# Improved Delay-and-Sum Beamforming Algorithm for

Authors: M. Klemm; I. J.  Craddock; J. A.  Leendertz; A. Preece; R. Benjamin
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

We have evaluated a modified delay-and-sum (DAS) beamforming algorithm for breast cancer detection with a microwave
radar-based system. The improved DAS algorithm uses an additional weight factor calculated at each focal point to improve image
quality. These weights essentially represent the quality of preprocessing and coherent radar operation. Using a multistatic UWB
radar system based on a hemispherical antenna array, we present experimental detection of 7 mm and 10 mm phantom tumours.
We show that the new proposed DAS algorithm improves signal-to-clutter ratio in focused images by 2.65 dB for 10 mm tumour,
and by 4.4 dB for 7 mm tumour.


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## Body

## 1. Introduction

X-ray mammography
is currently the most common
technique used in breast cancer screening. It employs ionising radiation,
requires uncomfortable compression of the breast during the examination, and is
of limited value for younger women. These limitations of mammography have
resulted in research into alternative methods for imaging breast cancer.Microwave radar-based imaging [1] is one of the more promising
candidates and has attracted the interest of a number of research groups around
the world.In radar-based imaging, the goal is to create a map of
microwave scattering, arising from the contrast in dielectric properties within
the breast. The radar approach originates from military and ground-penetrating
applications, and was proposed for breast cancer detection in the late nineties
independently by Benjamin in 1996 [2] and
Hagness et al.  in 1998 [3].The University of Bristol team is working on
multistatic ultrawideband (UWB) radar for breast cancer detection. Our radar
system is based on a real (as opposed to synthetic) aperture antenna array. We
have also developed a realistic 3D curved breast phantom with appropriate
electrical properties. Moreover, our experimental system was built in such a
way that it can be used directly with real breast cancer patients (clinical
trials have been recently commenced).


## 2. Measurement Setup

We have developed a microwave radar for breast cancer
detection, based on a curved hemispherical antenna array. In this paper, we
present results obtained using a second-generation symmetrical antenna array.
The new symmetrical antenna array, shown in Figure1, uses a recently
redesigned, smaller, stacked-patch antenna [4]. The array is formed around the lower part of a 78 mm
radius sphere, in four rows of four antennas. Antennas are aligned in rows and
columns; thus the array has two axes of symmetry.Symmetrical curved antenna array used in microwave
radar for breast cancer detection: (a) CAD model, (b) photograph of the
manufactured array. The array consists of sixteen UWB antennas populated on a
section of hemisphere.
(a)(b)During laboratory experiments, the array is first filled
with a matching medium, the spherical skin phantom (2 mm thick) is placed in
the correct position, and then we attach a tank to the top of the antenna array
to finally fill it with a breast fat equivalent liquid [5] (the same as the matching
medium). This setup represents truly a three-dimensional (3D) breast phantom.
The chest wall is not considered in our experiments. The electrical properties
of the tissue phantom are based on the published permittivity values for average
human breast. At the frequency of 6 GHz, materials have the following values.
(a) Breast fat/matching medium has permittivity close to 10 and attenuation of
0.8 dB/cm; (b) skin phantom has permittivity of 30 and attenuation of 16 dB/cm;
(c) tumour phantom has permittivity close to 50 and conductivity of 7 S/m. More
details about our breast phantom can be found in [6].The contrast between dielectric properties of breast
fat and tumour phantom materials is around 1 : 5. Recently published data in
[7], based on a large
clinical study, suggest that the contrast between healthy and malignant breast
tissues might also be lower. The lower contrast obviously poses a more challenging
radar detection problem.Our radar system operates in the multistatic mode.
With sixteen antennas in the array, one hundred and twenty (120) independent
radar measurements are recorded for processing (the monostatic measurement is
not performed). Measurements are performed in the frequency domain between 3
and 10 GHz using a standard vector network analyser (VNA). All recorded radar
signals are transformed into the time domain for further signal processing
(described in Section3).


