BRIEF NOTEJapanese Journal of Applied Physics 53, 07KF02 (2014)
http://dx.doi.org/10.7567/JJAP.53.07KF02

Comparison of spatial resolutions of parallel beamforming and diffraction tomography
in high frame rate echocardiography
Hideyuki Hasegawa1,2 and Hiroshi Kanai1,2*
1Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
E-mail: hasegawa@ecei.tohoku.ac.jp
2

Received November 28, 2013; accepted March 5, 2014; published online June 4, 2014
Echocardiography is a predominant modality for diagnosis of the heart by noninvasive and real-time observation of its cross-sectional images. In
addition, it has been recently shown that measurement of the rapid transition of myocardial contraction/relaxation and propagation of heart-wallvibration would be useful for assessment of myocardial function and viscoelasticity. However, such measurements require a higher frame rate ofseveral hundred hertz. Therefore, we realized high-frame-rate echocardiography using parallel beamforming with unfocused transmit beams. On
the other hand, there is another imaging method, namely, diffraction tomography. It has been reported that diffraction tomography realizes highspatial resolution. One of the problems of parallel beamforming with unfocused transmit beams is the degradation of spatial resolution; diffraction
tomography may alleviate this problem. In the present study, diffraction tomography was applied to high-frame-rate echocardiography with
unfocused transmit beams, and spatial resolutions realized by diffraction tomography and parallel beamforming were compared. Diffraction
tomography showed a spatial resolution (1.16 mm) similar to that (1.18 mm) of parallel beamforming in a relatively near region (44 mm in range
distance). In a deeper region (84 mm), diffraction tomography realized better spatial resolution (1.90 mm) than that (2.27 mm) of parallel
beamforming. However, still better spatial resolution (1.57 mm) was realized by parallel beamforming when it was used with a phase coherence
factor. The use of the phase coherence factor is not computationally intensive, and parallel beamforming with a phase coherence factor was shown
to be feasible in high-frame-rate echocardiography for the improvement of spatial resolution. © 2014 The Japan Society of Applied Physics

Recently, it has been reported that the propagation ofelectrical stimulation and spontaneous vibration of heart
muscle could be used for the evaluation of myocardial
viscoelasticity and viability.1,2) For the measurement of suchpropagation phenomena, higher temporal resolution ofseveral hundreds of hertz is necessary because the propagation speed is relatively high (up to several m/s).3) Also, it
has been shown that high-frame-rate echocardiography isuseful for the evaluation of myocardial function.4–6)
Various studies have been conducted on increasing theframe rate. Konofagou et al.7) and D’hooge et al.8) increased
the frame rate to above 200 Hz by reducing the size of theﬁeld of view and the total number of scan lines in an
ultrasonic image. Furthermore, Konofagou et al. introduced
an electrocardiogram (ECG)-gating technique in ultrasoundimaging to combine individual small sectors into a large ﬁeldof view.9) In this method, the lateral size of a sector(corresponding to the number of scan lines), which is
obtained in one acquisition, is narrowed to achieve a higher
frame rate of about 500 Hz. By measuring a number of small
sectors during the corresponding number of cardiac cycles,
the measured small sectors are combined into a large sector
format based on the ECG-gating. Although this method
achieves a frame rate of about 500 Hz, which is muchhigher than the conventional frame rate of several tens ofhertz, measurements for a number of cardiac cycles are
required.
To obtain the number of scan lines required to construct a
B-mode image with fewer transmissions, parallel beamforming has been proposed.10) Recently, parallel beamformingwith plane-wave transmissions has been intensively used intransient elastography11) with a linear array probe. However,in echocardiography, the size of the aperture is very limited
because it is necessary to insonify ultrasound from a narrow
acoustic window between ribs. To solve this problem, werealized high-frame-rate echocardiography by parallel beamforming with diverging wave transmissions, and an ultrasound image of the heart with a full lateral ﬁeld of view of 90
degrees was obtained at a very high frame rate of 316 Hz.12,13)

On the other hand, there is another imaging method,
namely, diffraction tomography,14,15) which is reported toyield ultrasound images with high spatial resolution. In thepresent study, spatial resolutions of parallel beamforming(PBF) and diffraction tomography (DT) were examined forthe improvement of spatial resolution in high-frame-rateechocardiography with unfocused transmit beams.
Currently, DT can be used with plane-wave transmission15,16) or synthetic aperture imaging.14) To use DT withdiverging wave transmission, a diverging wave must be
assumed to be a plane wave. Consequently, the diverging
angle should be less than 15 degrees, but images obtained by
DT with diverging beams were degraded compared withthose obtained by DT with plane waves.17) Therefore, in thepresent study, spatial resolutions of PBF and DT werecompared using a single plane-wave transmission.
In parallel receive beamforming,12,13) ultrasound echo
signals fsi(t)g received by individual transducer elements
are summed after applying appropriate delays dependingon the relative geometry between each element i ði ¼
0; 1; :::; L  1Þ and a point of interest p ¼ ðr; Þ. The timedelay ¸i(p) applied to the signal received by the i-thtransducer element is expressed as
i ðpÞ ¼ tr ðpÞ" #0:5
1 2
L1 2
2
þ
r cos  þ r sin   x i 
;
c0
2
ð1Þ
where c0 is the speed of sound and ¸tr(p) is the propagation
time delay of a transmit beam from the transducer array to thepoint of interest p. In the case of plane-wave transmission,¸tr(p) is expressed astr ðpÞ ¼

r cosð  Þ
;
c0

ð2Þ

where © is the direction of the transmit beam.
The beamformed RF signal g(p) at spatial point p is
obtained as

