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IOP PUBLISHING INVERSE PROBLEMS
Inverse Problems 26 (2010) 015005 (18pp) doi:10.1088/0266-5611/26/1/015005
An approximation error approach for compensating
for modelling errors between the radiative transfer
equation and the diffusion approximation in diffuse
optical tomography
T Tarvainen1,2, V Kolehmainen1, A Pulkkinen1,3, M Vauhkonen1,
M Schweiger2, S R Arridge2and J P Kaipio1,4
1Department of Physics, University of Kuopio, PO Box 1627, 70211 Kuopio, Finland
2Department of Computer Science, University College London, Gower Street,
London WC1E 6BT, UK
3Sunnybrook Research Institute, Sunnybrook Health Sciences Centre, 2075 Bayview Ave.,
Toronto, ON, M4N 3M5, Canada
4Department of Mathematics, University of Auckland, Private Bag 92019, Auckland Mail
Centre, Auckland 1142, New Zealand
E-mail: Tanja.Tarvainen@uku.fi
Received 1 June 2009, in final form 23 October 2009
Published 11 December 2009
Online at stacks.iop.org/IP/26/015005
Abstract
In this paper, we investigate the applicability of the Bayesian approximation
error approach to compensate for the discrepancy of the diffusion
approximation in diffuse optical tomography close to the light sources and
in weakly scattering subdomains. While the approximation error approach has
earlier been shown to be a feasible approach to compensating for discretization
errors, uncertain boundary data and geometry, the ability of the approach
to recover from using a qualitatively incorrect physical model has not been
contested. In the case of weakly scattering subdomains and close to sources,
the radiative transfer equation is commonly considered to be the most accurate
model for light scattering in turbid media. In this paper, we construct the
approximation error statistics based on predictions of the radiative transfer and
diffusion models. In addition, we investigate the combined approximation
errors due to using the diffusion approximation and using a very low-
dimensional approximation in the forward problem. We show that recovery is
feasible in the sense that with the approximation error model the reconstructions
with a low-dimensional diffusion approximation are essentially as good as with
using a very high-dimensional radiative transfer model.
(Some figures in this article are in colour only in the electronic version)
0266-5611/10/015005+18$30.00 © 2010 IOP Publishing Ltd Printed in the UK 1
Inverse Problems 26 (2010) 015005 T Tarvainen et al
1. Introduction
Diffuse optical tomography (DOT) is a relatively new non-invasive imaging method in which
images of the optical properties of the medium are estimated based on measurements of near-
infrared light on the surface of the object [1]. It has potential applications in medical imaging,
for example in breast cancer detection, monitoring of infant brain tissue oxygenation level and
functional brain activation studies; for reviews see [1,5].
In the inverse problem of DOT, absorption and scattering distributions within the object
are reconstructed. The iterative solution of this problem requires repetitive solutions of the
forward problem. Therefore, it is important to have a computationally feasible forward model
that describes light propagation within the medium accurately. The forward problem in
DOT is to solve the amount of measured light when the amount of applied light and optical
properties of the object are given. The light propagation in tissues is usually described through
transport theory and the most often used forward model is the diffusion approximation (DA)
to the radiative transfer equation (RTE). The DA is basically a special case of the first-order
spherical harmonics approximation to the RTE, and thus it has some limitations. First, the
medium is assumed to be scattering dominated, and second, light propagation cannot be
modelled accurately close to the collimated light sources and boundaries [1,2,26,28]. A
typical low-scattering region is the cerebrospinal fluid which surrounds the brain and fills the
ventricles.
Since the DOT image reconstruction problem is ill-posed, it tolerates measurement and
modelling errors poorly. Such modelling errors arise, for example, from using approximate
forward models which are unable to describe the measurements with adequate accuracy
[2,8,26,28]. Similarly, a model reduction by using too coarse discretization in the solution of
the forward problem can cause errors to the solution [4,6,7]. Further, in some applications,
the object geometry and locations of the light sources and detectors on the boundary of the
domain are not known exactly [9,23].
Modelling errors can be avoided by using forward models which describe the
measurements accurately. Unfortunately, this approach is sometimes computationally too
expensive and cannot be applied to practical applications. Another approach is to use strong
regularization in the solution of the inverse problem. In this case, the measurements are
implicitly assumed to be noisier than they are, and the properties of the prior model dominate
the solution of the inverse problem. Thus, for example, in the case of a smoothness prior, the
estimates may become over smoothed.
Recently, a Bayesian approximation error approach for the treatment of modelling errors
has been proposed [15,16]. In the Bayesian framework, all variables are modelled as random
in nature. In the approximation error approach, a model for the approximate statistics of the
model predictions is constructed and this model is employed when the overall posterior model is
laid down. In [4], the approximation error approach was utilized in DOT image reconstruction.
In that paper, the interplay between mesh density and measurement accuracy was investigated.