## 3. Focusing Algorithm

### 3.1. Extraction of Tumour Response from Measured Radar Data

The first step of signal processing deals with the
extraction of the tumour response from the raw measured data. This process must
be performed before equalisation and beamforming algorithms will be applied.
When a monostatic synthetic aperture radar is used for breast cancer detection,
tumour extraction aims at removing strong skin reflection from measured data.
This is usually performed by simple subtraction from the averaged skin
reflection signal (see [8]),
or by more sophisticated algorithms as presented in [9].The approach we use to extract the tumour response is
different. In our multistatic real aperture array, the measured response
contains not only strong skin reflections, but also reflections from other
mechanical parts of the array as well as antenna coupling signals. All these
undesired signals are usually of greater amplitude than that of the tumour
response. To subtract all unwanted signals, we physically rotate the antenna array
around its center and perform a second radar measurement. This target
displacement method is commonly used in radar cross-section measurements
[10, 11] to subtract undesired
signals.Rotation gives us two sets of measured data, in which
undesired signals such as antenna coupling or skin reflections are almost
identical and appear at the same time position; therefore they can be
eliminated. In contrast, a tumour response will appear at different time
position in these two measured sets (unless it is on the axis of rotation).
Applicability of this technique will depend on the homogeneity of the breast
within a given angle defined by rotation. We therefore assume that within the
angle of array rotation, (a) distance between antennas and skin remains
unchanged, (b) skin properties and thickness are the same, and (c) normal
breast tissue properties do not change. For more details about the performance
of this tumour extraction technique, please refer to [12].


### 3.2. Preprocessing (Equalisation)

Before applying the focusing algorithm, we have to
perform a preprocessing step. This process aims at the equalisation of
scattered tumour responses for different antenna pairs. Ideal preprocessing
would result in all received pulses being of the same shape and amplitude, and
perfectly time-aligned. In our preprocessing, the following steps are
performed: (1) extraction of the tumour response from measured data (see
[12]), (2)
equalisation of tissue losses, and (3) equalisation of radial spread of the
spherical wavefront. In the work reported herein for simplicity we do not
account for the frequency dependence of the tissue losses nor for the frequency-dependent
radiation patterns of the antennas.


### 3.3. Standard Delay-and-Sum Algorithm

Delay-and-sum (DAS) beamforming is a basic and
well-known method [13, 14]. First, we perform the preprocessing steps described
above. Next, appropriate time-delays Ti for all received signals are computed. The
time-delay Ti for a given transmitting and receiving antenna
is calculated based on the antenna's position, position of the focal point r=(x,y,z),
as well as an estimate of average wave propagation speed, which in our case is
simply assumed to be constant across the band.During the focusing, the focal point moves from one
position to another within the breast, resulting in spatial beamforming. At
each location, all time-shifted responses are coherently summed and integrated.
Integration is performed on the windowed signal, and the length of the
integration window is chosen according to the system bandwidth. A three-dimensional
(3D) map of scattered energy is formed in this way. The main advantage of the
DAS algorithm is its simplicity, robustness, and short computation time.Essentially, the scattered energy at the given focal
point within the breast volume can be expressed as(1)Fe(x,y,z)=∫0τ(∑i=1Mwi(x,y,z)⋅yi(t−Ti(x,y,z)))2dt, where M=N(N−1)/2 (N is the number of antennas in the array), wi is the location-dependent weight calculated
during preprocessing, yi is the measured radar signal, and Ti is the time-delay. τ is the length of the integration window,
chosen according to the system bandwidth. Due to the antenna effects and
dispersion, the integration window we utilize following coherent summation is
50 percent longer than the duration of the synthetic input pulse and equals 0.55
nanosecond. We have investigated the dependence of the window length on
focusing quality, and this value gave the best results.


### 3.4. Improved Delay-and-Sum Algorithm

The improved DAS algorithm uses an additional
weighting factorQF (quality factor), compared to the standard
DAS. QF can be interpreted as a quality factor of the
coherent focusing algorithm. It is calculated in three steps. Firstly, for each
focal point, we plot a curve of energy collection during the coherent signal
summation. An example of such a measured curve at a focal point containing a
tumour response is presented in Figure 2.Figure 2
Example of the curve of energy collection (measured
data).Next, the energy collection curve is rescaled by
normalising it to the standard deviation of energy,σe,
for all radar signals used in the summation. Normalisation is actually
performed using multiplication by 1/(1+σe),
since in the ideal case of equal energy in all (equalised) measured radar
signals, σe=0. This may be thought of as a heuristic
scaling of the data to give greater weight to those signals that, following
equalisation, more closely resemble the desired case of equal energy. The
utility of this heuristic weighting is evident from the results presented in
the following sections.In a last step, we estimate the coefficients of a
second-order polynomial (y=ax2+bx+c), which is the least-square fit of the
normalised curve of coherent energy collection. The choice of the second-order
polynomial comes from the fact that a curve of energy collection during a
perfect coherent signal summation would follow a quadratic curve. Then, we
assume that QF=a.
An order in which the signals are taken should not change the shape of the
curve, assuming the perfect equalisation. In practise, due to a nonideal
equalisation, the shape of the curve of energy collection will slightly depend
on the order in which signals are summed. However, it has negligible effect on
an a value since the final value of summed energy
is constant; a is calculated using least-square fitting.
Finally, the characteristic equation of the improved DAS algorithm is expressed
as(2)Fe(x,y,z)=QF(x,y,z)⋅∫0τ(∑i=1Mwi(x,y,z)⋅yi(t−Ti(x,y,z)))2dt.In the following section, we will present the
experimental results of phantom tumours detection, and discuss the new DAS
algorithm.


