Characteristics of Minimum Variance Beamformer for Frequency and

Plane-wave Compounding

Ryoya Kozai

, Norio Tagawa

1 a

, Masasumi Yoshizawa

and Takasuke Irie

1,3

Electrical Engineering and Computer Science, Tokyo Metropolitan University, Hino-shi, Tokyo, Japan

Tokyo Metropolitan College of Industrial Technology, Arakawa-ku, Tokyo, Japan

Microsonic Co. Ltd., Kokubunji-shi, Tokyo, Japan

Keywords:

Ultrasonic Imaging, Minimum Variance Beamforming, Frequency Compounding, High Resolution.

Abstract:

Recently, coherent plane-wave compounding (CPWC) that achieves high spatiotemporal resolution has been

studied actively as a spatial compounding beamformer. Further, various frequency compounding methods have

been proposed for reducing speckle noise. We already proposed the method called frequency and plane-wave

compounding minimum variance distortionless response (FPWC-MVDR), which achieves high spatial reso-

lution imaging by simultaneously optimizing frequency and spatial compounding based on minimum variance

scheme. In the algorithm of this method, the data-compounded-on-receive MVDR (DCR-MVDR) principle

developed for CPWC is extended and applied. In this study, we analyze the features and characteristics of

FPWC-MVDR and the weaknesses to be solved in the future through experiments.

1 INTRODUCTION

In order to realize high-resolution ultrasound imag-

ing, it is necessary to increase the amplitude of the tar-

get signals and reduce the noise level. Beamforming

is the standard method to realize it. Recent work on

ultrasound beamforming has made particular progress

in improving lateral resolution. Receive beamform-

ing is the basis of array signal processing using an

array transducer composed of multiple transducer el-

ements, and is a technology that generates spatial res-

olution by forming the beam directivity. The most

basic beamforming method is Delay-and-Sum (DAS)

(Thomenius, 1996). In this method, the delays of the

received signals of each element are aligned before

they are added. However, this technique is strongly

limited by the beamwidth, which is determined by the

aperture width of the transducer array.

Various beamformings have been proposed to

form narrower beams (Camacho et al., 2009), (Li

and Li, 2003), (Matrone et al., 2015). Among

them, much study has been done on minimum dis-

persion distortion-free response (MVDR) beamform-

ing (Capon, 1969), (Vignom and Burcher, 2008), (Asl

and Mahloojifar, 2012). MVDR is a method to min-

imize the noise power by minimizing the variance of

https://orcid.org/0000-0003-0212-9265

beamforming while guaranteeing the all-pass charac-

teristics of the target signal. The variance can be

minimized by adaptively calculating the weights for

adding the received echoes using the data variance-

covariance matrix. As a result, the signal level other

than the target is reduced, and the improvement in lat-

eral resolution and signal-to-noise ratio (SNR) can be

expected. In a conventional system, the true variance-

covariance matrix cannot be known, and it is esti-

mated from the received echo. Therefore, various es-

timation methods have been proposed as approximate

calculations. The most popular of these is subarray

averaging (Synnevag et al., 2007). In this method,

the sensor array is ﬁrst divided into subarrays, and for

each, the tensor product of the vectors whose com-

ponents are the echoes of each subarray element is

calculated. Then, a variance-covariance matrix is es-

timated by averaging them. However, to prevent the

variance-covariance matrix from becoming singular,

the subarray length must be less than half its origi-

nal length. This will signiﬁcantly reduce the image

resolution. In addition, these processes have the dis-

advantage that real-time imaging is difﬁcult due to the

high computational cost.

Nguyen et al. proposed the Data-Compounded-

on-Receive (DCR-MVDR) beamforming (Nguyen

and Prager, 2018) to solve these problems. This is

a method that efﬁciently applies MVDR to Coherent

Kozai, R., Tagawa, N., Yoshizawa, M. and Irie, T.

Characteristics of Minimum Variance Beamformer for Frequency and Plane-wave Compounding.

DOI: 10.5220/0010300404230430

In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 4: VISAPP, pages

423-430

ISBN: 978-989-758-488-6

423

Plane-Wave Compounding (CPWC) (Montaldo et al.,

2009) and achieves both temporal resolution and spa-

tial resolution. CPWC is a method that can trans-

mit and receive plane waves while changing the an-

gle and perform receive beamforming to obtain the

same image resolution as conventional focus beam

transmission in a short time. IDCR-MVDR requires

a variance-covariance matrix of echoes for changes

in plane wave transmit angle, which does not require

subarray averaging and is calculated using a snapshot

from the echo vectors received by the array. There-

fore, processing can be performed without reducing

the matrix size.

