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Improving axial resolution of medical ultrasoundimages by using noise-robust broadband filter
based on singular value decomposition
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Japanese Journal of Applied Physics 61, SG1061 (2022)
https://doi.org/10.35848/1347-4065/ac5a2cImproving axial resolution of medical ultrasound images by using noise-robust
broadband filter based on singular value decomposition
Kenta Kawamata1, Shohei Mori2* , Mototaka Arakawa1,2 , and Hiroshi Kanai1,2
1Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan2*

E-mail: shohei.mori.a2@tohoku.ac.jp

Received November 2, 2021; revised February 25, 2022; accepted March 2, 2022; published online May 31, 2022
Improving spatial resolution is a crucial issue in medical ultrasound. One of the improving methods is the post-processing of the received
ultrasound RF signal. In the present paper, we proposed a design method for a noise-robust broadband filter based on the singular value
decomposition of the received RF signal. To design a noise-robust filter, we proposed a logical method to determine the optimal truncated order ofsingular values, which was validated by applying the filter to noise-contaminated signals. Furthermore, the proposed filter applied to the wire
phantom resulted in a better axial resolution than that obtained without the filter and with our previously designed Wiener filter.
© 2022 The Japan Society of Applied Physics

1. Introduction

Ultrasound diagnosis is superior to other diagnostic modalities in terms of real-time use and non-invasiveness, and it isinexpensive. Therefore, it is useful in repeated diagnoses.
However, the image resolution of diagnostic ultrasoundsystems is more degraded than the other modalities; therefore, improving the spatial resolution has been an important
issue in the observation of tissue structures in vivo. Recently,
ultrasound systems have been applied for the observation of
microscopic biological tissues such as the structures of
myocardial ﬁber.1–4) Improving the spatial resolution in
ultrasound imaging will expand the possibilities of ultrasounddiagnosis. The determinants of the spatial resolution in
ultrasound systems depend on the bandwidth around thecenter frequency and the aperture size of the transducer. The
axial resolution of a general-purpose ultrasound system ismainly determined by the ultrasound wavelength; for example, it is approximately 0.4 mm when four cycles of a
7.5 MHz pulse wave are typically transmitted.
Many methods have been developed to improve spatial
resolution. One of these methods is the spatial compounding
method.5–7) This is an image-synthesizing method that uses
different images obtained by emitting ultrasound beams withdifferent angles to the same region. Therefore, the method
requires the acquisition of multiple images, which deteriorates the temporal resolution.
Many other methods for improving the spatial resolution insignal post-processing have also been proposed. For example,
pulse compression,8,9) frequency-domain processing,10,11) deconvolution techniques,12–17) and inversed ﬁltering.18–25) In
our previous study,26) Kageyama et al. designed a Wiener
ﬁlter27) by estimating the signal-to-noise ratio (SNR) from thereceived RF signals in the frequency domain. As the result, an
improvement in the measurement accuracy of the intimamedia complex thickness of the posterior wall of the carotid
artery has been attained.
In the present study, we propose a method to design anoise-robust broadband ﬁlter based on the singular value
decomposition (SVD), which is a type of principal component
analysis and is often compared with digital Fourier transform(DFT), which is a linear decomposition in the frequency
domain. SVD can be used for various purposes because thesingular value distribution obtained by SVD represents thefeatures of the received RF signal. SVD has been widely usedin medical ultrasound, especially in the ﬁeld of blood ﬂow
imaging.28–36) When SVD is used to identify artifacts or blood
ﬂow components from the acquired signal,37–39) the spatial
information of the signal and the temporal information in theframe direction are separated. In the present study, because thepurpose is to improve the axial resolution of medical
ultrasound images, only the information of the signal in the
beam (depth) direction for a single frame is used for SVD.
When SVD is used for broadband ﬁltering, the singularvalues must be truncated in an appropriate order to suppress
noise components. The truncated singular value decomposition (TSVD) has been widely used in medical ultrasound, forexample, to reduce artifacts37) and to identify blood ﬂow
components.38,39) However, the determination of the truncatedorder of singular values remains a difﬁcult challenge. Various
methods have been proposed to determine the truncated orderdepending on the purpose of using TSVD; the L-curve
method40,41) and the generalized cross-validation method42)
are commonly known as the properties of Tikhonov’s
regularized solution and have been widely used not only inthe ultrasonic ﬁeld. The stored energy of singular values33)
and the covariance matrix of singular values28) are often used
to identify blood ﬂow components and reduce artifacts.
We aim to design a broadband ﬁlter that is robust to noise.
Therefore, we examined a method for logically determining
the truncated order of the singular values according to the
SNR through simulation experiments in which noise iscontaminated on the waveform acquired for the wire.43)
However, in our previous study,43) the proposed method
for the ﬁlter design has not been fully validated, that is, thepropagation attenuation of ultrasound in each ﬁltering target
has not been considered, and the method for evaluating the
SNR of the target has not been proposed.
In the present paper, we proposed a design method for anoise-robust ﬁlter for high-axial resolution observation using
TSVD and proposed a logical method to determine theoptimal truncated order of the singular values. The designed
ﬁlter was applied to the ultrasound received signals from thewire phantom, and the axial resolution due to the ﬁlter
performance was quantitatively evaluated by the full width at
half maximum (FWHM).

