tip-06221-2010 1 discretization error analysis and
TRANSCRIPT
TIP-06221-2010 1
Discretization Error Analysis andAdaptive Meshing Algorithms for
Fluorescence Diffuse Optical Tomography inthe Presence of Measurement Noise
Lu Zhou, and Birsen Yazıcıā, Senior Member, IEEE
AbstractāQuantitatively accurate Fluorescence Diffuse Opti-cal Tomographic (FDOT) image reconstruction is a computa-tionally demanding problem that requires repeated numericalsolutions of two coupled partial differential equations and anassociated inverse problem. Recently, adaptive finite elementmethods have been explored to reduce the computation require-ments of the FDOT image reconstruction. However, existingapproaches ignore the ubiquitous presence of noise in boundarymeasurements. In this paper, we analyze the effect of finiteelement discretization on the FDOT forward and inverse prob-lems in the presence of measurement noise and develop noveladaptive meshing algorithms for FDOT that take into accountnoise statistics.
We formulate the FDOT inverse problem as an optimizationproblem in the maximum a posteriori framework to estimatethe fluorophore concentration in a bounded domain. We usethe Mean-Square-Error (MSE) between the exact solution andthe discretized solution as a figure of merit to evaluate theimage reconstruction accuracy, and derive an upper bound onthe MSE which depends on the forward and inverse problemdiscretization parameters, noise statistics, a priori information offluorophore concentration, source and detector geometry, as wellas background optical properties. Next, we use this error boundto develop adaptive meshing algorithms for the FDOT forwardand inverse problems to reduce the MSE due to discretization inthe reconstructed images. Finally, we present a set of numericalsimulations to illustrate the practical advantages of our adaptivemeshing algorithms for FDOT image reconstruction.
Index TermsāFluorescence diffuse optical tomography, adap-tive meshing algorithms, error analysis.
I. INTRODUCTION
FLUORESCENCE Diffuse Optical Tomography (FDOT)is an emerging molecular imaging modality with applica-
tions in small animal and deep tissue imaging [1], [2]. FDOTuses visible or near infrared light to reconstruct the concentra-tion, pharmacokinetics, as well as the life time of fluorophoresinjected into the tissue. Similar to its analogue Diffuse Optical
Copyright (c) 2009 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].
Manuscript received May 28, 2010; revised August 7, 2010. This workwas supported by US Army Medical Research-W81XWH-04-1-0559 andby the Center for Subsurface Sensing and Imaging Systems, under theEngineering Research Centers Program of the National Science Foundation(Award Number EEC-9986821). Asterisk indicates corresponding author.
L. Zhou is with Bloomberg L.P., New York, NY, USA.*B. Yazıcı is with the Departments of Electrical, Computer and Systems
Engineering and Biomedical Engineering, Rensselaer Polytechnic Institute,Troy, NY, USA (e-mail: [email protected]).
Tomography (DOT), FDOT poses a computationally intenseimaging problem. This stems from the requirement of nu-merically solving both the forward problem, comprised of aset of diffusion equations, and the inverse problem, which istypically represented by a nonlinear integral equation. Thenumerical solutions of FDOT forward and inverse problemscontain error due to discretization, and this discretizationerror together with the measurement noise deteriorate the finalreconstruction accuracy. Thus, the discretization presents atradeoff between the accuracy and the computational efficiencyof the image reconstruction. To improve the reconstructionaccuracy, one can reduce the mesh size and increase thenumber of discretization points. However, this also increasesthe size of the discretized forward and inverse problems,thereby decreasing the computational efficiency of the imagereconstruction. Recently, a number of adaptive discretizationtechniques for the Partial Differential Equation (PDE) basedinverse coefficient problems have been developed [3]ā[16].However, these approaches ignore the presence of noise inboundary measurements. In this paper, we analyze the effect offinite element discretization on the FDOT forward and inverseproblems in the presence of measurement noise and developnovel adaptive meshing algorithms that take into account noisestatistics.
There is extensive research on the analysis of discretizationerror in the numerical solutions of PDEs [17]ā[22]. However,in the area of PDE-based inverse coefficient problems, wherethe objective is to estimate primarily the coefficients of PDEs,relatively little has been published (see [3]ā[7]). As an applica-tion of the error analysis, Beilina et al. derived an a posteriorierror estimate and developed an adaptive meshing method forthe solution of an inverse acoustic scattering problem [8],[9]. In the area of FDOT, in [10], Bangerth et al. formulatedthe image reconstruction problem as a PDE-constrained opti-mization problem, and employed a mesh refinement methodsuggested in a dual weighted residual framework [3]. In [12],to achieve fast and robust parameter mapping between theadaptively refined/derefined meshes of forward and inverseproblems, Lee et al. developed an algorithm to identify andresolve the intersections of tetrahedral finite elements. In [11],this algorithm was utilized in FDOT reconstruction within adual adaptive meshing scheme in which the meshes for theforward and inverse problems are independently refined basedon an a posteriori error estimate. In our previous work [13],
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[15], we presented a Finite Element Method (FEM) basedapproach to analyze the effect of discretization on the accuracyof DOT and FDOT reconstructions under the assumption thatthe measurements are noise-free. These studies further led tothe development of new adaptive mesh generation algorithmsfor these two imaging modalities, that can effectively reducethe error due to discretization [14], [16]. Although mostof these studies [8]ā[10], [13]ā[16] take into account theinterdependence of forward and inverse problems and theirproposed adaptive meshing methods can effectively reduce thediscretization error in the reconstructed images, the effect ofmeasurement noise was not considered in the error analysisand adaptive meshing schemes.
In this paper, we focus on analyzing the effect of mea-surement noise in the FDOT forward and inverse problemdiscretizations, and develop adaptive meshing algorithms thattake into account noise statistics and can effectively reduce thediscretization error. We assume that FDOT boundary measure-ments are collected using a continuous wave (CW) imagingsystem. We model the forward problem of FDOT by a pair ofdiffusion equations at the excitation and emission wavelengths,and use FEM to solve these equations. We formulate the FDOTinverse problem as an optimization problem in the MaximumA Posteriori (MAP) framework to estimate the fluorophoreconcentration, and use the Mean-Square-Error (MSE) betweenthe exact solution and the discretized solution of the inverseproblem as a figure of merit to assess the error due todiscretization. We analyze the effect of discretization on thetwo components of the MSE, namely the bias and varianceof the reconstructed image, and derive upper bounds for thesequantities. These upper bounds depend on the forward andinverse problem discretization parameters, noise statistics, apriori information of fluorophore concentration, source anddetector geometry, as well as background optical properties.We next utilize these upper bounds to design local error indica-tors to use in the adaptive discretization of the FDOT forwardand inverse problems. Unlike the algorithms in [16], the newadaptive meshing algorithms take into account noise statisticsand a priori information of fluorophore concentration. The nu-merical simulation results show that the new adaptive meshingalgorithms can effectively improve the reconstruction accuracyand resolution when noise statistics are taken into accountas compared to the uniform meshing and adaptive meshingalgorithms presented in [16]. Similar results are also reportedin [23] where we compare the accuracy of reconstruction fordifferent meshing schemes using real measurements from aphantom experiment.
The outline of the paper is as follows: In Section II,III and IV, we introduce the FDOT forward and inverseproblems, and their discretizations, respectively. In Section V,we derive the upper bounds for the bias, variance and MSE ofthe reconstructed image. In Section VI, we present adaptivemeshing algorithms for FDOT forward and inverse problemsbased on the results in Section V. In Section VII, we presentsimulation results to demonstrate the performance of adaptivemeshing algorithms. Finally, in Section VIII, we conclude ourdiscussion.
TABLE IDEFINITION OF FUNCTION SPACES AND NORMS.
Notation ExplanationC(Ī©) Space of continuous functions on Ī©Lā(Ī©) Lā(Ī©) = f |ess supĪ©|f(x)| < ā Lp(Ī©) Lp(Ī©) = f | (
ā«Ī© |f(x)|pdx)1/p < ā , p ā [1,ā)
Hp(Ī©) Hp(Ī©) = f | (ā
|z|ā¤p ā„Dzwfā„20)1/2 < ā , p ā [1,ā)
ā„fā„0 The L2(Ī©) norm of fā„fā„p The Hp(Ī©) norm of fā„fā„ā The Lā(Ī©) norm of fā„fā„0,m The L2 norm of f over the mth finite element Ī©m
ā„fā„p,m The Hp norm of f over the mth finite element Ī©m
ā„fā„ā,m The Lā norm of f over the mth finite element Ī©m
II. FDOT FORWARD PROBLEM
A. Notational Conventions
Throughout the paper, we use capital cursive letters (A)for operators and bold capital letters (Ī£) for matrices. Wedenote functions by lowercase letters (g and Ļ etc.) and theirfinite dimensional approximations by corresponding uppercaseletters (G and Ī¦ etc.). We use bold to denote vectorizedquantities such as r, Ī. Table I provides a summary of keyvariables and function spaces used throughout the paper.
B. Diffusion Model for Light Propagation
We assume that the CW light sources are used to estimatethe fluorophore concentration in a bounded domain Ī© ā R3.Therefore, we use a pair of coupled frequency-domain dif-fusion equations, with modulation frequency Ļ = 0, andthe corresponding boundary conditions on āĪ© to model lightpropagation [24]:
āā Ā·D(r)āĻx(r, ri) + Āµax(r)Ļx(r, ri) = Si(r),
r ā Ī©, (1)
2D(r)āĻx(r, ri)
ān+ ĻĻx(r, ri) = 0, r ā āĪ©, (2)
āā Ā·D(r)āĻm(r, ri) + Āµam(r)Ļm(r, ri)
= Ļx(r, ri)Ī·Āµaxf (r), r ā Ī©, (3)
2D(r)āĻm(r, ri)
ān+ ĻĻm(r, ri) = 0, r ā āĪ©, (4)
where Ļx and Ļm are the photon densities at the excitation andemission wavelengths, respectively. D is the isotropic diffusioncoefficient. Āµax and Āµam are the absorption coefficients of themedium at the excitation and emission wavelengths, respec-tively. Ī· and Āµaxf are the quantum efficiency and absorptioncoefficient of the fluorophore. Ļ is a parameter governing theinternal reflection at the boundary āĪ©, and ā/ān denotesthe directional derivative along the unit normal vector on thedomain boundary. Si is the ith excitation source, modeled bya Gaussian function centered at the source position ri, fori = 1, Ā· Ā· Ā· , NS , where NS is the number of sources. Note thatsince Ļ = 0, we drop the frequency dependency of Ļx andĻm to simplify our notation.
