Minimum variance adaptive beamforming
applied to medical ultrasound imaging
Johan-Fredrik Synnevåg
Andreas Austeng
Sverre Holm
Department of Informatics, University of Oslo
P.O. Box 1080, N-0316 Oslo, Norway
Abstract— We have applied the minimum variance beamformer to medical ultrasound imaging and shown significant
improvement in image quality compared to delay-and-sum.
Reduced mainlobe width and suppression of sidelobes is
demonstrated on both simulated and experimental RF data of
closely spaced wire targets, resulting in increased resolution
and contrast. The method has been applied to experimental
RF data from a heart-phantom, demonstrating improved
definition of the ventricular walls. We have evaluated the
beamformers sensitivity to velocity errors and shown that
reliable amplitude estimates are achieved if proper regularization is applied.
I. I NTRODUCTION
Delay-and-sum (DAS) beamforming is the standard
technique in medical ultrasound imaging. An image is
formed by transmitting a narrow beam in a number
of angles and dynamically delaying and summing the
received signals from all channels. The large sidelobes of
the DAS beamformer can be suppressed using aperture
shading, resulting in increased contrast at the expense of
resolution. In contrast to the predetermined shading in
DAS, adaptive beamformers use the recorded wavefield
to compute the aperture weights. By suppressing interfering signals from off-axis directions and allowing large
sidelobes in directions where there is no received energy,
the adaptive beamformers can increase resolution.
The minimum variance (MV) adaptive beamformer [1]
and subspace-based methods have mostly been studied
in narrowband applications. Extensions to broadband
imaging include preprocessing with focusing- and spatial resampling filters, allowing narrowband methods to
be used on broadband data [2], [3]. We have applied
the MV beamformer to medical ultrasound imaging by
prefocusing in the direction of the transmitted beam – as
the delay-step in DAS – and replaced the summing with
the MV method. Similar methods have been used by
Mann and Walker [4], and Sasso and Cohen-Bacrie [5]
in medical ultrasound imaging. The former use a constrained adaptive beamformer on experimental data of
a single point target and a cyst phantom demonstrating
improved contrast and resolution, whereas the latter use
an MV beamfomer on a simulated data-set, showing
improved contrast in the final image. We demonstrate
resolution improvement and sidelobe suppression on
both simulated and experimental RF data of closely
spaced wire targets, and show improvement in the image
of a heart-phantom obtained from experimental RF data.
We also evaluate robustness of the beamformer to errors
in acoustic velocity, and show that reliable amplitude
estimates are achieved by regularization.
II. M ETHOD
A. Broadband adaptive beamforming
We assume an array of M elements, each recording a
signal xi (t). For a single reflector, under the assumption of ideal steering, the ith time-delayed channel is
described as
P
xi ( t ) = s d ( t ) +
∑ s p (t) ∗ gip (t) + ni (t),
(1)
p=1
where sd (t) is the reflected signal we are estimating,
s p (t) is interfering (off-axis) signal p, gip (t) accounts
for the propagation-loss and delay from reflector p to
sensor i, and ni (t) is noise on channel i. We want to find
the optimal set of sensor weights, w = [w1 . . . w M ],
to suppress noise and off-axis signals. For narrowband
signals the optimal w is found by solving [1]
min wH Rw
w
subject to wH a = 1,
(2)
where a is the steering vector of complex exponentials containing the time-delays required to focus in a
specific direction, and R is the covariance matrix of
the observations. For broadband signals the time-delays
cannot be expressed as complex exponentials, so we first
apply appropriate delays on each sensor to focus in the
direction of the transmitted beam. We can then compute
(3) with steeringvector a = [1 1 . . . 1] T . The solution to
(2) is
R−1 a
w = H −1 .