The DA was assumed to describe light propagation with a reasonable accuracy and it was used
as the true physical model for light transport. It was shown that, using the approximation error
method, it is possible to use mesh densities that would be unacceptable with a conventional
error model. The approach was extended to three dimensions and applied for the compensation
for modelling errors caused by domain truncation and a coarse computation mesh in [17] where
it was also tested with real measurements. In addition, the approximation error approach has
been utilized in DOT in reconstruction of absorption in the presence of anisotropies [8] and
compensating geometric mismodelling [9]. Outside optical imaging, the approximation error
approach has been utilized in electrical resistance tomography [15,18] and has been verified
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Inverse Problems 26 (2010) 015005 T Tarvainen et al
also to work with real measurements [19,20]. Further, the approach has been extended to
time-dependent inverse problems in [12,13].
In this paper, we investigate the applicability of the Bayesian approximation error approach
to compensate for the discrepancy of diffusion approximation close to the light sources and
in cases in which a turbid medium contains low-scattering subdomains. The RTE is used
as the true physical model for light transport, and the approximation error statistics are
constructed based on predictions of the radiative transfer and diffusion models. In addition,
approximation errors due to using a very low-dimensional approximation in the forward
problem are investigated.
The rest of the paper is organized as follows. The DOT forward problem is described
in section 2. The inverse problem in view of statistical inversion theory and the Bayesian
approximation error approach are reviewed in section 3. Simulation results are shown in
section 4, and the conclusions are given in section 5.
2. Forward problem
The forward problem in DOT is to solve the measurable data when the optical properties of the
medium and the amount of input light are given. Light propagation in biological material is
usually described through transport theory which can be modelled through stochastic methods,
such as Monte Carlo, or deterministic methods which are based on describing particle transport
with integro-differential equations. In the following, we consider the latter approach.
Let us denote the distribution of optical parameters by x(r) and let us denote the
measurements by a finite-dimensional vector y∈Rm. In DOT, the observation model with an
additive noise model is of the form
y=A(x(r )) +e, (1)
where Ais the forward model for light transport which maps the optical parameters to the
measurable data and edenotes the noise. In this paper, two forward models, namely the
RTE and the DA, are considered. Typically, the parameter x(r) is approximated in a finite-
dimensional basis and the continuous model (1) is replaced by an approximate equation
y=Ah(x) +e, (2)
where xis a discretized parameter distribution and h>0 is a mesh parameter controlling
the level of the discretization. In this paper, the finite element method (FEM) is used in the
construction of the approximate model Ah(x) :Rn→Rm.
2.1. Radiative transfer equation
A widely accepted model for light transport in tissues is the radiative transfer equation [14].
The RTE is a one-speed approximation of the transport equation, and thus it assumes that the
energy (or speed) of the particles does not change in collisions and that the refractive index is
constant within the medium.
Let ⊂Rn,n =2 or 3 denote the physical domain and let ∂ denote the boundary
of the domain. Further, let ˆ
s∈Sn−1denote the unit vector in the direction of interest. The
frequency domain RTE is of the form
iω
cφ(r, ˆ
s) +ˆ
s·∇φ(r, ˆ
s) +(μs+μa)φ(r, ˆ
s)
=μsSn−1
(ˆ
s·ˆ
s)φ(r, ˆ
s)dˆ
s+q(r, ˆ
s), r ∈, (3)
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Inverse Problems 26 (2010) 015005 T Tarvainen et al
where cis the speed of light in the medium, i is the imaginary unit, ωis the angular modulation
frequency of the input signal, and μa=μa(r ) and μs=μs(r) are the absorption and scattering
coefficients of the medium, respectively. Furthermore, φ(r,ˆ
s) is the radiance, (ˆ
s·ˆ
s)is the
scattering phase function and q(r, ˆ
s) is the source inside . The scattering phase function
(ˆ
s·ˆ
s)describes the probability that a photon with an initial direction ˆ
swill have a direction
ˆ
safter a scattering event. In DOT, the most usual phase function is the Henyey–Greenstein
scattering function [11] which in a two-dimensional case is of the form
(ˆ
s·ˆ
s)=1
2π
1−g2
(1+g2−2gˆ
s·ˆ
s)(4)
and in a three-dimensional case is of the form
(ˆ
s·ˆ
s)=1
4π
1−g2
(1+g2−2gˆ
s·ˆ
s)3/2.(5)
The scattering shape parameter gdefines the shape of the probability density and it takes
values between −1<g<1.
In DOT, we use the RTE boundary condition which assumes that no photons travel in an
inward direction at the boundary ∂ except at the source position j⊂∂, thus
φ(r, ˆ
s) =φ0(r, ˆ
s), r ∈j,ˆ
s·ˆ
n<0
0,r∈∂\j,ˆ
s·ˆ
n<0,(6)
where ˆ
nis an outward unit normal and φ0(r, ˆ
s) is a boundary source [1,27]. This boundary
condition implies that once a photon escapes the domain it does not re-enter it.