We make use of the adjoint problem associated with (3)and (4) to express the relationship between the fluorophore
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concentration and the measurements [25]:
āā Ā·D(r)āgām(r, rj) + Āµam(r)gām(r, rj) = 0, r ā Ī©, (5)
2D(r)āgām(r, rj)
ān+ Ļgām(r, rj) = Sā
j (r), r ā āĪ©, (6)
where gām(r, rj) is the solution of the adjoint problem for thejth adjoint source Sā
j located at the detector position rj ā āĪ©,for j = 1, Ā· Ā· Ā· , ND, where ND is the number of detectors.
Given NS sources and ND detectors, we define Īi,j to bethe measurement obtained by the jth detector, j = 1, . . . , ND,due to the ith source, i = 1, . . . , NS . Using (1)-(2) and (5)-(6),we write Īi,j as followings:
Īi,j =
ā«Ī©
gām(r, rj)Ļx(r, ri)Ī·Āµaxf (r)dr
=
ā«Ī©
gāj (r)Ļi(r)Āµ(r)dr, (7)
where we define gāj (r) := gām(r, rj) and Ļi(r) := Ļx(r, ri),suppressing the excitation (x) and emission (m) wavelengthsdependency of these functions to simplify our notation.
We define Āµ(r) := Ī·Āµaxf (r), and refer to Āµ(r) as thefluorophore concentration, the quantity to be reconstructed.Then we group the individual measurements into the followingvector:
Ī := [Ī1,1, . . . ,Ī1,ND,Ī2,1, . . . ,ĪNS ,ND
]T, (8)
and define the vector valued operator A : L2(Ī©) ā RNSND
as
(AĀµ)ij :=ā«Ī©
aij(r)Āµ(r)dr, (9)
where aij(r) := gāj (r)Ļi(r).Combining (7), (8) and (9), we write Ī = AĀµ, where the
integration is understood elementwise.
C. Iterative LinearizationThe integral equation in (9) is nonlinear in Āµ due to the
dependency of Ļi and gāj to Āµaxf . We use the Born approxima-tion to linearize (9) around a known background fluorophoreconcentration Āµ0. Note that the Born approximation is validwhen the perturbation of the fluorophore absorption coefficientis relatively small as compared to the known backgroundabsorption coefficient [24], [26].
Let Ļ0i and gā,0j be the solutions of (1)-(2) and (5)-(6) forĀµ = Āµ0, and a0ij(r) = Ļ0i (r)g
ā,0j (r). Then, the model (9)
becomes
(A0Āµ)ij :=
ā«Ī©
a0ij(r)Āµ(r)dr,
or Ī = A0Āµ, where a0ij , i = 1, . . . , NS and j = 1, . . . , ND,is the kernel of A0. Note that A0 is now linear in Āµ.
The solution at each linearization step can be iterativelyrefined based on the following model:
Ī = AkĀµk+1,
where Āµk+1 is the estimate of the fluorophore concentrationat the (k + 1)th iteration and
(AkĀµ)ij :=
ā«Ī©
akij(r)Āµ(r)dr,
with akij = Ļki (r)gā,kj (r), Ļki (r) and gā,kj (r) are computed
based on the fluorophore concentration Āµk estimated at kth
iteration. Note that we drop 0 and k superscripts on Ļ0i , gā,0j ,Ļki , and gā,kj for the rest of paper to simplify our notation.
III. FDOT INVERSE PROBLEM FORMULATION
A. Models for Measurement Noise and Fluorophore Concen-tration
We assume that the measurements are contaminated byadditive noise and write:
Ī = A0Āµ+ Īµ, (10)
where Īµ = [Īµ1,1, . . . , Īµ1,ND , Īµ2,1, . . . , ĪµNS ,ND ]T is the noise
vector. Without loss of generality, we assume that the compo-nents of the noise vector are mutually statistically independentGaussian random variables with zero-mean and known vari-ance Ļ2
Īµ,ij , for i = 1, . . . , NS and j = 1, . . . , ND. We denotethe covariance matrix of Īµ with
Ī£Īµ = diag([Ļ2Īµ,11, . . . , Ļ
2Īµ,1ND
, Ļ2Īµ,21, . . . , Ļ
2Īµ,NSND
]T).
We model the fluorophore concentration image as a Gaus-sian random field and assume that it is statistically independentof the noise. Furthermore, we assume that the fluorophoreconcentration Āµ has mean Āµ0 equal to the known backgroundfluorophore concentration. Without loss of generality, we as-sume that Āµ(r) and Āµ(r), r = r, are mutually statisticallyindependent. Note that at the (k+1)th iteration of the iterativereconstruction, we assume Āµk+1 has mean Āµk, the estimateobtained at the kth iteration. Thus, we define
E[Āµ(r)] = Āµ0(r),
CovĀµĀµ(r, r) = E [[Āµ(r)ā Āµ0(r)][Āµ(r)ā Āµ0(r)]]
=: Īŗ(r)Ī“(r ā r),
where E denotes expectation and Īŗ(r) ā„ 0 for all r ā Ī©.
B. The Maximum A Posteriori Estimators for FluorophoreConcentration
We consider the MAP estimator for Āµ which is given by thefollowing constrained minimization problem:
ĀµMAP = minĀµāL2(Ī©)
[JLH (Āµ) + JPR (Āµ)] , (11)
where
JLH(Āµ) =
NS ,NDāi,j
1
Ļ2Īµ,ij
[Īi,j ā (A0Āµ)i,j ]2,
JPR(Āµ) =
ā«Ī©
1
Īŗ(r)[Āµ(r)ā Āµ0(r)]
2dr.
Taking the Gateaux derivative of (11) with respect to Āµ,and setting it equal to zero, we obtain an integral equationfor which the MAP estimate, ĀµMAP , of the fluorophoreconcentration satisfies:(
Aā0Ī£
ā1Īµ A0ĀµMAP
)(r) +
ĀµMAP (r)
Īŗ(r)=
(Aā
0Ī£ā1Īµ Ī
)(r)
+Āµ0(r)
Īŗ(r), (12)
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where Aā0 : RNSND ā L2(Ī©) is the adjoint operator of
A0 [15].Note that when the a priori information on the fluorophore
concentration is not available, one can formulate the inverseproblem in the Maximum Likelihood (ML) framework withJPR(Āµ) = 0. In that case, the ML estimate, ĀµML, satisfies(
Aā0Ī£
ā1Īµ A0ĀµML
)(r) =
(Aā
0Ī£ā1Īµ Ī
)(r).
Since A0 and Aā0 are both compact [27], we consider the
following regularized form for the solution of ĀµML:((Aā
0Ī£ā1Īµ A0 +
Ī»0Ļ2Īµ,max
I)ĀµML
)(r) =
(Aā
0Ī£ā1Īµ Ī
)(r),
where I : L2(Ī©) ā L2(Ī©) is the identity operator, Ļ2Īµ,max
is the maximum value of Ļ2Īµ,ij , for i = 1, . . . , NS and j =
1, . . . , NS , and Ī»0 is a small positive constant. Note that thereare several methods for choosing appropriate regularizationparameters Ī»0 (see [28]ā[32]). In this paper, we assume thatĪ»0 is appropriately chosen based on the spectral decompositionof the operator Aā
0A0 [32].To simplify our notation, we define the operator B :
L2(Ī©) ā L2(Ī©) as
B := Aā0Ī£
ā1Īµ A0,
and express (12) as follows:
(BĀµMAP ) (r) +ĀµMAP (r)
Īŗ(r)=
(Aā
0Ī£ā1Īµ Ī
)(r) +
Āµ0(r)
Īŗ(r). (13)
We use the Galerkin method [27] to solve the integralequation defined in (13). Thus, we first define the variationalform of (13):
FMAP (Ļ, ĀµMAP ) =(Ļ,Aā
0Ī£ā1Īµ Ī
)+ (Ļ,
Āµ0
Īŗ), (14)
where
FMAP (Ļ, Āµ) := (Ļ,BĀµ) + (Ļ,Āµ
Īŗ), (15)
(Ā·, Ā·) denotes inner product in L2(Ī©) and Ļ is any testfunction in L2(Ī©). Then, the FDOT inverse problem involvesrecovering ĀµMAP based on (14).
Similarly, the ML estimate satisfies the following variationalform:
FML(Ļ, ĀµML) =(Ļ,Aā
0Ī£ā1Īµ Ī
), (16)
where
FML(Ļ, Āµ) := (Ļ,BĀµ) + (Ļ,Ī»0Āµ
Ļ2Īµ,max
),
and Ļ is any test function in L2(Ī©).Finally we note that a unique solution for the inverse
problem (14) or (16) exists when Īŗ and Ī»0 are appropriatelychosen [15].
IV. DISCRETIZATION OF THE FORWARD AND INVERSEPROBLEMS
In the following subsections, we, first, discuss the vari-ational formulation and finite element discretization of theforward problem to obtain a finite-dimensional approximationof the forward problem solution. Next, we use these finiteelement solutions of the forward problem in the inverse prob-lem formulation and discuss the discretization of the resultingapproximate inverse problem using the Galerkin method.