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where f [MHz] denotes the frequency. The received signal
ywire, d (n ) from the wire is Fourier transformed, and its
frequency spectrum Ywire, d ( f ) is compensated by Ad ( f ) in
the frequency domain as

2. Principles and filter designing
2.1. Measurement of the transfer function of theultrasonic transmission/reception system

A block diagram of the ultrasonic transmission/reception
system is shown in Fig. 1. When a transfer function hd (n ) isused in the ﬁltering process, a point spread function (PSF) isoften deﬁned as the transfer function18,20) based on thefollowing principle, where n denotes the discrete time. From
Fig. 1, the observed received signal yd (n ) from an object at
depth d [mm] is expressed as follows:
yd (n) = z d (n) + ed (n)= xd (n) * hd (n) * ad (n) + ed (n) ,(1 )where * denotes the convolution operation; z d (n ) is thereceived signal without contamination owing to the noise
component ed (n ); xd (n ) is the true reﬂectivity, and ad (n ) is
the propagation attenuation of ultrasound for the measurement target.
When the target is a wire positioned at a depth d in water,
the SNR can be assumed to be sufﬁciently high. Furthermore,when the diameter of the wire is sufﬁciently small comparedwith the wavelength of the transmitted wave, xd (n ) can beassumed to be an impulse d (n ). That is,
ed (n) » 0,
x d (n ) = d (n - d ).

(2 )
(3 )

By substituting Eqs. (2) and (3) into Eq. (1),ywire, d (n) = zwire, d (n)= d (n - d ) * h d (n ) * a d (n )= h d (n - d ) * a d (n ) ,(4 )¢ , d ( f ) = Ywire, d ( f ) · Ad ( f ) ,
Ywire

(6 )

¢ , d ( f ) is the corrected frequency characteristics ofwhere Ywire
the RF signal of the wire, and Ad ( f ) is given by44)
Ad ( f ) = e-8.686 · A · d · f= 10-A · d · f ,(7 )where 8.686 = 20 (log10 e), and A [dB/ (mm·MHz)] is theattenuation coefﬁcient in the medium along to the pathway tothe measurement target. Then, the inverse Fourier transform¢ d (n ) as
is applied to obtain the compensated RF signal ywire,y¢ wire, d (n) = IDFT [Y ¢ wire, d ( f )].

