Eigenspace-Based Minimum Variance Adaptive Beamformer
Combined with Delay Multiply and Sum:
Experimental Study
Moein Mozaffarzadeha , Ali Mahloojifara , Mohammadreza Nasiriavanakib , and Mahdi Oroojia
a

arXiv:1710.01767v1 [eess.SP] 4 Oct 2017

b

Department of Biomedical Engineering, Tarbiat Modares University, Tehran, Iran
Wayne State University, Bioengineering Department, Detroit, Michigan, United States
ABSTRACT

Delay and sum (DAS) is the most common beamforming algorithm in linear-array photoacoustic imaging (PAI)
as a result of its simple implementation. However, it leads to a low resolution and high sidelobes. Delay multiply
and sum (DMAS) was used to address the incapabilities of DAS, providing a higher image quality. However,
the resolution improvement is not well enough compared to eigenspace-based minimum variance (EIBMV). In
this paper, the EIBMV beamformer has been combined with DMAS algebra, called EIBMV-DMAS, using the
expansion of DMAS algorithm. The proposed method is used as the reconstruction algorithm in linear-array
PAI. EIBMV-DMAS is experimentally evaluated where the quantitative and qualitative results show that it
outperforms DAS, DMAS and EIBMV. The proposed method degrades the sidelobes for about 365 %, 221 %
and 40 %, compared to DAS, DMAS and EIBMV, respectively. Moreover, EIBMV-DMAS improves the SNR
about 158 %, 63 % and 20 %, respectively.
Keywords: Photoacoustic imaging, beamforming, delay multiply and sum, eigenspace-based minimum variance,
linear-array imaging.

1. INTRODUCTION
Photoacoustic imaging (PAI), unlike the X-ray which uses an ionizing radiation, is a nonionizing medical imaging
modality, providing the resolution of Ultrasound (US) imaging and the contrast of optical imaging.1 Photoacosutic (PA) signals are generated as a result of a laser illumination toward the target where PA waves are
propagated based on the thermoelastic effect and thermal gradient.2 PAI is a multiscale imaging modality that
has been used in different fields of study such as tumor detection3 and functional imaging.4
Since there is a high similarity between US and PA detected signals, many of beamforming algorithms used in
US imaging can be used in PAI.5, 6 Moreover, integrating these two imaging modalities has been a challenge
over the past few years.7, 8 The most prevalent beamformer in PAI is delay and sum (DAS). While it leads to
a low quality image as a result of its nonadaptiveness and blindness, it has been used extensively in PAI due
to its simple implementation.9 To address the incapabilities of DAS, adaptive beamformers, such as minimum
variance (MV), are commonly employed. MV has been investigated over the past few years, especially in US
imaging.10–12 Delay and standard deviation (DASD) beamforming algorithm was proposed to address the poor
appearance of interventional devices such as guide wires, needles, and catheters in US images.13
Matrone et al. proposed a new beamforming algorithm, called delay multiply and sum (DMAS), as a beamforming technique for linear array US imaging.14 Initially, DMAS was used for breast cancer detection, introduced
by Lim et al.15 DMAS has been used with multi-line transmission (MLT) for high frame-rate US imaging.16
Double stage (DS-DMAS), in which two stages of DMAS is used in order to achieve higher contrast and resolution compared to DMAS, was proposed based on the expansion of DMAS.17, 18 The lack of a high resolution
in DMAS was addressed using combining MV with DMAS, and it was shown that in the expansion of DMAS,
there are some terms which can be interpreted as DAS. These terms can be replaced by MV in order to improve
the resolution while the contrast is retained.19–21 A modified version of coherence factor (MCF) was introduced
Further author information: (Send correspondence to A. Mahloojifar)
Ali Mahloojifar: E-mail: mahlooji@modares.ac.ir, +98 21 82883304

based on DMAS, and it was shown that it provides higher contrast compared to conventional CF.22
In this paper, a novel algorithm based on combination of DMAS and eigenspace-based MV(EIBMV) is proposed.
The algorithm is called EIBMV-DMAS, and we have numerically evaluated this algorithm before, and the results
can be found in reference.20 The expansion of DMAS algorithm was used, and it was shown that in each term
of the expansion, there is a DAS algebra. Since DAS algorithm is a non-adaptive beamformer and leads to low
resolution images, we proposed to use EIBMV instead of the existing DAS in the expansion of DMAS. Here,
we evaluate the proposed method experimentally. The experimental results show that EIBMV-DMAS provides
resolution improvement and levels of sidelobe reduction, compared to DMAS and EIBMV, at the expense of
higher computational burden.
The rest of the paper is as follows. In section 2, the concept of beamforming in linear-array imaging and the proposed method are presented. Section 3 contains the experimental results and the necessary evaluation. Finally,
the conclusion is presented in section 4.