## 3.1. Extraction of Tumour Response from Measured Radar Data

The first step of signal processing deals with the
extraction of the tumour response from the raw measured data. This process must
be performed before equalisation and beamforming algorithms will be applied.
When a monostatic synthetic aperture radar is used for breast cancer detection,
tumour extraction aims at removing strong skin reflection from measured data.
This is usually performed by simple subtraction from the averaged skin
reflection signal (see [8]),
or by more sophisticated algorithms as presented in [9].The approach we use to extract the tumour response is
different. In our multistatic real aperture array, the measured response
contains not only strong skin reflections, but also reflections from other
mechanical parts of the array as well as antenna coupling signals. All these
undesired signals are usually of greater amplitude than that of the tumour
response. To subtract all unwanted signals, we physically rotate the antenna array
around its center and perform a second radar measurement. This target
displacement method is commonly used in radar cross-section measurements
[10, 11] to subtract undesired
signals.Rotation gives us two sets of measured data, in which
undesired signals such as antenna coupling or skin reflections are almost
identical and appear at the same time position; therefore they can be
eliminated. In contrast, a tumour response will appear at different time
position in these two measured sets (unless it is on the axis of rotation).
Applicability of this technique will depend on the homogeneity of the breast
within a given angle defined by rotation. We therefore assume that within the
angle of array rotation, (a) distance between antennas and skin remains
unchanged, (b) skin properties and thickness are the same, and (c) normal
breast tissue properties do not change. For more details about the performance
of this tumour extraction technique, please refer to [12].


## 3.2. Preprocessing (Equalisation)

Before applying the focusing algorithm, we have to
perform a preprocessing step. This process aims at the equalisation of
scattered tumour responses for different antenna pairs. Ideal preprocessing
would result in all received pulses being of the same shape and amplitude, and
perfectly time-aligned. In our preprocessing, the following steps are
performed: (1) extraction of the tumour response from measured data (see
[12]), (2)
equalisation of tissue losses, and (3) equalisation of radial spread of the
spherical wavefront. In the work reported herein for simplicity we do not
account for the frequency dependence of the tissue losses nor for the frequency-dependent
radiation patterns of the antennas.


## 3.3. Standard Delay-and-Sum Algorithm

Delay-and-sum (DAS) beamforming is a basic and
well-known method [13, 14]. First, we perform the preprocessing steps described
above. Next, appropriate time-delays Ti for all received signals are computed. The
time-delay Ti for a given transmitting and receiving antenna
is calculated based on the antenna's position, position of the focal point r=(x,y,z),
as well as an estimate of average wave propagation speed, which in our case is
simply assumed to be constant across the band.During the focusing, the focal point moves from one
position to another within the breast, resulting in spatial beamforming. At
each location, all time-shifted responses are coherently summed and integrated.
Integration is performed on the windowed signal, and the length of the
integration window is chosen according to the system bandwidth. A three-dimensional
(3D) map of scattered energy is formed in this way. The main advantage of the
DAS algorithm is its simplicity, robustness, and short computation time.Essentially, the scattered energy at the given focal
point within the breast volume can be expressed as(1)Fe(x,y,z)=∫0τ(∑i=1Mwi(x,y,z)⋅yi(t−Ti(x,y,z)))2dt, where M=N(N−1)/2 (N is the number of antennas in the array), wi is the location-dependent weight calculated
during preprocessing, yi is the measured radar signal, and Ti is the time-delay. τ is the length of the integration window,
chosen according to the system bandwidth. Due to the antenna effects and
dispersion, the integration window we utilize following coherent summation is
50 percent longer than the duration of the synthetic input pulse and equals 0.55
nanosecond. We have investigated the dependence of the window length on
focusing quality, and this value gave the best results.


