Download Wavelet-based Model Reduction Applied to Fluorescence Diffuse

Wavelet-based Model Reduction Applied to Fluorescence Diffuse
Optical Tomography
Anne Frassati, Anabela Da Silva, Jean-Marc Dinten, Didier Georges

Abstract— This paper is devoted to the Fluorescence Diffuse
Optical Tomography. This inverse problem (physical parameter
estimation problem) relies on an iterative algorithm based on
repetitive solutions of a set of partial differential equations. The
main goal of the paper is to present a wavelet-based model
reduction technique applied to the solution of the partial
differential equations involved in the problem, with the
objective of computation complexity reduction. The
effectiveness of the approach is then illustrated on a practical
example.
I.
INTRODUCTION
A
large number of fields uses the principle of tomography,
such as in geophysics with petroleum research or
sediments study [1]. Tomography is a numerical technique
which aims at reconstructing some physical properties inside
an object or a domain on the basis of measurements
performed around the object. From the viewpoint of
automatic control, tomography is a system identification
problem, since it consists in the estimation of physical
parameters (defining the properties of the object) appearing
in a set of Partial Differential Equations (PDEs) from
measurements (the so-called inverse problem). An inverse
problem resolution is usually based on the minimization of
the error between some measured outputs and some outputs
generated by a parameterized PDE model in the least-square
sense (infinite-dimensional output error identification). The
solution of such a problem relies mainly on the solution of a
set of parameterized PDEs (the so-called forward model)
coupled to the solution of a set of adjoint PDEs (the socalled adjoint problem expressing the sensitivities of the
unknown physical parameters with respect to the least-square
output error cost function). More precisely, here the focus is
made on near infrared optical tomography, and its principal
application on cancer research and small animal imaging [2].
More specifically, fluorescence-enhanced Diffuse Optical
Tomography (fDOT), by tagging the regions of interest with
target-specific fluorescing molecular probes, enables the
estimation of three-dimensional (3D) locations and
geometries of targeted areas such as tumors and can
potentially be a quantitative imaging modality [3].
Manuscript received October 12, 2006.
A. Frassati (corresponding author: (33) 4 38 78 44 28; fax: (33) 4 38 78
57 87; e-mail: [email protected]), A. Da Silva and J.-M. Dinten are with
LETI-CEA-MINATEC Recherche Technologique, 17 rue des Martyrs F38054 Grenoble Cedex 9, France.
D. Georges is with Control Systems Department - GIPSA-lab (former
LAG, Laboratoire d'Automatique de Grenoble), UMR 5216, BP 46, F38402
Saint
Martin
d'Hères
Cedex,
France
(e-mail:
[email protected]).
fDOT requires both an accurate forward model of coupled
excitation and emission light propagation through highly
scattering media and a robust method for inverting emission
measurements and reconstructing interior optical property
maps of the tissues and, in fine, the fluorochromes local
concentration. Two concurrent approaches are currently
under investigation for solving the forward problem. The
first one refers to a simplified formulation in order to
establish analytical solutions and is particularly interesting
for treating large number of unknows problem. Nevertheless,
its application is usually restricted to simplified geometries
such as semi-infinite media, slabs or cylinders. Some
attempts to apply it to realistic geometries have recently been
carried out [4]-[5]-[6], with the restrictive hypothesis of a
homogeneous scattering and absorbing medium. The
numerical resolution of the diffusion equation, by the Finite
Element Method (FEM), has no restriction considering the
geometry or homogeneity of the system, but is well-known to
be time and memory consuming. The resolution of an adjoint
problem [7]-[8]-[9], that allows a complete linearization of
the problem, considerably speeds up the treatment and
allows a full 3D resolution. Nevertheless, because the
accuracy of the results is still mesh dependent, the
dimensions of the matrices involved are still large and the
resolution of the problem is still long.
In the present work, we focus on a description of a
knowledge based model and its reduction. To that purpose, a
parametric identification problem is to be solved and the
Finite Element Method (FEM) is used. The principal
objective of the present work is to reduce the model in order
to speed up the forward model computation. A
multiresolution technique that uses a wavelet decomposition
is chosen in that way [10]. This decomposition has been
applied as basis functions for efficient choice of solutions of
PDEs, such as the Maxwell’s equations in the domain of
electromagnetics for example [11]-[12]-[13]. All the
matrices appearing in the discretized version of the PDEs are
projected onto an orthonormal wavelet basis, leading to more
sparse matrices. The inversion of these matrices is then
easier to carry out, and the computation time is obviously
reduced. Moreover, in order to take the best advantage of
this projection, the dimensions of these new matrices are
reduced according to the wavelet’s properties. This
compression leads to the reduction of a factor 2x2 of the
initial dimension that speeds even more the inversion
computation. Section II is devoted to a brief description of
the forward and inverse problems and their resolutions. In
section III a review of wavelets is presented, and their
application on large linear system reduction in section IV.
This section explains in what manner wavelets reduce the
number of equations, and the modification that implies.
Finally we discuss the results and conclude.
II. DESCRIPTION OF THE MODELS
In this section, we briefly recall the basic equations
governing light propagation through diffusive media. The
discussion is restricted to the frequency domain although the
results may be readily applied to the time-domain, by taking
into account the time evolution in the equations or via
Fourier transforms of the results. Some details on the FEM
implementation chosen to solve the problem are also
provided.
A. General Formulation of the Forward Problem
Suppose a diffusive medium with total volume  , and
bounded by the surface  . Suppose an intensity-modulated
point source located at the position rs inside the diffusive
medium with intensity Q0(rs) and frequency ω. The
fluorochromes present within the medium are excited by the
incident Diffuse Photon Density Wave (DPDW) ux at the
excitation wavelength x. Each excited fluorescent particule
then acts as a secondary point source and gives birth to a
fluorescent
DPDW,
propagating
with
a
larger
wavelengthm. The propagation of light through highly
scattering media is commonly modeled, within the Diffusion
Approximation, by the following set of weakly coupled
PDEs:
.(  D x (r)u x (r, rs ))  k x (r )u x (r, rs )  Qo (rs )
(1)