A. Forward Problem Discretization
We express the forward problem defined in (1)-(2) and (5)-(6) in variational forms and next apply the FEM to discretizeand solve the resulting problems. To do so, we first multiply(1) and (5) by two test functions Ī¾1 ā H1(Ī©) and Ī¾2 ā H1(Ī©),respectively, and apply Greenās theorem to the second-orderderivative terms. Then, using the boundary conditions in (2)and (6), we obtainā«
Ī©
(āĪ¾1 Ā·DāĻi + ĀµaxĪ¾1Ļi)dr +1
2Ļ
ā«āĪ©
Ī¾1Ļidl
=
ā«Ī©
Ī¾1Sidr, (17)ā«Ī©
(āĪ¾2 Ā·Dāgāj + ĀµamĪ¾2g
āj
)dr +
1
2Ļ
ā«āĪ©
Ī¾2gāj dl
=1
2Ļ
ā«āĪ©
Ī¾2Sāj dl. (18)
Let Lk denote the piecewise linear Lagrange basis func-tions used in discretizing the forward problem. We defineYi ā H1(Ī©), i = 1, . . . , NS , as the finite dimensionalsubspace spanned by Lk, k = 1, . . . , Ni. Note that Lkare associated with the set of points rp, p = 1, . . . , Ni,on Ī©. We further let Ī©ni denote the corresponding set ofelements used to discretize Ī©, for n = 1, . . . , N i
ā; such thatāŖNiā
n Ī©ni = Ī© and hni is the diameter of the smallest ball thatcontains the nth element. Similarly, we define Y ā
j ā H1(Ī©),j = 1, . . . , ND, as the finite-dimensional subspace spannedby Lk, k = 1, . . . , Nj , which are associated with the setof points rp, p = 1, . . . , Nj . We further let Ī©mj denotethe corresponding set of elements used to discretize Ī©, form = 1, . . . , Nāj
ā ; such thatāŖNāj
ām Ī©mj = Ī© and hmj is the
diameter of the smallest ball that contains the mth. Next,we replace Ī¾1, Ļi in (17) and Ī¾2, gāj in (18) by their finite-dimensional approximations defined as
Ī1(r) :=
Niāk=1
pkLk(r), Ī¦i :=āNi
k=1 ckLk(r), (19)
Ī2(r) :=
Njāk=1
pkLk(r), Gāj :=
āNj
k=1 dkLk(r), (20)
and obtain the matrix equations
Mci = qi, (21)Mādā
j = qāj , (22)
for coefficients ci = [c1, c2, ..., cNi ]T and dā
j =[d1, d2, ..., dNj ]
T .
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In (21) and (22), M and Mā are the finite element matricesand qi and qā
j are the load vectors resulting from the finiteelement discretization of the forward problem.
B. Inverse Problem Discretization
For the inverse problem discretization, we first substituteĪ¦i and Gā
j into the operators A0, Aā0 and B to obtain the
approximate operators denoted by A0, A0
āand B. We, next,
substitute A0, A0
āand B into (14) and (15) to obtain an
approximate inverse problem formulation:
FMAP (Ļ, ĀµMAP ) = (Ļ, A0
āĪ£ā1Īµ Ī) + (Ļ,
Āµ0
Īŗ), (23)
for all Ļ ā L2(Ī©), with
FMAP (Ļ, Āµ) := (Ļ, BĀµ) + (Ļ,Āµ
Īŗ),
where ĀµMAP ā L2(Ī©) is the solution of (23).Next, we define the finite-dimensional subspace V (Ī©) ā
L2(Ī©) spanned by the first-order Lagrange basis functionsLk, k = 1, . . . , N , which are associated with the set ofpoints rp, p = 1, . . . , N , on Ī©. We use Ī©t, t =1, . . . , Nā, to denote the corresponding set of elements usedto discretize Ī© with vertices at rp, p = 1, . . . , N , such thatāŖNā
t Ī©t = Ī©, and ht is the diameter of the smallest ball thatcontains the tth element. We substitute ĀµMAP and Ļ in (23)with their finite-dimensional approximations ĀµD
MAP ā V (Ī©)and ĪØ ā V (Ī©) defined, respectively, by
ĀµDMAP (r) :=
Nāk=1
mkLk(r), ĪØ(r) :=
Nāk=1
pkLk(r), (24)
and obtain the following fully discretized inverse problemformulation:
FMAP (ĪØ, ĀµDMAP ) = (ĪØ, A0
āĪ£ā1Īµ Ī) + (ĪØ,
Āµ0
Īŗ). (25)
The resulting inverse problem formulation can be expressedas the following matrix-vector equation:
FNm = GN , (26)
where m = [m1, Ā· Ā· Ā· ,mN ]T represents the unknown coef-ficients in the finite approximation of ĀµD
MAP , and FN andGN are the finite element matrix and the load vector resultingfrom (25).
V. ANALYSIS OF THE DISCRETIZATION ERROR IN THEPRESENCE OF MEASUREMENT NOISE AND A PrioriINFORMATION ON FLUOROPHORE CONCENTRATION
In this section, we analyze the effect of forward andinverse problem discretizations on the accuracy of FDOTreconstruction in the presence of measurement noise and apriori information of the fluorophore concentration. In thisrespect, we quantify the error in the mean square sense, andderive an upper bound for the MSE in FDOT reconstructiondue to discretization. Next, we discuss the case of the MLestimate, as well as the case involving correlated noise and apriori fluorophore concentration models. Finally, we commenton the implications of the MSE bound for the discretizationsof the FDOT forward and inverse problems.
A. Error Bound on the MSE due to Discretization
We are interested in quantifying the difference between theexact estimate ĀµMAP and the estimate ĀµD
MAP obtained afterforward and inverse problem discretizations. Thus, we define
eMAP (r) := ĀµMAP (r)ā ĀµDMAP (r),
and quantify the difference between ĀµMAP and ĀµDMAP in term
of the MSE defined as follows:
MSE[ĀµDMAP ] :=
ā«Ī©
E[|eMAP (r)|2
]dr. (27)
We further express (27) as
MSE[ĀµDMAP ] = Bias2[ĀµD
MAP ] + Var[ĀµDMAP ],
where
Bias2[ĀµDMAP ] :=
ā«Ī©
|E [eMAP (r)]|2 dr, (28)
Var[ĀµDMAP ] :=
ā«Ī©
E[|eMAP (r)ā E[eMAP (r)]|2
]dr.
(29)
We refer to Bias[ĀµDMAP ] as the bias of ĀµD
MAP with respect tothe exact MAP estimate ĀµMAP and Var[ĀµD
MAP ] as the varianceof ĀµD
MAP .In Theorem 1 and 2, we present upper bounds for
Bias[ĀµDMAP ] and Var[ĀµD
MAP ]; and next use these bounds todevelop new adaptive meshing algorithms for FDOT in thefollowing sections.
Theorem 1:Consider the Galerkin projection of the variationalproblems (17), (18) and (23) described in Section IV.Let ĀµMAP (r) := E[ĀµMAP (r)]. Then,1) ĀµMAP satisfies the following variational problem:
FMAP (Ļ, ĀµMAP ) = (Ļ,Aā0Ī£
ā1Īµ Ī) + (Ļ,
Āµ0
Īŗ),
for all Ļ ā L2(Ī©), where Ī = E[Ī] = A0Āµ0 and2) Assume that ĀµMAP ā H1(Ī©), then
Bias2[ĀµDMAP ] ā¤ CB [B1 +B2 +B3]
2, (30)
where
B1 =
NSāi=1
Niā,NDān,j
(F 1ijā„gāj ĀµMAP ā„0,ni
+F 2ijā„gāj ā„ā,ni
)ā„Ļiā„1,nihni,
B2 =
NDāj=1
Nājā ,NSām,i
(F 1ijā„ĻiĀµMAP ā„0,mj
+F 2ijā„Ļiā„ā,mj
)ā„gāj ā„1,mjhmj ,
B3 =
Nāāt=1
NS ,NDāi,j
I1ijā„GājĪ¦iā„0,t + I2t
Ā· ā„ĀµMAP ā„1,tht,
TIP-06221-2010 6
with
F 1ij =
2ā„Īŗā„āā„gājĻiā„0Ļ2Īµ,ij
, F 2ij =
ā„Īŗā„ā|Īi,j |Ļ2Īµ,ij
,
I1ij =ā„Īŗā„āā„Gā
jĪ¦iā„0Ļ2Īµ,ij
, I2t = ā„Īŗā„āā„ā„ā„ā„ 1Īŗ
ā„ā„ā„ā„ā,t
,
and CB is a constant independent of the discretiza-tion parameters hni, hmj and ht.
Proof:See Appendix A. Theorem 2:
Consider the Galerkin projection of the variationalproblems (17), (18) and (23) described in Section IV.Let Ļij ā L2(Ī©) be the solution of the followingvariational problem:
FMAP (Ļ, Ļij) = (Ļ,Aā0Ī£
ā1Īµ eij),
for all Ļ ā L2(Ī©), and eij ā RNSND is the[ND(i ā 1) + j]th column vector of the NSND ĆNSND identity matrix. Assume Ļij ā H1(Ī©), thenVar[ĀµD
MAP ] satisfies the following inequality:
Var[ĀµDMAP ] ā¤ CV [V1 + V2 + V3]
2, (31)
where
V1 =
NSāi=1
Niā,NDān,j
F 1ij
NS ,NDāiā²,jā²
ā„ā„gājDiā²jā²Ļiā²jā²ā„ā„0,ni
+ F 3ijDijā„gāj ā„ā,ni
ā„Ļiā„1,nihni,
V2 =
NDāj=1
Nājā ,NSām,i
F 1ij
NS ,NDāiā²,jā²
ā„ĻiDiā²jā²Ļiā²jā²ā„0,mj
+ F 3ijDijā„Ļiā„ā,mj
ā„gāj ā„1,mjhmj ,
V3 =
Nāāt=1
NS ,NDāi,j
I1ijā„GājĪ¦iā„0,t + I2t
Ā·
NS ,NDāi,j
ā„DijĻijā„1,t
ht,
with
F 1ij =
2ā„Īŗā„āā„gājĻiā„0Ļ2Īµ,ij
, F 3ij =
ā„Īŗā„āĻ2Īµ,ij
,
I1ij =ā„Īŗā„āā„Gā
jĪ¦iā„0Ļ2Īµ,ij
, I2t = ā„Īŗā„āā„ā„ā„ā„ 1Īŗ
ā„ā„ā„ā„ā,t
,
Dij =
[Ļ2Īµ,ij +
ā«Ī©
Īŗ(r)aā,0ij (r)a0ij(r)dr
]1/2,
and CV is a constant independent of the discretiza-tion parameters hni, hmj and ht.
Proof:See Appendix B. We note that, for the ML estimate, ĀµML, of the fluorophore
concentration, the upper bounds for Bias2[ĀµDML] and Var[ĀµD
ML]
are given as in (30) and (31) with Dij = ĻĪµ,ij and Īŗ(r) =Ļ2Īµ,max/Ī»0. We also note that, when the measurements are
noise-free, the inverse problem formulation in this work canbe reduced to the one in [15], by setting Ļ2
Īµ,ij = 1 for i =1, . . . , NS and j = 1, . . . , ND, and Īŗ(r) = Ī»1. Then, the errorbound for Bias2[ĀµD
MAP ] can be reduced to the combinationof the two error bounds in [15], and Var[ĀµD
MAP ] vanishes asshown in the proof of Theorem 2.