2. MATERIALS AND METHODS
In this section, the concept of the proposed algorithm is explained. When PA signals are detected by a linear
array of US transducer, beamforming algorithms such as DAS can be used to reconstruct the image from the
detected PA signals. Formula of DAS, as the most common beamformer, can be written as follows:
yDAS (k) =

M
X
i=1

xi (k − ∆i ),

(1)

where yDAS (k) is the output of the beamformer, k is the time index, M is the number of elements of array, and
xi (k) and ∆i are the detected signals and the corresponding time delay for detector i, respectively. DAS results
in low quality images due to its blindness and also the limited available angle view in linear-array imaging. To
address this problem, DMAS was introduced in which the same as DAS, corresponding sample for each element
is calculated, but samples are combinatorially coupled and multiplied before summation.14 DMAS formula is
given as follows:
M
−1 X
M
X
yDM AS (k) =
xi (k − ∆i )xj (k − ∆j ).
(2)
i=1 j=i+1

The dimensionally squared problem of (2) was also addressed using following equations:14
q
1 6 i 6 j 6 M.
x̂ij (k) = sign[xi (k − ∆i )xj (k − ∆j )] |xi (k − ∆i )xj (k − ∆j )|, for
yDM AS (k) =

M
−1
X

M
X

x̂ij (k).

(3)

(4)

i=1 j=i+1

Performing sign, absolute and square root after coupling procedure in (3) and (4), which requires (M 2 − M )/2
computations for each pixel, result in a slow imaging. To put it more simply, sometimes these library functions require many clock cycles, causing improper timing performance of DMAS algorithm. Applying following
procedure to the received PA signals reduces the computational number of the sign, absolute and square root
operations to M for each pixel:23
p
x̄i (k) = sign[xi (k)] xi (k)
for
1 6 i 6 M,
(5)
x̂ij (k) = x̄i (k)x̄j (k)

for

1 6 i 6 j 6 M.

(6)

DMAS beamformer can be considered as a beaming algorithm with a correlation process. In other words, the
output of this beamformer is the spatial coherence of detected PA signals, and it is a non-linear beamforming

Figure 1: Schematic of the experimental setup.
algorithm. The expansion of DMAS algorithm can be written as follows:
h
i
xid (k)xjd (k) = x1d (k) x2d (k) + x3d (k) + x4d (k) + ... + xM d (k)
{z
}
|
i=1 j=i+1
first term
i
h
i
h
i h
+ x2d (k) x3d (k) + x4d (k) + ... + xM d (k) +... + x(M −2)d (k) x(M −1)d (k) + xM d (k) + x(M −1)d (k).xM d (k) .
{z
}
{z
} |
{z
}
|
|

yDM AS (k) =

M
−1
X

M
X

second term

(M-2)th term

(M-1)th term

(7)
where xid (k) and xjd (k) are delayed detected signals for element i and j, respectively. As can be seen, there is
a DAS in every terms of the expansion, and it can be used to modify the DMAS beamformer. In every terms
of (7), there exists a summation procedure which is a type of DAS algorithm. It is proposed to use EIBMV
adaptive beamformer for each term instead of DAS. In other words, since DAS is a non-adaptive beamformer
and considers all the calculated samples for each element of the array the same as each other, consequently, the
acquired image by each term is a low quality image with high levels of sidelobe and broad mainlobe. In order
to use EIBMV instead of each DAS in the expansion, we need to carry out some modifications and prepare the
expansion in (7) to adopt the EIBMV. The necessary modifications are extensively explained in reference21 and
are out of the scope of this paper.
The output of EIBMV-DMAS can be written as follows:
yEIBM V −DM AS (k) =
!