In DOT, the measurable quantity is the exitance J+(r) on the boundary of the domain.
Utilizing the boundary condition (6), it can be written as [3]
J+(r) =Sn−1
(ˆ
s·ˆ
n)φ(r, ˆ
s)dˆ
s=ˆ
s·ˆ
n>0
(ˆ
s·ˆ
n)φ(r, ˆ
s)dˆ
s, r ∈∂. (7)
2.2. Diffusion approximation
In DOT, the diffusion approximation is more commonly used because of its computational
simplicity. In the DA framework, the radiance is approximated by
φ(r, ˆ
s) ≈1
|Sn−1|(r) −n
|Sn−1|ˆ
s·(κ∇(r )), (8)
where (r) is the photon density which is defined as
(r) =Sn−1
φ(r, ˆ
s)dˆ
s(9)
and nis the dimension of the domain (n=2,3). Further, parameter κ=(n(μa+μ
s))−1is
the diffusion coefficient where μ
s=(1−g1)μsis the reduced scattering coefficient, and g1
is the mean of the cosine of the scattering angle [1,15]. In the case of the Henyey–Greenstein
scattering function, equations (4) and (5), we have g1=g. By inserting the approximation (8)
and similar approximations written for the source term and phase function into
equation (3) and following the derivation in [1,14,24], we obtain
−∇ · κ∇(r) +μa(r) +iω
c(r) =q0(r), (10)
where q0(r) is the source inside . Equation (10) is the frequency domain version of the DA.
The boundary condition (6) cannot be expressed in terms of variables of the diffusion
approximation. Instead it is often replaced by an approximation that the total inward directed
4
Inverse Problems 26 (2010) 015005 T Tarvainen et al
photon current is zero. Further, by taking into account the mismatch between the refractive
indices of the medium and surrounding medium, a Robin-type boundary condition can be
derived [1]. The boundary condition can be written as
(r) +1
2γn
κξ ∂(r)
∂ˆ
n=⎧
⎨
⎩
Is
γn
,r∈i
0,r∈∂ \i,
(11)
where Isis a diffuse boundary current at the source position j⊂∂,γnis a dimension-
dependent constant which takes values γ2=1/π and γ3=1/4, and ξis a parameter governing
the internal reflection at the boundary ∂ [15,24]. In the case of matched refractive index,
ξ=1. Further, the exitance is written as [1,24]
J+(r) =−κ∂(r)
∂ˆ
n=2γn
A(r). (12)
3. Inverse problem
In this section, the statistical approach to the DOT inverse problem and approximation error
approach are reviewed. Further details can be found in [4,15].
In the inverse problem of DOT, the optical parameters of the object xare estimated based
on light transport measurements yon the surface of the object. In this paper, the estimated
optical parameters are the absorption and scattering coefficients (μa,μ
s) within the medium.
3.1. Bayesian approach with a conventional error model
In the Bayesian approach, the inverse problem is treated as a problem of statistical inference.
All variables are modelled as random variables and the measurements are used to determine
the posterior probability density of the parameters of primary interest.
Let us assume that xand yare random variables in finite-dimensional spaces Rnand Rm
called parameter and data space, respectively. The joint probability density of xand ycan be
written in terms of conditional probability densities as
π(x,y) =π(x)π(y|x) =π(y)π(x|y). (13)
The solution of the inverse problem is the posterior probability density π(x|y) which according
to equation (13)isoftheform
π(x|y) =π(x)π(y|x)
π(y) ,(14)
where π(x) is the prior probability density and π(y|x) is the likelihood density. Equation (14)
is the well-known Bayes’ formula. It is often written in the non-normalized form
π(x|y) ∝π(y|x)π(x). (15)
If we assume that the noise eand the unknown xare mutually independent, formula (2) leads
to likelihood density
π(y|x) =πnoise(y −Ah(x)), (16)
where πnoise is the probability distribution of the noise e. If the unknown xand the measurement
errors can be modelled as Gaussian random variables, we have
x∼N(x∗,
x), e ∼N(e∗,
e),
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Inverse Problems 26 (2010) 015005 T Tarvainen et al
where x∗∈Rnand e∗∈Rmare the means, and x∈Rn×nand e∈Rm×mare the covariance
matrices. In this case, the posterior density (15) becomes
π(x|y) ∝exp −1
2(y −Ah(x) −e∗)T−1
e(y −Ah(x) −e∗)−1
2(x −x∗)T−1
x(x −x∗).
(17)
The practical solution for the inverse problem is obtained by calculating point estimates from
the posterior density. Since we are targeting at computationally efficient inverse problem
solvers, we consider here only the MAP estimate. It is obtained as
xMAP =arg max
x{π(x|y)}
=arg min
x{Le(y −Ah(x) −e∗)2+Lx(x −x∗)2},(18)
where the Cholesky factorizations are −1
e=LT
eLeand −1
x=LT
xLx. In the following
sections, we refer to the solution of (18) as the MAP estimate with the conventional error
model approach. For more information about statistical inversion theory, see e.g. [15].