.(  Dm (r )u m (r, rs ))  k m (r )u m (r, rs )   (r )u x (r, rs )
The first equation represents the propagation of excitation
light (subscript x), from the point-like source, located at
position rs, to the fluorochrome at a position r inside the
diffusive volume  . The second models the generation and
propagation of the emitted light (subscript m). The
coefficient kx (respectively: km) is the total complex
absorption at the excitation (resp.: emission) wavelength,
defined as k x (r)   ax (r)  i c , c is the speed of light
inside the medium, µax is the absorption coefficient due to the
presence of both the local chromophores and fluorochromes
concentrations. Dx (resp.: Dm) is the diffusion coefficient at
the excitation (resp.: emission) wavelength. The function
r involved in the fluorescence source term is the
conversion factor of the excitation light at a point r into
fluorescence light and depends on intrinsic properties of the
fluorochromes such as the fluorescence life-time, the
quantum yield and also their local concentration. Hence, this
parameter is precisely the parameter to be recovered by
reconstruction. In so far as this parameter is directly
proportional to the local concentration, it allows the
recollection of the position of the fluorochromes as well as
the quantification.
These equations are subject to the Robin boundary
conditions, to be verified at all the system boundaries  :
n.( D x (r)u x (r, rs ))   x u x (r, rs )  0 r  

n.( Dm (r )u m (r, rs ))   m u m (r, rs )  0
(2)
n is the outward normal unitary vector to the surface,  u
is a multiplicative factor, self-consistently accounting for
index mismatching at the boundaries and depending on the
Fresnel reflection coefficients [14]. The physical quantity
representing the actual measurement is the output flux, or
exitance [W/m2], proportional to the DPDW according to
(2), measured at the boundary of the diffusive medium:
m (r,  )   Dm (r)n.um (r,  )   mum (r,  ) r  
(3)
B. Perturbation approach
Most of reconstruction techniques within the Diffusion
Approximation rely on a first order perturbation approach,
that is the measured DPDW u (=[x,m]) is decomposed
into a Taylor series truncated at first order: u  u .
The deviation u to the initial estimate has to be small
and is supposed to be only due to a deviation on the
estimation of all or part of the parameters to be reconstructed
Dx  Dx , Dm  Dm , k x  k x , k m  k m ,    .
C. Resolution of the Adjoint Problem
The Adjoint Method [15] has been widely used in other
domains of sciences and has been recently applied to the
fDOT because it allows the reduction of the number of
systems of equations that have to be solved. Let g x , g m and
g xm be the functions chosen to obey the following set of
equations:
.(  Dx g *x )  k x g *x  q x