X
M
M
X
wi,new xid (k)
wj (k)xjd (k) − wi (k)x2id (k) ,
i=1

|

j=1

{z

ith term

(8)

}

where wi,new is the calculated weight for each term in (8), wi is the calculated weights for each term in (7)
while the steering vector in the formula of EIBMV is a vector of ones. It should be noticed that when there is a
multiplication, resulting in the squared dimension, the method mentioned in (3) and (4) is used to prevent the
squared dimension

3. EXPERIMENTAL RESULTS
A linear-array of PAI system was used to detect the PA waves, and the major components of system include an
ultrasound data acquisition system, Vantage 128 Verasonics (Verasonics, Inc., Redmond, WA), a Q- switched
Nd:YAG laser (EverGreen Laser, Double-pulse Nd: YAG system) with a pulse repetition rate of 25 Hz, wavelength 532 nm and a pulse width of 10 ns. A transducer array (L7-4, Philips Healthcare) with 128 elements and
5.2 M Hz central frequency was used as a receiver. A function generator is used to synchronize all operations
(i.e., laser firings and PA signal recording). The data sampling rate was 20.8320 M Hz. The schematic of the
designed system is presented in Figure 1. A wire has been used as the phantom of imaging where the surface

Figure 2: Experimental images using the data obtained with wire targets. (a) DAS, (b) DMAS, (c) EIBMV, (d)
EIBMV-DMAS. All images are shown with a dynamic range of 60 dB.
DAS
DMAS
EIBMV
EIBMV-DMAS

Depth= 22 mm

0

-100
-150
-200
-250
-15

0

Power [dB]

Power [dB]

-50

DAS
DMAS
EIBMV
EIBMV-DMAS

Depth= 22 mm

-50

-100

-150

-10

-5

0

5

10

15

-1.5

Lateral distance [mm]

-1

-0.5

0

0.5

1

Lateral distance [mm]

(a)

(b)

Figure 3: (a) Lateral variations of DAS, DMAS, EIBMV and EIBMV-DMAS at the depth of 22 mm of the
images shown in Figure 2. (b) A zoomed version of (a) for a better comparison.
of the array was perpendicular to the imaging phantom. Thus, it is expected to see a cross section of the wire,
which would be like a point. The reconstructed images using the concerned beamformers are shown in Figure 2
where DAS leads to a low quality image having a low resolution along with high sidelobes. Moreover, the effects
of the noise of the imaging system and artifacts affect the reconstructed image. DMAS results in a higher noise
and artifacts suppression and improves the image quality as shown in Figure 2(b). However, the resolution is
not good enough in comparison with EIBMV, which can be perceived by comparing the images shown in Figure
2(b) and Figure 2(c). Using EIBMV-DMAS results in a higher resolution, and lower artifacts and sidelobes
compared to other concerned beamformers, as shown in Figure 2(d). To have a more precise evaluation, the
lateral variations of the images shown in Figure 2, are shown in Figure 3(a) along with a zoomed version (Figure
3(b)). The level of sidelobes for DAS, DMAS, EIBMV and EIBMV-DMAS are about -38 dB, -55 dB, -127
dB and -177 dB, respectively. To put it more simply, EIBMV-DMAS outperform DAS, DMAS and EIBMV
in the term of sidelobes about 139 dB, 122 dB and 50 dB, respectively. The narrower width of the mainlobe,
resulting from EIBMV-DMAS compared to other beamformers, can be clearly seen in Figure 3(b). Although the
resolution enhancement of EIBMV-DMAS is not significant compared to EIBMV, the sidelobes degradation is
considerable. To have a quantitative comparison, the signal-to-noise ratio (SNR) has been calculated using the
formula explained in reference.17 The calculated SNRs can be seen in Table 1. As shown, the proposed method
results in a higher SNR compared to other beamformers where it outperforms DAS, DMAS and EIBMV of about
65 dB, 41 dB and 18 dB, respectively. All the experimental results show the superiority of the proposed method.

4. CONCLUSION
In this paper, the combination of DMAS beamformer and EIBMV adaptive beamforming algorithm has been
experimentally examined. The proposed method was quantitatively and qualitatively evaluated, and it was

Table 1: SNR (dB) values at the depth of 22 mm.
Depth(mm)
DAS DMAS EIBMV EIBMV-DMAS
22
41.76
65.24
88.88
106.96
shown that it outperforms DMAS and EIBMV in the terms of sidelobes, width of mainlobe and SNR. EIBMVDMAS improved the sidelobes, in comparison with DAS, DMAS and EIBMV, for about 139 dB, 122 dB and
50 dB, respectively. Moreover, SNR was improved by the proposed method of about 158 %, 63 % and 20 %,
respectively.

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