3.2. Approximation error approach
Let us assume that the continuous model x(r) → A(x(r)), equation (1), can be approximated
by a densely discretized finite-dimensional model
RN→Rm,x→ Aδ(x), δ > 0small.
Thus, the discretized model, that is exact within the measurement accuracy, is of the form
y=Aδ(x) +e. (19)
In the approximation error approach, the observation model is written in the form
y=Ah(x) +(Aδ(x ) −Ah(x))+e
=Ah(x) +ε(x) +e, (20)
where Ah(x) is the reduced model and ε(x) is the modelling error [4]. The modelling error
describes the discrepancy between the accurate forward model and the reduced model, i.e. a
model that is an approximation to the accurate physical model and/or a model with a coarser
discretization or a smaller computational domain.
In the approximation error approach, a Gaussian approximation is constructed for ε, and
the total error n=ε+eis approximated by a Gaussian distribution, thus
ε∼N(ε∗,
ε), n ∼N(n∗,
n).
Further, if we assume that xand εare mutually independent, we get an approximation that is
referred to as the enhanced error model [15]
π(x|y) ∝exp−1
2(y −Ah(x) −ε∗−e∗)T−1
n(y −Ah(x) −ε∗−e∗)
−1
2(x −x∗)T−1
x(x −x∗).(21)
The MAP estimate with the enhanced error model is obtained as [4]
xMAP =arg max
x{π(x|y)}
=arg min
x{Lε+e(y −Ah(x) −ε∗−e∗)2+Lx(x −x∗)2},(22)
where LT
ε+eLε+e=(ε+e)−1=−1
n. In the following, we refer to the solution of (22)as
the MAP estimate with the approximation error model approach.
6
Inverse Problems 26 (2010) 015005 T Tarvainen et al
3.2.1. Computation of the approximation error. The mean and the covariance of the
approximation error, ε∗and εrespectively, can be simulated as follows. We draw a set of
samples {x(), =1,...,r}from the prior distribution π(x) and compute the corresponding
approximation errors
ε
h=Aδ(x() )−Ah(x()). (23)
The mean and covariance of the approximation error are estimated from the samples as
ε∗=1
r
r
=1
ε()
h(24)
ε=1
r−1
r
=1
ε()
hε()T
h−ε∗εT
∗.(25)
The statistics of the approximation error was computed in [4] using a proper smoothness prior
model [15] as the prior distribution and drawing samples from it. In [19], another approach was
chosen, and the mean and covariance of the approximation error were computed by simulating
samples in a cylindrical domain including randomly generated spheres. In this work, a similar
approach as in [4] is chosen. It is described in more detail later in section 4.2.
We note that the mean and the covariance of the approximation error are computed before
the measurements and they are valid for the employed geometry and other parameters as long
as the employed prior model can be assumed to be feasible. Thus, all factors in (22) can also be
precomputed and the computational burden is (essentially) the same as when the conventional
likelihood model is used.
4. Results
We investigated the validity of the approximation error approach to compensate for modelling
errors between the RTE and the DA. Further, we made reconstructions using two different
density forward model discretizations to evaluate the capability of the approximation error
method to compensate for model reduction by coarse discretization.
Both the RTE and DA were numerically solved using the finite element method. The
finite element (FE) solution of the RTE was implemented as in [25] and the FE solution of the
DA was implemented as in [4,24]. The data were simulated using the Monte Carlo method.
4.1. Set-up of the simulations
4.1.1. Data generation. In simulations, circular domains which contained different
inclusions were investigated. Two different size domains were considered: 40 mm in diameter
and 20 mm in diameter. In both cases, 16 equally spaced sources and detectors were located
on the boundary of the domain. The modulation frequency of the input signal was chosen as
100 MHz, and the refractive indices of the background medium, inclusion, and the surrounding
medium were nin =1.
In both cases, three different internal structures were investigated: a medium with
a highly absorbing inclusion and a low-scattering inclusion (case A), a medium with a
highly absorbing inclusion, a large low-scattering inclusion and a highly scattering inclusion
(case B), and a medium with a highly absorbing inclusion and a low-scattering layer (case C). In
all simulations, the background absorption and scattering coefficients were μa=0.025 mm−1
and μs=2mm
−1, respectively. Further, in all simulations the scattering shape parameter was
7
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Tab l e 1 . The absorption μa(mm−1)and scattering μs(mm−1)coefficients of the background
medium and internal structures for different tests.
