Japanese Journal of Applied Physics
Vol. 46, No. 7B, 2007, pp. 4813–4819#2007 The Japan Society of Applied Physics

Accurate Estimation of Scattering Strength Distribution by Simultaneous Receptionof Ultrasonic Echoes with Multichannel Transducer Array
Yusaku A BE, Hideyuki H ASEGAWA, and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 24, 2006; accepted April 4, 2007; published online July 26, 2007)

Recently, there have been several studies on ultrasonic cross-sectional imaging based on simultaneous reception of echo
signals with an array transducer without scanning ultrasonic beams during transmission. In those studies, parallel processing
was applied to create an image from a data set simultaneously received by the array. However, the lateral resolution of theparallel processing is not high. In this study, in order to improve the spatial resolution of parallel processing, the least-squares
estimation and the truncated singular value decomposition (tSVD) are applied to the echo signals from two wire targets
simultaneously received with a multichannel transducer array. We introduced a weighting for correcting the eﬀect of thedirectivity of the elements of the array. The experimental results show a higher lateral resolution of the tSVD method withweighting than that of conventional parallel processing. The axial resolution is also improved by considering the ﬁnite
duration of the transmitted ultrasonic pulse. A typical application of this method is nondestructive evaluation, that is, thedetection of cavities and cracks in welded metal structures. [DOI: 10.1143/JJAP.46.4813]
KEYWORDS: synthetic aperture, parallel processing, truncated singular value decomposition, simultaneous reception, lateral
resolution, frame rate

1.

Introduction

In some methods reported in literature,1–5) echo signals are
simultaneously received by all elements of a linear arraytransducer without forming or scanning ultrasonic beams
during transmission. To create an image from a data set
simultaneously received by the array, the parallel processing
technique6) is used. In this method, multiple receiving beams
are formed simultaneously within a wide transmitted beam(or spherical wave insoniﬁcation). This imaging procedure
achieves a high frame rate which cannot be realized byconventional B-mode imaging with scanning of thin ultrasonic beams during transmission. Such methods based on
simultaneous reception has been applied to ﬂow imaging,1)
the measurement of two-dimensional motion vectors,2,3) andreal-time three-dimensional imaging.4,5) Furthermore, synthetic aperture processing,7) which generates a wide eﬀective
aperture by means of several transmissions, can be combinedwith parallel processing to improve the spatial resolution.
Parallel beam forming (PBF), which is performed bymeans of time-delay focusing, achieves spatially matchedﬁltering,5,8) which maximizes the signal-to-noise ratio (SNR)at each point. However, the lateral resolution of the PBF islower than that of conventional phased array beam forming
because the directivity of the transmitting beam is weak inparallel processing.
Some methods for inverse ﬁltering of simultaneously
received echoes have been introduced in order to improve
the lateral resolution.9,10) These methods are based on thediscrete propagation model of ultrasonic pulse echoes and on
solving the inverse problem in order to estimate thescattering strength distribution from received signals.11,12)
However, this inverse problem is diﬃcult to solve undersome conditions. In such a situation, small errors inmeasured data produce large errors in the estimatedresults.13) For the application of these inverse ﬁltering
methods, weighting to the receiving aperture should beconsidered. Because the elements of a transducer array havedirectivity, the contributions of the received signals to theestimation are not equal. If the diﬀerence in the amplitudesof received signals is not considered in the propagation
model, it induces large errors in the estimation.
In this study, to increase the lateral resolution, thescattering strength distribution in the region of interest(ROI) is estimated by applying the least squares (LS)method and truncated singular value decomposition (tSVD)
to the simultaneously received echo signals from two wire
targets during one transmission. The spatial resolution in theseparation of the scatterers is compared with that of PBF,
and the eﬀects of weighting and truncation are investigated.
A typical application of this method is nondestructive
evaluation (NDE), that is, the detection of cavities andcracks in welded metal structures. To investigate otherapplications, we analyzed the inﬂuences of noise on theseestimators.
2.