*
*

.(  Dm g m )  k m g m  qm

*
.(  Dx g *xm )  k x g *xm  g m


(4)
* stands for “complex conjugated”, q x   (r  rs ) is a
point source located at rs, q m   (r  rd ) is a point source
located at a detector position rd. Using Robin boundary
conditions (2), and applying the Green’s theorem, the
following expression can readily be demonstrated:

*
u m   g m
Dm u m d 3 r 


 g xmDx u x d
*

*
3
r



 g mk mu m d
3
r
 g xmk x u x d
*
3
r
(5)

*
gm
 u x d 3 r

The problem is then reduced to the resolution of five
PDEs (Eqs (1) and (4)). The functions ux and um, on the one
*
hand, and g *x , g m
and g *xm , on the other, can then be
stored, and do not need to be recalculated anytime the
parameters Dx , Dm , k x , k m ,   are updated.
D. Finite Element Matrix Formulation
As the medium is inhomogeneous, a numerical resolution
is definitely required for solving (1) and (4), and the FEM is
a method of choice. Within the FEM, the volume object 
is divided into a limited number of elements, joined at
vertices. The different quantities are approximated by
piecewise quadratic functions. We used the Galerkin
projection method to express the equations into a weak form
and the PDEs can be reformulated into a linear compact
matrix form:
AU  Q ; A T G  Δ
(6)
is the symbol for “transposed matrix”. U is the vector
representing the solutions to the forward problem, and G
those of the adjoint problem:
T
 u (r, rs ) 
 g (r , r ) g xm (rd , r )
U(r, rs )   x
; G (rd , r )   x d
(7)

0
g m (rd , r ) 
u m (r, rs )

A is a matrix accounting for the different optical
parameters:


T
A(r )   .(  D(r ))  K (r )

0I 33 
 D x (r )I 33

 D(r ) 66   0I
Dm (r )I 33 
33



0 
 k (r )
 K (r )
 x

22



(
r
)
k
m (r )


(8)
(9)



minimal polynome of A: K k (, c)  span c, c,...,  k 1c is the
Krylov subspace at iteration k, generated by the vector c.
Arnoldi method consists in constructing an orthonormal
basis v1 ,..., vk  of K k ( , b) at each iteration k:
v1  b b ,
v j  wj
wj
with w j 1  v j  (h1 j v1  ... h jj v j )
and hij  vi v j .
Then the least squares method consists in minimizing
x which gives the initialization of following iteration. We
stop when k=kA. It is thus still time consuming for the
resolution of our problem involving large dimension
matrices.
In section III, an improving method, based on the
reduction of this model, is proposed.
E. Resolution of the Inverse Problem
When discretizing the medium into elementary volumes,
(11) readily appears as a linear function of the parameters
D, K , representing the set of unknowns
Dx , Dm , k x , k m ,   . In order to introduce expression
(11) in a reconstruction scheme, it is usual to reformulate the
problem in terms of a weight matrix:
δU  W D δD  W K δK
One can demonstrate [8] that the vector δU representing
the variation on both the excitation and emission signals can
also be written with the following compact formulation:
δU(rd , rs )   G T (rd , r)δA(r)U(r, rs )d
The Generalized Minimal RESidual method (GMRES
[16]) is used for the forward and adjoint problems resolution
(6). This method is optimal, but iterative. It is derived from
the Arnoldi process for constructing an orthogonal basis of
Krylov subspaces. The idea is that the solution of a linear
system Ax=b, where A is sparse and large, is situed in a
Krylov subspace, whose dimension is the degree kA of
b  z with z  Kk (, b) and we obtain an approached solution
 0 

with 
 . Q and Δ are the vector and matrix
62
 0 
accounting for the source terms:

