 0.016, 0.013, 0.032,
and 0.078 m=s. The averages of the tissue velocities in
the range of 0 ≤ vy ≤ 1.5 m=s estimated using the speckle
tracking method were 0.851 (0.057), 0.389 (0.049), −0.564(0.054), and −1.146 (0.055) m=s when vx = 1.0, 0.5, −0.5,
and −1.0 m=s, respectively. In contrast, the averages using

07HF03-7© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

(a)Fig. 10. Tissue motion velocity in the beam-scan direction at a depth of
4 cm estimated by the conventional speckle-tracking method and the FCC
technique in the experimental study using an elasticity phantom. The truevelocities in the beam-scan direction are 1.0 and 0.5 m=s. The tissue motionin the y direction ranges from 0 to 1.5 m=s and no deformation occurs. The
ultrasound beam-spot width at the depth of 4 cm is 0.6 and 1.9 mm in the x
and y directions, respectively.

(b)Fig. 9. Tissue motion velocity in the beam-scan direction at a depth of
4 cm estimated using the conventional speckle-tracking method and the FCC
technique in the simulation study. The true velocities in the x direction are(a) 1.0 and 0.5 m=s and (b) −0.5 and −1.0 m=s. The tissue motion in the y
direction ranges from 0 to 1.5 m=s and deformation caused by random
motion exists. The ultrasound beam-spot width is 0.6 and 1.9 mm in the x andy directions, respectively.

the FCC technique were 0.993 (0.042), 0.506 (0.043),−0.471 (0.055), and −1.077 (0.053) m=s. In all cases, theestimated velocities using the conventional method and the
FCC technique are statistically signiﬁcant (P < 10−10).
When vx = 1.0, 0.5, −0.5, and −1.0 m=s, in the range of0 ≤ vy ≤ 1.5 m=s the employment of the FCC techniquesuppressed the RMSE of the estimated velocity to 9.8, 11, 42,
and 52%, respectively, of that calculated using the conventional speckle-tracking method. This result indicates that theexistence of deformation will suppress the performance of
the FCC technique; however, the employment of the FCC
technique is beneﬁcial to compensate for the underestimationof the conventional speckle-tracking method caused by outof-plane motion and deformation.
3.3 Investigation of the FCC technique in an experimental
study

Figure 10 shows the tissue motion velocity in the beam-scandirection at a depth of 4 cm estimated by the conventional
speckle-tracking method and the FCC technique in anexperimental study. In this study, we used an elasticity
phantom. No random motion existed and the ultrasoundbeam-spot width at a depth of 4 cm was 0.6 and 1.9 mm in thex and y directions, respectively. Each average velocity wascalculated using 10 samples.

In the study using an elasticity phantom with scatterers,
the RMSEs of the averaged velocities estimated using theconventional speckle-tracking method were 0.169 and
0.131 m=s when vx = 1.0 and 0.5 m=s, respectively. The y
components of the tissue motion velocity ranged from 0 to
1.5 m=s. In contrast, the RMSEs using the FCC techniquewere 0.057 and 0.037 m=s. The averages of the tissue
velocities in the range of 0 ≤ vy ≤ 1.5 m=s estimated using
the conventional method were 0.865 (0.073) and 0.401(0.052) m=s when vx = 1.0 and 0.5 m=s, respectively. In
contrast, the averages using the FCC technique were 0.952(0.063) and 0.476 (0.045) m=s. When vx = 1.0 and 0.5 m=s,
in the range of 0 ≤ vy ≤ 1.5 m=s the employment of the FCC
technique not only suppressed the RMSE of the estimatedvelocity to 34 and 28%, respectively, of that calculated using
the conventional speckle-tracking method but also decreased
the standard deviation of the estimated velocity.
Figure 11 shows the tissue motion velocity and its error
in the beam-scan direction in a 2D measurement plane
estimated using the conventional speckle-tracking method
and the FCC technique in an experimental study using anelasticity phantom, where the true velocity in the x direction
vx was 0.5 m=s. The tissue motion velocity in the y direction
vy was 0, 0.25, and 0.5 m=s in the ranges of x < −3.24 mm,−3.24 ≤ x ≤ 3.24 mm, and 3.24 mm < x, respectively. The
grid size of the measurement region was 1.08 and 1.0 mm inthe x and y directions, respectively. We also employed the 2D
spatial median ﬁlter with a kernel size of 3 × 3.
We investigated the error of the proposed method in
estimating the tissue motion velocity in the x direction using
the RMSE. The RMSEs of the estimated velocities using theconventional speckle-tracking method were 0.006, 0.018, and0.052 m=s when vy = 0, 0.25, and 0.5 m=s, respectively. Sixty
samples were used for the stochastic investigation in eachcase for vy. In contrast, the RMSEs using the FCC techniquewere 0.009, 0.012, and 0.042 m=s. The averages of the tissue
velocities estimated using the speckle tracking algorithm
were 0.500 (0.006), 0.488 (0.014), and 0.470 (0.042) m=s
when vy = 0, 0.25, and 0.5 m=s, respectively. In contrast, theaverages using the FCC technique were 0.507 (0.006), 0.499