Principles

2.1 Discrete model of pulse echo measurement
To apply inverse ﬁltering, a linear discrete model of pulse
echo measurement is employed.12) As shown in Fig. 1, thescattering strength distribution in the region of interest (ROI)
is discretized in an M  N matrix, O. The scattering strength
at position ðxm ; zn Þ (m ¼ 0; 1; . . . ; M  1; n ¼ 0; 1; . . . ; N 1) is represented by an element ðOÞm;n of the matrix. An
ultrasonic pulse emitted from a transmitting position, xT , isinsoniﬁed to the entire ROI and is reﬂected by scatterers in
the ROI. The echo is received at positions fxRj g ð j ¼ 0;
1; . . . ; L  1Þ of L elements simultaneously. The sampled Klength echo signals, fyj ðkÞg ðk ¼ 0; 1; . . . ; K  1Þ, obtained atreceiving position xRj are represented by a K-dimensional
vector, yj ¼ ½yj ð0Þyj ð1Þ    yj ðK  1ÞT , where T denotes atranspose. The amplitudes of the received signals varydepending on the receiving positions because of attenuation,
spreading loss, and the directivity of an element. Therefore,
in this paper, the weighting function, fwðm;nÞ
g, is introducedj
to show the change in the amplitude of the received signal.
Thus, the received signal is expressed asE-mail address: hkanai@ecei.tohoku.ac.jp

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Y. ABE et al.

yjway, the N-dimensional row vectors fðOÞm;: g of O in Fig. 1
are arranged to form an MN-dimensional vector, o ¼ ½ðOÞ0;:ðOÞ1;:    ðOÞM1;: T . Using the propagation matrix P,
eq. (2.1) is simply rewritten as

xR j

xR0

y ¼ Po þ e;transmitting
position ( xT , 0 )N imageelements

x

xRL-1

O

where e is a KL-dimensional vector corresponding to
additive noise.

region ofinterestz(O)m, :

(xm , zn )M image elements
Fig. 1. Schematic view of the ultrasonic measurement system.

yj ¼

M
1 X
N 1
X

ð2:6Þ

(m,n)

hj

wðm;nÞ
hðm;nÞ
ðOÞm;n ;
j
j

ð2:1Þ

m¼0 n¼0

where the K-dimensional vector hðm;nÞ
jshows the discrete
impulse response which corresponds to the delay due topropagation from the transmitting point ðxT ; 0Þ to a receiving
point ðxRj ; 0Þ via point ðxm ; zn Þ. The integral ratio of the
propagation delay, kjðm;nÞ , to the sampling period, Ts , is given
by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðxT  xm Þ2 þ z2n þ ðxm  xRj Þ2 þ z2n
ðm;nÞ
kj
¼
;
ð2:2Þ
cTs
where c is the speed of sound in water. The K-dimensional, is given byvector, hðm;nÞj
1 if k ¼ kjðm;nÞ given by eq. (2.2),
ðm;nÞ
ð2:3Þ
ðhj Þk ¼
0 otherwise.
The transmitted ultrasonic pulse is not the impulse, and it
has a ﬁnite pulse duration. The ultrasonic pulse transmitted
from a transducer is expressed by the convolution of thepulse waveform, sðkÞ, which is determined by the frequency
response of the transducer, and the delayed impulse, hðm;nÞ
.
j
Therefore, let S be a K  K lower triangular matrix showing
the waveform of the transmitted pulse, sðkÞ, such that
1
0
sð0Þ
0
0
...
0
B sð1Þ
sð0Þ
0
...
0 C
C
B
C
B
B
.. C
..
C: ð2:4Þ
.
S¼B
sð2Þ
sð1Þ
sð0Þ
.
C
B
C
B
..
..
..
..
C
B
@
.
0 A
.
.
.
sðK  1Þ sðK  2Þ

...

sð1Þ

2.2 Estimation of scattering strength distribution
Let us regard the estimation of the scattering strength
distribution as an inverse problem. We follow Lingvall’s
method14) and the inverse ﬁltering is applied as presented
below. It is assumed that noise, e, is white and has nocorrelation among the receiving positions fxRj g. The leastsquares solution of eq. (2.6), denoted by o^ LS , is given byminimizing the power of noise e, JLS , as2
JLS ¼ y  Po ¼ yT y  2oT PT y þ oT PT Po: ð2:7Þ
Setting the partial derivative of JLS with respect to oT tozero, the normal equation is obtained as
PT Po ¼ PT y:
The solution of eq. (2.8) is given byo^ LS ¼ ðPT PÞ1 PT y:

The K  N matrix,
hðm;N1Þ, is formed for all combinations of j and m. Thejpropagation matrix, P, which is a KL  MN matrix isconstructed as
10p0;1. . . p0;M1
p0;0
C
B
p1;1
. . . p1;M1 C
B p1;0
C
B
ð2:5Þ
P¼B .
C:
..
..
..
C
B ..
.
.
.
A
@
pL1;0 pL1;1 . . . pL1;M1
The L vectors y0 ; . . . ; yL1 are stacked on each other to form
a KL-dimensional vector, y ¼ ½yT0 yT1    yTL1 T . In a similar

ð2:9Þ

In this study, eq. (2.6) is over-determined (KL > MN). If
matrix P is full rank, a unique LS solution exists.
The singular value decomposition (SVD) of matrix P isgiven by
P ¼ UVT ;ð2:10Þ

where U and V are KL  KL and MN  MN orthogonal
matrices composed of the eigenvectors of PPT and PT P,
respectively. Using the SVD, matrices U and V are
determined uniquely.  ¼ diag½1 2    MN  is a KL 
MN diagonal matrix and i is the i-th singular value, where1  2      MN . The singular values are the squareroots of the eigenvalues of both PPT and PT P. In the tSVD,
the nonsigniﬁcant singular values are set to zero. Using thetSVD, the pseudo-inverse matrix of P is given byþ TPþ
r ¼ Vr U ;ð2:11Þ

1 1
10    0 is a MN  KL
where þr ¼ diag½1 2    rdiagonal matrix, and r is the truncation order. By substituting eq. (2.11) into eq. (2.9), the tSVD estimate, o^ tSVD , isgiven byo^ tSVD ¼ Pþ
r y:

sð0Þ

pj;m ¼ S½wðm;0Þhjðm;0Þ    wðm;N1Þ

j
j

ð2:8Þ

ð2:12Þ

Using the vectors, fhðm;nÞ
g, in eq. (2.3) as a focusing delay,jthe PBF estimation at point ðxm ; zn Þ is given byL1X^ PBF Þm;n ¼ðhðm;nÞ
ÞT yj :
ð2:13Þ
ðOjj¼0^ PBF , ðhðm;nÞ ÞT , and yj are rearranged in the
The valuables O
jsame way as eq. (2.5), and the PBF estimation is obtained aso^ PBF ¼ PT y:ð2:14Þ

The PBF estimation is obtained by the product of thetranspose of P and the received signals y. In the PBF, thewaveform of the transmit pulse sðkÞ is not considered (S ¼ I,

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Y. ABE et al.

I: identity matrix) in propagation matrix P in eq. (2.14) inorder to perform only the coherent summation of thereceived signals. As shown in eq. (2.14), the least-squares
solution, o^ LS , in eq. (2.9) becomes identical to o^ SA whenmatrix P is orthogonal.

trigger for transmitting
ultrasonic pulse

function generator
RF burst pulse

digitaloscilloscope

power AMP

2.3 Consideration of eﬀects of noise in estimation
To compare the eﬀects of noise e, eqs. (2.9), (2.12), and(2.14) are resolved into the true components and the noise
components by substituting eq. (2.6) into the respective
equations.15)
The LS estimation is rewritten as

pre-AMPtransmitting
transducerxTo^ LS ¼ ðPT PÞ1 PT ðPo þ eÞzreceivingtransducer
x

xRj

metal
wires

¼ o þ ðPT PÞ1 PT eT
¼ o þ Vþ
MN U e:ð2:15Þ

If the SNR is inﬁnite (e ¼ 0), LS estimate o^ LS is equal to theactual scattering strength distribution, o. However, in thecase of a ﬁnite SNR, higher-order singular values are verysmall, particularly when the condition number, 1 =MN , isTlarge. Then, the matrix 2-norm of Vþ
MN U in eq. (2.15) is
16)
given byT
þkVþ
MN U k2 ¼ kMN k2 ¼