Qo (rs ) (r  rs )
Q(r, rs ) 21  

0




0

Δ(r , r )   (rd  r )

 d
0
 (rd  r )


The vector δU representing, according to the perturbation
approach, the deviation on both the excitation and emission
measured signals to reference excitation and emission
signals, thus only depends on the resolution of the forward
and adjoint problems (6), and on the parameters to be
reconstructed D, K  .
(10)
(12)
Each column of δU represents the measurement on the
whole set of detectors for a given source point. The
Algebraic Reconstruction Technique (ART) is chosen here
since it allows efficient storage of the huge matrices W .
III. WAVELETS AND MULTIRESOLUTION: BASIC DEFINITIONS
AND PROPERTIES
Each time the variation on the parameters to be
reconstructed are updated, only the matrix δA has to be
recalculated. By using the Green’s theorem:
δU(rd , rs )   T G T (rd , r)δD (r )U(r, rs )d

 G T (rd , r )δK (r)U(r, rs )d

(11)
The concept of a wavelet, introduced for the first time in
the 1980s by the geophysicist J. Morlet [17], has been
largely applied for signal decomposition and approximation
[10]-[11]-[12]-[13]-[18]. A brief didactical introduction to
the concept of wavelets, important to understand their
particular application in this paper, is presented in this
section.
If L2(R) is the space of finite energy signals, a
multiresolution analysis is a succession of subspaces Vj in
L2(R) which verify the following conditions:
(i) j  , V j 1  V j
(ii) lim j V j  0 and lim j V j is dense in L2(R)
(iii) s(t ) V j  s(t / 2) V j 1
(iv) Φ(t ) , “scaling function”, such that (t  k ), k  
forms an orthonormal basis of V0.
This implies that
is an
2  j / 2 (t / 2 j  k ), k  
orthonormal basis of Vj.
The multiresolution analysis of a signal is the calculation
of its orthogonal projections on the subspaces Vj. When the
resolution level j gradually increases, the projection of a
signal s(t) on Vj gives more and more rough approximations
of this signal. The difference between the two
approximations projV s(t ) and projV s(t ) is the information

j 1

This implies that (t ) , “wavelet function”, such that
 j/2
This basis can also be represented in terms of a
corresponding matrix, noted M and represented
schematically on Fig. 2. Each row corresponds to the 2 3
different values of, respectively, from row 1 to 4, the
functions (t / 2 2  k ) 2 ( t  0;1 ; k  0;1;2;3 ) and, from
j
of details which were available at scale 2j-1 but lost at scale
2j. These details are contained in a subspace Wj such as
V j 1  V j  W j .
2
Fig. 1. Schematic of a Haar wavelet basis (23 vectors of 23 points)

(t / 2 j  k ), k   is an orthonormal basis of Wj.
The decomposition of a signal on the subspaces Vj and Wj
can be simplified by the following representation:
...
V3
V2
W4
V1
W3
V0
W2
W1
V0  V1  W1  V2  W2  W1  V3  W3  W2  W1  ... 
row
5
to
8,
the
functions
(t / 2 2  k )
2
( t  0;1 ; k  0;1;2;3 ). The values are coded on a gray
level scale: the maximum value (in black) is 1
minimum (in white)  1
2 and the
2 . For example, the first point in
the first row gives the value (t  0; k  0) 2  1 2 , while
the second point in the fifth row gives the value
(t  1; k  0) 2   1 2 .
row 1
row 5
 Wj
j 1 
The first approximation of a signal s(t) is given by its
projection on V1 and the first lost details will be found in W1.
For the purpose of simplicity, in all that follows, only the
first order resolution is taken into account.
A variety of different wavelets can be used for
multiresolution. To fix the ideas, the Haar’s wavelet is
considered here. This one is defined by the following basis
functions:
1 if 0  t  1
(t )  
0 otherwise
(13)
1 if 0  t  0.5