Finally, we can combine Theorem 1 and 2 to obtain anupper bound for MSE[ĀµD
MAP ] using the fact that Bi and Vi,for i = 1, 2, 3, are all positive terms:
MSE[ĀµDMAP ] = Bias2[ĀµD
MAP ] + Var[ĀµDMAP ]
ā¤ CM
[3ā
i=1
Bi +3ā
i=1
Vi
]2
, (32)
where CM = maxCB , CV , and Bi and Vi, i = 1, 2, 3, aredefined in Theorem 1 and 2.
B. General Models for Noise and Fluorophore Concentration
For a more general a priori second order statistical modelof the fluorophore concentration, we consider CovĀµĀµ(r, r) =Īŗ(r, r) as the kernel of a positive definite operator K :L2(Ī©) ā L2(Ī©). Similarly, for a general noise model, weassume that the covariance matrix, Ī£Īµ, of the measurementnoise is a positive definite matrix that is not necessarilydiagonal. Then, the variational form of the inverse problemformulation becomes
FMAP (Ļ, ĀµMAP ) =(Ļ,Aā
0Ī£ā1Īµ Ī
)+ (Ļ,Kā1Āµ0),
where
FMAP (Ļ, Āµ) := (Ļ,BĀµ) + (Ļ,Kā1Āµ).
In this case, it can be shown that the upper bounds forBias2[ĀµD
MAP ] and Var[ĀµDMAP ] are also of the same forms as
in (30) and (31) with new coefficients given by
F 1ij =
2
ā„Īŗā1ā„ā
NSNDāp=1
(Ī£ā1Īµ
)p,(iā1)ND+j
ā„gājĻiā„0,
F 2ij =
1
ā„Īŗā1ā„ā
NSNDāp=1
(Ī£ā1Īµ
)p,(iā1)ND+j
|Īi,j |,
F 3ij =
1
ā„Īŗā1ā„ā
NSNDāp=1
(Ī£ā1Īµ
)p,(iā1)ND+j
,
I1ij =1
ā„Īŗā1ā„ā
NSNDāp=1
(Ī£ā1Īµ
)p,(iā1)ND+j
ā„GājĪ¦iā„0,
I2t =1
ā„Īŗā1ā„āā„Īŗ0ā„0 ā„Īŗ0ā„ā,t ,
Dij =[(Ī£Īµ)(iā1)ND+j,(iā1)ND+j
+
ā«Ī©
ā«Ī©
Īŗ(r, r)aā,0ij (r)a0ij(r)drdr
]1/2,
TIP-06221-2010 7
where Īŗā1 denotes the kernel of the operator Kā1 : L2(Ī©) āL2(Ī©), i.e.,
(Kā1Āµ)(r) =
ā«Ī©
Īŗā1(r, r)Āµ(r)dr,
and Īŗ0(r) is the Kolmogorov decomposition of Īŗā1(r, r)[33]; (Ī£Īµ)p,q and
(Ī£ā1Īµ
)p,q
, for p, q = 1, . . . , NSND, denotethe entries on the pth row and the qth column of Ī£Īµ andĪ£ā1Īµ , respectively. Clearly, these coefficients reduce to those in
Theorem 1 and 2 when the independent noise and fluorophoreconcentration models are considered in FDOT reconstruction.
C. Implications of Theorem 1 and 2 for Discretizations ofForward and Inverse Problems
In this subsection, we discuss the implications of the errorbounds given in Theorem 1 and 2 for the discretization of theforward and inverse problems of FDOT.
Equation (30) in Theorem 1 presents an upper bound forBias2[ĀµD
MAP ], which takes into account the noise statisticsand the a priori information of fluorophore concentration, inaddition to the factors such as the interdependence betweenthe forward and inverse problem solutions, the source-detectorconfiguration, and their positions with respect to the fluo-rophore heterogeneity. In this error bound, B1 and B2 repre-sent the contribution from the forward problem discretization.To keep these quantities small, the mesh parameters hni andhmj of the nth and mth elements in the meshes used in solvingĪ¦i and Gā
j , respectively, have to be chosen small when theircorresponding scaling factors
NDāj=1
(F 1ijā„gāj ĀµMAP ā„0,ni + F 2
ijā„gāj ā„ā,ni
)ā„Ļiā„1,ni,
andNSāi=1
(F 1ijā„ĻiĀµMAP ā„0,mj + F 2
ijā„Ļiā„ā,mj
)ā„gāj ā„1,mj ,
are large on those elements. Further examination of these fac-tors suggests an adaptive refinement scheme within each mesh,because ā„gāj ĀµMAP ā„0,ni, ā„gāj ā„ā,ni, ā„Ļiā„1,ni, ā„ĻiĀµMAP ā„0,mj ,ā„Ļiā„ā,mj , and ā„gāj ā„1,mj all vary within the mesh. This meshrefinement scheme is similar to the one suggested by Theorem1 in our previous work [15]: For the ith source or the jth
detector, it refines the mesh close to that source or detector,as well as around the fluorophore heterogeneity and otherdetectors or sources. At the same time, the coefficients F 1
ij =2ā„gājĻiā„0ā„Īŗā„ā/Ļ2
Īµ,ij and F 2ij = |Īi,j |ā„Īŗā„ā/Ļ2
Īµ,ij in B1 andB2 may vary for different source-detector pairs. To keep B1
and B2 low, one has to generate finer meshes for the source-detector pairs with smaller noise variances (higher F 1
ij andF 2ij), as compared to those pairs with larger noise variances.
In this respect, the error bound in Theorem 1 suggests a newadaptive mesh refinement scheme across different meshes insolving Ī¦i and Gā
j based on the measurements and the noisestatistics. We note that this is a major difference betweenthe implications of the error bounds in this paper and thosepresented in our previous work [15].
In B3, which corresponds to the contribution from theinverse problem discretization, the discretization parameter htof the inverse mesh is not only scaled by the inverse problemsolution ā„ĀµMAP ā„1,t, but also scaled by the finite elementsolutions of the forward problem, the noise variance, and thea priori information of the fluorophore concentration:
NS ,NDāi,j
I1ijā„GājĪ¦iā„0,t + I2t ,
where I1ij = ā„Īŗā„āā„GājĪ¦iā„0/Ļ2
Īµ,ij , and I2t = ā„Īŗā„ā ā„1/Īŗā„ā,t.This result also suggests a new adaptive meshing criteria forthe inverse problem, not only based on the forward and inverseproblem solutions, but also based on the noise statistics anda priori information of the fluorophore concentration. Morespecifically, to keep B3 low, one has to refine the mesh aroundthe heterogeneity of fluorophore concentration, the source-detector pairs with low noise variances, as well as the regionwhere the fluorophore concentration has low variance.
Equation (31) in Theorem 2 shows the effect of the forwardand inverse problem discretizations, the a priori informationof the fluorophore concentration, as well as the noise onVar[ĀµD
MAP ].In this error bound, V1 and V2 correspond to the contribution
from the forward problem discretization, and V3 correspondsto the contribution from the inverse problem discretization.This error bound has a similar form as the one in (30), butĪij is replaced with the standard deviation, Dij , of the (i, j)th
measurement, and ĀµMAP is replaced with DijĻij , where Ļijis the image reconstructed by the imaging system using thebasis vector eij in the measurement space RNSND . This resultindicates that Var[ĀµD
MAP ] is independent of the fluorophoreconcentration, but depends explicitly on the noise statistics,as well as the factors related to the imaging geometry andthe background optical properties, which are incorporated intothe error bound through the functions Ļij . More specifically,DijĻij indicates where ĀµMAP may have high variance due tothe (i, j)th measurement. Therefore, to keep the error boundin (31) low, one has to refine the mesh in the region whereDijĻij , has high value, in addition to the region close to thesources and detectors.
VI. ADAPTIVE MESHING ALGORITHMS IN THE PRESENCEOF MEASUREMENT NOISE AND A Priori INFORMATION ON
FLUOROPHORE CONCENTRATION
In this section, based on Theorem 1 and 2 given in Sec-tion V, we present two new adaptive meshing algorithmsfor FDOT forward and inverse problems. Taking the noisestatistics and a priori information on fluorophore concentrationinto account and using MSE[ĀµD
MAP ] as a figure of merit,these algorithms adaptively discretize the FDOT problem tominimize the error due to discretization in the mean squaresense. In the following, we first give the error indicator basedon the error bound in (32) for each element in the mesh usedin solving the forward or inverse problem, then we describethe steps of the algorithms in detail.
For the forward problem discretization, we aim to minimizethe summation
ā2i=1[Bi + Vi] in (32). Rearranging the terms
TIP-06221-2010 8
in this summation, we obtain
2āi=1
[Bi + Vi] =
NSāi=1
Niāā
n=1
Īµif (n) +
NDāj=1
Nājāā
m=1
Īµjf (m), (33)
where
Īµif (n) :=
NDāj=1
F 1ij
ā„gāj ĀµMAP ā„0,ni +NS ,NDāiā²,jā²
ā„gājDiā²jā²Ļiā²jā²ā„0,ni
+
(F 2ij + F 3
ijDij
)ā„gāj ā„ā,ni
ā„Ļiā„1,nihni, (34)
Īµjf (m) :=
NSāi=1
F 1ij
ā„ĻiĀµMAP ā„0,mj +
NS ,NDāiā²,jā²
ā„ĻiDiā²jā²Ļiā²jā²ā„0,mj
+(F 2ij + F 3
ijDij
)ā„Ļiā„ā,mj
ā„gāj ā„1,mjhmj . (35)
Each Īµif (n) (or Īµjf (m)) entails the contribution of the nth
(or mth) element of the mesh used in solving Ī¦i (or Gāj ), to
the MSE. Equation (33) shows that, the contribution of theforward problem discretization to MSE can be expressed as asummation of the contribution of each element in all meshesused in solving Ī¦i and Gā
j . Thus, we use Īµif (n) and Īµjf (m)as the error indicators in adaptive mesh refinement for theforward problem solution.
