# Fast and Robust Reconstruction for Fluorescence Molecular Tomography viaL1-2 Regularization

Authors: Haibo Zhang; Guohua Geng; Xiaodong Wang; Xuan Qu; Yuqing Hou; Xiaowei He
Journal: BioMed Research International
 (2016)

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## Abstract

Sparse reconstruction inspired by compressed sensing has attracted considerable attention in fluorescence molecular tomography (FMT). However, the columns of system matrix used for FMT reconstruction tend to be highly coherent, which meansL1 minimization may not produce the sparsest solution. In this paper, we propose a novel reconstruction method by minimization of the difference of L1 and L2 norms. To solve the nonconvex L1-2 minimization problem, an iterative method based on the difference of convex algorithm (DCA) is presented. In each DCA iteration, the update of solution involves an L1 minimization subproblem, which is solved by the alternating direction method of multipliers with an adaptive penalty. We investigated the performance of the proposed method with both simulated data andin vivo experimental data. The results demonstrate that the DCA for L1-2 minimization outperforms the representative algorithms for L1, L2, L1/2, and L0 when the system matrix is highly coherent.


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## Body

## 1. Introduction

Fluorescence molecular tomography (FMT) has become a promising molecular imaging modality since it has the ability to provide localization and quantitative analysis of the fluorescent probe for preclinical research [1, 2]. However, FMT reconstruction suffers from high ill-posedness due to the insufficiency of external measurements, which is caused by high absorption and scattering in photon propagation through biological tissues [3].To alleviate the ill-posedness of FMT, somea priori information, such as anatomical information, optical properties, permissible region, and sparsity of target distribution, has been successfully incorporated in FMT reconstruction [4–7]. In addition, many regularization techniques have also been devoted to get an accurate and stable solution. Conventionally, L2 norm regularizer is a common penalty term in spite of its over-smoothness and results with lower resolution [8]. Another common regularizer is L0 norm, which is nondeterministic polynomial (NP) hard and can be solved by a greedy approach such as the orthogonal matching pursuit (OMP) [9]. Inspired by compressive sensing (CS) theory, the L1 norm regularizer as the convex relaxation of L0 has become a widely used sparsity-inducing norm for FMT reconstruction [10–13]. However, L1 norm regularizer is not always providing the sparsest solution for the inverse problem of FMT [14]. This gives way to nonconvex Lp (0<p<1) norm regularizer, which has been applied to optical tomography and was found to have better results than L1 does [15]. Some comparative studies show that nonconvex Lp (0<p<1) norm regularizer with p near 1/2 performs the best result among regularizers of Lp (0<p<1) norm [16].Recently, a new nonconvex regularizer namedL1-2 has been proposed and produced better solution than Lp (p=1/2) norm regularizer when the sensing matrix was large and highly coherent [17, 18]. The Magnetic Resonance Imaging (MRI) image recovery tests have also indicated that L1-2 norm regularizer outperforms L1/2 and L1 for highly coherent matrix [17]. Meanwhile, the columns of system matrix used for FMT reconstruction are also highly coherent with the finite element computing framework [19].In this paper, newL1-2 norm regularization was proposed to improve the FMT imaging. In our method, a difference of convex algorithm (DCA) was presented to solve the nonconvex L1-2 minimization problem. And the alternating direction method of multipliers (ADMM) with an adaptive penalty was used to solve the subproblem with fast convergence for each DCA iteration. The performance of the proposed method was validated with simulated data andin vivo experimental data.The outline of this paper is as follows. Section2 elaborates the forward model and L1-2 norm regularization algorithm. Section 3 demonstrates the feasibility and effectiveness of the method with both simulated data andin vivo experimental data. Finally, we conclude the paper and discuss relevant issues in Section 4.


## 2. Methods

### 2.1. Light Propagation Model

As an approximation to Radiative Transfer Equation (RTE), the Diffusion Approximation associated with Robin boundary conditions has been widely used for modeling the light transportation in biological tissues [20, 21]. For steady-state FMT with point excitation sources, the coupled diffusion equations can be presented as follows:(1)∇·Dexr∇Φexr-μa,exrΦex=-Θδr-rs∇·Demr∇Φemr-μa,emrΦem=-Φexrημafr,where subscript ex and em denote excitation light and emission light, respectively. r∈Ω is the domain under consideration. Dex=1/3(μa,ex+μs,ex′) and Dem=1/3(μa,em+μs,em′) are diffusion coefficients with μa,ex, μa,em as absorption coefficients for excitation and emission wavelengths, g is the anisotropy parameter, and μs,ex′, μs,em′ are the reduced scattering coefficients. Φex and Φem denote the photon density. ημaf is the unknown fluorescent yield to be reconstructed. Using the finite elements method (FEM), (1) can be linearly discretized as follows:(2)KexΦex=bexKemΦem=FX,where Kex and Kem denote the system matrix at excitation and emission wavelengths, respectively. The symmetric matrix F is obtained by discretizing the unknown fluorescent yield distribution. The final linear relationship between the unknown fluorescence yield x and the measured surface data ϕ can be obtained as follows:(3)Ax=ϕ,where A is M×N linear system matrix which is large-sized and ill-posed.