(8 )¢ d (n) in a vectorNext, to convolve the ﬁlter with ywire,
matrix form, let us deﬁne a (2N + 1) ´ (2M + 1) matrix¢ d (n ) is converted into an analytic signal by
Ywire, d , where ywire,¢ d (n )} are arranged
the Hilbert transform in advance and {ywire,by shifting the sample points one by one, as shown in Fig. 2.
Thus, the element at the i th row and j th column(0  i  2N , 0  j  2M , N > M ) of Ywire, d is given by¢ d ( p - M - N + i + j),
Ywire, d (i , j ) = ywire,(9 )¢ d (n ). To design
where p is the positive peak position of ywire,the (2M + 1)-dimensional ﬁlter vector fd , the height 2N + 1
of Ywire, d should be set to a size that allows the ﬁlter to be
convolved with the entire range of echoes from the wire at a
depth d, that is, M + Lh + 1 < N , where 2Lh + 1 is thewhere ywire, d (n ) and zwire, d (n ) correspond to yd (n ) and z d (n )of Eq. (1), respectively. The PSF is deﬁned as the transfer
function hd (n ) and is obtained from the received signal
ywire, d (n ) as described below.
2.2. Designing of broadband filter based on SVD
We aim to design a ﬁlter so that the ﬁltered signal becomes
an impulse. However, this ﬁlter ampliﬁes the noise component at the same time. Therefore, we designed a ﬁlter toimprove the axial resolution while suppressing noise.
First, it is necessary to consider the propagation attenuationad (n ) for the received signal ywire, d (n ) from the wire at a
depth d since the received signal yd (n ) is convolved with¢ d (n) afterad (n ) as shown in Eq. (1). The received signal ywire,compensating for ad (n ) is obtained as follows:
The propagation attenuation ad (n ) is depth-dependent andexhibits a frequency response. The frequency response of the
propagation attenuation, Ad ( f ), is deﬁned by the Fouriertransform of ad (n ) as,
Ad ( f ) = DFT [ad (n)] ,(5 )Fig. 1. Block diagram of the ultrasonic transmission/reception system inthe time domain.

Fig. 2. (Color online) (a) Received RF signal of thin wire and (b) storage
method of Ywire, d .

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K. Kawamata et al.
(b)

(a)

Figure 3. (Color online) (a) Evaluation index bd, r (R ) of the noise robustness of the ﬁlter to determine the optimal truncated order and (b) determined
optimal truncated order R̂d, r of singular values for each SNR r at depth d = 30 mm.

effective length of the impulse response hd (n ), as shown in
Fig. 2.
As described at the beginning of Sect. 2.2, since we aim
for the ﬁltered signal to be impulsive, a (2N + 1)-dimensional model vector md of an ideal impulse is prepared as an
output model. Moreover, since the average power of thesignal should not be changed without or with the ﬁlter, theamplitude of the output impulse is deﬁned by⎧1⎪
m d (i ) = 2 M + 1
⎨
⎪0
⎩

p+M

¢ d (n)]2
å [ ywire,(i = N + 1 )n=p-M

singular values should be truncated by an appropriate orderconsidering this trade-off relationship so that the noise
components are suppressed and the resultant impulse component becomes a broad frequency range to improve the axial
resolution.
Thus, let us introduce a (2M + 1) ´ (2N + 1) matrix S-R 1
where the components higher than the order R are truncatedto zero in S-1 as follows:-1å R (i , j ) =.

(i ¹ N + 1 )(10)¢ d (n) around
That is, the root of the average power of ywire,n = p is set as the amplitude of md at the center position of
i = N + 1 in the row direction.
The ﬁlter vector fd , which is applied at depth d, is designed
to minimize the following squared error gd between the ﬁlteredoutput waveform Ywire, d fd from the impulse model vector md :gd = ∣ Ywire, d fd - m d ∣2 .

(11)

Using the least-squares method, the ﬁlter f̂d that minimizes gd
is given byfˆd = VS-1UHm d ,(12)where U and V are the (2N + 1) ´ (2N + 1) unitary matrixof the left singular vectors and the (2M + 1) ´ (2M + 1)
unitary matrix of right singular vectors, respectively, obtainedby SVD of Ywire, d , that is,
Ywire, d = USVH,(13)

where S is a (2N + 1) ´ (2M + 1) matrix given byS = ⎡D⎤,⎣O⎦(15)

Substituting S-R 1 into S-1 in Eq. (12), the (2M + 1)-dimensional ﬁlter f̂d , R is determined such that the singular values ofthe order higher than R are truncated asfˆd , R = Vå-R 1UHm d ,(16)which shows a ﬁlter that emphasizes only the components
below the truncated order R and suppresses the components
above R.
Let us deﬁne the (2N + 1) ´ (2M + 1) matrix Ytarget, d ,which stores the received RF signals to be ﬁltered in the same
way as Ywire, d in Eq. (9). By applying the ﬁlter f̂d , R designed
around depth d, the (2N + 1)-dimensional output vector y’out, dis given by¢ d = Ytarget, d fˆd , R.
yout,

(17)