As discussed in Section V, since both theorems suggestan adaptive refinement scheme across all meshes used insolving Ī¦i and Gā
j , the new adaptive meshing algorithm limitsthe total number of nodes in all forward problem meshes,instead of separately limiting the number of nodes in eachmesh used for solving Ī¦i or Gā
j . For the adaptive refinementprocess, the algorithm is initiated with a set of coarse uniformmeshes. With each sweep of refinement and for each sourceor detector, it computes the error indicator Īµif (n) or Īµjf (m)on every element and the average value Īµf of the errorindicators on all elements in all meshes. Every element withĪµif (n) > Īµf or Īµjf (m) > Īµf is refined thereafter. By doing so,the resulting meshes provide spatially varying resolution notonly within each mesh, but also among all forward problemmeshes. The algorithm is stopped when the total number ofnodes in all forward problem meshes reaches a predeterminedallowable limit. Algorithm 1 describes the detailed steps ofthis refinement process in the form of a pseudocode.
For the inverse problem discretization, we aim to minimizethe summation [B3 + V3] in (32). Rearranging the terms inthis summation, we obtain
B3 + V3 =
Nāāt=1
Īµi(t),
Algorithm 1 The pseudocode of the adaptive meshing algo-rithm for the forward problem.
ā Generate the initial uniform meshes for all forwardproblems:
(āi, N iā), ā
i =āŖNi
ān=1ān, i = 1, . . . , NS , and
(āāj ,Nājā ), āāj =
āŖNājā
m=1ām, j = 1, . . . , ND
ā Set the maximum number of nodes Nfmax in all meshes
while number of nodes in all meshes less than Nfmax
for i = 1, . . . , NS and j = 1, . . . , ND
for each element ān ā āi with mesh parameterhni or ām ā āāj with mesh parameter hmj
if first linearization Use analytical solutions for Ļi and gāj and
a priori information about ĀµMAP and Ļijto compute Īµif (n) in (34) or Īµjf (m) in (35)
else Use current solution updates Ī¦i, Gā
j , ĀµDMAP
and Ī ij to compute Īµif (n) in (34) or Īµjf (m)
in (35)end
end Compute Īµf Refine the elements with Īµif (n) > Īµf orĪµjf (m) > Īµf
Update the mesh āi, i = 1, . . . , NS , andājā, j = 1, . . . , ND.
endā Solve for Ī¦i, i = 1, . . . , NS , and Gā
j , j = 1, . . . , ND
Algorithm 2 The pseudocode of the adaptive meshing algo-rithm for the inverse problem.
ā Generate an initial uniform mesh:(ā,Nā), ā =
āŖNā
t=1ātā Set the maximum number of nodes N i
max
while number of nodes N less than N imax
for each element āt ā ā with mesh size parameterht
if first linearization Use current solution updates Ī¦i, Gā
j anda priori information about ĀµMAP , Ļij tocompute Īµi(t) in (36)
else Use current solution updates Ī¦i, Gā
j , ĀµDMAP
and Ī ij to compute Īµi(t) in (36)end
Compute Īµi Refine the elements with Īµi(t) > Īµi Update the mesh ā
endā Solve for ĀµD
MAP and Ī ij
where
Īµi(t) :=
NS ,NDāi,j
I1ijā„GājĪ¦iā„0,t + I2t
Ā·
ā„ĀµMAP ā„1,t +NS ,NDā
i,j
ā„DijĻijā„1,t
ht.(36)
TIP-06221-2010 9
Clearly, Īµi(t) is the contribution of the tth element of the mesh,used in solving the inverse problem, to the MSE. Therefore, weuse Īµi(t) as the error indicators in adaptive mesh refinementfor the inverse problem solution.
Our new algorithm for the inverse problem starts from acoarse uniform mesh. In each sweep of the refinement, itcomputes Īµi for each element and the average value Īµi forall elements, and refines those elements with Īµi(t) > Īµi.The algorithm stops when the total number of nodes exceedsa predetermined allowable limit. Algorithm 2 describes thedetailed steps of this refinement process in the form of apseudocode.
The practical implementations of both algorithms requireseveral adjustments: Since Ļi, gāj , ĀµMAP , and Ļij in (34), (35)and (36) can not be computed exactly, we use the analyticalsolution of the diffusion equation on an unbounded domainto approximate gāj and Ļi [16] and a priori information aboutĀµMAP and Ļij in the first iteration and the updated finite-dimensional solutions thereafter. Also note that in both Algo-rithm 1 and 2, one needs to solve for Ļij , i = 1, 2, . . . , NS ,and j = 1, 2, . . . , ND. Ļijs can be computed once numericallygiven the source-detector geometry and background opticalproperties, and stored for the computation of the error indica-tors in adaptive mesh refinement.
Finally, for the forward and inverse problem discretiza-tions, the computational complexity of our adaptive meshingalgorithms can be reduced from O(NDN
iā), O(NSN
ājā ) or
O(NSNDNā) to O(N iā), O(Nāj
ā ) or O(Nā), respectively,by using approximations similar to those given in our previousworks [14], [16]. With these modifications, our new adaptivemeshing algorithms have the same computational complexityas that of the conventional method.
VII. NUMERICAL SIMULATION
In this section, we demonstrate the implications of our erroranalysis and the performance of our new adaptive meshingalgorithms in a set of numerical simulations. We primarilyfocus on showing the effect of measurement noise on thediscretization as well as the FDOT image reconstruction inthe simulation study. For the effect of a priori information,see [34] for a detailed simulation study.
In the following sections, we first describe the setup of oursimulation study, then we present the results of adaptive meshgeneration and FDOT image reconstruction.
A. Simulation Setup
In the numerical simulation study, we considered a 6 cmĆ 6 cm Ć 3 cm cubic domain Ī© shown in Fig. 1. We setthe homogeneous background absorption coefficient Āµaxe =Āµame = 0.05 cmā1 and diffusion coefficient D = 0.0410cm for both excitation and emission wavelengths, and set therefractive index mismatch parameter Ļ = 3 for the boundaryāĪ©. At the center of the domain, we placed a fluorophoreheterogeneity with 3 mm radius and constant absorption coef-ficient Āµaxf = 0.015 cmā1. In the rest of the domain, we as-sumed Āµaxf = 0. To reconstruct the fluorophore concentrationimage, we placed 36 sources and 36 detectors evenly on two
Fig. 1. The optical domain and source-detector configuration for thesimulation study. The squares and triangles denote the detectors and sources,respectively.
6 Ć 6 grids at the bottom and top surfaces of the domain, asshown in Fig. 1. We simulated both the excitation and emissionlight fields by solving the coupled diffusion equations (1)and (3) with their corresponding boundary conditions (2) and(4), using the parameters above on a fine uniform grid with81Ć81Ć41 nodes.
To simulate measurement noise, we considered a shot-noise model described in [35]. When a sufficiently largenumber of photons are detected, the Poisson distribution canbe approximated by a Gaussian distribution with the varianceproportional to the magnitude of the measurements. In thiscase, the variance, Ļ2
Īµ,ij , of each noise component is given byĪ±|Ī0,ij |, where Ī0,ij is the noise-free measurement obtainedat the jth detector due to the ith source. We define the Signal-to-Noise-Ratio (SNR) of the measurements as
SNRij = 10 log10|Ī0,ij |2
Ļ2Īµ,ij
= 10 log10|Ī0,ij |Ī±
.
Note that, each measurement, Īij , has a different SNR propor-tional to log10 |Ī0,ij |. We simulated the noise Īµ for 3 differentvalues of Ī±: 5Ć10ā11, 1Ć10ā9 and 5Ć10ā9, correspondingto approximately 40, 26 and 20 dB average SNR over all mea-surements Īij , i = 1, . . . , NS and j = 1, . . . , ND. For eachvalue of Ī±, we generated 100 different realizations of noiseand obtain three sets of noise contaminated measurements withapproximately 1%, 5% and 10% noise.
In the FDOT reconstruction, we considered a simplifieda priori model for the fluorophore concentration, and setĪŗ(r) = Īŗ0, where Īŗ0 = 5 Ć 10ā6 is a constant chosenempirically. Finally, we note that we performed our simulationstudy using deal.II finite element C++ library [36] and usedhexahedral finite elements with trilinear Lagrange basis func-tions to discretize both the forward and inverse problems. Weused the Gaussian quadrature method to evaluate the integralsin the variational problems (17), (18) and (25), as well asin calculating the function norms of the finite dimensionalsolutions on an element.
B. Simulation Results - Mesh Generation
We used three different types of coarse meshes: uniformmeshes, the adaptive meshes generated by our previous al-gorithms in [16], and the adaptive meshes generated by ournew algorithms described in Section VI, to reconstruct the
TIP-06221-2010 10
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(a) The adaptive mesh with 3289 nodes generated by ournew algorithm for the detector located at (-2.5,-2.5,1.5) forthe 1% noise case.
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(b) The adaptive mesh with 18876 nodes generated by ournew algorithm for the detector located at (-0.5,-0.5,1.5) forthe 1% noise case.
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(c) The adaptive mesh with 3588 nodes generated by ournew algorithm for the detector located at (-2.5,-2.5,1.5) forthe 10% noise case.
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(d) The adaptive mesh with 18054 nodes generated by ournew algorithm for the detector located at (-0.5,-0.5,1.5) forthe 10% noise case.
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(e) The adaptive mesh with 8304 nodes generated by ourprevious algorithm for the detector located at (-2.5,-2.5,1.5).
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(f) The adaptive mesh with 7973 nodes generated by ourprevious algorithm for the detector located at (-0.5,-0.5,1.5).
Fig. 2. Examples of the adaptive meshes for the forward problem used in the simulation study. The mesh is cut through to show the mesh structure inside.
fluorophore concentration image. For the forward problem,the total number of nodes in the meshes used to solve allĪ¦i and Gā
j , i = 1, . . . , NS and j = 1, . . . , ND, ranges from500,000 to 650,000 (roughly 7000 to 9000 for each mesh);and for the inverse problem, it ranges from 2000 to 3000.Note that the uniform meshes used in solving the forward andinverse problems have 25Ć25Ć13 nodes and 17Ć17Ć9 nodes,respectively.
