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ISSC 2005, Dublin. September 1–2
A Contrast Source Inversion Scheme for Imaging
Acoustic Contrast
Koen W.A. van Dongen†and William M.D. Wright∗
Ultrasonics Research Group
Department of Electrical and Electronic Engineering
University College Cork, Cork
IRELAND
E-mail: †koen@rennes.ucc.ie ∗bill.wright@ucc.ie
Abstract —In this paper we present an inversion method which
allows the reconstruction of density and compressibility profiles from
measured scattered acoustic pressure and velocity data. The imaging
method is based on a complete three-dimensional vectorial descrip-
tion of the acoustic wave fields. A conjugate gradient scheme is used
to solve the corresponding integral equations iteratively. In order
to regularize the inverse problem, a contrast source formulation is
used. Good results are obtained with this novel imaging method
when tested on synthetic data.
I Introduction
Acoustic wave fields are used in a wide range of
applications to probe the interior of objects in a
nondestructive and noninvasive manner. The scale
on which this takes place varies from the kilometer
range in the oil and gas industry down to the mil-
limeter range in medical applications. One of the
medical applications we are interested in is hyper-
thermia cancer treatment. In this case, acoustic
wave fields are used to measure during or directly
after the treatment the acoustic medium parame-
ters in and around the tumor in order to investi-
gate the effect of the treatment. Next, changes in
the medium parameters of the tissue are related to
changes in temperature[1]. Consequently, there is
a demand for imaging methods which reconstruct
these velocity profiles with high accuracy.
In this paper we investigate a novel method to
localize and characterize three dimensional objects
from measured scattered acoustic pressure and ve-
locity wave fields. Contrast functions are used
to describe the variations present in the acoustic
medium parameters; the volume density of mass
and the compressibility. The scattering mecha-
nism is formulated via two integral equations of
the second type, one for the pressure and one for
the velocity wave field.
During the forward problem, the integral equa-
tions are solved for known incident wave fields,
known contrast functions and unknown scattered
wave fields. The inverse problem refers to the situ-
ation where a solution is obtained for the unknown
contrast function for known incident and measured
scattered wave fields[2]. It is well known that
this inverse problem is ill-posed. Therefore, many
imaging methods rely on reducing the number of
unknowns by describing the scattering mechanism
as a function of a single contrast function, e.g.
changes in speed of sound only. In this case, a
scalar formulation of the problem is sufficient.
In this paper, we discuss a novel technique based
on a vectorial description of the wave field prob-
lem. This method allows the reconstruction of
both the density and the compressibility from mea-
sured acoustic data by using minimization method
based on a conjugate gradient inversion scheme.
In a standard conjugate gradient inversion scheme
the L2-norm of a single error function is minimized
iteratively. This error function is defined as the
residual of the measured scattered wave field mi-
nus the computed scattered wave field based on
the approximate solution. In order to regularize
the problem, a contrast source formulation is used
which splits the error functional into two parts.
Note that, similar problems play an important
role in electromagnetic scattering problems. Here,
contrasts are defined via changes in the complex
permittivity and/or magnetic permeability. In this
area, good results are recently obtained by Van den
Berg and Abubakar[3] by using a contrast source
formulations similar to the one formulated in this
paper.
II Forward Problem
The formulation of the scattering problem is done
in the temporal Laplace domain with Laplace pa-
rameter ˆs. Results for the frequency domain Ω
are obtained by taking the limit ˆs→ −iω, with
i2=−1 and ωthe temporal angular frequency.
The symbol ”ˆ” on top of a parameter is used to
show its temporal dependency. Positions in the
spatial domain R3are notated as a vector xmor
xn, with {m, n}= 1,2 or 3.