To compensate for the sound pressure characteristics of the¢ d is normalized byreceived RF signal owing to the depth, yout,
the spatially averaged power around d of the received RF
signal yd as⎛ E [∣ yd ∣2 ] ⎞ ¢yout, d = ⎜y.
¢ d ∣2 ] ⎟ out ¢ d
E [∣ yout,
⎝
⎠

(14)

D is a (2M + 1) ´ (2M + 1) diagonal matrix withs1, s2,...,s2M + 1
fordiagonal
elements,
andthe[2 (N - M )] ´ (2M + 1) matrix O is a zero matrix. The j
th diagonal element sj of D is a singular value, and is apositive real number ordered as s1  s2    s2M + 1  0.
When all orders of the singular values are used for the ﬁlter
design, the resultant ﬁlter f̂d becomes the ﬁlter obtained fromthe inverse of the PSF and the axial resolution with the ﬁlter
improves. At the same time, however, a ﬁlter with such hightruncate order is unstable to noise because the high-ordersingular values are dominated by noise components. Thus, the⎧1 /si, j (1  i = j  R)
0
(R + 1  i = j  2 M + 1 ) .
⎨0(i ¹ j )⎩(18)

The (2N + 1) L -dimensional ﬁnal output waveform vector
yout along the ultrasonic beam is obtained by arranging{yout, d } obtained for L sections along the depth direction asy⎡ out, d1 ⎤⎢ yout, d 2 ⎥yout = ⎢, ⎥⎢y⎥⎣ out, dL ⎦(19)

where L denotes the number of sections.

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Fig. 4. (Color online) (1) Waveforms without the ﬁlter, (2) output waveforms when the singular value was used with the full rank, and (3) output waveforms
when the singular value was truncated at the optimal order by the proposed method, where (a) noise was not contaminated, (b) r = 50 dB, and (c) r = 20 dB
at depth d = 30 mm. (The underlined numbers in the ﬁgure indicate the FWHM.)

To consider the SNR r, the noise-contaminated signal
ynoisy, d , r, q (n) is generated by adding the Gaussian white noise¢ d (n ):er, q (n ) on ywire,¢ d (n) + er, q (n) ,ynoisy, d , r, q (n) = ywire,(20)

where Q sets of the white noises {er, q (n )} were independently
generated on the computer for each SNR r, and q (q = 1 ~ Q)shows the set number of the generated white noises. Let us
prepare a (2N + 1) ´ (2M + 1) matrix Ynoisy, d , r, q which storesthe noise-contaminated signal ynoisy, d , r, q (n) in the same way as
the matrix Ywire, d of Fig. 2(b) as
Ynoisy, d , r, q (i , j ) = ynoisy, d , r, q ( p - M - N + i + j ).
Fig. 5. (Color online) Structure of the wire phantom.

(21)

Then, we deﬁne the set of the following index bd, r (R) for
each depth d and each SNR r to evaluate the noise robustnessof the designed ﬁlter:2.3. Logical method for determining the truncatedorder of singular values

bd , r (R) = E q[∣ Ynoisy, d , r, q fˆd , R - m d ∣2 ] ,As described in Sect. 2.2, the high-order singular values of
Ywire, d are dominated by noise components. Thus, we propose
a logical method to determine the optimal truncated orderR̂d,r of the singular values considering the SNR rtarget of themeasurement target at a depth d as follows:(22)

where Eq [·] shows the average operation on q, and bd, r (R)evaluates the mean squared difference of the result ofapplying the ﬁlter f̂d , R to Ynoisy, d , r, q from the ideal impulse
model md . The designed ﬁlter f̂d , R is robust to noise if bd, r (R)is small. Thus, the optimal truncated order R̂d,r is determined

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to minimize bd, r (R) as follows:Rˆ d , r = argmin bd , r (R).