(t )   1 if 0.5  t  1
0 otherwise

(14)
23 points
Fig. 2. Schematic matrix representation of a wavelet basis
When the matrix M is multiplied by a signal s(t), the
resulting signal can be decomposed into two parts: the first
part is the approximation s1(t) of s(t), the second is the lost
details between s(t) and s1(t). According to the wavelet basis
definition, s(t) should have a number of points equal to 2n, n
being an integer.
The wavelet matrix M can be applied to a two dimensions
signal. If an image I of size 2n×2n is considered (see the
classical example (“Lena”), illustrated on Fig. 3), MIMT
provides four images of size 2n-1×2n-1 (Fig. 4), one is the first
approximation of I, and the other three are respectively the
vertical, horizontal and diagonal details.
For the first order resolution, the corresponding
orthonormal basis is composed by V1 and W1: V0  V1  W1 .
Half of this basis is composed by successive translations of
(t), the second half by translations of (t) (Fig. 1).
Fig. 3. Original image (512×512 points)
In what follows, we are going into details in order to see
how the wavelets are integrated in the resolution by the finite
element method of the diffusion equations.
a)
b)
A. Novel formulation of the forward problem
To summarize this method, let us consider a linear
equation as (6). Let M be the Haar wavelet matrix, defined
in section III. M can be defined from the subspaces V1  W1
for example, or
c)
d)
Fig. 4. Results of the projection of the image on Haar wavelet basis
(first resolution): a) first approximation (256×256 points); b)
vertical details; c) horizontal details; d) diagonal details.
 W j as recommended in [11]-[12].
j 1
Haar’s wavelets are considered. According to the wavelets
properties, M is an orthonormal matrix and MMT=
MTM=Id (Id stands for the identity matrix), and one can
reformulate (6) as follows:
MAM T MU  MQ
One readily understands that the main interest of the
wavelets decomposition is a reduction of the dimensions of
the signal, leading to a large number of applications in image
compression. In the present context, we intend to apply the
wavelets decomposition in a non conventional way in order
to reduce a model. The system to be resolved is described by
(6). The matrix A to be inverted is a huge matrix with typical
dimensions 213×213. Its inversion is thus time and memory
consuming, even by using efficient algorithms such as the socalled GMRES. This method is optimal, but is still time
consuming for the resolution of our problem. In the
following section, an improving method is proposed.
IV. COUPLING FINITE ELEMENT METHOD / WAVELETS
In this section, we propose to apply a model reduction by
applying the wavelets decomposition to matrix A. This one is
then treated just like if it was a 2D image. A change-of-basis
is performed: A is projected onto the orthonormal wavelet
basis. The resulting matrix A’ is more sparse and easier to
invert. The dimension of A’ is then reduced by applying the
multiresolution principle exposed in section III.
The following diagram (Fig. 5) shows the different steps
of our image reconstruction procedure. The model reduction
by using the wavelets decomposition is introduced before the
forward model classical resolution.
Diffusion equation
Definition of finite
elements mesh
Resolution of
equation: modelling of
luminous flux
Model reduction
Application of
wavelets: change
of basis and
multiresolution
 A' U'  Q'
U’=MU becomes the new unknown, Q’=MQ represents
vector Q in the new basis, and A’=MAMT is the projection
of A in the wavelet basis. By considering once again the
properties of the matrix M, the vector of unknowns U is:
U  M T U'
(16)
After this transformation, the resulting matrix A’ is sparse,
and many of its elements are negligible. At this stage, the
dimension of A’ could be reduced by selecting a threshold
value [11]-[12]-[13]: elements in A’ with magnitudes smaller
than this threshold could be discarded. This method
considerably reduces the computational effort of the
resolution, essentially if the original matrix A is dense.
As mentioned, one can go a step forward and speed up
even more the treatment by using the multiresolution
principle, in order to reduce the dimensions of the matrices