## 3.4. Improved Delay-and-Sum Algorithm

The improved DAS algorithm uses an additional
weighting factorQF (quality factor), compared to the standard
DAS. QF can be interpreted as a quality factor of the
coherent focusing algorithm. It is calculated in three steps. Firstly, for each
focal point, we plot a curve of energy collection during the coherent signal
summation. An example of such a measured curve at a focal point containing a
tumour response is presented in Figure 2.Figure 2
Example of the curve of energy collection (measured
data).Next, the energy collection curve is rescaled by
normalising it to the standard deviation of energy,σe,
for all radar signals used in the summation. Normalisation is actually
performed using multiplication by 1/(1+σe),
since in the ideal case of equal energy in all (equalised) measured radar
signals, σe=0. This may be thought of as a heuristic
scaling of the data to give greater weight to those signals that, following
equalisation, more closely resemble the desired case of equal energy. The
utility of this heuristic weighting is evident from the results presented in
the following sections.In a last step, we estimate the coefficients of a
second-order polynomial (y=ax2+bx+c), which is the least-square fit of the
normalised curve of coherent energy collection. The choice of the second-order
polynomial comes from the fact that a curve of energy collection during a
perfect coherent signal summation would follow a quadratic curve. Then, we
assume that QF=a.
An order in which the signals are taken should not change the shape of the
curve, assuming the perfect equalisation. In practise, due to a nonideal
equalisation, the shape of the curve of energy collection will slightly depend
on the order in which signals are summed. However, it has negligible effect on
an a value since the final value of summed energy
is constant; a is calculated using least-square fitting.
Finally, the characteristic equation of the improved DAS algorithm is expressed
as(2)Fe(x,y,z)=QF(x,y,z)⋅∫0τ(∑i=1Mwi(x,y,z)⋅yi(t−Ti(x,y,z)))2dt.In the following section, we will present the
experimental results of phantom tumours detection, and discuss the new DAS
algorithm.


## 4. Experimental Detection Results

This section presents the experimental results of
tumour detection using our curved antenna array and 3D breast phantom. Focusing
results for standard DAS algorithm are compared to those for the improved DAS,
and differences between both algorithms are discussed. Results are presented
for tumours of two different sizes located at different positions: (a) 10 mm
spherical tumour located at positionPT(x=20,y=20,z=−20), (b) 7 mm spherical tumour located at
position PT(x=20,y=10,z=−10). All coordinates are quoted in
millimeter.