For the forward problem, the examples of the adaptivemeshes generated for the detectors located at (ā2.5,ā2.5, 1.5)and (ā0.5,ā0.5, 1.5) in the 1% and 10% noise cases are
shown in Fig. 2. Figs. 2(a) - 2(d) show the meshes generatedby our new algorithm. We observe there are more nodes in themeshes for the detector located at (ā0.5,ā0.5, 1.5) than in themeshes for the detector located at (ā2.5,ā2.5, 1.5). Figs. 2(e)and 2(f) show the corresponding meshes generated by ourprevious algorithm, and these two meshes have approximatelysame number of nodes. We plotted the relationship betweenthe number of nodes in the meshes generated by our new andprevious algorithms for a certain source or detector and thedistance of that source or detector to the center of the fluo-rophore heterogeneity in Fig. 3. Note that for the sources and
TIP-06221-2010 11
1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
x 104
Distance of the source or detector to the hetergeneity (cm)
Num
ber
of n
odes
in th
e m
esh
New algorithm for 1% noise levelNew algorithm for 10% noise levelPevious algorithm in [6]
Fig. 3. The relationship between the number of nodes in the forward adaptivemesh for a certain source (or detector) and the distance of the source (ordetector) to the center of the fluorophore heterogeneity in 1% and 10% noisecases.
ā3
ā2
ā1
0
1
2
3
ā3ā2
ā10
12
3
ā1
0
1
x
y
z
4
6
8
10
12
14
16x 10
ā4
Fig. 4. The cross-section of the imageāNS ,ND
i,j DijĻij , reconstructedusing coarse uniform meshes using data with 10% noise.
detectors which have the same distance to the heterogeneity,we plotted the average number of nodes in the correspondingmeshes. For our new algorithm, we observe that the closer thesources or detectors to the heterogeneity, the larger the numberof nodes is in the associated meshes. This can be explainedwith the fact that for those source-detector pairs closer tothe heterogeneity, the measurements have higher SNR. Asa result, our new algorithm generates finer meshes for thesesource-detector pairs, so that the accuracy of the correspondingforward problem solutions can match the accuracy of themeasurements. This result in our forward problem meshes withvarying resolution for different source-detector pairs unlike ourprevious algorithm.
We note the difference between the adaptive meshes gen-erated for the cases of 1% and 10% measurement noise inFigs. 2(a) - 2(d), which illustrates the impact of the noiselevel on the forward adaptive meshes generated by our newalgorithm. Since the value of coefficient Dij increases as Ļ2
Īµ,ijincreases, the images DijĻij , i = 1, . . . , NS , j = 1, . . . , ND,have more contribution to the mesh refinement when the noiselevel is high. Fig. 4 shows the cross-section of
āi,j DijĻij
for the 10% noise case. These results indicate that our newalgorithm can refine the meshes adaptively according to themeasurement noise level, while our previous algorithm gener-
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(a) The adaptive mesh with 2721 nodes generated by ournew algorithm for the inverse problem for the 1% noise case.
ā5
0
5
ā5
0
5
ā2
0
2
x
yz
(b) The adaptive mesh with 2785 nodes generated by ournew algorithm for the inverse problem for the 10% noisecase.
ā5
0
5
ā5
0
5
ā2
0
2
x
y
z
(c) The adaptive mesh with 2652 nodes generated by ourprevious algorithm for the inverse problem.
Fig. 5. Examples of the adaptive meshes for the inverse problem used in thesimulation study. The mesh is cut through to show the mesh structure inside.
ates the same mesh for different noise levels.Fig. 5 shows sample adaptive meshes for the inverse prob-
lem. Figs. 5(a) and 5(b) show two different meshes generatedby our new algorithm for 1% and 10% noise levels, respec-tively. These meshes are refined around the fluorophore het-erogeneity as well as around the nearby sources and detectors.This shows that our new algorithm takes into account the noisestatistics, and adaptively refines the mesh according to thenoise level. On the other hand, our previous algorithm neglectsthe impact of noise on the discretization, and generates thesame mesh for different noise levels as shown in Fig. 5(c).
TIP-06221-2010 12
TABLE IIMEAN-SQUARE-ERROR, BIAS AND VARIANCE OF THE IMAGES RECONSTRUCTED BY USING DIFFERENT MESHES IN THE SIMULATION STUDY.
Noise Level Images Bias2 (Ć10ā4) Var (Ć10ā6) MSE (Ć10ā4)
1%ĀµDMAP,U 3.295 100% 0.037 100% 3.295 100%
ĀµDMAP,NA 0.857 26.00% 0.009 23.49% 0.857 26.00%ĀµDMAP,A 2.040 61.93% 0.038 101.45% 2.041 61.93%
5%ĀµDMAP,U 3.251 100% 0.893 100% 3.260 100%
ĀµDMAP,NA 0.753 23.16% 0.199 22.27% 0.755 23.16%ĀµDMAP,A 2.028 62.38% 0.990 110.92% 2.038 62.52%
10%ĀµDMAP,U 3.217 100% 3.670 100% 3.253 100%
ĀµDMAP,NA 0.768 23.89% 0.872 23.77% 0.777 23.89%ĀµDMAP,A 2.045 63.57% 4.301 117.19% 2.088 64.18%
C. Simulation Results - Image Reconstruction
In this part of the simulation study, we considered 3 setsof reconstructions using 3 sets of measurements at differentnoise levels. To obtain the exact solutions of the forward andinverse problems, we solved the forward and inverse problemson a fine mesh with 61 Ć 61 Ć 31 nodes. We assumed that theerror due to discretization in the resulting image, denoted byĀµMAP , is negligible with respect to the images reconstructedusing the three types of coarse meshes; and used this image asa baseline to compute the MSE. In each reconstruction set, weused ĀµD
MAP,U , ĀµDMAP,A, and ĀµD
MAP,NA to denote the imagesreconstructed using the coarse uniform meshes, the adaptivemeshes generated by our previous algorithm in [16] and thenew algorithms described in Section VI, respectively.
We calculated the bias, variance and the MSE of the recon-structed images for each set of reconstructions by averagingall reconstructed image samples for 100 realizations of noise.The results are tabulated in Table II. Additionally, we tabulatedthe percentage of each quantity as compared to the one of theimages, ĀµD
MAP,U , reconstructed by using the coarse uniformmeshes: The left column is the absolute value, and the rightcolumn is the corresponding percentage. The results in Table IIshow that the bias squares of the images, reconstructed usingdifferent types of meshes, remain at a fixed level when thenoise level changes, while the variances of the images increaseas the noise level increases. The bias square, the variance aswell as the MSE of ĀµD
MAP,NA are approximately reduced by75% as compared to ĀµD
MAP,U , when our new algorithm is used.On the other hand, our previous algorithm in [16] providesabout 40% reduction in the bias square, but no reduction inthe variance of ĀµD
MAP,A with respect to ĀµDMAP,U .
Figs 6 and 7 show the cross-section of the sample imagesat z = 0 plane reconstructed using different types of mesheswhen the noise level is 1% and 10%. The cross-section ofthe baseline images are shown in Fig. 6(a) and Fig. 7(a).We observe that the variability of images in Fig. 7 is morevisible as compared to that of the images in Fig. 6 due toincreased noise level in the measurements. The shape of thesmall fluorophore heterogeneity is better resolved in ĀµD
MAP,A
and ĀµDMAP,NA as compared to the one in ĀµD
MAP,U , due to thespatially varying resolution provided by the adaptive meshes.Additionally, we observe a higher variability in ĀµD
MAP,A
than that of ĀµDMAP,NA in Fig. 7(d) and Fig. 7(c), while the
difference between the images in Fig. 6(d) and Fig. 6(c) is
not as noticeable due to lower noise level. These observationscan be seen more clearly in Fig. 8, where the reconstructedimages along the y-axis on z = 0 plane are shown for 1% and10% noise cases. The solid lines in Fig. 8(a) and Fig. 8(b)represent the baseline image ĀµMAP which is assumed to havenegligible error. We observe that the image, ĀµD
MAP,NA, is thebest approximation to ĀµMAP in all three reconstructed images,which has higher response at the center of the fluorophoreheterogeneity and lower background variation, as comparedto those of ĀµD
MAP,U and ĀµDMAP,A.
In summary, the simulation study shows that1) The new adaptive meshing algorithms can adaptively
discretize the FDOT forward and inverse problems ac-cording to the noise level, and, unlike the algorithmsin [16], generates the forward problem mesh with vary-ing resolution for different source-detector pairs.
2) The new adaptive meshing algorithms can effectivelyreduce the bias, variance, as well as the MSE of thereconstructed images with respect to the uniform mesh-ing scheme while keeping the sizes of the discretizedforward and inverse problems under predetermined al-lowable limits.
3) As compared to our previous adaptive meshing algo-rithms [16], the new algorithms is more effective inreducing the variance and MSE of reconstructed images,particularly for high levels of measurement noise.
VIII. CONCLUSION
In this work, we analyzed the effect of discretization onthe accuracy of FDOT reconstruction in the presence ofmeasurement noise. We formulated the FDOT inverse prob-lem as an optimization problem in the MAP framework toestimate the fluorophore concentration in a bounded domain.To quantitatively assess the accuracy of FDOT reconstruction,we first defined the MSE between the exact solution and thediscretized solution of the inverse problem. We, then, identifiedtwo components of the MSE: the bias and the variance of thereconstructed image, and derived an upper bound for eachcomponent. These upper bounds identify the key factors thatdetermine the extent to which the forward and inverse problemdiscretizations can affect the accuracy of FDOT reconstruction.These factors include the noise statistics and the a prioriinformation of fluorophore concentration in addition to theinterdependence between the forward and inverse problem
TIP-06221-2010 13
ā3
ā2
ā1
01
2
3
ā3ā2
ā10
12
3
ā5
0
5
10
15
x 10ā3
x
z=0
y
ā5
0
5
10
15x 10
ā3
(a) The baseline image.
ā3
ā2
ā1
01
2
3
ā3ā2
ā10
12
3
ā5
0
5
10
15
x 10ā3
x
z=0
y
ā5
0
5
10
15x 10
ā3
(b) The image reconstructed using the uniform meshes.
ā3
ā2
ā1
01
2
3
ā3ā2
ā10
12
3
ā5
0
5
10
15
x 10ā3
x
z=0
y
ā5
0
5
10
15x 10
ā3
(c) The image reconstructed using the adaptive meshesgenerated by our new algorithms.