(3)
a R a
In practice the covariance matrix is replaced by the
sample covariance matrix
R̂ =
1 H
xx ,
N
(4)
where x = [ x1 (t) . . . x M (t)]T and N is the number of
samples used in the estimation. In narrowband applications we find one set of weights for the whole timeseries, x(t), and there may be a sufficient number of
samples to compute a good estimate of the covariance
matrix. In broadband beamforming the optimal set of
weights changes constantly with time, forcing us to
compute the sample covariance matrix from one or only
a few time-samples per channel. If we use (4) directly
to compute R̂, we will get an ill-conditioned matrix not
suited for the inversion required in (3). Instead we divide
the array into overlapping subarrays which are averaged
to obtain the covariance matrix estimate, a technique
called spatial smoothing [6]. The spatial sample covariance matrix for one time-sample, k, is obtained by first
forming a new observation matrix


x1 [k ] . . . x M− L+1 [k ]
 x2 [k ] . . .
x M− L [k ] 


(5)
x̃k =  .
,
..

 ..
.
x L [k] . . .
x M [k]
where L is the number of elements in each subarray. The
covariance matrix for sample k is then computed as
1
R̃k =
x̃ x̃H .
M−L+1 k k
(6)
By replacing R with R̃ in (3) we see that the number
of coefficients in w is reduced, which will limit the
degrees of freedom to suppress off-axis signal and noise.
However, we use all the original channels to obtain the
amplitude estimate in a specific direction for sample k,
given by


xl
M− L+1


..
(7)
Ak (φ) = ∑ wH 
,
.
l =1
xl + L−1
so we use the full spatial extent of the array in the beamforming. Alternatively, the power estimate is computed
as
Pk (φ) = | Ak (φ)|2 .
(8)
Preprocessing by spatial smoothing is normally used
to decorrelate signals in narrowband applications, enabling use of adaptive methods on strongly or perfectly
correlated (coherent) sources. We can expect that closely
spaced reflectors are highly correlated in medical ultrasound imaging, as the reflected signals are simply
delayed and scaled versions of the transmitted beam.
However, coherent signals are resolvable in broadband
adaptive imaging, as long as data are processed coherently and not split into frequency sub-bands [7]. In
broadband applications spatial smoothing preprocessing
is rather a means to obtain an accurate and wellconditioned covariance matrix suited for inversion.
B. Robust adaptive beamforming
Because of the high resolution of the minimum variance beamformer at high signal-to-noise ratios, a large
number of scan lines may be required to avoid angular undersampling and to ensure reliable amplitude
estimates. A common way to increase robustness of the
beamformer is to add a constant, ǫ, to the diagonal of the
covariance matrix before evaluating (3), replacing R with
R + ǫI. This gives a wider mainlobe and ensures that reflections slightly off axis are passed without attenuation.
The resolution is thereby also decreased. Several methods exist where appropriate values of ǫ are found based
on uncertainty of the model parameters [8]. Adding a
constant to the diagonal of the covariance matrix can be
seen as increasing the noise level in the recorded data
before finding the optimal aperture shading, assuming
the noise is white. As white noise becomes dominant,
the minimum variance solution approaches conventional
beamforming with uniform shading. This can be seen
if the recorded wavefield is simply white noise. The
covariance matrix is then the identity matrix multiplied
by a constant. From (3) we get
w=
1
(σ 2 I)−1 a
a
= H = a,
N
a a
aH (σ 2 I)−1 a
(9)
where σ 2 determines the variance of the noise. This is
the DAS solution with uniform weights. The solution
to (9) is called the quiescent response of the adaptive
beamformer. By selecting
ǫ = δtr{R}I,
(10)
where δ is a constant and tr{·} is the trace operator,
the calculated aperture shading is balanced between the
optimal adaptive solution and the quiescent response.
III. R ESULTS
A. Simulated data
We have simulated a 96 element, 4 MHz transducer using Field II [9], imaging a number of pairwise reflectors
located at depths 30-80 mm. The reflectors were separated by 2 mm. We simulated all individual transmitter
and receiver configurations and applied full dynamic focus on both transmission and reception. White, gaussian
noise was added to each scan-line before beamforming.