Combining reciprocity[4] with the acoustic wave
field equations results in an expression for the total
pressure wave field ˆptot(xm) and the total veloc-
ity wave field ˆvtot
i(xm) for {i, j}= 1,2 or 3 which
reads
ˆptot(xm) = ˆpinc(xm) + ˆpsct(xm),(1)
ˆvtot
i(xm) = ˆvinc
i(xm) + ˆvsct
i(xm),(2)
where ˆpinc (xm) and ˆvinc
k(xm) are the incident pres-
sure and velocity fields and where ˆpsct(xm) and
ˆvsct
i(xm) refer to the scattered pressure and veloc-
ity fields. In the presence of acoustic contrasts in
the homogenous background medium these scat-
tered wave fields equal
ˆpsct (xm) = ˆ
Gpq(xm, xn)∆ˆη(xn)ˆptot (xn)
+ˆ
Gpf
i(xm, xn)∆ˆ
ζ(xn)ˆvtot
i(xn),(3)
ˆvsct
i(xm) = ˆ
Gvq
i(xm, xn)∆ˆη(xn)ˆptot (xn)
+ˆ
Gvf
i,j (xm, xn)∆ˆ
ζ(xn)ˆvtot
j(xn),(4)
where the contrasts functions ∆ˆη(xn) and ∆ˆ
ζ(xn)
are defined by variations in the acoustic medium
parameters compressibility κand density ρof the
background medium (bg) and the object medium
(obj) via
∆ˆη(xn) = ˆs∆κ(xn) = ˆsκbg −κob j(xn),(5)
∆ˆ
ζ(xn) = ˆs∆ρ(xn) = ˆsρbg −ρobj (xn).(6)
The Green’s tensor functions shown in equa-
tions (3) and (4) are defined as follows
ˆ
Gpq(xm, xn)ˆq(xn) = ζbg ˆ
G(xm, xn)∗ˆq(xn),(7)
ˆ
Gpf
i(xm, xn)ˆ
fi(xn) = −∂i
hˆ
G(xm, xn)∗ˆ
fi(xn)i,(8)
ˆ
Gvq
i(xm, xn)ˆq(xn) = −∂ihˆ
G(xm, xn)∗ˆq(xn)i,(9)
ˆ
Gvf
i,j (xm, xn)ˆ
fj(xn) = 1
ζbg {
∂i∂jhˆ
G(xm, xn)∗ˆ
fj(xn)i
+δi,j δ(xm−xn)∗ˆ
fj(xn)o,(10)
with ∂ithe spatial derivative in the xidirection,
δi,j Kronecker’s delta function, δ(xm−xn) the im-
pulse response function, ˆ
G(xm, xn) the scalar form
of Green’s function, ˆq(xn) a volume density of in-
jection rate source and ˆ
fj(xn) a volume density
of force. Note that, f(xm, xn)∗g(xn) refers to a
convolution of the functions f(xn) and g(xn) over
the spatial domain D. A solution for the forward
problem is obtained by using a standard conjugate
gradient inversion scheme, as described by Klein-
man and Van den Berg[5].
III Inverse Problem
The aim is to compute the contrast functions of
the acoustic medium parameters compressibility κ
and density ρby solving the set of integral equa-
tions stated in equations (1)-(10) for known mea-
sured scattered acoustic wave fields. Due to the
ill-posed nature of the inverse problem they can
not be solved with a standard conjugate gradi-
ent inversion scheme and regularization is required.
This regularization is obtained by using a contrast
source formulation.
a) Contrast Source Formulation
Examining equations (3) and (4) reveals that the
sources generating the scattered wave fields are
defined as the product of two contrast functions
with corresponding total wave fields. Combining
these terms lead to an expression for the contrast
sources, ˆwp
k(xm) and ˆwv
i,k (xm), which read
ˆwp
k(xm) = ˆs∆κ(xm)ˆptot
k(xm),(11)
ˆwv
i,k (xm) = ˆs∆ρ(xm)ˆvtot
i,k (xm),(12)
where the subscript k= 1,...,K refers to the spa-
tial position of the transmitter which generates the
incident wave field. Consequently, equations (3)
and (4) can be formulated as
ˆpsct
k(xm) = ˆ
Gpq (xm, xn) ˆwp
k(xn)
+ˆ
Gpf
i(xm, xn) ˆwv
i,k (xn),(13)
ˆvsct
i,k (xm) = ˆ
Gvq
i(xm, xn) ˆwp
k(xn)
+ˆ
Gvf
i,j (xm, xn) ˆwv
j,k (xn).(14)
Combining equations (11)-(14) leads to a reformu-
lation of the contrast sources as follows
ˆwp
k(xm) = ˆs∆κ(xm)ˆpinc
k(xm)
+ ˆs∆κ(xm)ˆ
Gpq (xm, xn) ˆwp
k(xn)
+ ˆs∆κ(xm)ˆ
Gpf
i(xm, xn) ˆwv
i,k (xn),(15)
ˆwv
i,k (xm) = ˆs∆ρ(xm)ˆvinc