### 4.1. 10 mm Spherical Phantom Tumour

In Figure3, we present 3D focusing results for a 10 mm spherical phantom tumour located at the position PT(x=20,y=20,z=−20).
Figures present −3dB contour maps of
scattered energy, when the focusing was performed using the standard DAS
algorithm (Figure 3(a)) and the improved
DAS algorithm (Figure 3(b)).Detection results of a 10 mm spherical phantom tumour:
(a) standard DAS, (b) improved DAS withQF=a.
3D figures present −3dB contour map of scattered energy.
(a)(b)As we can see in Figure3(a), there are several
scatterers present in the image when focusing using standard DAS algorithm. The
strongest scatterer within the entire 3D volume is located at position PT(x=15,y=18,z=−18), and it is associated with tumour
response. In Figure 3, we have also indicated locations of two other strong
scatterers located at positions P1(x=−21,y=9,z=−18) and P2(x=−18,y=−21,z=−27). In
Figures 4(a) and 4(b), we
present 2D focusing results for standard DAS on the horizontal planes (z-axis)
containing the PT, P1 (both at z=−18), and P2(z=−27) signals associated with clutter. The 2D
contour plots show signal energy on a linear scale, normalised to the maximum
in the entire 3D volume (values below 0.1 are rendered as white). The skin
location at each plane is presented as a black circle. From Figure 4(a), for
the plane containing tumour we can relatively easily recognise the focused
tumour response (PT) and the nearby twin tumour response (at x=18, y=9; the  twintarget
response is due to tumour extraction
method—mechanical array rotation by 10 degrees). In the same figure, we can
also see that the strong clutter at position P1 and the weaker clutter (blue patches) at other
positions arose from imperfect extraction of tumour response. Figure 4(b)
presents the 2D focusing result through the plane (z=−27) containing strong clutter scatterer at P2.2D focusing results for standard and improved DAS
algorithms, for different horizontal planes along thez-axis: (a) standard DAS
(z=−18), (b) standard DAS (z=−27), (c) improved DAS (z=−18), (d) improved DAS
(z=−27). 2D contour plots show signal energy on a linear scale, normalised to
the maximum in the 3D volume; values below 0.1 are rendered as white.
(a)(b)(c)(d)Significantly better detection results were obtained
using improved DAS algorithm presented herein. 3D and 2D focusing results for
the improved DAS are presented in Figures3(b) and
4(c)-4(d),
respectively. The 3D contour map of scattered energy contains only the tumour
response (PT) and the twin tumour response. Unlike the
image obtained using standard DAS, there are no other clutter scatterers
visible. Signal-to-clutter ratio, defined as the ratio between energy of the
tumour response and the strongest clutter energy within a single 3D dataset,
was improved from 1.25 dB for standard DAS to 3.9 dB for improved DAS (2.65 dB
improvement).The same improved performance is observed in the 2D
results shown in Figures4(c) and
4(d). In the horizontal plane containing PT,
the tumour response clearly stands out, and very little clutter exists in the
image. In the plane containing P2,
clutter is also significantly suppressed.Results described
above (Figures3 and
4) have shown the improved tumour detection of the new DAS
algorithm, which uses additional weight QF,
compared to standard DAS. In what follows, we will analyse the improved DAS
algorithm and explain why it provides better results. To do so, we will go
through all steps of the new algorithm at the three focal points (PT, P1, P2) mentioned earlier in this section.After extraction of the tumour response from measured
data (by mechanical array rotation), resultant signals are being preprocessed
and time-aligned. This initial step is identical for standard and improved DAS
algorithms. Then, all pulsed signals (120 signals for our radar) are coherently
summed. During this process, the curve of coherent energy collection is
obtained, at each focal point within the focusing volume. This curve is
presented in Figures5(a)–5(c) for focal points PT, P1, P2,
respectively. The final value obtained after summation of all radar signals is
equal to the focused energy Fe in standard DAS.Curves of energy collection at focal pointsPT, P1, and P2 (see Figure 6(a)): (a) curve for PT as in standard DAS, (b) curve for P1 as in standard DAS, (c) curve for P2 in standard DAS, (d) curve for PT as in improved DAS together with the fitted
polynomial, (e) curve for P1 in the improved DAS together with the fitted
polynomial, (f) curve for P2 in the improved DAS together with the fitted
polynomial.
(a)(b)(c)(d)(e)(f)If we assume that the focused energy for tumour
location using standard DAS algorithm is equal to unityFs(PT)=1, focused energy values at focal points P1 and P2 equal  Fs(P1)=0.88 and Fs(P2)=0.85. Next, in the improved DAS algorithm,
we calculate the standard deviation of energy σe for all radar signals, and recalculate energy
collection curves by normalising them to σe.
The rationale to do so is based on the fact that after the initial
preprocessing equalisation step, all radar signals should have similar energy.We can see the rescaled (normalised) curves in Figures5(d)–5(f) (solid blue curves) for locations PT, P1, P2,
respectively. We observe that after this normalisation the results have
improved since the curves for P1, P2 have significantly smaller amplitudes than for PT.
Since clutter signals cannot be thought of as being totally uncorrelated, we do
not however simply use σe as the weight factor, but we apply additional
criteria related to the coherent summation of radar signals as follows.As known in the ideal coherent summation of scattered
pulses, the energy collection curve should follow a parabola(y=x2). Therefore, to check the “quality”
of coherent addition of radar signals in our system, we perform a second-order
polynomial (y=ax2+bx+c) fitting (in the least-square sense) to the
measured energy collection curves. This process is performed on the normalised
curves. Results of polynomial fitting are shown in Figures
5(d)–5(f) (dashed black curves), and
the constant a associated with x2 equals a=0.167 for PT, a=0.071 for P1,
and a=0.028 for P2.
Then, assuming that QF=a (see (2)), focused energy using improved DAS
at our example points of interest is  Fs(PT)=1, Fs(P1)=0.38, and Fs(P2)=0.14. The result of applying this process
throughout the entire volume is shown in 3D and 2D by Figures 3
and 4,
respectively.