(23)

R

In this procedure, R̂d,r is independently determined for each
depth d and each SNR r.
Gaussian white noise was generated on the computer using
the Box–Muller method,45) and the number Q of trials togenerate white noise was set to 200. The power of thegenerated white noise er, q (n ) in Eq. (20) was set such that the
SNR r ranged from 0 to 70 dB with 0.05 dB interval and R̂d,rwas determined for each r.
Figure 3(a) shows the evaluation index bd, r (R) of the noise
robustness of the ﬁlter calculated for wires at depth
d = 30 mm. As shown in Fig. 3(a), bd, r (R) rapidly increased
for R beyond R̂d,r. This conﬁrms that the ﬁltered result wassensitive to noise and it was important to determine theoptimal truncated order R̂d,r in the ﬁlter design.
Figure 3(b) shows the optimal truncated order R̂d,r forvarious SNR r at depth d = 30 mm. As shown in Fig. 3(b),R̂d,r approximately linearly shifted to lower orders as the
SNR r logarithmically decreased. Thus, it is important totruncate the singular values at different orders R̂d,r according
to the SNR r.
2.4. Method for evaluating SNR in actual caseIn the proposed method, the truncated order R̂d, rtarget isdetermined locally and adaptively according to the SNR
rtarget of the ﬁltering target. In this section, we describe amethod for the evaluation of the SNR. In the present paper,by assuming that the ﬁltering target is motionless between
frames, we evaluate the SNR r (n , k ) for the received RF
signal yj, k (n ) of the j th frame along the k th beam to beﬁltered as follows:r (n , k ) = 10 log10

PS (n , k )[dB] ,P N (n , k )

(24)

where
PS (n , k ) =n+M1å (E j[yj,k (n¢)])2 ,2M + 1 n¢= n - M

P N (n , k ) =n+M1å E j[(yj,k (n¢)
2M + 1 n¢= n - M- E j [yj, k (n¢)])2 ].

(a)

(25)

(26)

This SNR evaluation is also used for the preliminary
determination of the optimal truncated order R̂d,r ofEq. (23) for each depth d and SNR r. In the ﬁltering
procedure, rtarget (n , k ) is estimated by Eq. (24), and thenR̂d, rtarget is determined from Eq. (23) as the optimal truncatedorder for ﬁltering. When the ﬁlter is applied to an object thatmoves between frames, the SNR evaluation of Eq. (24)should be replaced, which is our future work.
3. Experiments and results
3.1. Filtering for received signal from wire in water
(b)

(c)

Fig. 6. (Color online) (a) B-mode image of the wire phantom, (b)evaluated result of SNR rtarget , and (c) optimal truncated order R̂d, rtarget
estimated based on the SNR r.

To evaluate the robustness of the designed ﬁlter, the designed
ﬁlter f̂d , R of Eq. (16) was applied to the received signal Ywire, dfrom the wire in the water. A linear probe connected toultrasound diagnostic equipment (SSD-6500; ALOKA,
Japan) was used. The center frequency of the transmitted
wave was 7.5 MHz, and the sampling frequency was
40 MHz. The focus was set to a depth of 20 mm. The matrix
size of Ywire, d in the ﬁlter design was set to
2N + 1 = 101 (1.9 mm) and 2M + 1 = 53 (1.0 mm), thatis, the ﬁlter length was 53 (1.0 mm). The attenuation in water
can be assumed to be zero in the frequency range and depthof the ultrasound used in this experiment.46)
Figure 4(1) shows the received RF signal ywire, d (n ) from awire and the noise-contaminated signals ynoisy, d ,50,1 (n ) andynoisy, d ,20,1 (n ) with SNRs of 50 and 20 dB at depth
d = 30 mm, respectively. Figure 4(2) shows the waveforms
with the ﬁlter f̂d , R obtained with full rank (R = 53) of
singular values. Figure 4(3) shows the waveform with ﬁlter
fˆd , Rˆ designed by truncating the singular values at the optimal
truncated order R̂d,r. Each envelope is plotted as a black line,
and the FWHM is shown in the ﬁgure.
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Fig. 7. (Color online) (1) B-mode images and (2) the envelopes of the red dashed line (a) without the ﬁlter, (b) with the Wiener ﬁlter, (c) with the proposedﬁlter without considering attenuation, and (d) with the proposed ﬁlter considering attenuation. (The underlined numbers in the ﬁgure indicate the FWHM).