1
MN

:

ð2:16Þ

The norm of the second term in eq. (2.15), which shows themagnitude of error, becomes enormous when 1 =MN isr
MNrlarge.
zﬄﬄﬄﬄ}|ﬄﬄﬄﬄ{ zﬄﬄﬄﬄ}|ﬄﬄﬄﬄ{
Using the MN  MN matrix Ir ¼ diag½1; . . . ; 1; 0; . . . ; 0,
the tSVD estimation is rewritten aso^ tSVD ¼ Pþ
r ðPo þ eÞ
T
T
¼ Vþ
r U ðUV o þ eÞT
¼ VIr VT o þ Vþ
r U e:

ð2:17Þ

T
The matrix 2-norm of Vþr U is 1=r . Thus, the truncation
reduces the magnitude of error in eq. (2.17) to r =MN ofthat in eq. (2.16). For medical application, the SNR is low
and the truncation order r should be set to a lower order.
Then, the bias error, o  VIr VT o, is increased due to the
low-order truncation. As a result, the spatial resolution
becomes low.
In the same way as above, eq. (2.14) is rewritten aso^ PBF ¼ PT ðPo þ eÞ
¼ VT UT ðUVT o þ eÞ
¼ VVT o þ VT UT e;

ð2:18Þ

where  ¼ T  is a MN  MN matrix composed of theeigenvalues of PT P. Even if the SNR is inﬁnite, bias error iscaused because matrix  is not equal to the identity matrix.
However, the true scatter strength distribution o and noise e
are multiplied by matrices with the 2-norm of kVVT k2 ¼
12 and kVT UT k2 ¼ 1 , respectively. The magnitude of theﬁrst term is more ampliﬁed than that of the second term.
This means that the PBF enhances the echo signal from eachpoint (xm ; zn ). Thus, the PBF is eﬀective when the SNR is
low.

PC
GP-IB

water

auto XYZ stage

Fig. 2. Experimental setup for acquisition of simultaneously received
signals using two ultrasonic transducers.

3.

Experimental Procedure

3.1 Data acquisition
The experimental setup is illustrated in Fig. 2. The
multichannel simultaneous reception by an array transducer
is simulated with two single ﬂat transducers (diameter
D ¼ 1 mm) implementing the following data acquisition
sequence. An ultrasonic pulse (center frequency fc ¼ 10MHz) is emitted from one of the transducers at the position
xT . The length of the Fresnel zone is D2 =4 ¼ 1:6 mm. Twometal wires with a diameter of 30 mm, which can beconsidered as point scatterers, are set in the water tank. The
distance between two wires is 1 mm and the wires are at thesame depth and symmetric with respect to the z-axis. The
ultrasonic pulse is scattered by the wires and is received by
the other transducer at the position xRj . The acquisition of thereceived signals, fyj g, is triggered by the transmitting pulse.
The receiving transducer is moved to the next receiving
point xRjþ1 , using an x-stage, for the next transmission.
3.2 Experimental setup
The relative positions of the simulated array and the ROI
are shown in Fig. 3. Figure 4 shows the correspondence
among the sampled RF data points, and their positions in the
ROI. The transmitting point, the center of the simulated
array, and the center of the ROI are set on the z-axis. The
ROI consists of M  N ¼ 11  80 sampled points, whose
intervals are xm ¼ 200 mm and zn ¼ c=ð2 fs Þ ¼ 14:88 mm(c ¼ 1, 488 m/s at 21.0  C17)) along the x-axis and z-axis,
respectively. The number of reception points is L ¼ 61, andtheir interval is xR ¼ 200 mm. The number of samples at
each receiving position is K ¼ 125 and the sampling
frequency is fs ¼ 50 MHz.
The impulse responses fhðm;nÞ
g are determined from thejabove parameters using eqs. (2.2) and (2.3).
3.3 Determination of weighting function
The weighting function, fwðm;nÞ
g, in eq. (2.1) is deterj
mined from the directivity of the transducers. The directivityof ﬂat transducer is expressed as J1 ðka sin Þ ;ð3:1ÞDðÞ ¼ 2ka sin  4815

Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

12 mm

Y. ABE et al.

xTtransducer
array
x

xR j

xm

x
θR

θT
46.4 mmzn
scattererzFig. 5. Incident angle T and received angle R for point ðxm ; zn Þ.