Background Inclusion 1
Inclusion
2/layer Inclusion 3
μaμsμaμsμaμsμaμs
Case A: 0.025 2 0.05 2 0.025 0.005 – –
Case B: 0.025 2 0.05 2 0.025 0.001 0.025 4
Case C: 0.025 2 0.05 2 0.01 0.02 – –
g=0 throughout the domain. We note that in biological material the parameter gis close to 1
rather than 0. However, due to the computational complexity of the RTE with highly peaked
forward scattering we decided to use value g=0, especially since we wanted to use the RTE as
a reference method in reconstructions. The simulated absorption and scattering distributions
are shown on the top rows of figures 4–9, and the optical properties are summarized in table 1.
The data were generated using a Monte Carlo simulation. In the Monte Carlo simulation,
a photon packet method, developed in [21] and extended in [10], was modified to allow
computation in complex inhomogeneous geometries. This was achieved by depicting the
computational domain with triangular elements. In the case of the 40 mm diameter domain,
5×107photon packets were simulated, and in the case of the 20 mm diameter domain, 2 ×
107photon packets were simulated. A review of the Monte Carlo method is given in
appendix A.1.
4.1.2. Inverse problem meshes. For the representation of the absorption and scattering,
both domains dwith diameter d=40 mm and diameter d=20 mm were divided into
K=1296 triangles d,k. The absorption and scattering coefficients were represented in a
piecewise constant basis
μa(r) ≈
K
k=1
μakχ(μa)
k(r) (26)
μs(r) ≈
K
k=1
μskχ(μs)
k(r), (27)
where χk(r) is the characteristic function of the subdomain d,k. This mesh was used in all
of the reconstructions for the representation of the absorption and scattering.
In both size domains, the forward solutions were computed in FE meshes with two
different density discretizations using the DA as the forward model. The fine meshes h1
40
and h1
20 consisted of 8237 nodes and 16 024 elements and the coarse meshes h2
40 and h2
20
consisted of 843 nodes and 1579 elements. Both the MAP estimate with the conventional
error model and the MAP estimate with the approximation error model were solved using all
of the discretization meshes.
For comparison, we calculated reconstructions in which the RTE solution in the fine
meshes was used as the light transport model. In the RTE reconstructions, the angular
discretization consisted of 16 angular directions.
4.2. Computation of the approximation error
The distribution of the approximation error was computed for both the fine mesh and the coarse
mesh as described in section 3.2 for both size domains d,d=40 mm and d=20 mm.
8
Inverse Problems 26 (2010) 015005 T Tarvainen et al
0 1020304050
0
0.04
0.08
0.12
0.16
0 1020304050
0
0.04
0.08
0.12
0.16
λ
λ
d = 40 mm: d = 20 mm:
Figure 1. 50 largest eigenvalues λof the approximation error covariance matrix εfor the dense
mesh (∗) and coarse mesh (◦) in the case of 40 mm in diameter domain (left image) and 20 mm in
diameter domain (right image).
We computed the statistics of the approximation error using the same approach as in [4].
Thus, we used the proper smoothness prior model as the prior distribution π(x) and draw
2500 samples from it. The proper smoothness prior was constructed similarly as described in
[4]. The marginal distributions of the optical properties of the specified pixels were
π(μa)∝exp−1
2(μa−μa∗)T−1
μa(μa−μa∗)(28)
π(μs)∝exp−1
2(μs−μs∗)T−1
μs(μs−μs∗),(29)
where μa∗=0.025 mm−1and μs∗=2mm
−1. Further, μa=diagσ2
μawhere
σμa=0.023/3mm
−1and μs=diagσ2
μswhere σμs=2/3mm
−1. This choice
of absorption and scattering distributions corresponds to the assumption that the optical
parameters at the specified pixels are with prior probability 0.99 in the given range. The
absorption and scattering distributions were chosen such as they typically are in medical
applications of DOT.
The approximation errors were computed using equation (23) where the accurate forward
model Aδwas the RTE solution in the fine mesh h1
dand the forward model Ahwas the DA for
the meshes hp
d,p =1,2. The means and the covariances of the approximation errors were
computed using equations (24) and (25), respectively.
Figure 1shows 50 largest eigenvalues of the approximation error covariance matrices.
As can be seen, the eigenvalues decrease quite quickly. This indicates that the problem is
behaving well in the sense that one could describe error between the models and discretization
using less than 50 eigenvectors.
Figure 2shows the covariance structure of the conventional error model and the enhanced
error model. As can bee seen, the conventional error model covariance is a diagonal matrix.
The enhanced error model covariance matrices, on the other hand, show complicated structure.
This indicates that one can expect different reconstructions depending on which error model
covariance is used.