### 4.2. 7 mm Spherical Phantom Tumour

This example presents the detection of a smaller 7 mm
spherical tumour phantom. In Figure6, we present 3D focusing results for the 6 mm spherical phantom tumour located at the position PT(x=20,y=10,z=−10).
Figures present −3 dB contour maps of
scattered energy, when focusing was performed using the standard DAS algorithm
(Figure 6(a)) and the improved DAS algorithm (Figure 6(b)).Detection results of a 7 mm spherical phantom tumour:
(a) standard DAS, (b) improved DAS withQF=a. 3D figures present −3 dB contour
map of scattered energy.
(a)(b)As we can see in Figure6(a), by using standard DAS
algorithm there are multiple scatterers present in the image. As we did previously
for the 10 mm tumour, again this time we concentrate the attention on three
focal points: spherical phantom tumour located at PT(x=21,y=9,z=−6), the strongest clutter scatterer at P1(x=−24,y=−15,z=−27), and another strong clutter at P2(x=18,y=−9,z=−33). A significantly better image, with
clearly visible tumour scatterer at PT and no other clutter targets, is presented in
Figure 6(b) for the improved DAS algorithm. Signal-to-clutter ratio was
improved from 0.8 dB for standard DAS to 5.2 dB for improved DAS, providing 4.4 dB better performance using a proposed algorithm.Looking at all 2D focused images (Figure7) for
standard DAS, we can observe that clutter strength generally increases closer
to skin. This observation is confirmed when looking at locations of focal
points investigated above. In our 3D breast phantom, the 2 mm skin layer has a
radius  rskin=59 mm. The true tumour response at PT is located 35 mm away from the skin (rPT=24 mm), the strongest clutter signal at P1 is 20 mm away from the skin (rP1=39 mm), and another strong clutter at P2 is also 20 mm away from the skin (rP2=39 mm). As we can see, all the strong clutter
signals are located closer to skin than to the tumour.2D focusing results for standard and improved DAS
algorithms, for different horizontal planes along thez-axis: (a) standard DAS
(z=−9), (b) standard DAS (z=−27), (c) standard DAS (z=−33), (d) improved DAS
(z=−6), (e) improved DAS (z=−27), (f) improved DAS (z=−33). 2D contour plots
show signal energy on a linear scale, normalised to the maximum in the 3D
volume; values below 0.1 are rendered as white.
(a)(b)(c)(d)(e)(f)In Figure8, curves of coherent energy collection are
presented for focal points PT, P1, P2.
Plots associated with standard DAS algorithm are shown in Figures
8(a)–8(c) for focal points PT, P1, P2,
respectively. The value obtained after summation of all radar signals is equal
to the focused energy Fe in standard DAS. Next, in the improved DAS
algorithm we calculate the standard deviation of energy σe for all radar signals, and recalculate energy
collection curves by normalising them to σe.
Resultant normalised curves for locations PT, P1,
and P2 are depicted in Figures 8(d)–8(f)
(solid blue curves), respectively.Curves of energy collection at focal pointsPT, P1, and P2 (see Figure 6(a)): (a) curve for PT as in standard DAS, (a) curve for P1 as in standard DAS, (c) curve for P2 as  in standard DAS, (d) curve for PT as in improved DAS together with the fitted
polynomial, (e) curve for P1 in the improved DAS together with the fitted
polynomial, (f) curve for P2 in the improved DAS together with the fitted
polynomial.
(a)(b)(c)(d)(e)(f)We observe that, after normalisation, the curves forP1, P2 have significantly dropped compared to PT,
due to the higher values of standard deviation of the energy content of the
radar signals. Next, we performed the second-order polynomial fitting on the
normalised energy collection curves to obtain the weight factor QF=a.
Results of polynomial fitting are shown as dashed black curve in Figures
8(d)–8(f). The constant a associated with x2 equals a=0.094 for PT, a=−0.01 for P1,
and a=−0.049 for P2.
Interestingly, due to a noncoherent signal summation for focal points P1 and P2, a has not only lower absolute value than for PT,
but has also negative sign. The focused energy Fe using improved DAS algorithm (as in
(2)) will become negative for focal points, where QF=a<0,
additionally improving imaging results.