involved in the resolution. The matrix M is constructed with
orthonormal vectors of the basis V1  W1 . The finite elements
mesh is chosen such that the size of the matrix A is 2n×2n, n
integer (in our problem, n is typically equal to 13 or 14). In
the present technique, the matrix A is considered as an
image, and the same process as for “Lena’s” photograph is
applied. Each element of A can be considered as a pixel of a
2D-image.
The projection A’ of A on the Haar wavelet basis M is
composed of four submatrices of size 2n-1×2n-1, the first one
representing the approximation (Fig. 6), and the three others
the details (horizontal, vertical and diagonal details) (Fig. 7).
=
Reconstruction
Image
Fig. 6. Original matrix A (16×16 points)
Fig. 5. schematic of algorithm and introduction of wavelets
(15)
Fig. 7. Simplified matrix A’: top left: first approximation of A (8×8
points); top right: vertical details; bottom left: horizontal details;
bottom right: diagonal details.
we suggest ordering the nodes according to the vertical
dimension first, then according to the two horizontal
dimensions. Another type of rearrangement can be defined
and discussed, but we found out this one is appropriate.
Then, the following convention is applied in order to
define the resulting reduced mesh:
--The nodes of the original mesh are considered by pair
(indices 2×i-1 and 2×i, i=1 to 2n-1), and the mean value of
their coordinates is calculated.
--If two consecutive nodes of the original mesh do not
belong to the same space area, the one that corresponds to
the highest value in the diagonal of A is kept.
V. RESULTS AND DISCUSSION
We note A1 the top left part of A’ (approximation).
In the same way as A, we calculate the projection of the
vector Q in the same wavelet basis: Q’=MQ (length 2n). The
new vector Q’ contains two subvectors of size 2n-1, the first
one representing the approximation of Q, the second one the
details.
We note Q1 the first part of Q’ (approximation).
This matrix decomposition is restricted to the first order,
and this is justified by the shape of the sparse matrix A’. The
model is then reduced by using the approximation matrix A1
instead of the whole matrix A.
B. Reduction of the system and resolution of the new
diffusion equation
The reduced equation to be solved is:
A1U1=Q1
(17)
The reduced vector of unknowns U1 has twice less points
than the original vector U.
In the present case, each element of the original matrix A
refers to a specific space coordinate (x,y,z), or a specific
node of the finite element mesh. A new finite element mesh
has then to be defined, corresponding to the coordinates of
the elements of the resulting matrix A1.
When the approximated result U1 is obtained, it has
indeed twice less points than the original. The appropriate
corresponding 3D coordinate system has to be chosen in
order to preserve the spatial coherence between two matrix
elements. For instance, if one considers a 2D-image such as
Lena’s photograph (Fig. 3), with its linear coordinate system
In order to test the robustness of the algorithm, we
proceeded to experiments on calibrated objects (phantoms),
with optical properties Dx , Dm , k x , k m  known and
supposed to be constant, by using a tomographer designed in
our laboratory.
The only quantity to be reconstructed here is the
distribution of β. All other parameters are supposed known
and constant.
A. Experimental setup
The experiments have been performed with the
tomographer described in [4] (Fig. 8). The optical system is
composed of a laser source (690 nm, 25 mW) for
illumination and a CCD camera (Orca ER, Hamamatsu) for
detection. The source is guided to the object via an optical
fibre. The movements of the scanning fibre are driven by two
translating plates (Microcontrol) and monitored with a
computer. The CCD camera is focused at the top surface of
the object. For fluorescence light detection, a filter (high
pass RG9, Schott) is placed in front of the camera.
xi , yi  , i  1,..., 2 n , after the wavelet transform (Fig. 4) the
coordinate system corresponding to the first order
approximation would be given by the mean values between
two consecutive coordinates of the original system
( x2i1  x2i )