07HF03-8© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

(a)

(b)

(c)

(d)

(e)
Fig. 11. Estimated tissue motion velocity in the x direction vx, i.e., in the beam-scan direction, in a 2D measurement plane (a) using the conventional
speckle-tracking algorithm and (b) using the FCC technique in the experimental study, where an elasticity phantom is used. The true tissue velocity in the
beam-scan direction vx is 0.5 m=s. Error in estimated tissue motion velocity in the x direction vx (c) using the conventional speckle-tracking algorithm and(d) using the FCC technique in the experimental study. (e) In this setting, the tissue motion in the y direction vy, i.e., the out-of-plane motion, is 0, 0.25, and
0.5 m=s in the ranges of x < −3.24 mm, −3.24 ≤ x ≤ 3.24 mm, and 3.24 mm < x, respectively. The grid size of the measurement region is 1.08 and 1 mm in thex and y directions, respectively. The ultrasound beam-spot size at a depth of 4 cm is 0.6 and 1.9 mm in the x and y directions, respectively.

(0.012), and 0.494 (0.042) m=s. In all cases, the estimatedvelocities using the conventional method and the FCC
technique were statistically signiﬁcant (P < 0.002). The
conventional speckle-tracking method increasingly underestimated the lateral tissue motion velocity vx as the out-ofplane motion vy progressively increased. In contrast, the FCC
technique succeeded in estimating the lateral tissue motion
velocity with an error of less than 0.007 m=s. This result
supports the view that the FCC technique improves theperformance of an ultrasound cross-correlation technique inmedical diagnosis, for example, in the assessment of cardiac
function.
4.

Discussion

As described in Sect. 3.1, the beam-spot interval on the tissue
becomes narrow when the lateral tissue velocity vx has a large
positive value. Because a narrow beam-spot interval on thetissue results in high accuracy in interpolating crosscorrelation values, when vx = 1.0 m=s, the accuracy ininterpolating the cross-correlation value becomes the greatest.
This is the most probable reason why the FCC technique

has a high performance when vx is positive, as shown in
Figs. 6(a), 7(a), and 9(a). When vx is negative, i.e., Figs. 6(b),7(b), and 9(b), the time diﬀerence between a reference
window and the best-match comparison window is smaller
than that when of vx is positive, because the tissue movement
at vx < 0 m=s is opposite to the beam scan direction.
Therefore, the error in the cross-correlation caused by theout-of-plane motion and deformation should be small whenvx is negative, resulting in the high performance of theconventional speckle-tracking method at vx < 0 m=s, as
shown in Figs. 6(b), 7(b), and 9(b).
When vx = 1.0 m=s, the existence of deformation suppressed the performance of the FCC technique, as shown in
Figs. 6(a) and 9(a). This deterioration in the performance of
the FCC technique might have been caused by introducing
the additional assumption expressed in Eq. (6). When no
deformation exists, the FCC technique introduces only asingle and valid assumption as expressed in Eq. (6), namelythat the cross-correlation coeﬃcient depends on the valueof the beam overlap area. Therefore, the FCC techniquesucceeded in compensating for the error involved in the

07HF03-9© 2015 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 54, 07HF03 (2015)