## 4.1. 10 mm Spherical Phantom Tumour

In Figure3, we present 3D focusing results for a 10 mm spherical phantom tumour located at the position PT(x=20,y=20,z=−20).
Figures present −3dB contour maps of
scattered energy, when the focusing was performed using the standard DAS
algorithm (Figure 3(a)) and the improved
DAS algorithm (Figure 3(b)).Detection results of a 10 mm spherical phantom tumour:
(a) standard DAS, (b) improved DAS withQF=a.
3D figures present −3dB contour map of scattered energy.
(a)(b)As we can see in Figure3(a), there are several
scatterers present in the image when focusing using standard DAS algorithm. The
strongest scatterer within the entire 3D volume is located at position PT(x=15,y=18,z=−18), and it is associated with tumour
response. In Figure 3, we have also indicated locations of two other strong
scatterers located at positions P1(x=−21,y=9,z=−18) and P2(x=−18,y=−21,z=−27). In
Figures 4(a) and 4(b), we
present 2D focusing results for standard DAS on the horizontal planes (z-axis)
containing the PT, P1 (both at z=−18), and P2(z=−27) signals associated with clutter. The 2D
contour plots show signal energy on a linear scale, normalised to the maximum
in the entire 3D volume (values below 0.1 are rendered as white). The skin
location at each plane is presented as a black circle. From Figure 4(a), for
the plane containing tumour we can relatively easily recognise the focused
tumour response (PT) and the nearby twin tumour response (at x=18, y=9; the  twintarget
response is due to tumour extraction
method—mechanical array rotation by 10 degrees). In the same figure, we can
also see that the strong clutter at position P1 and the weaker clutter (blue patches) at other
positions arose from imperfect extraction of tumour response. Figure 4(b)
presents the 2D focusing result through the plane (z=−27) containing strong clutter scatterer at P2.2D focusing results for standard and improved DAS
algorithms, for different horizontal planes along thez-axis: (a) standard DAS
(z=−18), (b) standard DAS (z=−27), (c) improved DAS (z=−18), (d) improved DAS
(z=−27). 2D contour plots show signal energy on a linear scale, normalised to
the maximum in the 3D volume; values below 0.1 are rendered as white.
(a)(b)(c)(d)Significantly better detection results were obtained
using improved DAS algorithm presented herein. 3D and 2D focusing results for
the improved DAS are presented in Figures3(b) and
4(c)-4(d),
respectively. The 3D contour map of scattered energy contains only the tumour
response (PT) and the twin tumour response. Unlike the
image obtained using standard DAS, there are no other clutter scatterers
visible. Signal-to-clutter ratio, defined as the ratio between energy of the
tumour response and the strongest clutter energy within a single 3D dataset,
was improved from 1.25 dB for standard DAS to 3.9 dB for improved DAS (2.65 dB
improvement).The same improved performance is observed in the 2D
results shown in Figures4(c) and
4(d). In the horizontal plane containing PT,
the tumour response clearly stands out, and very little clutter exists in the
image. In the plane containing P2,
clutter is also significantly suppressed.Results described
above (Figures3 and
4) have shown the improved tumour detection of the new DAS
algorithm, which uses additional weight QF,
compared to standard DAS. In what follows, we will analyse the improved DAS
algorithm and explain why it provides better results. To do so, we will go
through all steps of the new algorithm at the three focal points (PT, P1, P2) mentioned earlier in this section.After extraction of the tumour response from measured
data (by mechanical array rotation), resultant signals are being preprocessed
and time-aligned. This initial step is identical for standard and improved DAS
algorithms. Then, all pulsed signals (120 signals for our radar) are coherently
summed. During this process, the curve of coherent energy collection is
obtained, at each focal point within the focusing volume. This curve is
presented in Figures5(a)–5(c) for focal points PT, P1, P2,
respectively. The final value obtained after summation of all radar signals is
equal to the focused energy Fe in standard DAS.Curves of energy collection at focal pointsPT, P1, and P2 (see Figure 6(a)): (a) curve for PT as in standard DAS, (b) curve for P1 as in standard DAS, (c) curve for P2 in standard DAS, (d) curve for PT as in improved DAS together with the fitted
polynomial, (e) curve for P1 in the improved DAS together with the fitted
polynomial, (f) curve for P2 in the improved DAS together with the fitted
polynomial.
(a)(b)(c)(d)(e)(f)If we assume that the focused energy for tumour
location using standard DAS algorithm is equal to unityFs(PT)=1, focused energy values at focal points P1 and P2 equal  Fs(P1)=0.88 and Fs(P2)=0.85. Next, in the improved DAS algorithm,
we calculate the standard deviation of energy σe for all radar signals, and recalculate energy
collection curves by normalising them to σe.
The rationale to do so is based on the fact that after the initial
preprocessing equalisation step, all radar signals should have similar energy.We can see the rescaled (normalised) curves in Figures5(d)–5(f) (solid blue curves) for locations PT, P1, P2,
respectively. We observe that after this normalisation the results have
improved since the curves for P1, P2 have significantly smaller amplitudes than for PT.
Since clutter signals cannot be thought of as being totally uncorrelated, we do
not however simply use σe as the weight factor, but we apply additional
criteria related to the coherent summation of radar signals as follows.As known in the ideal coherent summation of scattered
pulses, the energy collection curve should follow a parabola(y=x2). Therefore, to check the “quality”
of coherent addition of radar signals in our system, we perform a second-order
polynomial (y=ax2+bx+c) fitting (in the least-square sense) to the
measured energy collection curves. This process is performed on the normalised
curves. Results of polynomial fitting are shown in Figures
5(d)–5(f) (dashed black curves), and
the constant a associated with x2 equals a=0.167 for PT, a=0.071 for P1,
and a=0.028 for P2.
Then, assuming that QF=a (see (2)), focused energy using improved DAS
at our example points of interest is  Fs(PT)=1, Fs(P1)=0.38, and Fs(P2)=0.14. The result of applying this process
throughout the entire volume is shown in 3D and 2D by Figures 3
and 4,
respectively.