Figure 1 shows the images obtained with DAS and
MV beamforming over 55 dB dynamic range. We used
subarray length, L = 48, for the image in figure (b).
We see that the reflectors are better resolved by the MV
beamformer, and that the sidelobes present in the DAS
image have been suppressed. Figure 2 shows the lateral
resolution at depths 40 and 80 mm, where power is
plotted versus angle. We see that the mainlobe width
of the MV beamformer is less than 1/4 of DAS, and that
the maximum sidelobe level is reduced by 20 dB for the
closest targets and around 10 dB for the deeper. We see
MV
30
0
35
35
−10
40
40
45
45
50
50
Depth [mm]
Depth [mm]
DAS
30
55
60
DAS
MV
MV
(δ=1/(L*10)
0
−10
−20
Power [dB]
Power [dB]
−20
DAS
MV
MV
(δ=1/(L*10)
−30
−40
−30
−40
55
60
−50
−50
−60
65
65
−60
70
70
−70
6
75
75
80
80
8
10
12
14
Angle [degrees]
16
18
−70
6
8
( a)
−10
0
10
20
−10
0
[mm]
( a)
Fig. 1.
DAS
MV
0
−10
−20
−20
−30
−40
(b)
Fig. 3. Lateral resolution of DAS, MV and regularized MV beamformers at depths (a) 40 mm (b) 80 mm. Images were processed with
5% error in acoustic velocity.
−60
−60
( a)
16
18
DAS
MV
−40
−50
10
12
14
Angle [degrees]
power estimates of the regularized MV beamformer are
similar to DAS, but the mainlobe width is still narrower
and the sidelobes are comparable to the MV beamformer
without regularization. There will be a trade-off between
mainlobe width and robustness against velocity errors,
depending on the choice of δ. However, we see that
the sidelobes are rapidly falling towards the background
noise level also when regularization is used.
−30
−50
8
18
(b)
−10
−70
6
20
16
Simulated wire targets: (a) DAS (b) MV
Power [dB]
Power [dB]
0
10
[mm]
10
12
14
Angle [degrees]
−70
6
B. Experimental data I: Wire targets
8
10
12
14
Angle [degrees]
16
18
(b)
Fig. 2. Lateral resolution of DAS (solid) and MV (solid with diamonds)
beamformed images at different depths: (a) 40 mm (b) 80 mm. Plots
show estimated power vs. angle.
that the sidelobe level of the MV beamformer is rapidly
falling towards the background noise level. At 80 mm,
where resolution is worst, the targets are only resolved
by about 3 dB for DAS, but are resolved by 12 dB for MV.
Note that the ability to resolve closely spaced reflectors
will depend on the signal-to-noise ratio (SNR), which is
about 60 dB (peak SNR) for the closest reflectors and
decreasing with depth.
We have also looked at the MV beamformers sensitivity to small errors in acoustic velocity. We processed
the simulated data-set in figure 1 with 5% velocity error.
Figure 3 shows a comparison of DAS (solid line), MV
(solid with diamonds) and regularized MV (dashed) at
depth 40 and 80 mm. We see that the power estimates
from the MV beamformer are sensitive to errors, due to
the high resolution. For the regularized MV beamformer
1
. We see that the
we used equation (10) for δ = 10L
We recorded experimental RF data with a specially
programmed GE Vingmed ultrasound scanner using a
96 element, 3.5 MHz transducer driven at 4 MHz. Ascans from all transmitter and receiver combinations
were recorded, allowing full dynamic focus on image
formation. The wires were separated by 2 mm. Figure 4
shows the lateral resolution after beamforming. We see
that the mainlobe width is reduced and that the sidelobes are lower for the MV beamformer, confirming the
performance improvements shown in the simulations.