1.2 mm2 mmtion. The amplitude of ultrasound, pðdÞ, after propagating
through distance d is given byregion ofinterestzpðdÞ ¼

Fig. 3. Setup of the ROI and the transducer array.

xR0

x T = xR(L -1)/2

xR1

xR L -1xx0x1x(M -1)/2

ð3:5Þ

where p0 is the amplitude at the transmitting point, ð1=dÞ
shows the diﬀraction loss, ed shows the absorption, and 
is the absorption coeﬃcient of water. Then, djðm;nÞ is the
propagation distance from the transmitting point ðxT ; 0Þ to areceiving point ðxRj ; 0Þ via a point ðxm ; zn Þ in the ROI, anddmin is the minimum of djðm;nÞ . The attenuation term of theweighting function, Np , is determined by the ratio ofpðdjðm;nÞ Þ to pðdmin Þ as!
h 
ipðdjðm;nÞ Þ
dmin
ðm;nÞ

d
¼
Np ¼
exp

d
: ð3:6Þminjpðdmin Þdjðm;nÞ

∆ xR

z0
z1

1
p0 ed ;dxM -1

Using Np , the weighting function wðm;nÞ
, which includes
jthe attenuation factor, is given by
wðm;nÞ
¼ DðT Þ  DðR Þ  Np :
j
∆ zn

zN -1

In the situation shown in Fig. 3, the inﬂuence of theattenuation term is much smaller than that of directivity(min Np ¼ 0:97) because the attenuation in water is verysmall and the diﬀerence between djðm;nÞ and dmin is small.
Therefore, the weighting function is determined using
eq. (3.2).

∆ xm
sampled pointz
Fig. 4. Illustration of sampling of the ROI and the array.

where a is the radius of the transducer, k is the wave number,
and J1 is the ﬁrst-order Bessel function. Using the directivityDðÞ, the weighting function is deﬁned as
wðm;nÞ
¼ DðT Þ  DðR Þ;
j

ð3:7Þ

ð3:2Þ

where R is the receiving angle and T is the incident angle
for each point ðxm ; zn Þ, as shown in Fig. 5. These angles are
given byxm  xT
T ¼ tan1
;
ð3:3Þ
zn
xm  xRj
R ¼ tan1
:
ð3:4Þ
zn
The amplitudes of received signals are also aﬀected byattenuation during propagation. The attenuation is caused by
diﬀraction loss, absorption, and scattering. Since scattering
does not occur in water, the attenuation factor in theweighting function is determined by diﬀraction and absorp-

4.
4.1

Results

The case of neglecting the waveformof the transmitting pulse
Here we show the estimation results when the waveformof the transmitted pulse sðkÞ is not considered in propagation
matrix P in order to compare the results obtained by tSVD
with those obtained by PBF. The distribution of the singularvalues fi g of the propagation matrix P of the measurement
system used in this study is shown in Fig. 6. Matrix P is fullrank and the unique LS solution exists because all singularvalues are nonzero. In this study, the truncation order r is set
so that r =1 is equal to 20 dB.
The received signals, fyj g, from the two wires are shownin Fig. 7. The amplitude of the received signals is represented in a gray-scale image.
The estimation results of PBF, LS, and tSVD are shown in
Fig. 8. The amplitude of the estimated scattering strength isrepresented in color in a linear scale. To enable the4816

Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

Y. ABE et al.

PBF
PBF with weighting

0
1
normalized amplitude

–20
–30

σi / σ

1

[dB]

–10

–40
–50
–60

without weightingwith weighting

–70
–80

(a)

0.8
0.6
0.4
0.2
0
–0.2

–1

–0.5

1 100 200 300 400 500 600 700 800 900
normalized amplitude

normalized
amplitude
1

63
63.5
64

–1

–0.5

4
–4
–2
0
2
receiving position xR j [mm]–16Fig. 7. Received signals fyj g from two wire targets.
46.4
46.8

0

0.5

1

PBF
tSVD
tSVD with weighting

64.5
65
–6

1

(b)

0
normalized amplitude

time from the trigger [ µs]

62.5

1
0.8
0.6
0.4
0.2
0
–0.2

0.5

PBF
LS
LS with weighting

order iFig. 6. Singular value distribution of propagation matrix P. The transmitted pulse is not considered.