2 , ( y2i 1  y2i ) 2 , i  1,..., 2 n1 .
Applying this principle to a matrix representing intensity
values corresponding to 3D space coordinates that,
moreover, are not distributed on a regular cubic mesh, is not
that trivial. First of all, a rearrangement of the elements is
needed. To have the best coherence in wavelet projection,
Fig. 8. Schematic of the experimental setup.
A solid semi-cylindrical phantom (Fig. 9) has been
designed for the experiment.
The diameter is approximately 4 cm and the length is 6.35
cm. It is composed with a mixture of epoxy resin
(Solloplast), titanium dioxide powder (Sigma-Aldrich) as the
scatterers, and black India ink (Dalbe) as the absorber. The
index of refraction is estimated to be 1.54, the diffusion
coefficient 10 cm-1, and the absorption coefficient 0.2 cm-1.
In order to easily introduce fluorescent inclusions, four holes
have been drilled, at different heights (Fig.9).
To model the fluorescent region, we considered two thin
glass tubes (external diameter: 1.5 mm, internal diameter: 1
mm, length: 3 cm), each one filled with 310-3 cm3 of
commercial fluorescent dyes (Alexa 750, 10 μM, Molecular
Probes), introduced in the phantom at 12 mm height.
detector. Among the 10241344 possible pixels, we first
perform a binning of the data and then select only a grid of
1616 detectors. Each point of this new detector screen
collects the light that comes from a given point of the surface
of the object. According to geometrical optics laws, we have
considered instead the resulting projection of these detectors
to the physical surface of the object (Fig. 10).
(a)
(b)
Fig. 10. Results 3D after the reconstruction; (a) without wavelets
decomposition; (b) with Haar wavelets decomposition. Crosses:
different positions of the source; Dots: positions of the detectors.
(a)
Fig. 9. Photograph (top) and schematics of the resin phantom (units
mm) with two capillary tubes filled with Alexa750 fluorophores
inside.
A standard experiment consists in a series of three sets of
data acquisition:
--A first scan is performed with no object inside the
imaging chamber in order to precisely locate the positions of
the sources. In this experiment, 129 source at different
positions are considered;
--The phantom is then placed inside the chamber and a
second scan is carried out in order to acquire the excitation
signal;
--The filter is then added and the emission signal is
finally acquired.
B. Reduction of the model
Each pixel of the CCD camera represents a possible
(b)
(c)
Fig. 11. 2D Results (on horizontal plans) after the reconstruction;
(a) without decomposition; (b) with wavelets decomposition; (c)
absolute value of the difference between (a) and (b).
The FEM is used to solve the 129 forward problems
corresponding to the source different positions and the
1616 adjoint problems corresponding to the chosen
detectors of the CCD camera. Once the forward and adjoint
model solutions computed (6), the inverse problem is solved
by using a classical ART algorithm. The reconstruction of
the distribution of the parameter , proportional to the
concentration of the fluorochromes, is represented, with a
gray level scale, on Fig. 10. (a) (3D view) and Fig. 11. (a)
(2D views). For the purpose of the multiresolution process,
we considered a mesh with 213 nodes, leading to a matrix A
with 213213 elements. The projection on the Haar wavelet
Fig.
1.
basis is then performed (15). For the forward and the adjoint
Magnetization as
resolutions, only the first order matrix approximation
a function model
of
applied
field.
A1, with dimension 212212, and the corresponding source
Note that “Fig.”
vector Q1, with dimension 1212, are considered. We
is abbreviated.
proceeded then to the numerical resolution (6) of the reduced
There is a period
after the figure
forward problem and of the reduced adjoint problem by the
number, followed
GMRES inversion of (17). The application of this
by two spaces. It
multiresolution process basically provides a resolution of (6)
is good practice
to explain that
the is three times faster than the classical one. The matrices
significance Uof and G issued from, respectively, the forward and adjoint
the figure in the
problems are introduced in a classical ART algorithm in
caption.
order to solve (12). Note that in the present case, all the
optical parameters Dx , Dm , k x , k m  are kept constant, and
only the spatial distribution of the concentration of the
fluorochromes is reconstructed, via the parameter  (r ) . The
results are presented in Fig. 10. (b) and Fig. 11. (b).