4.3. Reconstructions
We calculated the MAP estimates with the conventional error model and the MAP estimates
with the approximation error model for different cases using the meshes described in
section 4.1. To improve the numerical stability of the reconstruction algorithm, the data
9
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Figure 2. Signed standard deviation of conventional error model covariance sign(e)·√|e|
(left column) and enhanced error model covariance sign(ε+e)·√|ε+e|for the fine mesh (centre
column) and for the coarse mesh (right column). On the top row are images for case 1 (domain
with 40 mm in diameter) and on the bottom row are images for case 2 (domain with 20 mm in
diameter).
and solution spaces were scaled similarly as in [22,25]. Thus, in the data space we used
logarithm of amplitude and phase shift as measurables using the transformation
˜
y=˜
yA
˜
yϕ=ln|y|
arg(y)=Re
Imln y. (30)
In the solution space we used the transformation
˜μa=μa
¯μa
,˜μs=μs
¯μs
,(31)
where ¯μaand ¯μsare the means of the absorption and scattering, respectively. With the
transformations, the MAP estimate with the conventional error model, equation (18), is
obtained as
˜
xMAP =arg min
˜
x{Le(˜
y−˜
Ah(¯
x˜
x) −e∗)2+L˜
x(˜
x−˜
x∗)2},(32)
where ˜
Ais the forward model which maps the absorption and scattering parameters to
the scaled measurables. Similarly, the MAP estimate with the approximation error model,
equation (22), is obtained as
˜
xMAP =arg min
˜
x{Lε+e(˜
y−˜
Ah(¯
x˜
x) −ε∗−e∗)2+L˜
x(˜
x−˜
x∗)2}.(33)
The choice of prior in reconstruction step does not need to be the same as that used in the
generation of samples for the approximation error. In this case, we used a proper smoothness
prior defined on the scaled parameters, with dimensionless mean 1 and variance 1/3.
The minimization problems (32) and (33) were solved with a Gauss–Newton algorithm
which was equipped with a line search algorithm for the determination of the step length and
a positivity constraint for the estimated parameters. For comparison, we calculated the MAP
estimates using the conventional error model, functional (32), and the RTE as the forward
model ˜
Ain the fine meshes h1
40 and h1
20.
4.3.1. Case 1: domain with 40 mm in diameter. As the first case, we considered simulations
in a circular domain with 40 mm in diameter. The target absorption and scattering distributions
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Inverse Problems 26 (2010) 015005 T Tarvainen et al
1 3 5
0.01
0.03
0.05
1 3 5
0.01
0.03
0.05
1 3 5
0.01
0.03
0.05
1 3 5
0.01
0.03
0.05
μa
μa
μa
μa
μs
μs
μs
μs
Figure 3. Objective functions for case 1.A (two inclusions) as a function of scattering and
absorption computed in the fine mesh h1
40 (top row) and in the coarse mesh h2
40 (bottom row). The
results using the conventional error model functional (32) are on the left and the results using the
approximation error model functional (33) are on the right.
of different situations are shown on the top row of figures 4–6and the optical properties are
given in table 1.
To investigate the minimization problem, we first considered the domain with a highly
absorbing inclusion and a low-scattering inclusion (case 1.A). We computed the norms (32)
and (33) using simulated data ˜
y, and forward solution ˜
Ahwhich was calculated with different
optical properties. The forward solution ˜
Ahwas calculated for all combinations of 20
homogeneous absorption values on the range μa∈[0.001,0.05] mm−1and 20 homogeneous
scattering values on the range μs∈[0,5] mm−1. The obtained objective functions are shown in
figure 3. As can be seen, the objective function of the conventional error model shows a
different profile than the objective function of the approximation error model. The objective
function of the approximation error model shows a more clearly defined minimum than
the conventional one. This is especially evident in the case of the coarse mesh h2
40.The
results indicate that the approximation error model approach can be expected to improve the
reconstructions compared to the MAP estimates with the conventional error model.
The reconstructed absorption and scattering distributions computed using the conventional
error model and the RTE as the forward model are shown in figure 4. The DA reconstructions
with the conventional error model are shown in figure 5, and the DA reconstructions with the
approximation error model are shown in figure 6.
As can be seen from figures 4–6, the DA reconstructions with the approximation
error approach are as good as the reconstructions with the RTE. Furthermore, the DA
reconstructions with the conventional error model in the fine mesh are almost as good
as the RTE reconstructions. However, the DA reconstructions in the coarse mesh with
the conventional error model differ from the other approaches clearly. In that case, the
reconstructions are unclear and the inclusions cannot be distinguished.
11
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Figure 4. Reconstructed absorption and scattering coefficients for case 1.A (first and second
columns), case 1.B (third and fourth columns) and case 1.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the reconstructions obtained
using the RTE as the forward model in the fine mesh h1
40 (bottom row).
Figure 5. Reconstructed absorption and scattering coefficients for case 1.A (first and second
columns), case 1.B (third and fourth columns) and case 1.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the MAP estimates with the
conventional error model approach obtained using the DA as the forward model in the fine mesh
h1
40 (second row) and coarse mesh h2
40 (third row).