When the noise was not contaminated [Fig. 4(a)], thesignal from the wire became an ideal impulse by using thefull rank of the singular values [Fig. 4(a-2)]. When the noise
was contaminated with r = 20 dB [Fig. 4(b)], the wire could

not be observed at all when the full rank of the singularvalues was used [Fig. 4(c-2)]. For r = 50 dB and using thefull rank of singular values [Fig. 4(b-2)], the wire could beobserved; however, a large noise component appeared in theSG1061-6© 2022 The Japan Society of Applied Physics

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Table I. FWHM of the enveloped signals without and with the ﬁlter in
Fig. 7, and the improvement rate of FWHM compared to the original for the
beam C.
FWHM [mm](a) Original(b) By Wiener ﬁlter(c) By proposed ﬁlter
without considering
attenuation(d) By proposed ﬁlter
fˆ30, Rˆ with considering
attenuation

Shallower wire(d = 28.6 mm )Deeper wire(d = 29.6 mm )0.260 (–)
0.227 (13%)
0.244 (7%)

0.273 (–)
0.202 (27%)
0.267 (3%)

0.132 (49%)

0.141 (49%)

Fig. 9. (Color online) The FWHM of the enveloped signals obtained by the
proposed ﬁltering for each SNR h.
Table II. FWHM of the enveloped signals without and with the proposedﬁlter in Fig. 7, and the improvement rate of FWHM compared to the originalfor the beams A–F.
FWHM [mm]Beam No.
A
B
C
D
E
F

Shallower
DeeperShallower
DeeperShallower
DeeperShallower
DeeperShallower
DeeperShallower
Deeper(a) Original(b) Proposed ﬁlter fˆ30, Rˆ

0.256 (–)
0.270 (–)
0.267 (–)
0.250 (–)
0.260 (–)
0.273 (–)
0.268 (–)
0.277 (–)
0.348 (–)
0.279 (–)
0.270 (–)
0.277 (–)

0.116 (55%)
0.134 (51%)
0.134 (50%)
0.123 (51%)
0.132 (49%)
0.141 (49%)
0.156 (42%)
0.165 (41%)
0.144 (59%)
0.154 (45%)
0.128 (53%)
0.161 (42%)

surrounding area. On the other hand, even when the noise
was contaminated, by truncating the singular values with the
optimal truncated order R̂d,r , the signal from the wire could
be observed [Figs. 4(b-3) and 4(c-3)].
By comparing the axial resolution based on the FWHM, it
was conﬁrmed that the axial resolution was improved for all
SNRs by ﬁltering with the optimal truncated order R̂d,r[Fig. 4(3)] compared to that with no ﬁltering [Fig. 4(1)]. As
the SNR r became low, the singular values must be truncatedat a lower order. Therefore, when comparing the axial
resolution of Figs. 4(b-3) and 4(c-3), the FWHM ofFig. 4(c-3) was 0.109 mm, which was broader than the
FWHM of Fig. 4(b-3) of 0.054 mm. However, even in
Fig. 4(c-3), the wire position could be conﬁrmed, and the

(a)

(b)

(a)

(b)

Fig. 8. (Color online) The envelopes (a) without ﬁltering and (b) with the proposed ﬁltering for the beams (1) A and (2) F in Fig. 5.

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Fig. 10. (Color online) The envelope of the wire echo obtained by the
proposed ﬁltering for each of h = ¥ , 40, 30, 20, and 10 dB.

axial resolution was improved compared with Fig. 4(c-1) [theFWHM of Fig. 4(c-1) was 0.193 mm]. Thus, the ﬁlter
designed by the method in Sects. 2.2 and 2.3 is robust to
noise and improves the axial resolution.
3.2. Filtering for received signal from phantom
To evaluate the usefulness of the designed ﬁlter fˆd , Rˆ , the ﬁlter
was applied to a general purpose ultrasound phantom [Model
054GS by CIRS (USA)]. As shown in Figs. 5, 13 wires were
lined up inside the phantom, and the beams A–F shown by
the red arrows in Fig. 5 were used for evaluation. The two