C. Experimental data II: Heart-phantom
We then applied the MV beamformer to an experimental data-set from a heart-phantom obtained from
the Biomedical Ultrasound Laboratory, University of
Michigan [10]. Data were recorded with a 64 elemement
3.5 MHz transducer. We used subarray length, L = 32 for
the MV beamformer. Figure 5 shows the images obtained
with DAS and MV beamforming over 55 dB dynamic
range. We see that the resolution is generally better in
the MV beamformed image, where the “smearing” of
reflectors apparent in the DAS image is avoided due to
the increased resolution. The ventricular walls are better
defined, owing to the reduced mainlobe width and decreased sidelobe level, demonstrating the beamformers
capabilities on realistic images.
IV. D ISCUSSION
DAS
MV
Power [dB]
0
−20
−40
−60
−10
−5
0
5
Angle [degrees]
10
Fig. 4. Lateral resolution of DAS and MV beamformed images of wire
targets from experimental RF data at depth 57 mm.
DAS
40
50
60
Depth [mm]
80
90
100
110
−30
−20
−10
0
[mm]
10
20
30
40
10
20
30
40
( a)
MV
40
50
60
Depth [mm]
70
80
90
100
110
120
−40
V. C ONCLUSION
We have successfully applied the minimum variance beamformer to medical ultrasound imaging and
shown significant performance improvements compared
to delay-and-sum. Results have been demonstrated on
both simulated and experimental data. Evaluation of
robustness shows that reliable amplitude estimates are
achieved while still improving resolution. The method
has been demonstrated on RF data from a realistic image,
showing its potential in medical ultrasound imaging.
R EFERENCES
70
120
−40
Equation (9) shows that in the absence of modelerrors the MV beamformer will always perform equal
or better than DAS with uniform shading, provided that
the images are formed with sufficient angular sampling.
As the SNR decreases the adaptive solution approaches
DAS with uniform aperture shading. Regularizing the
MV method affects the ability to resolve closely spaced
targets, but the sidelobe level rapidly approaches the
background noise level, increasing contrast in the final
image. If the assumed acoustic velocity differs from
the actual, the accuracy of the amplitude (or power)
estimates is affected, but initial studies show that similar robustness against errors as DAS can be achieved,
provided that proper regularization is applied.
−30
−20
−10
0
[mm]
(b)
Fig. 5. Experimental RF data from a heart-phantom: (a) DAS (b) MV.
[1] J. Capon. High-resolution frequency-wavenumber spectrum analysis. Proc. IEEE, 57:1408–1418, August 1969.
[2] Jeffrey Krolik and David Swingler. The performance of minimax
spatial resampling filters for focusing wide-band arrays. IEEE
Transactions on Signal Processing, 39(8), August 1991.
[3] S. Sivanand, Jar-Ferr Yang, and M. Kaveh. Focusing filters
for wide-band direction finding. IEEE Transactions on Signal
Processing, 39(2), Februrary 1991.
[4] J. A. Mann and W. F. Walker. A constrained adaptive beamformer
for medical ultrasound: Initial results. Ultrasonics Symposium,
2002. Proceedings. 2002 IEEE, 2:1807–1810, October 2002.
[5] Magali Sasso and Claude Cohen-Bacrie. Medical ultrasound imaging using the fully adaptive beamformer. Acoustics, Speech and
Signal Processing, 2005. Proceedings (ICASSP ’05). IEEE International
Conference on, 2, March 2005.
[6] Tie-Jun Shan, Mati Wax, and Thomas Kailath. On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, 33(4), August 1985.
[7] H. Wang and M. Kaveh. Coherent signal-subspace processing for
the detection and estimation of angles of arrival of multiple wideband sources. IEEE Transaction on Acoustics, Speech, and Signal
Processing, 33(4), August 1985.
[8] Jian Li, P. Stoica, and Zhisong Wang. On robust Capon beamforming and diagonal loading. IEEE Transactions on Signal Processing,
51:1702–1715, July 2003.
[9] J. A. Jensen. Field: A program for simulating ultrasound systems.
Medical & Biological Engineering & Computing, 34:351–353, 1996.
[10] Ultrasound RF data-set heart from the Biomedical Ultrasonic Laboratory, University of Michigan.
Available at
http://bul.eecs.umich.edu/, September 2005.