On the other hand, a technique called frequency

compounding has also been studied as an approach

to improve image quality (Magnin et al., 1982),

(Mesurolle et al., 2006), (Cui and Liu, 2011), which is

a method of transmitting and receiving different fre-

quencies in the same path and averaging them to en-

hance the target signal and reduce artifact echo. This

takes advantage of the different interference patterns

depending on the frequency. This method is mainly

used for speckle reduction.

We have proposed the frequency and plane-wave

compounding minimum variance distortionless re-

sponse (FPWC-MVDR), which can realize high spa-

tial resolution imaging by optimizing compounding

of the frequency and plane waves simultaneously

(Kozai et al., 2020). DCR-MVDR principle devel-

oped for CPWC is extended and applied to the al-

gorithm of this method. The novelty of our ap-

proach is to optimize frequency compounding as well.

Traditional frequency compounding uses a constant

weighted average. However, this does not effectively

use frequency information for ultrasound imaging.

There may be appropriate averaging weights for each

frequency, depending on the pixel position to be im-

aged. FPWC-MVDR has adopted an approach that

optimizes frequency compounding by adaptively cal-

culating the averaging weights. In this paper, we show

various characteristics of FPWC-MVDR investigated

through experiments. In particular, the trend of fre-

quency weights offers interesting implications for fu-

ture work.

2 METHOD

2.1 Minimum Variance Distortionless

Response

In this paper, all observations are treated as analytic

signals obtained by quadrature detection of RF sig-

nals, that is, IQ (In-phase and Quadrature-phase) sig-

nals. First, we explain MVDR using conventional

sub-array averaging. Beamformer output is generally

expressed by the following exuation.

y(t) = w

x, (1)

where w is a weight vector,

is Hermitian transpo-

sition, and x is a received echo vector composed of

echoes acquired by all the elements, and has a time

delay corresponding to the imaging position and as-

sociated phase correction. The number of elements is

set to N. MVDR adaptively ﬁnds the weight vector

w by solving the following constrained optimization

problem (Mehdizadeh et al., 2012).

Minimize w

Rw, subjectto w

= 1, (2)

where R is the N × N variance-covariance matrix. w

can be obtained by solving this minimization problem

and the solution is obtained as follows:

ˆw =

−1

. (3)

In the subarray averaging process, an estimate of R

is found by partitioning x into N − L + 1 subarrays of

length L. The estimated value

R is as follows:

R =

N − L + 1

N−L+1

∑

l=1

, (4)

where x

≡ [x

l+1

,·· · ,x

l+L−1

]

. By this subarray

processing, the matrix size becomes L × L, hence the

size of w is also L. Therefore, the averaged echo for

each subarray ¯x

is deﬁned by the following equation.

¯x

l+L−1

∑

m=l

. (5)

By using the weights of Eq. 3, the out put of MVDR

for each pixel ˆy is as follows:

ˆy =

N−L+1

∑

l=1

ˆw

¯x

. (6)

2.2 Coherent Plane-wave Compounding

In CPWC processing, plane waves are transmitted and

received multiple times at different angles. When

transmitting M times at different angles and receiv-

ing with N elements, the following two-dimensional

array can be generated by performing time delay and

phase correction according to pixel position.

X =







1,1

1,2

··· x

1,N

2,1

2,2

··· x

2,N

M,1

M,2

··· x

M,N







, (7)

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

424

where x

i, j

is the echo received by element j for the

pulse transmitted at the angle i. Details of the de-

lay calculation are given in (Montaldo et al., 2009).

The following CPWC output is obtained by averag-

ing these matrix elements.

ˆy

CPWC

∑

i=1

∑

j=1

i, j

. (8)

2.3 Data-Coumpounded-on-Receive

MVDR

The DCR-MVDR uses the array information and an-

gle information of Eq. 7 because it transmits and

receives while changing the angle like CPWC. The

DCR-MVDR creates a snapshot from the array infor-

mation and adaptively determines the angle weights

without using the sub-array averaging in MVDR. The

snapshot is calculated as follows:

k,i

N − 1

∑

j=1, j6=k

i, j

. (9)

The variance-covariant matrix is deﬁned using the

snapshot of Eq. 9.