The reliability of the absorption and scattering estimates was evaluated by calculating the
relative estimation errors
˜
eμa=(μa−ˆμa)2d
μ2
ad·100,˜
eμs=(μs−ˆμs)2d
μ2
sd·100,(34)
where μaand μsare the original absorption and scattering coefficients, and ˆμaand ˆμsare the
estimated values. The estimation errors were calculated both for the MAP estimates with the
conventional error model and for the MAP estimates with the approximation error model. In
12
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Figure 6. Reconstructed absorption and scattering coefficients for case 1.A (first and second
columns), case 1.B (third and fourth columns) and case 1.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the MAP estimates with the
approximation error model approach obtained using the DA as the forward model in the fine mesh
h1
40 (second row) and coarse mesh h2
40 (third row).
Tab l e 2 . The relative estimation errors of absorption ˜
eμa(%)and scattering ˜
eμs(%)calculated for
the RTE, the DA with the conventional error model (DA-CEM) and the DA with the approximation
error model (DA-AEM) in the fine mesh h1
40 and coarse mesh h2
40.
Case 1.A Case 1.B Case 1.C
˜
eμa˜
eμs˜
eμa˜
eμs˜
eμa˜
eμs
h1
40: RTE 2.1 1.5 2.0 1.6 6.5 8.1
DA-CEM 2.2 1.8 4.3 7.9 7.5 8.3
DA-AEM 1.8 1.5 2.3 5.8 7.4 8.8
h2
40: DA-CEM: 21 18 30 39 29 27
DA-AEM 2.0 1.7 2.5 8.2 6.5 9.3
addition, the estimation error for the absorption and scattering estimates obtained using the
RTE as the light transport model were computed. The relative estimation errors are given
in table 2. As can be seen, the relative estimation errors of the RTE and the DA in the fine
mesh h1
40 are almost the same except for case 1.B (large low-scattering inclusion) where the
relative estimation errors of the DA are larger than the relative estimation errors of the RTE.
Furthermore, the DA with the approximation error approach gives slightly smaller values for
estimation errors than the DA with the conventional error model approach in cases 1.A and
1.C. In case 1.B, this difference is even more significant and the error of absorption is reduced
almost to the half. In the coarse mesh, the relative estimation errors of the approximation error
approach are significantly smaller than the relative estimation errors of estimates obtained with
the conventional error model approach. These results are consistent with the reconstructions
shown in figures 4–6.
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Inverse Problems 26 (2010) 015005 T Tarvainen et al
Figure 7. Reconstructed absorption and scattering coefficients for case 2.A (first and second
columns), case 2.B (third and fourth columns) and case 2.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the reconstructions obtained
using the RTE as the forward model in the fine mesh h1
20 (bottom row).
The results show that, if the approximation error approach is utilized, the DA
reconstructions both in the fine mesh and in the coarse mesh are as good as the RTE
reconstructions in the fine mesh. Based on the results, it also seems that although the medium
contains low-scattering regions, the DA with the conventional error model can (surprisingly)
be used to reconstruct them properly if dense discretization is used in the solution of the
forward problem. If the coarse discretization mesh is used, the DA with the conventional error
model fails to produce good quality reconstructions.
4.3.2. Case 2: domain with 20 mm in diameter. As the second case, we did simulations in
a circular domain with 20 mm in diameter. The target absorption and scattering distributions
of different situations are shown on the top row of figures 7–9and the optical properties are
given in table 1.
The reconstructed absorption and scattering distributions computed using the conventional
error model and the RTE as the forward model are shown in figure 7. The DA reconstructions
with the conventional error model are shown in figure 8and the DA reconstructions with the
approximation error model are shown in figure 9.
As can be seen from figures 7and 9, the DA reconstructions with the approximation error
approach are as good as the reconstructions with the RTE. In all cases, the inclusions can be
distinguished, although case 2.B which contains a large low-scattering region and case 2.C
which includes a low-scattering layer are extremely challenging. The DA reconstructions with
the conventional error model, figure 8, differ from the other approaches and the reconstructions
are unclear. Note difference of figures 5–8.
We computed the relative estimation errors for the estimates of the different approaches
using (34). The results are given in table 3. As can be seen, the relative estimation errors of
the DA with the conventional error model are larger than the relative estimation errors of the
RTE. This is especially evident when coarse discretization is used in the solution of the DA.
When the approximation error model approach is used, the error in the DA solution is reduced
to the level of the RTE estimation errors. These results are consistent with the reconstructions
shown in figures 7–9.
14
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Figure 8. Reconstructed absorption and scattering coefficients for case 2.A (first and second
columns), case 2.B (third and fourth columns) and case 2.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the MAP estimates with the
conventional error model approach obtained using the DA as the forward model in the fine mesh
h1
20 (second row) and coarse mesh h2
20 (third row).
Figure 9. Reconstructed absorption and scattering coefficients for case 2.A (first and second
columns), case 2.B (third and fourth columns) and case 2.C (fifth and sixth columns). The
absorption coefficients are on the left and the scattering coefficients are on the right of each case.