wires were 0.25 mm apart for beam A and 4.0 mm apart for
beam F, respectively. The wire material was nylon with adiameter of 80 μm. The attenuation coefﬁcient A of thephantom was 0.07 dB/ (mm · MHz).
When applying the ﬁlter, we ﬁrst evaluated the SNR rtargetof the target, and then we determined the truncated orderR̂d, rtarget by referring to the optimal truncated order R̂d,r predetermined for each depth d and SNR r as described in
Sect. 2.3. The ﬁlter fˆd , Rˆ designed at a depth of d = 30 mm
was applied to the phantom data.
Figure 6 shows (a) the B-mode image of the wire phantom,(b) the evaluated SNR rtarget for each depth along each beam,and (c) the determined optimal truncated order R̂d, rtarget foreach depth and beam, respectively. In Fig. 6(b), the SNR washigh at the position where the wires existed because thesignal intensities from wires were strong. As the depth
became deeper, the SNR decreased due to the attenuationof ultrasonic waves. Thus, the SNR was conﬁrmed to bereasonably evaluated using Eq. (24).
In Fig. 6(c), the optimal truncated orders R̂d, rtarget of thesingular values were large at the positions where the SNR
was high. This means that the ﬁlter was designed using alarge number of singular values for regions with a high
SNR. Therefore, by determining R̂d, rtarget for each depth dand each SNR rtarget using our proposed method, the ﬁlter
was designed by considering the trade-off relationshipbetween the axial resolution improvement and noise suppression.
Figure 7 shows (1) the B-mode images and (2) the
envelopes on the red dashed line in the B-mode images(a) without the ﬁlter, (b) with the Wiener ﬁlter, (c) withthe proposed ﬁlter without considering attenuation[ad (n ) = d (n)], and (d) with the proposed ﬁlter fˆ30, Rˆ byconsidering attenuation, respectively. In the B-mode images,
the axial resolutions with the ﬁlters (b) and (d) were
improved compared to those without the ﬁlter. The
FWHM of the envelope of the wire echo in Fig. 7(2) isshown in Table I. The axial resolution of the shallower wire
at a depth of 28.6 mm was improved by 13% (from 0.260 to
0.227 mm) with the Wiener ﬁlter, while it was improved by
49% (from 0.260 to 0.132 mm) with the proposed ﬁlter byconsidering attenuation. For the deeper wire at a depth of
29.6 mm, the Wiener ﬁlter improved the axial resolution by
27% (from 0.273 to 0.202 mm), while considering attenuation in the proposed ﬁlter improved it by 49% (from 0.273
to 0.141 mm). Thus, it was conﬁrmed that the proposedﬁlter with considering attenuation improved the axial
resolution. In the proposed ﬁlter without considering attenuation [Fig. 7(c)], the axial resolution of both wires
was slightly improved (7% and 3%), which shows that thecorrection of the high-frequency components is essential
to widen the frequency characteristics in the ﬁlter design.
As shown in Figs. 7(c-2) and 7(d-2), with ﬁltering, somepeaks other than those at the wire positions were observed
such as at a depth of 29 mm. However, they were alreadyobserved without ﬁltering, as shown in Fig. 7(a-2). The
cause of this observation will be investigated in more detailin the future.
To observe the improvement of the axial resolution by the
proposed ﬁlter fˆ30, Rˆ in more detail, the FWHM and its
improvement rate are shown in Table II. The envelopes

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K. Kawamata et al.

(a)

(b)

Fig. 11. (Color online) (a) Results of the SNR evaluation and (b) the determined optimal truncated order at h = 30 dB.

without and with ﬁltering for the beams A and F are shown in
Fig. 8. As shown in Table II, the improvement in axial
resolution was nearly 50% for all beams, and the discrimination capability of wires was improved as shown in
Figs. 8(1-b) and 8(2-b).
3.3. Evaluation of the noise robustness of filter
To conﬁrm the noise robustness of the proposed ﬁlter, the RF
signals received from the phantom in Sect. 3.2 were
contaminated with white noise generated on the computer.
The power of the noise was set such that the ratio h of thesquared average brightness of the shallower wire echoes
along the red dashed line in Fig. 7(a-1) to the power of thenoise was from 10 to 50 dB (intervals: 5 dB).
Figure 9 shows the FWHM of the wire echo for all h , andFig. 10 shows the envelope of the wire echo forh = ¥ , 40, 30, 20, and 10 dB. As in the underwater wire
experiment in Sect. 3.1, the FWHM decreased as the noise
superposition level increased, but for SNRs, the axial
resolution was improved compared to that without the ﬁlter.
Figures 11(a) and 11(b) show the results of the SNR
evaluation and the determined optimal truncated order,
respectively, for h = 30 dB. By comparing Figs. 11(a) and11(b) for h = 30 dB to Figs. 6(b) and 6(c) for h = ¥ , theestimated SNR of Eq. (24) decreased as the power of thenoise increased, and the optimal truncated orders R̂d,rdecreased.
Figure 12 shows the B-mode images (a) without theﬁlter, (b) with the Wiener ﬁlter, and (c) with the proposedﬁlter for h = 30 dB, respectively. The FWHM of theenvelope of the wire, shown by the red arrow on each Bmode image, is summarized in Table III, where the values
in parentheses show the improvement in the axial resolution from that without ﬁltering. In the evaluation of the Bmode images and quantitative evaluation by FWHM, it was
conﬁrmed that the proposed ﬁlter increased the axial
resolution.
As shown in Table III, the improvement rates of the axial
resolution for the proposed ﬁlter decreased owing to thelowering of the truncated order, which was associated with alower SNR; however, the improvement in axial resolution
was maintained at nearly 40%. However, when the Wiener
ﬁlter was applied, the improvement rates of the axial
resolution signiﬁcantly decreased with a lower SNR, andthe rate was only 3% for h = 30 dB.