DCR

∑

k=1

+ εI, (10)

where ε is the diagonal loading parameter and I is an

identity matrix. Adjusting ε increases the diagonal

and makes the variance-covariance matrix more ro-

bust. ε is set relative to ∆ = Tr(

R)/L, where L is the

rank of

R indicating the ﬁrst term in the right hand

side of Eq. 10. The signal to which the weights are

applied is the vector u ≡ [u

,·· · ,u

]

, which is

obtained by adding the matrix components of Eq. 7 in

the array direction. Namely, each element u

is com-

puted as follows:

∑

j=1

i, j

, for i = 1, 2,·· · ,M. (11)

2.4 Frequency and Plane-wave

Compounding-MVDR

When M times are transmitted at different angles and

different frequencies are transmitted L times for each

angle, and N elements are used, the third-order tensor

data X can be generated (Fig. 1(a)). First, as shown in

Fig. 1(a), the matrix Y for each angle is extracted and

used as frequency and array information. Then, fre-

quency weights are calculated by DCR-MVDR pro-

cessing, and the weights are multiplied in the fre-

quency direction to compress the frequency informa-

tion (Fig. 1(b)). Where p

(i = 1,2, ··· ,N) represents a

(a)

(b)

(c)

Figure 1: FPWC-MVDR processing overview: (a) partion

third-order tensor data; (b) frequency compression; (c) an-

gular weights calculation and FPWC-MVDR output.

snapshot for each transmission angle deﬁned as Eq. 9.

The matrix Z ≡ [z

,·· · ,z

], where z

is generated

as in Fig. 1(b), is formed by performing the above pro-

cessing at all angles. Finally, by performing the same

processing as DCR-MVDR on this matrix, it becomes

the output of FPWC-MVDR (Fig. 1(c)).

3 EXPERIMENTS

3.1 Experiment Condition

In the experiments, the transmission and reception se-

quences were generated using an experimental plat-

form for medical ultrasound (RSYS0003, Microsonic

Inc., Japan) with a sampling rate of 31.25 MHz. The

number of transducer elements used for both trans-

mission and reception is 64, while the element pitch is

0.315 mm. Transmitted waves are restricted to 7-level

quantization. A linear array probe (T0-1599, Nihon

Dempa Kogyo Co., Ltd., Japan) was also used. The

center frequency of this probe is 7.5 MHz and its spe-

Characteristics of Minimum Variance Beamformer for Frequency and Plane-wave Compounding

425

Figure 2: Experimental setting for soft tissue-mimicking

phantom.

Table 1: Experimental parameters.

Parameter Value

Transmission wave FM-chirp pulse

Frequency band width 2 MHz

Chirp pulse duration 8 µs

Transmission center freq. Randomly determined

at 4 − 8 MHz

Transmission voltage 60 V

Apodization Hanning window

Measurement tenparature 24

◦

ciﬁc bandwidth is 70 %. The signal processing was

performed ofﬂine using MATLAB software.

Figure 2 shows the experimental setting. We

present the experimental results obtained using a

soft tissue-mimicking phantom (US-2 multi-purpose

phantom N-365; Kyoto Kagaku Co., Ltd., Japan),

with a speed of sound of 1,432 m/s (25

◦

C) and at-

tenuation of 0.59 dB/cm/MHz. As shown in Fig. 2,

the phantom contains six string wires with a diame-

ter of 0.1 mm. The distances between these wires are

1.0 mm, 2.0 mm, 3.0 mm, 4.0 mm and 5.0 mm, mea-

sured from the side closest to the phantom.

Table 1 shows the experimental setting. As for

transmission, plane FM-chirp waves were transmit-

ted 33 times while changing the angle in 0.5 deg.

Figure 3: Comparison of intensity distribution proﬁles in

the axial direction by regular and random changes of trans-

mission frequency band.

(a) (b)

(e)

Figure 4: B-mode images when targeting string : (a)

CPWC; (b) DCR-MVDR; (c) FC-DCR-MVDR; (d) FPWC-

MVDR; (e) Normalized FPWC-MVDR.

steps from −8.0 deg. to +8.0 deg.. In addition, we

changed the frequency at each angle and transmit-

ted 8 times. 64 elements were used for transmis-

sion and reception. First, it was necessary to consider

whether to change the transmission frequency band,

that is, the center frequency of the transmission wave,

at regular intervals or randomly. Therefore, the ﬁnite

element method (FEM) simulation under almost the

same transmission/reception conditions as the exper-

iment was performed using the engineering simula-

tion software OnScale. The object to be imaged is a

thin metal wire. Figure 3 shows an example of the in-

tensity distribution proﬁle obtained by FPWC-MVDR

using both frequency change methods. From this,

it can be seen that periodic changes cause artifacts

due to coherence between different frequencies. In

the following, a random frequency change is adopted.