ā3
ā2
ā1
01
2
3
ā3ā2
ā10
12
3
ā5
0
5
10
15
x 10ā3
x
z=0
y
ā5
0
5
10
15x 10
ā3
(d) The image reconstructed using the adaptive meshesgenerated by our previous algorithms.
Fig. 6. The cross-section of the reconstructed sample images on z = 0 plane in 1% noise case in the simulation study.
solutions, the source-detector configuration, and their positionswith respect to the fluorophore heterogeneity.
Based on these bounds, we developed new adaptive meshingalgorithms for the FDOT forward and inverse problems toreduce the MSE in reconstructed images. Unlike the al-gorithms in [16], these algorithms take into account noisestatistics, as well as a priori information on fluorophoreconcentration in the adaptive mesh refinement process. Wedemonstrated the performance of these algorithms in a set ofnumerical simulations. The simulation results showed that thenew algorithms generate adaptive forward meshes with varyingresolution not only within each mesh for a certain source-detector pair, but also across the meshes for all source-detectorpairs. Additionally, we showed that the meshes generated bythe new algorithms can effectively reduce both the bias andthe variance of the reconstructed images, thereby effectivelyreducing the total MSE as compared to the algorithms in [16],as well as the uniform meshing scheme for a fixed number ofnodes.
In our inverse problem formulation, the regularization aswell as the a priori information on fluorophore concentrationintroduce bias into the reconstructed images with respect tothe true fluorophore concentration. This type of error maysometimes overwhelm other types of errors in the recon-structed images. The regularization parameter and the varianceof fluorophore concentration provide a way to balance thebias and the error due to the measurement noise [37], [38].
In the development of our adaptive meshing algorithms, weassumed that the optimal regularization parameter as well asthe variance of the fluorophore concentration are known apriori, and therefore we kept them fixed for different meshingschemes. Taking into account the severely ill-posed nature ofthe FDOT inverse problem, we compared the performanceof the reconstruction for different meshing schemes withthe optimally regularized solution obtained with uniform finemeshing to isolate the error due to discretization only. How-ever, since the error bounds given in Theorem 1 and 2 takeinto account both the regularization and a priori information,it is possible to study the interplay between these parametersand the problem discretization, and to adaptively refine themesh while adjusting parameters to reduce the overall error inthe reconstructed images.
Finally, we note that the error analysis approach introducedin this work is not limited to FDOT imaging, and can beextended to analyze the error due to discretization in otherPDE-based inverse coefficient estimation problems, such asDOT, bioluminescence tomography, electrical impedance to-mography and microwave tomography.
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ā3
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APPENDIX APROOF OF THEOREM 1: UPPER BOUND FOR BIAS2[ĀµD
MAP ]
Let ĀµMAP (r) := E[ĀµMAP (r)] and ĀµDMAP (r) :=
E[ĀµDMAP (r)], then E[eMAP (r)] = ĀµMAP (r)ā ĀµD
MAP (r). Wecan further express Bias2[ĀµD
MAP ] in (28) as
Bias2[ĀµDMAP ] =
ā«Ī©
ā£ā£ĀµMAP (r)ā ĀµDMAP (r)
ā£ā£2 dr= ā„ĀµMAP (r)ā ĀµD
MAP (r)ā„20.
It is clear that Bias2[ĀµDMAP ] is the square of L2(Ī©) norm of
the difference between ĀµMAP and ĀµDMAP .
Taking the expectation on both sides of (14) and (25), andapplying Fubiniās theorem [39], we can show that ĀµMAP andĀµDMAP satisfy the following variational forms:
FMAP (Ļ, ĀµMAP ) = (Ļ,Aā0Ī£
ā1Īµ Ī) + (Ļ,
Āµ0
Īŗ), (37)
FMAP (ĪØ, ĀµDMAP ) = (ĪØ, A0
āĪ£ā1Īµ Ī) + (ĪØ,
Āµ0
Īŗ), (38)
where Ī = E[Ī] = A0Āµ0, according to the noise modeland the a priori model for fluorophore concentration. Further,we let ĀÆĀµMAP ā L2(Ī©) be the solution of the followingapproximate inverse problem:
FMAP (Ļ, ĀÆĀµMAP ) = (Ļ, A0
āĪ£ā1Īµ Ī) + (Ļ,
Āµ0
Īŗ),
for all Ļ ā L2(Ī©). Then we have
Bias2[ĀµDMAP ]
ā¤[ā„ĀµMAP ā ĀÆĀµMAP ā„0 + ā„ ĀÆĀµMAP ā ĀµD
MAP ā„0]2, (39)
by the triangular inequality.For the first term ā„ĀµMAP ā ĀÆĀµMAP ā„0 in (39), we follow the
similar procedures given in [15] and obtain
ā„ĀµMAP ā ĀÆĀµMAP ā„0 ā¤ā„Īŗā„ā[ā„ā„ā„(B ā B)ĀµMAP
ā„ā„ā„0
+ā„ā„ā„(A0
āāAā
0)Ī£ā1Īµ Ī
ā„ā„ā„0
]. (40)
TIP-06221-2010 16
For termā„ā„ā„(B ā B)ĀµMAP
ā„ā„ā„0, we have [13]ā„ā„ā„(B ā B)ĀµMAP
ā„ā„ā„0
ā2
ā„ā„ā„ā„ā„ā„NS ,NDā
i,j
gājĻi
Ļ2Īµ,ij
ā«Ī©
(gāj ei + Ļieāj )ĀµMAP dr
ā„ā„ā„ā„ā„ā„0
ā¤2
NS ,NDāi,j
ā„gājĻiā„0Ļ2Īµ,ij
ā«Ī©
|(gāj ei + Ļieāj )ĀµMAP |dr,
where ei = Ļi ā Ī¦i and eāj = gāj ā Gāj . Decomposing the
integral on Ī© into a summation of the integrals on the finiteelements Ī©mj , m = 1, Ā· Ā· Ā· , Nāj
ā , and Ī©ni, n = 1, Ā· Ā· Ā· , N iā,
which are used to discretize the forward problem, we arriveat
ā„(B ā B)ĀµMAP ā„0
ā¤2
NSāi=1
Niā,NDān,j
ā„gājĻiā„0Ļ2Īµ,ij
ā„gāj ĀµMAP ā„0,niā„eiā„0,ni
+
NDāj=1
Nājā ,NSām,i
ā„gājĻiā„0Ļ2Īµ,ij
ā„ĻiĀµMAP ā„0,mjā„eājā„0,mj
.
(41)
Similarly, for termā„ā„ā„(A0
āāAā
0)Ī£ā1Īµ Ī
ā„ā„ā„0, we have its upper
bound as ā„ā„ā„(A0
āāAā
0)Ī£ā1Īµ Ī
ā„ā„ā„0
ā
ā„ā„ā„ā„ā„ā„NS ,NDā
i,j
gāj ei + Ļieāj
Ļ2Īµ,ij
Īi,j
ā„ā„ā„ā„ā„ā„0
ā¤
NSāi=1
Niā,NDān,j
|Īi,j |Ļ2Īµ,ij
ā„gāj ā„ā,niā„eiā„0,ni
+
NDāj=1
Nājā ,NSām,i
|Īi,j |Ļ2Īµ,ij
ā„Ļiā„ā,mjā„eājā„0,mj
. (42)
In the end, substituting (41) and (42) into (40) and using thediscretization error bounds given by [40]
ā„eiā„0,ni ā¤ Cā„Ļiā„1,nihni,ā„eājā„0,mj ā¤ Cā„gāj ā„1,mjhmj ,
lead to B1 and B2 in Theorem 1.For the second term ā„ ĀÆĀµMAP ā ĀµD
MAP ā„0 in (39), we alsofollow the procedures in [15] and obtain
ā„ ĀÆĀµMAP ā ĀµDMAP ā„0 ā¤ā„Īŗā„ā
[ā„ā„ā„B(ĀÆĀµMAP ā Ļ)ā„ā„ā„0
+
ā„ā„ā„ā„ ĀÆĀµMAP ā Ļ
Īŗ
ā„ā„ā„ā„0
]. (43)
Let Ļ ā V (Ī©) be the interpolant of ĀÆĀµMAP and eĀµ := ĀÆĀµMAPā
Ļ be the interpolation error, we haveā„ā„ā„B(ĀÆĀµMAP ā Ļ)ā„ā„ā„0
=
ā„ā„ā„ā„ā„ā„NS ,NDā
i,j
Gāj (Ā·)Ī¦i(Ā·)Ļ2Īµ,ij
ā«Ī©
Gāj (r
ā²)Ī¦i(rā²)eĀµ(r
ā²)drā²
ā„ā„ā„ā„ā„ā„0
ā¤NS ,NDā
i,j
ā„GājĪ¦iā„0Ļ2Īµ,ij
ā«Ī©
|Gāj (r
ā²)Ī¦i(rā²)eĀµ(r
ā²)|drā²
ā¤NS ,NDā
i,j
ā„GājĪ¦iā„0Ļ2Īµ,ij
Nāāt=1
ā„GājĪ¦iā„0,tā„eĀµā„0,t, (44)
andā„ā„ā„ā„ ĀÆĀµMAP ā Ļ
Īŗ
ā„ā„ā„ā„0
ā¤Nāāt=1
ā„ā„ā„eĀµĪŗ
ā„ā„ā„0,t
ā¤Nāāt=1
ā„ā„ā„ā„ 1Īŗā„ā„ā„ā„ā,t
ā„eĀµā„0,t . (45)
Assume that ĀÆĀµMAP ā H1(Ī©). Then an upper bound for thediscretization error can be given by
ā„eĀµā„0,t ā¤ Cā„ ĀÆĀµMAP ā„1,tht.
Approximating ĀÆĀµMAP by ĀµMAP , and substituting (44), (45)and the discretization error bound into (43), we obtain B3 inTheorem 1.