H. Taki et al.

speckle tracking method almost perfectly under the condition
that cross-correlation coeﬃcients were calculated accurately.
In the presence of deformation, the FCC technique introduces
the assumption that the cross-correlation depends on thespatio-temporal distance given by Eq. (6). Equation (7)indicates that the FCC technique works correctly for any
value of β; however, there is no information regarding therelationship between the eﬀect of the passage of time on the
cross-correlation and the eﬀect of the horizontal distance.
Therefore, the additional assumption expressed in Eq. (6)has average validity, and the average validity of this equation
might have caused the suppression of the FCC performance.
However, the deformation severely reduces the estimation
accuracy of the conventional speckle-tracking method, andthus the employment of the FCC is beneﬁcial in the case of
deformation, as shown in Fig. 9. Future work should takeaccount of the beam-width variation caused by the measurement depth.
The proposed method can be applied in the case of using
a transmit beam. That is, this work includes no investigation
into the application to the high-frame-rate imaging method
using parallel beamforming with an unfocused transmit
beam.43) Simultaneous acquisition of the received signals of
all scan lines using parallel beamforming also suppresses theeﬀect of out-of-plane motion and deformation on lateralvelocity estimation.
5.

Conclusions

In the current study, we reported on the intrinsic error
involved in a speckle tracking method caused by out-of-planemotion and deformation. We proposed a compensation
method for this intrinsic error that requires no a priori
information, and we investigated the performance of the
proposed compensation method in simulation and experimental studies. When the beam-spot interval on the tissue
was half the beam-spot width or less, the RMSE of theestimated tissue motion velocity calculated using the
proposed method ranged from 6.4 to 34% of that using
the conventional speckle-tracking method. This study shows
the high potential of the proposed compensation method for
improving the estimation of tissue motion velocity using an
ultrasound cross-correlation technique in medical diagnosis,
such as in the assessment of cardiac function.
Acknowledgments

This work is partly supported by the Innovative Techno-Hub
for Integrated Medical Bio-Imaging of the Project forDeveloping Innovation Systems from the Ministry of
Education, Culture, Sports, Science and Technology, Japan(MEXT), and by MEXT=JSPS KAKENHI Grant Number
25870345.

1) G. E. Trahey, J. W. Allison, and O. T. Von Ramm, IEEE Trans. Biomed.
Eng. BME-34, 965 (1987).
2) P. M. Embree and W. D. O’Brien, IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 37, 176 (1990).
3) B. F. Osmanski, M. Pernot, G. Montaldo, A. Bel, E. Messas, and M. Tanter,
IEEE Trans. Med. Imaging 31, 1661 (2012).
4) H. Takahashi, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 53, 07KF08(2014).