Images from top to bottom: the simulated distributions (top row) and the MAP estimates with the
approximation error model approach obtained using the DA as the forward model in the fine mesh
h1
20 (second row) and coarse mesh h2
20 (third row).
The results show that when the domain is smaller, the DA with the conventional error
model does not give as good results as the RTE. Furthermore, the results show that in this
case, the approximation error approach can be used to compensate for the modelling errors
between the DA and the RTE. Thus, if the approximation error method is utilized, one gets as
15
Inverse Problems 26 (2010) 015005 T Tarvainen et al
Tab l e 3 . The relative estimation errors of absorption ˜
eμa(%)and scattering ˜
eμs(%)calculated for
the RTE, the DA with the conventional error model (DA-CEM) and the DA with the approximation
error model (DA-AEM) in the fine mesh h1
20 and coarse mesh h2
20.
Case 2.A Case 2.B Case 2.C
˜
eμa˜
eμs˜
eμa˜
eμs˜
eμa˜
eμs
h1
20: RTE 2.6 1.0 7.5 1.5 7.0 4.8
DA-CEM 4.3 1.8 6.0 8.6 8.5 9.1
DA-AEM 2.6 1.1 3.2 6.2 7.1 5.0
h2
20:DA-CEM252434312424
DA-AEM 2.8 1.3 4.6 6.4 5.3 5.2
good reconstructions using the DA as the light transport model both in the fine mesh and in
the coarse mesh as with the RTE in the fine mesh.
5. Conclusions
In this paper, the applicability of the Bayesian approximation error approach to compensate
for the discrepancy of the diffusion approximation in DOT close to the light sources and
in low-scattering subdomains was investigated. In addition, we investigated the combined
approximation errors due to using the DA and coarse discretization in the solution of the
forward problem.
The approximation error statistics were constructed based on predictions of the RTE in
a fine discretization and the DA in different density discretizations. The method was tested
with 2D simulations in different size circular domains including low-scattering regions.
The results show that, with the approximation error approach, the reconstructions with
a low-dimensional DA are essentially as good as with a high-dimensional RTE. Thus, the
approximation error approach can be used to compensate for the modelling errors between
the RTE and the DA in diffuse optical tomography. Furthermore, the results are also valid for
combined approximation errors due to using the DA and using coarse mesh in the forward
problem.
Acknowledgments
The authors would like to thank PhD Aku Sepp¨
anen (University of Kuopio) for valuable
conversations regarding prior distributions. This work was supported by the Academy of
Finland (projects 122499, 119270 and 213476), National Technology Agency of Finland
(TEKES), Finnish IT Center for Science (CSC) and EPSRC grant EP/E034950/1.
Appendix
A.1. Monte Carlo simulation
Photons are created on the computational domain boundary at the location of a light source.
Each new photon packet is created with an initial weight w=1 and an initial direction inward
into the domain. Length of propagation before scattering event is drawn from
s =− ln ξ
μa+μs
,ξ∼Unif(]0,1[). (A.1)
16
Inverse Problems 26 (2010) 015005 T Tarvainen et al
The photons are propagated from element to element by using line–line-intersection
testing. Each time a photon packet enters an element, the distance lwhich the photons can
travel inside the element before reaching its boundary is computed. Attenuation and phase
of the photon packet are computed in a continuous manner. When a photon packet with a
complex weight wmoves a distance d, the new weight wof the photon packet is calculated as
w=wexp −iω
c−μad.(A.2)
If a photon packet that has s lenters an element, the photons are propagated a distance
d=lto the interface between the two elements and s is updated to s=s −l.
When the two elements have a different μs,s is updated to s=(s −l)μc
sμn
s, where
the indices cand nrefer to the scattering coefficients of the current element the photon
packet is leaving and the next element the photon packet is entering. If a photon packet has
s < l, the photon packet is propagated as above for a distance d=s after which it is
scattered. A new propagation direction is drawn using the Henyey–Greenstein phase function,
equations (4) and (5), and a new propagation length is drawn from equation (A.1). After
drawing the new propagation direction and length, the distance lto the nearest boundary of
the current element is computed.
Propagation in this manner is continued until the photon packet leaves the computation
domain through a boundary or when |w|<w
tol. In the latter case, a roulette for survival of the
photon packet is spanned in an energy conserving way such that if ξ>1
m,ξ∼Unif(]0,1[)the
weight of the photon packet is updated to w=mw. Otherwise the photon packet is removed
from the computation and a new photon packet is generated as above. Values of wtol =0.001
and m=10 were used.
When the photon packet leaves the computation domain through the boundary, this specific
boundary element receives photon density due to the photons. For each photon packet that
passes through the boundary, the photon density is updated as
=+w
Nl,(A.3)
where wis the weight of the photon packet, Nis the number of photons computed and l is
the length of the boundary element.
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