In order to evaluate the performance of the frequency

compounding, we also transmitted once at each angle

using the total band of frequencies used. Due to the

experimental system’s limitations, the pulse width is

limited and is strongly affected by frequency depen-

dent attenuation (FDA), so a relatively low frequency

is used.

3.2 Performance Evaluation

We ﬁrst evaluate the beamformer performance. Fig-

ure 4 is a comparison of B-mode images. Figure 4(a)

and (b) are B-mode images of CPWC and DCR-

MVDR when the total frequency band is used. Fre-

quency Compound-DCR-MVDR (FC-DCR-MVDR)

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

426

(a)

(b)

Figure 5: Intensity distribution proﬁles: (a) on the axial

in the center of the B-mode image; (b) on the lateral line

through the foremost target.

Table 2: Axial resolution at the foremost target.

Axial width [mm]

Method -6 dB -10 dB

CPWC 0.44 0.57

DCR-MVDR 0.44 0.60

FC-DCR-MVDR 0.55 0.73

FPWC-MVDR 0.37 0.53

Norm. FPWC-MVDR 0.23 0.23

in Fig. 4(c) is a method in which DCR-MVDR is

combined with a frequency compounding at constant

weights, and the frequency compounding is not op-

timized. Figure 4(d) is a result of the proposed

method, FPWC-MVDR. Figure 4(e) is a B-mode im-

age that has been FPWC-MVDR processed after the

frequency dependent attenuation (FDA) compensa-

tion (Normalized FPWC-MVDR), considering that

frequency information is not used effectively due to

the FDA.

Table 3: Lateral resolution at the foremost target.

Lateral width [mm]

Method -6 dB -10 dB

CPWC 0.95 1.58

DCR-MVDR 0.63 0.95

FC-DCR-MVDR 0.63 0.95

FPWC-MVDR 0.63 0.95

Norm. FPWC-MVDR 0.63 0.95

(a)

(b)

Figure 6: Intensity distribution proﬁles when frequency

compounding is performed after plane wave compounding:

(a) on the axial in the center of the B-mode image; (b) on

the lateral line through the foremost target.

To check the performance in more detail, Fig. 5

shows the intensity distribution proﬁle across the

string at the foremost position. In addition, Tables 2

and 3 show the axis and lateral resolution at the fore-

most target. In DCR-MVDR and FC-DCR-MVDR,

we set ε = ∆. In FPWC-MVDR and Normalized

FPWC-MVDR, we set ε = ∆ for frequency weight

calculation and ε = 2∆ for angle weight calculation.

These results show that frequency compounding re-

duces axial side lobes. In addition, it can be con-

ﬁrmed that FPWC-MVDR, which adaptively obtains

the weights of the frequency compounding, can re-

alize high axial resolution. we can also conﬁrm that

compensating the FDA improves its performance.

3.3 Other Characteristics

First, the processing order of the frequency com-

pounding and the plane wave compounding was ex-

changed, and the effect was evaluated. Figure 6 shows

the results of comparing the intensity proﬁles. In this

ﬁgure, the method of conducting the plane wave com-

pounding ﬁrst is referred to as PFWC-MVDR. The

results show that the PFWC-MVDR has a slightly

lower axial resolution than the FPWC-MVDR, while

slightly improving the lateral sidelobe level. How-

ever, since the number of transmissions with the fre-

Characteristics of Minimum Variance Beamformer for Frequency and Plane-wave Compounding

427

(a)

(b)

Figure 7: Intensity distribution proﬁles when using the en-

tire frequency band (4-10 MHz): (a) on the axial in the cen-

ter of the B-mode image; (b) on the lateral line through the

foremost target.

quency changed is 8 less than the number of transmis-

sions with the azimuth direction changed with 33, it

is necessary to conﬁrm the effect in the future.

Subsequently, DCR-MVDR, FPWC-MVDR, and

Normalized FPWC-MVDR were executed using the

entire effective band of 4− 10 MHz of the used probe.