07KF02-1© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF02 (2014)

ð3Þ

5 mm

(b)

(c)

9.9 mmsðt  i ðpÞÞ:

0

5 mm

(a)
9.9 mm

5 mm
9.9 mmgðpÞ ¼L1X

H. Hasegawa and H. Kanai

Let us brieﬂy describe the principle of diffraction
tomography with plane-wave transmission.16) Assuming thescattering coefﬁcient ðx; zÞ at spatial point p ¼ ðr; Þ ¼ ðx; zÞdepends on the local ﬂuctuation of the bulk modulus andusing Born approximation, echo signal sðxm ; !Þ scattered atpoint p ¼ ðx; zÞ and received at position pm ¼ ðxm ; 0Þ of them-th transducer element is expressed as
ZZ 1sðxm ; !Þ ¼ k2
pi ðpÞ  ðpÞ  gðpm jpÞ dx dz;
ð4Þ
1

where k is the wavenumber (= ½/c0, ½: angular frequency).
In the present study, a plane wave was used as thetransmitted wave pi(p). A plane wave propagating in the
direction (qx, qz) is expressed aspi ðpÞ ¼ A  expfjðqx x þ qz zÞg;Fig. 1. B-mode images of phantom obtained by (a) parallel beamforming,(b) diffraction tomography, and (c) parallel beamforming with phasecoherence factor.

ð5Þ

jk2 A
¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2 k2  k2x

where A is an amplitude coefﬁcient. Assuming the scatteredwave from point p is a circular wave, which is expressed by
the Hankel function of the ﬁrst kind of order 0, the Greenfunction g(pm«p) is expressed as

ZZ 1ðx; zÞ
1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 exp½jfðkx  qx Þx þ ð k2  k2x  qz Þzg dx dz:
ð10Þgðpm jpÞ ¼ H0ð1Þ ðkjpm  pÞ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Zj 1 exp½jfkx ðxm  xÞ  k2  k2x zg
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ¼
dkx ; ð6Þ4 1
k2  k2x

The right-hand side of Eq. (10) corresponds to the 2-D
frequency spectrum ðkx  qx ; ðk2  k2x Þ1=2  qx Þ of thescattering coefﬁcient distribution gðx; zÞ. Therefore, Eq. (10)is expressed aswhere kx is the wavenumber in the x-direction. Equation (4)can be modiﬁed as
ZZ 1
jk2 A
ðx; zÞ  expfjðqx x þ qz zÞgsðxm ; !Þ ¼
41
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Z1
exp½jfuðxm  xÞ  k2  u2 zg
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

du dx dz:
k2  u2
1
ð7Þ

ð11Þ

By applying the Fourier transform to sðxm ; !Þ with respect
to xm, the 2-D frequency spectrum Sðkx ; !Þ of echo signals
received by individual transducer elements is expressed as
Z1sðxm ; !Þ  expðjkx xm Þ dxmSðkx ; !Þ ¼
1
ZZ 1
jk2 A
¼
ðx; zÞ  expfjðqx x þ qz zÞg4
1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Z1
expfjðux þ k2  u2 zÞg
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k 2  u2
Z1
1
expfjðu  kx Þxm g dxm du dx dz:
ð8Þ
1

Using Dirac’s delta function ¤(0)
Z1
expfjðkx  uÞxm g dxm ¼ 2ðkx  uÞ;
1Equation (8) is modiﬁed as
ZZ 1
jk2 A
Sðkx ; !Þ ¼
ðx; zÞ  expfjðqx x þ qz zÞg2
1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
expfjðkx x þ k2  k2x zÞg
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dx dz
k2  k2x

ð9Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
k2 ASðkx ; !Þ ¼ j pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  ðkx  qx ; k2  k2x  qx Þ:22
2 k  kx