0

1
0.8

(c)

0.6
0.4
0.2
0
–0.2

–1

–0.5

0

0.5

1

lateral position x m [mm]

47.2

(a)

depth z n [mm]

47.6
46.4

(b)

normalized
amplitude
1

Fig. 9. Proﬁle plots of the results shown in Fig. 8 at the depth of zn ¼
47:06 mm. (a) PBF estimation, (b) LS estimation, (c) tSVD estimation,with PBF estimation without weighting.

46.8
0

47.2

(c)

47.6
46.4

(d)
–1

46.8
47.2

(e)

47.6
–1

0

0
1 –1
lateral position x m [mm](f)1

Fig. 8. Results in the case of neglecting the transmitted pulse. (a) PBF^ PBF without weighting (wj;m ¼ 1). (b) Weighted PBF. (c) LSestimation O^ LS without weighting (wj;m ¼ 1). (d) Weighted LS. (e) tSVD
estimation O^ tSVD without weighting (wj;m ¼ 1). (f) Weighted tSVD.
estimation O

The proﬁles of these estimations along the x-axis at thedepth of zn ¼ 47:06 mm are shown in Fig. 9. The proﬁle of
the PBF with weighting becomes smoother than that without
weighting in Fig. 9(a). On the contrary, LS and tSVD
estimations do not become smooth with weighting, as shownin Figs. 9(b) and 9(c). The tSVD estimation has two
signiﬁcant peaks, which correspond to the positions of thewires. Furthermore, the diﬀerence between the peak values
and the lowest value between peaks is larger than that in
PBF, and the weighting enhances the diﬀerence. Therefore,
the tSVD method is stabler than the LS method, and theperformance of the tSVD with weighting is better than thatof the PBF for the separation of the neighboring scatterers.
4.2

comparison of diﬀerent estimations, these estimation results
are normalized by the maximum amplitudes of the respective estimates. The two wires are set at almost the same
depths because the wavefronts in Fig. 8(a) are aligned at a
certain depth. As shown in Figs. 8(c) and 8(d), the LS
estimation is strongly distorted due to the inﬂuence of thesmaller singular values in inverting propagation matrix P.
^ LS isFigures 8(e) and 8(f) show that the distortion of O
suppressed using tSVD.

The case of considering the waveformof the transmitted pulse
Now we show the estimation results when the waveformof the transmitted pulse sðkÞ is considered. The transmitting
pulse sðkÞ is measured by receiving an echo from a glass
plate. The measured waveform is shown in Fig. 10. The
singular value distribution of matrix P, in which sðkÞ isconsidered, is shown in Fig. 11. Compared with Fig. 6, thehigher-order singular values become very small because sðkÞis a narrow band signal in comparison with the impulse

4817Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

Y. ABE et al.

transducer

PBF
tSVD
tSVD with weighting

normalized amplitude

1

water
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1

0.5

0

1

1.5
time t [µs]

2

2.5

0
–10
[dB]

–20
–40
–50
–60
–70
–80

without weightingwith weighting

normalized amplitude

1–30

σi / σ

0.4
0.2
0
–0.5
0
0.5
lateral position x m [mm]1Fig. 13. Proﬁle plots of the results shown in Figs. 12(c) and 12(d) at thedepth of zn ¼ 46:89 mm.

Fig. 10. Waveform of transmitted pulse sðkÞ.

1 100 200 300 400 500 600 700 800 900
order iFig. 11. Singular value distribution of propagation matrix P. The transmitted pulse is considered.