## 4.2. 7 mm Spherical Phantom Tumour

This example presents the detection of a smaller 7 mm
spherical tumour phantom. In Figure6, we present 3D focusing results for the 6 mm spherical phantom tumour located at the position PT(x=20,y=10,z=−10).
Figures present −3 dB contour maps of
scattered energy, when focusing was performed using the standard DAS algorithm
(Figure 6(a)) and the improved DAS algorithm (Figure 6(b)).Detection results of a 7 mm spherical phantom tumour:
(a) standard DAS, (b) improved DAS withQF=a. 3D figures present −3 dB contour
map of scattered energy.
(a)(b)As we can see in Figure6(a), by using standard DAS
algorithm there are multiple scatterers present in the image. As we did previously
for the 10 mm tumour, again this time we concentrate the attention on three
focal points: spherical phantom tumour located at PT(x=21,y=9,z=−6), the strongest clutter scatterer at P1(x=−24,y=−15,z=−27), and another strong clutter at P2(x=18,y=−9,z=−33). A significantly better image, with
clearly visible tumour scatterer at PT and no other clutter targets, is presented in
Figure 6(b) for the improved DAS algorithm. Signal-to-clutter ratio was
improved from 0.8 dB for standard DAS to 5.2 dB for improved DAS, providing 4.4 dB better performance using a proposed algorithm.Looking at all 2D focused images (Figure7) for
standard DAS, we can observe that clutter strength generally increases closer
to skin. This observation is confirmed when looking at locations of focal
points investigated above. In our 3D breast phantom, the 2 mm skin layer has a
radius  rskin=59 mm. The true tumour response at PT is located 35 mm away from the skin (rPT=24 mm), the strongest clutter signal at P1 is 20 mm away from the skin (rP1=39 mm), and another strong clutter at P2 is also 20 mm away from the skin (rP2=39 mm). As we can see, all the strong clutter
signals are located closer to skin than to the tumour.2D focusing results for standard and improved DAS
algorithms, for different horizontal planes along thez-axis: (a) standard DAS
(z=−9), (b) standard DAS (z=−27), (c) standard DAS (z=−33), (d) improved DAS
(z=−6), (e) improved DAS (z=−27), (f) improved DAS (z=−33). 2D contour plots
show signal energy on a linear scale, normalised to the maximum in the 3D
volume; values below 0.1 are rendered as white.
(a)(b)(c)(d)(e)(f)In Figure8, curves of coherent energy collection are
presented for focal points PT, P1, P2.
Plots associated with standard DAS algorithm are shown in Figures
8(a)–8(c) for focal points PT, P1, P2,
respectively. The value obtained after summation of all radar signals is equal
to the focused energy Fe in standard DAS. Next, in the improved DAS
algorithm we calculate the standard deviation of energy σe for all radar signals, and recalculate energy
collection curves by normalising them to σe.
Resultant normalised curves for locations PT, P1,
and P2 are depicted in Figures 8(d)–8(f)
(solid blue curves), respectively.Curves of energy collection at focal pointsPT, P1, and P2 (see Figure 6(a)): (a) curve for PT as in standard DAS, (a) curve for P1 as in standard DAS, (c) curve for P2 as  in standard DAS, (d) curve for PT as in improved DAS together with the fitted
polynomial, (e) curve for P1 in the improved DAS together with the fitted
polynomial, (f) curve for P2 in the improved DAS together with the fitted
polynomial.
(a)(b)(c)(d)(e)(f)We observe that, after normalisation, the curves forP1, P2 have significantly dropped compared to PT,
due to the higher values of standard deviation of the energy content of the
radar signals. Next, we performed the second-order polynomial fitting on the
normalised energy collection curves to obtain the weight factor QF=a.
Results of polynomial fitting are shown as dashed black curve in Figures
8(d)–8(f). The constant a associated with x2 equals a=0.094 for PT, a=−0.01 for P1,
and a=−0.049 for P2.
Interestingly, due to a noncoherent signal summation for focal points P1 and P2, a has not only lower absolute value than for PT,
but has also negative sign. The focused energy Fe using improved DAS algorithm (as in
(2)) will become negative for focal points, where QF=a<0,
additionally improving imaging results.


## 5. Conclusions

In this paper, we have presented a modified
delay-and-sum (DAS) beamforming algorithm for breast cancer detection. The
improved DAS algorithm uses an additional weight factor calculated at each
focal point, to improve image quality. These weights essentially represent the
quality of the preprocessing step and the coherent radar operation. Using
measured data from a multistatic UWB radar system, we have presented
experimental detection of 7 mm and 10 mm tumours in a phantom. We have shown
that the proposed new DAS algorithm improves signal-to-clutter ratio in focused
images by 2.65 dB for 10 mm tumour, and by 4.4 dB for 7 mm tumour.Further, it may be noted that this
improvement in signal-to-clutter ratio is
comparable to that achieved [12] by a much more complex data-adaptive algorithm based
on robust Capon beamforming. Importantly, however, it requires significantly
less (order-of-magnitude) computation time.



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*Source: 761402-2008-06-05.xml*