5) J. Ophir, Ultrason. Imaging 13, 111 (1991).
6) M. O’Donnell, A. R. Skovoroda, B. M. Shapo, and S. Y. Emelianov, IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 41, 314 (1994).
7) J. Kybic and M. Unser, IEEE Trans. Image Process. 12, 1427 (2003).
8) H. Shida, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 51, 07GF05(2012).
9) S. W. Flax and M. O’Donnell, IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 35, 758 (1988).
10) G. C. Ng, S. S. Worrell, P. D. Freiburger, and G. E. Trahey, IEEE Trans.
Ultrason. Ferroelectr. Freq. Control 41, 631 (1994).
11) D. Asari, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 53, 07KF21(2014).
12) M. Sato, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 53, 07KF03(2014).
13) K. Kitamura, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 51, 07GF08(2012).
14) Y. Honjo, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phys. 51, 07GF06(2012).
15) H. Kanai and M. Tanaka, Jpn. J. Appl. Phys. 50, 07HA01 (2011).
16) L. Bohs and G. Trahey, IEEE Trans. Biomed. Eng. 38, 280 (1991).
17) V. Behar, D. Adam, P. Lysyansky, and Z. Friedman, Ultrasonics 43, 57(2004).
18) M. Lubinski, S. Emelianov, and M. O’Donnell, IEEE Trans. Ultrason.
Ferroelectr. Freq. Control 46, 82 (1999).
19) J. Crosby, B. Amundsen, T. Hergum, E. Remme, S. Langeland, and H.
Torp, Ultrasound Med. Biol. 35, 458 (2009).
20) M. Ledesma-Carbayo, J. Kybic, M. Desco, A. Santos, M. Suhling, P.
Hunziker, and M. Unser, IEEE Trans. Med. Imaging 24, 1113 (2005).
21) B. Zitová and J. Flusser, Image Vision Comput. 21, 977 (2003).
22) O. Goksel, H. Eskandari, and S. E. Salcudean, IEEE Trans. Med. Imaging
32, 408 (2013).
23) O. Goksel and S. E. Salcudean, IEEE Trans. Med. Imaging 30, 11 (2011).
24) H. Eskandari, S. E. Salcudean, R. Rohling, and J. Ohayon, Phys. Med. Biol.
53, 6569 (2008).
25) B. Heyde, R. Jasaityte, D. Barbosa, V. Robesyn, S. Bouchez, P. Wouters, F.
Maes, P. Claus, and J. D’Hooge, IEEE Trans. Med. Imaging 32, 449 (2013).
26) R. Z. Azar, A. Baghani, S. E. Salcudean, and R. Rohling, IEEE Trans.
Ultrason. Ferroelectr. Freq. Control 57, 2421 (2010).
27) J. M. Abeysekera, R. Zahiri-Azar, O. Goksel, and R. Rohling, Ultrasonics
52, 156 (2012).
28) L. de Isla, D. Balcones, C. Fernández-Golfín, P. Marcos-Alberca, C.
Almería, J. Rodrigo, C. Macaya, and J. Zamorano, J. Am. Soc.
Echocardiogr. 22, 325 (2009).
29) P. Reant, L. Barbot, C. Touche, M. Dijos, F. Arsac, X. Pillois, M. Landelle,
R. Roudaut, and S. Laﬁtte, J. Am. Soc. Echocardiogr. 25, 68 (2012).
30) F. Maﬀessanti, H.-J. Nesser, L. Weinert, R. Steringer-Mascherbauer, J. Niel,
W. Gorissen, L. Sugeng, R. M. Lang, and V. Mor-Avi, Am. J. Cardiol. 104,
1755 (2009).
31) K. Saito, H. Okura, N. Watanabe, A. Hayashida, K. Obase, K. Imai, T.
Maehama, T. Kawamoto, Y. Neishi, and K. Yoshida, J. Am. Soc.
Echocardiogr. 22, 1025 (2009).
32) R. Jasaityte, B. Heyde, V. Ferferieva, B. Amundsen, D. Barbosa, D.
Loeckx, G. Kiss, F. Orderud, P. Claus, H. Torp, and J. D’hooge, Int. J.
Cardiol. Imaging 28, 1049 (2012).
33) M. De Craene, S. Marchesseau, B. Heyde, H. Gao, M. Alessandrini, O.
Bernard, G. Piella, A. R. Porras, L. Tautz, A. Hennemuth, A. Prakosa, H.
Liebgott, O. Somphone, P. Allain, S. Makram Ebeid, H. Delingette, M.
Sermesant, J. D’hooge, and E. Saloux, IEEE Trans. Med. Imaging 32, 1632
(2013).
34) H. Taki, T. Sakamoto, M. Yamakawa, T. Shiina, and T. Sato, Proc. IEEE
Ultrasonics Symp., 2012, p. 2563.
35) B. H. Briggs, Handbook for MAP (ICSU, Paris, 1984) Vol. 13, p. 166.
36) M. Yamamoto, T. Sato, T. Tsuda, S. Fukao, and S. Kato, Pure Appl.
Geophys. 130, 605 (1989).
37) D. A. Holdsworth and I. M. Reid, Ann. Geophys. 22, 3829 (2004).
38) F. Viola and W. F. Walker, IEEE Trans. Ultrason. Ferroelectr. Freq. Control
50, 392 (2003).
39) J. Luo and E. E. Konofagou, IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 57, 1347 (2010).
40) J. Udesen, F. Gran, K. L. Hansen, J. A. Jensen, C. Thomsen, and M. B.
Nielsen, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1729 (2008).
41) R. Zahiri-Azar and S. E. Salcudean, IEEE Trans. Biomed. Eng. 53, 1990
(2006).
42) H. Taki, K. Taki, T. Sakamoto, M. Yamakawa, T. Shiina, M. Kudo, and T.
Sato, IEEE Trans. Med. Imaging 31, 417 (2012).
43) H. Hasegawa and H. Kanai, Jpn. J. Appl. Phys. 53, 07KF02 (2014).

07HF03-10© 2015 The Japan Society of Applied Physics