In this study, the optimal truncated order R̂d,r wasdetermined for each SNR r, and the proposed ﬁlter was
optimized for each SNR rtarget of the measurement target.
Therefore, this ﬁlter design can be applied to the low SNR
cases. On the other hand, although we designed the Wiener
ﬁlter considering the SNR of the target, the weighting control
parameter, which determined how much the SNR was
reﬂected in the ﬁlter design, was ﬁxed to a constant value
because a method to determine the parameter had not yet
been established.
3.4. Discussion about depth dependence of the
proposed filter
Figure 13 shows the superimposition of the received RF
signals {ywire, d (n )} acquired by placing a tungsten wire with adiameter of 10 m m at a depth d in water and moving the wirein the range from d = 5 to 55 mm at an interval of 1 mm.
The focus was set to a depth of 20 mm. The amplitudes of thereceived signals {ywire, d (n )} used in the ﬁlter design changedin the depth direction due to the sound pressure characteristics.
In the present paper, the ﬁlter was designed only for a
depth of d = 30 mm to focus on the evaluation of the noise
robustness of the proposed ﬁlter. Although the transfer
function of the ultrasonic transmission/reception system isdepth-dependent as shown in Fig. 13, the principle of the
proposed ﬁlter design can be extended and applied at
different depths.
4. Conclusions

In the present paper, we proposed a broadband ﬁlter based on
TSVD to improve the axial resolution of medical ultrasoundimages. We proposed a logical method to determine thetruncated order of the singular values based on the SNR todesign a noise-robust ﬁlter. Basic experiments targeting wires
in water conﬁrmed that the proposed method could be used todesign a noise-robust ﬁlter. Then, the usefulness of the
proposed ﬁlter was conﬁrmed through an experiment using
a wire phantom as the measurement target. The axial
resolution of the proposed ﬁlter was improved by 36% on
average of the FWHM of the wires in the phantom comparedwith our previously designed Wiener ﬁlter under high SNR
conditions. Furthermore, phantom evaluations under lower
SNR conditions demonstrated the robustness of the proposedﬁlter.

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K. Kawamata et al.

(a)Fig. 13. Received RF signals {ywire, d (n )} from the wire in the water.

Acknowledgments

This work was supported in part by JSPS KAKENHI
19KK0100.

(b)

ORCID iDs

Shohei Mori https://orcid.org/0000-0002-5494-1055
Mototaka Arakawa
https://orcid.org/0000-0001-9386645X
Hiroshi Kanai https://orcid.org/0000-0002-6567-1687(c)Fig. 12. (Color online) B-mode images (a) without the ﬁlter, (b) with the
Wiener ﬁlter, and (c) with the proposed ﬁlter at h = 30 dB.

Table III. FWHM of the enveloped signals without and with the ﬁlter in
Fig. 12, and the improvement rate of FWHM compared to the original.

(a) Original(b) By Wiener ﬁlter(c) By proposed ﬁlter fˆ30, Rˆ

FWHM [mm]h = ¥ (no noise)h = 30 dB

0.260 (–)
0.227 (13%)
0.132 (49%)

0.266 (–)
0.259 (3%)
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