i,k (xm)
+ ˆs∆ρ(xm)ˆ
Gvq
i(xm, xn) ˆwp
k(xn)
+ ˆs∆ρ(xm)ˆ
Gvf
i,j (xm, xn) ˆwv
j,k(xn).(16)
Based on the formulations used for equa-
tions (11)-(16), two sets of error functionals are
defined. The first one is referred to as the error
in the data equation and is based on the residual
of the measured scattered wave fields minus the
scattered wave fields based on the approximated
contrast sources. Hence, at the Nth iteration step
this error functional reads
ˆrp
S;N;k(xm) = ˆpsct
k(xm)−ˆ
Gpq
S(xm, xn) ˆwp
N;k(xn)
−ˆ
Gpf
i(xm, xn) ˆwv
N;i,k(xn),(17)
ˆrv
S;N;i,k(xm) = ˆvsct
i,k (xm)−ˆ
Gvq
i(xm, xn) ˆwp
N;k(xn)
−ˆ
Gvf
i,j (xm, xn)wv
N;j,k(xn),(18)
with Sthe spatial domain containing the transduc-
ers. The second one is referred to as the error in
the object equation and is based on the error in
the computed contrast sources. Hence, this error
functional equals
ˆrp
D;N;k(xm) = ˆs∆κN(xm)ˆpinc
k(xm)−ˆwp
N;k(xm)
+ ˆs∆κ(xm)Gpqwp
k(xm)
+ ˆs∆κN(xm)ˆ
Gpf
iˆwv
N;i,k(xn),(19)
ˆrv
D;N;i,k(xm) = ˆs∆ρN(xm)ˆvinc
i,k (xm)−ˆwv
N;i,k(xm)
+ ˆs∆ρ(xm)Gvq
iwp
k(xm)
+ ˆs∆ρN(xm)ˆ
Gvf
i,j ˆwv
N;j,k(xn),(20)
with Dthe spatial domain containing the contrast
and hence the contrast sources. Next, we define
the L2-norm of the four-dimensional vector repre-
senting the acoustic wave fields in the spatial do-
main Sas
ˆpk(xm)
ˆvi,k(xm)
2
S
=X
xm∈S,k,ΩZˆpk(xm) [ ˆpk(xm)]∗
+X
i
ˆvi,k(xm) [ ˆvi,k (xm)]∗,(21)
and the L2-norm of a four-dimensional vector rep-
resenting the contrast sources in the spatial do-
main Das
ˆwp
k(xm)
ˆwv
i,k (xm)
2
D
=X
xm∈D,k,Ω(ˆwp
k(xm) [ ˆwp
k(xm)]∗
+X
i
Zˆwv
i,k(xm)ˆwv
i,k(xm)∗),(22)
where the constant Zis used to correct for differ-
ences in dimensions and is defined as
Z=κbg
ρbg .(23)
Finally, we define the normalized error functional
χ6= 0
χ= 0
xsrc
mxrec
m
x1
x2
D
S
Fig. 1: The setup used for fan beam measurements
ErrNas
ErrN=
ˆrp
S;N;k(xm)
ˆrv
S;N;i,k(xm)
2
S
ˆpsct
k(xm)
ˆvsct
i,k (xm)
2
S
+
ˆrp
D;N;k(xm)
ˆrv
D;N;i,k(xm)
2
D
∆ˆηN−1(xm)ˆpinc
k(xm)
∆ˆ
ζN−1(xm)ˆvinc
i,k (xm)
2
D
,(24)
with ∆ˆηN(xm) and ∆ˆ
ζN(xm) based on the approx-
imated medium parameters of the objects. These
medium parameters are obtained via direct mini-
mization of the cost functionals in equations (19)
and (20), consequently
κobj
N(xm) = κbg −P
k,Ω
ℜhˆsˆptot
N;k(xm) ˆwp
N;k(xm)i
P
k,Ωˆsˆptot
N;k(xm)
2,
(25)
ρobj
N(xm) = ρbg −P
i,k,Ω
ℜhˆsˆvtot
N;i,k(xm) ˆwv
N;i,k(xm)i
P
i,k,Ωˆsˆvtot
N;i,k(xm)
2.
(26)
Consequently, a conjugate gradient minimiza-
tions scheme is used to construct the contrast
sources. After each update of the contrast sources,
the contrast function is obtained via direct mini-
mization.
IV Results and Discussion
The integral formulation shown in equations (1)-
(10) is applied to the situation shown in Fig. 1.
Table 1: The medium parameters compressibility,
κ, density, ρ, and speed of sound, c, for various
tissues [6], [7], [8].
κ ρ c
[10−9(Pa)−1] [kg/m3] [m/s]
T= 37 ◦C 0.36648 1056.6 1607
T= 45 ◦C 0.10533 1053.3 1612
T= 50 ◦C 0.36571 1051.0 1613
Here, 36 transducers (K= 36) are positioned on
the spatial circular domain Swhich encloses the
circular cylindrical domain Dcontaining the ob-
jects. Data is obtained for the situation where
one transducer acts as a transmitter while the re-
mainder act as receivers. A complete data set
is obtained by rotating the transducer system in
equiangular steps over 360◦around the object.