Basically, what is to be noticed is that the main information
on the location of the fluorochromes is preserved with a
computation time much smaller.
VI. CONCLUSION
In the present paper, a method of model reduction,
borrowed from the domain of signal processing, has been
successfully applied to the problem of fluorescence enhanced
diffuse optical tomography. A wavelet decomposition is
applied to the model matrix issued from the linearization of
the forward problem, obtained here by using the finite
element method.
This method is interesting in the present case because it
allows a compression of the data with a limited lost of
information. We restricted the study to the reduction of the
forward model, but one can easily apply the procedure to the
adjoint problem, and to the first order approximation.
In our future work, we intend to exploit the matrix of the
details as well, in order to keep all possible information.
Higher orders of approximation will also be examined. And
finally, more complex objects will be studied experimentally,
in order to test the robustness of the technique with noisier
data, non homogeneous media and more complex
geometries. A 3D measurement system of the surface of a
small animal body should allow us to test the algorithm in
vivo.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Y. Desaubies, A. Tarantola, and J. Zinn-Justin., Geophysical
Tomography. North-Holland Eds., 1990.
R. Weissleder, V. Ntziachristos, “Shedding light onto live molecular
targets”, Nature Medecine, vol. 9, no. 1, pp. 123-128, Jan. 2003.
V. Ntziachristos, E.A. Schellenberger, J. Ripoll, D. Yessayan, E.
Graves, A. Bogdanov, L. Josephson, R. Weissleder , “Visualization of
antitumor treatment by means of fluorescence molecular tomography
with an annexin V-Cy5.5 conjugate”, P.N.A.S., vol. 101, no. 33, pp.
12294-12299, Aug. 2004.
L. Hervé, A. Koenig, A. Da Silva, M. Berger, J. Boutet, J.M. Dinten,
P. Peltié, P. Rizo, “Non Contact Fluorescence Diffuse Optical
Tomography of Heterogeneous Media”, submitted to Applied Optics.
R.B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental
fluorescence tomography of tissues with noncontact measurements”,
IEEE Transactions On Medical Imaging, vol. 23, no. 4, pp. 492-500,
Apr. 2004.
J. Ripoll, and V. Ntziachristos, “From finite to infinite volumes:
Removal of boundaries in diffuse wave imaging”, Phys. Rev. Letters
96 (17), May. 2006.
M. J. Eppstein, F. Fedele, J. Laible, C. Zhang, A. Godavarty, and E.
M. Sevick-Muraca, “A Comparison of Exact and Approximate
Adjoint Sensitivities in Fluorescence Tomography”, IEEE
Transactions On Medical Imaging, vol. 22, no. 10, pp. 1215-1223,
Oct. 2003.
F. Fedele, J.P. Laible, M.J. Eppstein, “Coupled complex adjoint
sensitivities for frequency-domain fluorescence tomography: theory
and vectorized implementation”, Journal of Computational Physics,
vol. 187, pp. 597–619, May 2003.
A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite
element based tomography for fluorescence optical imaging in tissue”,
Optics Express, vol. 12, no. 22, pp. 5402-5417, Nov. 2004.
S. G. Mallat, “A wavelet tour of signal processing”, San Diego,
Academic Press, 1999.
M.N.O. Sadiku, C.M. Akujuobi, R.C. Garcia, “An introduction to
wavelets in electromagnetics”, IEEE Microwave Magazine, pp.63-72,
Jun. 2005.
R.L. Wagner, W.C. Chew, “A study of wavelets for the solution of
electromagnetic integral equations”, IEEE Transactions on Antennas
and Propagation, vol. 43, no. 8, pp. 802-810, Aug. 1995.
Z. Xiang, Y. Lu, “An effective wavelet matrix transform approach for
efficient solutions of electromagnetic integral equations”, IEEE
Transactions on Antennas and Propagation, vol. 45, no. 8, pp. 12051213, Aug. 1997.
R.C. Haskell, L.O. Svaasand, T.T. Tsay, T.C. Feng, M.S. McAdams,
B.J. Tromberg, “Boundary conditions for the diffusion equation in
radiative transfer”, J. Opt. Soc. Am. A, vol. 11, no. 10, pp. 2727-2741,
Oct. 1994.
G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint equations
and perturbation algorithms in Nonlinear problems, CRC Press,
Boca Raton, FL, 1996.
Y. Saad, M.H. Schultz, “GMRES: A Generalized Minimal Residual
algorithm for solving nonsymmetric linear systems”, SIAM
J.Sci.Stat.Comput., vol. 7, no. 3, pp. 856-869, July 1986.
A. Grossmann, J. Morlet, “Decomposition of Hardy Functions Into
Square Integrable Wavelets of Constant Shape”, SIAM J. On
Mathematical Analysis 15 (4): pp.723-736, 1984.
G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and
numerical algorithms I”, Communications on Pure and Applied
Mathematics, vol. XLIV, pp. 141-183, Mar. 1991.