The results are shown in Fig. 7. Since the frequency

components of the received echo are approximately

7 MHz or less, the performance is degraded com-

pared to the case where the 4 − 8 MHz band is used

(Fig. 5). Normalized FPWC-MVDR compensates

for the FDA attenuation by the normalization pro-

cess, but the sidelobe increases especially in the lat-

eral direction. Therefore, the frequency weights at the

foremost target position were investigated. Compar-

ing the weights of both determined by Normalized

FPWC-MVDR shown in Fig. 8, it can be seen that

the high frequency band is heavily used in the case of

4 − 10 MHz. Since the high frequency band is gen-

erally used heavily, there is a possibility that the low

SNR in the high frequency band may have an adverse

effect.

What is interesting about the result of these

weights are that the tendency of the magnitude of the

weights with respect to the frequency are V-shaped,

that is, the weights are inversely proportional to the

frequency in the low frequency band. An interesting

(a)

(b)

Figure 8: Absolute value of weights used for frequency

compounding: (a) when using the 4 − 8 MHz band, (b)

when using the 4 − 10 MHz band.

result is that lower frequencies are used more heav-

ily than frequencies near the center. Therefore, the

frequency weighting (amplitude and phase) by Nor-

malized FPWC-MVDR was evaluated by FEM simu-

lation using OnScale. Figure 9 shows the characteris-

tics obtained around the target at a position of 15 mm

in the axial direction and 2.24 mm in the lateral direc-

tion. From these, it can be seen that the closer to the

target, the sharper the V-shape with the larger weight.

It can also be conﬁrmed that the phase of the weight is

inverted with respect to the center of the entire band,

but the relationship of this characteristic with the tar-

get position is not so clear.

4 CONCLUSIONS

In this study, we evaluated the performance and

characteristics of the receive beamforming method

FPWC-MVDR that we recently proposed by opti-

mally integration of frequency and spatial compound-

ing. Previously, we were studying methods for detect-

ing the positions of a small number of reﬂectors with

high resolution (Tagawa and Zhu, 2018), (Zhu and

Tagawa, 2019a), (Zhu and Tagawa, 2019b), (Nguyen

et al., 2020), and they were suitable for anomaly de-

tection such as tumor imaging. On the other hand,

the technical feature of these methods was to realize

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

428

(a)

(b)

(c)

(d)

Figure 9: Differences in frequency compounding weights

around the imaging target position: (a) axial weight magni-

tude; (b) axial weight phase; (c) lateral weight magnitude;

(d) lateral weight phase.

the resolution of the wavelength level of the carrier by

utilizing various frequencies for transmission and re-

ception. The FPWC-MVDR featured in this study is

also the result of applying the viewpoint of effective

use of multiple frequencies to receive beamforming.

From the evaluation results shown in this paper,

it was found that the frequency characteristics of the

weights used in the frequency compounding are par-

ticularly interesting. This result seems to indicate

that the addition of echoes with carrier frequencies far

from the center of the band is required to effectively

cancel the off-axis signal. In the future, we will an-

alyze the meaning and inevitability of the character-

istics of this weight through detailed simulations and

imaging experiments of various objects.

In the FPWC-MVDR, considering the reduction

of the calculation amount, the echo data is ﬁrst re-

duced in the frequency direction, the data amount is

reduced, and then the plane wave transmission an-

gle is reduced to output. If importance is placed

on the optimality, it is desirable to calculate a two-

dimensional weight map of frequency and transmis-

sion angle and reduce the three-dimensional tensor

data at once. On the other hand, assuming a case

where a high frame rate image such as a cardiac echo

is required, a method capable of extracting the car-

rier phase more efﬁciently is desirable. In the fre-

quency compounding in this study, pulses of differ-

ent carrier waves were transmitted multiple times and

the corresponding echoes were used. However, the

FPWC-MVDR proposed in this study can also be ex-

ecuted by extracting and analyzing narrowband com-

ponents by signal processing from the echo for one

wideband transmission. We have already begun re-

search on this method. In addition, the realization of

adaptive beamforming by deep learning is effective

in reducing the amount of calculation, and there are

many reports (Luijten et al., 2019). This idea is also

applicable to our FPWC-MVDR. We plan to proceed

with further studies in terms of both computational

complexity and efﬁciency.

Research on improving image quality by reducing

speckles and noise is also being enthusiastically pur-

sued in the ﬁeld of optics (Leo et al., 2014), (Shah-

doosti and Rahemi, 2019), (Jeon et al., 2018). In re-

cent years, many studies on super-resolution technol-

ogy based on deep learning have been conducted on

various images (Yang et al., 2019), (Ouyang et al.,

2018), (Huang et al., 2018). Based on the results

there, it is important to improve the image quality of

ultrasonic images from a wider perspective.

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