46.4
depth z n [mm]

0.6

–0.2
–1

normalized amplitude

normalized amplitude

glass

0.8

46.8

normalized
amplitude

47.2

(a)

47.6
46.4

(b)

1

47.2

1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1

without weightingwith weighting

(a)
46.6

46.8

47

47.2

47.4

47.6

without weightingwith weighting

(b)
46.6

46.8

47

47.2

47.4

47.6

depth z n [mm]Fig. 14. Proﬁle plots of the tSVD estimation along z-axis at xm ¼ 0:4
mm. (a) Results in the case of neglecting the transmitted pulse shown in
Figs. 8(e) and 8(f). (b) Results in the case of considering the transmitted
pulse shown in Figs. 12(c) and 12(d).

0

46.8

1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1

–1

(c)

47.6
–1

0

1 –1
lateral position x m

(d)
0
[mm]

1

Fig. 12. Results in the case of considering transmitted pulse. (a) LS^ LS without weighting (wj;m ¼ 1). (b) Weighted LS. (c) tSVD
estimation O^ tSVD without weighting (wj;m ¼ 1). (d) Weighted tSVD.
estimation O

assumed in §4.1. As in §4.1, the truncation order r is set to
be a value of the order at r =1 ¼ 20 dB.
The estimation results of LS and tSVD are shown in
Fig. 12. The distortions of LS estimations shown in
Figs. 12(a) and 12(b) are stronger than those in Figs. 8(c)and 8(d). As shown in Figs. 12(c) and 12(d), the strong noise
ampliﬁcation is suppressed using tSVD. The proﬁles of
estimations of tSVD along the x-axis at the depth ofzn ¼ 46:89 mm are shown in Fig. 13 compared with the PBF
proﬁle shown in Fig. 9(a). Using tSVD and weighting, ahigher lateral resolution was achieved.
The proﬁles along the z-axis at xm ¼ 0:4 mm ofFigs. 8(e), 8(f), 12(c), and 12(d) are shown in Figs. 14(a)and 14(b), respectively. The transmitted pulse component isdirectly appears in Fig. 14(a). On the other hand, the pulse
duration becomes short when considering sðkÞ in matrix P, as
shown in Fig. 14(b) [the estimate became close to an
impulse (= scattering strength)].

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Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

Y. ABE et al.

–10center frequency of 10 MHz, which corresponds to theimprovement in the axial resolution by the convolution of
the transmitted pulse and the impulse response.

–20

5.

–30

In this study, we constructed an experimental setup for thesimulation of multichannel simultaneous reception of echo
signals by a linear array transducer using two single-element
ultrasonic transducers. The echo signals from two ﬁne wires
were acquired, and the scattering strength distribution in the
ROI was estimated. Compared with the PBF, the tSVD
method with weighting was eﬀective in separating theneighboring scatterers. The results showed a potential of the
proposed method for B-mode imaging with a high spatial
resolution at a high frame rate. It is considered that the tSVD
method is eﬀective for NDE applications. In medical
applications, the SNR is low and a lower-order truncation,
which leads to a degradation of spatial resolution, isnecessary.

0
power spectrum [dB]

(a)

–40
–50
–60

0

5

10

15

20

25

0
power spectrum [dB]

(b)
–10
–20
–30
–40
–50
–60

without weightingwith weighting
0

5

10

15

20

25

power spectrum [dB]

0

(c)

–10
–20
–30
–40
–50
–60

Conclusions

without weightingwith weighting
0

5

10
15
frequency [MHz]20

25

Fig. 15. (a) Power spectrum of the transmitted pulse sðkÞ. (b) Power
spectra of the proﬁles shown in Fig. 14(a). (b) Power spectra of the
proﬁles shown in Fig. 14(b).

The power spectra of these proﬁles along the z-axis areshown in Fig. 15. Figure 15(a) shows the power spectrum of
the transmitted pulse sðkÞ. The bandwidth of the estimate
without considering sðkÞ shown in Fig. 15(b) is similar tothat of in Fig. 15(a). As shown in Fig. 15(c), the bandwidthof the estimate considering sðkÞ is broadened around the1) S. I. Nikolov and J. A. Jensen: IEEE Trans. Ultrason. Ferroelectr. Freq.
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