Synthetic data has been computed based on
the synthetic contrast function shown in Fig. 2(a)
and 2(b). The medium parameters, see Table 1,
correspond to the situation where in human liver
tissue at 37 oC one lesion is heated to 45 oC and
two lesions are heated to 50 oC. The three dimen-
sional volume contains 32 ×32 ×4 elements of size
2.5×2.5×2.5 mm3. Each temporal signal con-
tains 16 points, equally spread over the frequency
domain Ω = [5,156] k Hz.
An additional data set ˆpsct
k(xm) and ˆvsct
i,k (xm) is
computed by adding 10% white noise to the mea-
sured scattered wave fields, hence
ˆpsct
k(xm) = ˆpsct
k(xm) + rmax ˆpsct
k(xm),(27)
ˆvsct
i,k (xm) = ˆvsct
i,k (xm) + rmax ˆvsct
i,k (xm),(28)
with random rational number r∈[−0.1,0.1],
max (|ˆpsct
k(xm)|) the maximum value present in
the set of all absolute values of ˆpsct
k(xm) and
max ˆvsct
i,k (xm)the maximum value present in
the set of all absolute values of ˆvsct
i,k (xm).
In Fig. 2(c)-2(f) the results obtained with the
contrast source formulation are shown. A velocity
profile based on the synthetic and reconstructed
medium parameters is shown in Fig. 3. The mini-
mization process was stopped after 100 iterations.
At this stage, further decrease in the error func-
tional can be neglected as can been observed from
Fig. 4.
A personal computer containing 1 GB of RAM
has been used to perform the computations in
approximately half a day. In the computational
scheme there are present a large number of loops
which can be performed in parallelized way. This
makes the application ideal to be performed on a
computer cluster in order to reduce the computa-
tional time.
By comparing the reconstructed profiles (com-
pressibility, density and velocity) with the syn-
thetic profiles it can be shown that the maximum
relative error is below 0.3%, even in presence of
10% white noise. Finding a numerical solution for
the noise is difficult, since noise is not covered in
the set of equations describing the scattering mech-
anism. Consequently, the error functional has a
slower converging rate when compared with a noise
free signal. In the reconstructed images, the pres-
ence of noise is reflected by the large amount of
small fluctuations in the reconstructed background
medium surrounding the objects. Preprocessing
the signal with various filtering techniques could
be a solution to this problem.
V Conclusion
A novel imaging method has been presented which
allows the reconstruction of both density and
compressibility profiles from measured scattered
acoustic pressure and velocity data. The imaging
method is based on a complete three-dimensional
vectorial description of the acoustic wave fields. In
order to regularize the inverse problem, a contrast
source formulation is used. Good results are ob-
tained with this novel imaging method when tested
on synthetic data with and without 10% white
noise. The results show that the maximum error
is less than 0.3% in all the reconstructed images.
Acknowledgments
This work was financially supported via a Marie
Curie Intra-European fellowship. We also grate-
fully acknowledge all the fruitful discussions we
had with Conor Brennan and Konstantin M. Bo-
grachev. Finally, we would like to thank the Boole
Centre for Research in Informatics, University Col-
lege Cork, Ireland, for the usage of their computer
facility.
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00.05
0
0.05
0
0.5
1
x 10−12
x [m]
∆κsyn(x,y)
y [m] 0
0.2
0.4
0.6
0.8
1
x 10−12
(a)
00.05
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0.05
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00.05
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∆ρCS(x,y)
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2
3
4
5
(d)
00.05
0
0.05
0
0.5
1
x 10−12
x [m]
∆κCS(x,y) (noise)
y [m] 0
0.2
0.4
0.6
0.8
1
x 10−12
(e)
00.05
0
0.05
0
2
4
x [m]
∆ρCS(x,y) (noise)
y [m] 0
1
2
3
4
5
(f)
Fig. 2: A cross section of the synthetic and reconstructed contrast functions.
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00.05
0
0.05
1608
1610
1612
x [m]
csyn(x,y)
y [m]
1608
1609
1610
1611
1612
(a)
00.05
0
0.05
1608
1610
1612
x [m]
cCS(x,y)
y [m]
1608
1609
1610
1611
1612
(b)
00.05
0
0.05
1608
1610
1612
x [m]
cCS(x,y) (noise)
y [m]
1608
1609
1610
1611
1612
(c)
0 0.02 0.04 0.06
0
0.02
0.04
0.06
| csyn − cCS | / csyn
x [m]
y [m]
0
0.5
1
1.5
x 10−3
(d)
0 0.02 0.04 0.06
0
0.02
0.04
0.06
| csyn − cCS | / csyn (noise)
x [m]
y [m]
0
5
10
15
x 10−4
(e)
Fig. 3: A cross section of the synthetic and reconstructed velocity profiles and their relative errors.
0 20 40 60 80 100
10−4
10−3
10−2
10−1
100
Nit
Err
Err (noise)
Err
Fig. 4: The error functional E rr.