APPENDIX BPROOF OF THEOREM 2: UPPER BOUND FOR VAR[ĀµD
MAP ]
We express Var[ĀµDMAP ] in (29) using ĀµMAP , ĀµD
MAP , ĀµMAP
and ĀµMAP as
Var[ĀµDMAP ] :=
ā«Ī©
E[ ā£ā£ĀµMAP (r)ā ĀµD
MAP (r)
ā[ĀµMAP (r)ā ĀµDMAP (r)]
ā£ā£2] dr=
ā«Ī©
E[|[ĀµMAP (r)ā ĀµMAP (r)]
ā[ĀµDMAP (r)ā ĀµD
MAP (r)]ā£ā£2] dr.
Subtracting (37) and (38) from (14) and (25), respectively, weobtain
FMAP (Ļ, ĀµMAP (r)ā ĀµMAP ) = (Ļ,Aā0Ī£
ā1Īµ (Īā Ī)), (46)
FMAP (ĪØ, ĀµDMAP (r)ā ĀµD
MAP ) = (ĪØ, A0
āĪ£ā1Īµ (Īā Ī)). (47)
Let eij ā RNSND , i = 1, . . . , NS and j = 1, . . . , ND, bethe unit vector given by
eij = [0, . . . , 1, . . . , 0]T ,
with only non-zero entry at [(iā 1)ND + j]th position. Then,we define Ļij(r) ā L2(Ī©) and Ī ij(r) ā V (Ī©) as the solutionsof the following variational problems:
FMAP (Ļ, Ļij) = (Ļ,Aā0Ī£
ā1Īµ eij),
FMAP (ĪØ,Ī ij) = (ĪØ, A0
āĪ£ā1Īµ eij),
for all Ļ ā L2(Ī©) and ĪØ ā V (Ī©). Clearly, Ļij and Ī ij arethe solutions of (46) and (47) given the unit base vector eijin the measurement space RNSND . Due to the linearity of the
TIP-06221-2010 17
bilinear form, any solutions of (46) and (47) can be given asthe linear combinations of Ļijs and Ī ijs, respectively.
We further define Ļ(r) and Ī (r) as
Ļ(r) = [Ļ(r)11, . . . , Ļ(r)1NS, Ļ(r)21, . . . , Ļ(r)NSND
]T ,
Ī (r) = [Ī (r)11, . . . , Ī (r)1NS , Ī (r)21, . . . , Ī (r)NSND ]T .
Then ĀµMAP (r) ā ĀµMAP (r) and ĀµDMAP (r) ā ĀµD
MAP (r) canbe given as
ĀµMAP (r)ā ĀµMAP (r) = Ļ(r)T (Īā Ī), (48)ĀµDMAP (r)ā ĀµD
MAP (r) = Ī (r)T (Īā Ī). (49)
Substituting (48) and (49) into (46) and (47), we have
Var[ĀµDMAP ]
=
ā«Ī©
E[ā£ā£ā£[Ļ(r)āĪ (r)]
T(Īā Ī)
ā£ā£ā£2] dr=
ā«Ī©
[Ļ(r)āĪ (r)]T E
[(Īā Ī)(Īā Ī)T
]Ā· [Ļ(r)āĪ (r)] dr
=
ā«Ī©
[Ļ(r)āĪ (r)]TĪ£Ī [Ļ(r)āĪ (r)] dr, (50)
where Ī£Ī is the covariance matrix of Ī. From the propertyof Gaussian random field, it can be shown that A0Āµ in ourmeasurement model (10) is a multivariate Gaussian randomvariable statistically independent to the noise Īµ. More specif-ically, using the Fubiniās theorem, we can derive the mean ofA0Āµ and the covariance between each pair of its entries:
E [A0Āµ] = A0Āµ0,
Cov [(A0Āµ)ij , (A0Āµ)kl] =
ā«Ī©
Īŗ(r)aā,0ij (r)a0kl(r)dr.
Then, due to the independence between the noise and fluo-rophore concentration, Ī£Ī can be obtained as the sum of thecovariance matrices of A0Āµ and noise Īµ. We use
(Ī£Ī
)p,q
,for p = 1, . . . , NSND, and q = 1, . . . , NSND, to denote theentry at the pth row and the qth column of Ī£Ī, then we have(
Ī£Ī)p,q
= Ī“pqĻ2Īµ,ij +
ā«Ī©
Īŗ(r)aā,0ij (r)a0kl(r)dr,
where Ī“pq is the Kronecker delta function and the indices i,j, k, l, p and q have the following relationship:
p = (iā 1)ND + j, q = (k ā 1)ND + l,
for i, k = 1, . . . , NS , j, l = 1, . . . , ND, and p, q =1, . . . , NSND.
In this respect, we can express (50) as
Var[ĀµDMAP ]
=
ā«Ī©
NS ,NDāi,j
(Ī£Ī
)(iā1)ND+j,(iā1)ND+j
Ā· |Ļij(r)āĪ ij(r)|2 dr
+
ā«Ī©
NS ,NDāi,j
NS ,NDāk,l,kl =ij
(Ī£Ī
)(iā1)ND+j,(kā1)ND+l
Ā· [Ļij(r)āĪ ij(r)] [Ļkl(r)āĪ kl(r)] dr.
Applying Cauchy-Schwarz inequality, we obtain
Var[ĀµDMAP ]
ā¤NS ,NDā
i,j
(Ī£Ī
)(iā1)ND+j,(iā1)ND+j
ā„Ļij(r)āĪ ij(r)ā„20
+
NS ,NDāi,j
NS ,NDāk,l,kl =ij
(Ī£Ī
)(iā1)ND+j,(kā1)ND+l
Ā· ā„Ļij(r)āĪ ij(r)ā„0 ā„Ļkl(r)āĪ kl(r)ā„0 .
Since Ī£Ī is positive semi-definite, we have [41]ā£ā£ā£(Ī£Ī)p,q
ā£ā£ā£ ā¤ ā£ā£ā£(Ī£Ī)p,p
ā£ā£ā£1/2 ā£ā£ā£(Ī£Ī)q,q
ā£ā£ā£1/2 ,and
Var[ĀµDMAP ]
ā¤NS ,NDā
i,j
(Ī£Ī
)(iā1)ND+j,(iā1)ND+j
ā„Ļij(r)āĪ ij(r)ā„20
+
NS ,ND,NS ,NDāi,j,k,l,ij =kl
ā£ā£ā£(Ī£Ī)(iā1)ND+j,(iā1)ND+j
ā£ā£ā£1/2Ā·ā£ā£ā£(Ī£Ī
)(kā1)ND+l,(kā1)ND+l
ā£ā£ā£1/2Ā· ā„Ļij(r)āĪ ij(r)ā„0 ā„Ļkl(r)āĪ kl(r)ā„0
ā¤
NS ,NDāi,j
ā£ā£ā£(Ī£Ī)(iā1)ND+j,(iā1)ND+j
ā£ā£ā£1/2Ā·ā„Ļij(r)āĪ ij(r)ā„0]2 . (51)
Note that when the measurements are noise-free and thefluorophore concentration is also deterministic, we have Ī£Ī =E[(Īā Ī)(Īā Ī
]= 0, thereby Var[ĀµD
MAP ] = 0.For each ā„Ļij(r) ā Ī ij(r)ā„0 in (51), we let Ī = eij and
further assume Ļij ā H1(Ī©). Following a similar approachas in Appendix A, we obtain an upper bound for ā„Ļij(r) āĪ ij(r)ā„0 given by
ā„Ļij(r)āĪ ij(r)ā„0 ā¤ C
Ā·
NSāiā²=1
Niā²ā ,NDān,jā²
2ā„Īŗā„āā„gājā²Ļiā²ā„0Ļ2Īµ,iā²jā²
ā„gājā²Ļijā„0,niā²
Ā·ā„Ļiā²ā„1,niā²hniā² +ā„Īŗā„āĻ2Īµ,ij
Niāā
n=1
ā„gāj ā„ā,niā„Ļiā„1,nihni
+
NDājā²=1
Nājā²ā ,NSām,iā²
2ā„Īŗā„āā„gājā²Ļiā²ā„0Ļ2Īµ,iā²jā²
ā„Ļiā²Ļijā„0,mjā²
Ā·ā„gājā²ā„1,mjā²hmjā² +ā„Īŗā„āĻ2Īµ,ij
Nājāā
m=1
ā„Ļiā„ā,mj
+
Nāāt=1
NS ,NDāiā²,jā²
ā„Īŗā„āā„Gājā²Ī¦iā²ā„0ā„Gā
jā²Ī¦iā²ā„0,tĻ2Īµ,iā²jā²
+ā„Īŗā„āā„ā„ā„ā„ 1Īŗ
ā„ā„ā„ā„ā,t
)ā„Ļijā„1,tht
]. (52)
TIP-06221-2010 18
Finally, letting Dij =ā£ā£ā£(Ī£Ī
)(iā1)ND+j,(iā1)ND+j
ā£ā£ā£1/2 andsubstituting (52) into (51), we arrive at (31) in Theorem 2.
Lu Zhou received the M.S. degree in biomedi-cal engineering from Tsinghua University, China,and the Ph.D degree in electrical engineering fromRensselaer Polytechnic Institute, Troy, NY, in 2006and 2010, respectively. He currently is a softwareengineer at Bloomberg L.P., New York, NY.
Birsen Yazıcı received BS degrees in ElectricalEngineering and Mathematics in 1988 from BogaziciUniversity, Istanbul Turkey and MS and Ph.D. de-grees in Mathematics and Electrical Engineeringboth from Purdue University, W. Lafayette IN, in1990 and 1994, respectively. From September 1994until 2000, she was a research engineer at theGeneral Electric Company Global Research Center,Schenectady NY. During her tenure in industry,she worked on radar, transportation, industrial andmedical imaging systems. Her work on industrial
systems received the 2nd best paper award in 1997 given by IEEE Transactionsin Industrial Applications. From 2001 to June 2003, she was an assistant pro-fessor at Drexel University, Electrical and Computer Engineering Department.In Fall 2003, she joined Rensselaer Polytechnic Institute where she is currentlyan associate professor in the Department of Electrical, Computer and SystemsEngineering and in the Department of Biomedical Engineering.
Prof. Yazıcıās research interests span the areas of statistical signal process-ing, inverse problems in imaging, image reconstruction, biomedical optics,radar and X-ray imaging. She currently serves as an associate editor forthe IEEE Transactions on Image Processing and SIAM Journal on ImagingScience. She is the recipient of the Rensselaer Polytechnic Institute 2007School of Engineering Research Excellence Award. She holds 11 US patents.