By applying the inverse Fourier transform to Eq. (11), thescattering coefﬁcient distribution gðx; zÞ is estimated.
An ultrasound imaging phantom (CIRS 54GS) was
measured using modiﬁed ultrasonic diagnostic equipment(Hitachi-Aloka ¡-10) with a 3.75 MHz phased array probe.
Ultrasonic echoes received by individual transducer elements
were acquired at a sampling frequency of 30 MHz at a 12-bit
resolution.
Spatial resolutions of PBF and DT were compared byconstructing B-mode images from one transmission of aplane wave at a steering angle of 0°. Figures 1(a) and 1(b)show B-mode images of the phantom obtained by PBF andDT, respectively. Lateral amplitude proﬁles at wire strings
located at range distances of 44, 64, and 84 mm are shown in
Figs. 2(a)–2(c). Only in Fig. 2(1-a), lateral amplitude proﬁles
obtained by rectangular (red) and Hamming (green) receive
apodization are shown. By comparing Figs. 2(1-b) and 2(2-b), we found that the amplitude proﬁle obtained by DT
corresponds to that obtained by PBF with rectangular receive
apodization. Therefore, lateral amplitude proﬁles obtained by
PBF with rectangular receive apodization were comparedwith those obtained by DT. As can be seen from the full
widths at half-maximum of the amplitude proﬁles listed inTable I, spatial resolutions of PBF and DT are similar in ashallow region (44 mm), but the spatial resolution of DT isslightly better than that of PBF in deeper regions. In DT,
Eq. (11) is calculated for each frequency component
contained in the received ultrasonic echo signal by Fouriertransform. After the calculation of Eq. (11), all the acquired
frequency components are combined by inverse Fouriertransform. Therefore, DT can utilize the frequency band of

07KF02-2© 2014 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 53, 07KF02 (2014)

H. Hasegawa and H. Kanai

(1)

(2)
1
0.8

0.6

0.6

0.4

0.4

0.2
0
1

0.2

(b)

0.8
0.6
0.4
0.2
0
1

(c)

0

0
1

(b)

0.8
0.6
0.4
0.2

0.8

0.6

0.6

0.4

0.4

−20

−40

−60

0
1

0.8

0.2
0

PBF
DT
PBF with PCF(a)amplitude [dB](a)normalized echo amplitude

normalized echo amplitude

1
0.8

(c)

0

2

4
6
8
lateral distance [mm]

10

12

0.2
0

2

4
6
8
lateral distance [mm]

10

12

0

0

2

4
6
8
lateral distance [mm]1012Fig. 2. (Color online) Results obtained by (1) parallel beamforming and(2) diffraction tomography. Lateral amplitude proﬁles (red: rectangular
receive apodization, green: Hamming receive apodization) at (a) 44, (b) 64,
and (c) 84 mm.

Table I. Lateral spatial resolutions of PBF, DT, and PBF with PCF(unit: mm).
Depth
44

64

84
2.27

PBF

1.18

1.79

DT

1.16

1.70

1.90

PBF with PCF

0.79

1.16

1.57

the pulsed ultrasound used for imaging, and a spatial
resolution similar to that of PBF can be obtained.
On the other hand, PBF can be used with a weighting
method, namely, phase coherence factor (PCF),18) which
evaluates the variance in phases of echo signals received byindividual elements after the operation of delay compensation, as in the conventional delay and sum beamforming
process. PCF can suppress echoes from scatterers except forthat scattered from a focal point. Also, PCF can suppress
echo signals with low signal-to-noise ratios (SNRs) because
the phases of echo signals with low SNRs tend to ﬂuctuate
randomly. The random ﬂuctuation of phase increases thephase variance, and PCF becomes low. As a result, PCF also
improves the SNR of the obtained ultrasonic image by
suppressing echo signals with low SNRs. PCF can be usedwith low computational load and would be useful also for the
improvement of spatial resolution in PBF. Figure 1(c) showsa B-mode image of the phantom obtained by PBF with PCF,
and Fig. 3 shows lateral amplitude proﬁles at a range distanceof 44 mm obtained by PBF, DT, and PBF with PCF. The full
widths at half-maximum of the amplitude proﬁles in Fig. 3
are listed in Table I. As can be seen in Fig. 3, PBF with PCF
achieved much better spatial resolution than did PBF and DT.
Actually, DT achieved slightly better spatial resolution thanPBF, but DT requires the 2-D Fourier transform and inverse
2-D Fourier transform. Although fast Fourier transform canbe used for enhancing computational efﬁciency, PCF does
not need a high computational load and, therefore, PBF with
PCF would be useful for high temporal and spatial resolution
echocardiography. One of the problems in the use of PCF is
the suppression of speckle echoes from a diffuse scattering

Fig. 3. (Color online) Lateral amplitude proﬁles at a range distance of
44 mm obtained by the various methods.

medium, such as echoes from the inside of the heart wall.
This problem should be solved in future work to enable theuse of high-frame-rate echocardiography with PCF.
High-frame-rate echocardiography contributes to themeasurement of cardiac dynamics, which must be measured
to assess myocardial function and viscoelasticity. In thepresent study, parallel beamforming and diffraction tomography were examined for the improvement of spatial
resolution in high-frame-rate echocardiography using unfocused transmit beams. Diffraction tomography improved
the spatial resolution in a deeper region compared withparallel beamforming alone. However, parallel beamformingwith the phase coherence factor further improved the spatial
resolution. The calculation of the phase coherence factor
did not require extensive computational load, and parallel
beamforming with the phase coherence factor was considered
to be feasible for realizing high temporal and spatial
resolution echocardiography.

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07KF02-3© 2014 The Japan Society of Applied Physics