### 2.2. Inverse Reconstruction of FMT by DCA-L1-2 Algorithm

The CS theory provides sufficient conditions for the exact recovery of the sparse signals from limited number of measurements. One commonly used concept is the mutual coherence [17, 22] which is defined as(4)μA=maxi≠j⁡apT·aqap2·aq2,where ap and aq are different columns of A. The mutual coherence of system matrix A derived by FEM method is always as high as above 90% [19]. In the highly coherent regime of CS, Lp (p=1/2) and L1-2 norm regularizers are expected to yield the sparest solution that L1 regularization always fails to [17, 18].Recently, a DCA-L1-2 algorithm was proposed and theoretical properties of L1-2 minimization have been proved in papers [17, 23]. Considering the advantages of L1-2 minimization, we converted linear matrix equation (3) into the following unconstrained optimization problem:(5)minx∈Rn⁡12Ax-ϕ22+λx1-x2,where λ>0 is a regularization parameter which is usually empirically selected and x1-x2 denotes the L1-2 regularization operator.To resolve minimization problem (5), the difference of convex algorithm (DCA) [24] which is a descent method without line search was used. Equation (5) can be decomposed into DC decomposition as F(x)=G(x)-H(x), where(6)Gx=12Ax-ϕ22+λx1,Hx=λx2.In (6), x2 is differentiable with gradient x/x2. An iterative scheme was used to solve F(x) as follows:(7)xn+1=argminx∈Rn⁡12Ax-ϕ22+λx1-x-xn,xnxn2.In each DCA iteration, there is aL1-regularized convex subproblem that needs to be solved:(8)minx∈Rn⁡12xTATAx+xTϕ+λxnxn2Tx+λx1.We use the augmented Lagrangian method and transform (8) into the following:(9)Lδx,y,u=12xTATAx+ATϕ+λxnxn2Tx+λy1+uTx-y+δ2x-y22.The subproblem is solved by minimizingLδ with respect to x, minimizing Lδ with respect to y, and updating u successively. In order to solve (9) with a fast speed of convergence, an ADMM strategy with an adaptive penalty [25] was utilized as follows:(10)xk+1=argminx⁡Lδkx,yl,ul,yk+1=argminz⁡Lδkxl+1,y,ul,uk+1=ul+δkxl+1-yl+1,δk+1=min⁡δmax,ρδk.In the above iterations, the update ofy is based on the soft-thresholding operator, where(11)Sx,ri=sgnximax⁡xi-r,0.Meanwhile, the penaltyδ was updated as an adaptive form as follows:(12)δk+1=min⁡δmax,ρδk,where δmax is an upper bound of {δk} and ρ is defined as follows:(13)ρ=ρ0,ifδyk+1-ykxk+1<ε1,otherwise,where ρ0≥1 is a constant.Algorithm1 presents the iterative process of DCA-L1-2 algorithm for FMT reconstruction. To begin with the iteration, the initial value x1 was set as L1 subproblem.Algorithm 1:DCA-L1-2 algorithm for FMT.
Initializion: set x0=0, x1≠0, εouter>0, εinner>0, and n=1 While x n - x n - 1 > ε o u t e r setz=xnxn2, x0=0, x1=xn, i=1, yi=xi, ui=0 Whilexk-xk-1>εinner xk+1=ATA+δkI-1(δkyk-z-uk) yk+1=shrinkxk+1+ukδk,λδk uk+1=uk+δk(xk+1-yk+1) δk+1=min⁡(δmax,ρδk) k=k+1 end n=n+1 xn=xk end


## 2.1. Light Propagation Model

As an approximation to Radiative Transfer Equation (RTE), the Diffusion Approximation associated with Robin boundary conditions has been widely used for modeling the light transportation in biological tissues [20, 21]. For steady-state FMT with point excitation sources, the coupled diffusion equations can be presented as follows:(1)∇·Dexr∇Φexr-μa,exrΦex=-Θδr-rs∇·Demr∇Φemr-μa,emrΦem=-Φexrημafr,where subscript ex and em denote excitation light and emission light, respectively. r∈Ω is the domain under consideration. Dex=1/3(μa,ex+μs,ex′) and Dem=1/3(μa,em+μs,em′) are diffusion coefficients with μa,ex, μa,em as absorption coefficients for excitation and emission wavelengths, g is the anisotropy parameter, and μs,ex′, μs,em′ are the reduced scattering coefficients. Φex and Φem denote the photon density. ημaf is the unknown fluorescent yield to be reconstructed. Using the finite elements method (FEM), (1) can be linearly discretized as follows:(2)KexΦex=bexKemΦem=FX,where Kex and Kem denote the system matrix at excitation and emission wavelengths, respectively. The symmetric matrix F is obtained by discretizing the unknown fluorescent yield distribution. The final linear relationship between the unknown fluorescence yield x and the measured surface data ϕ can be obtained as follows:(3)Ax=ϕ,where A is M×N linear system matrix which is large-sized and ill-posed.


## 2.2. Inverse Reconstruction of FMT by DCA-L1-2 Algorithm

The CS theory provides sufficient conditions for the exact recovery of the sparse signals from limited number of measurements. One commonly used concept is the mutual coherence [17, 22] which is defined as(4)μA=maxi≠j⁡apT·aqap2·aq2,where ap and aq are different columns of A. The mutual coherence of system matrix A derived by FEM method is always as high as above 90% [19]. In the highly coherent regime of CS, Lp (p=1/2) and L1-2 norm regularizers are expected to yield the sparest solution that L1 regularization always fails to [17, 18].Recently, a DCA-L1-2 algorithm was proposed and theoretical properties of L1-2 minimization have been proved in papers [17, 23]. Considering the advantages of L1-2 minimization, we converted linear matrix equation (3) into the following unconstrained optimization problem:(5)minx∈Rn⁡12Ax-ϕ22+λx1-x2,where λ>0 is a regularization parameter which is usually empirically selected and x1-x2 denotes the L1-2 regularization operator.To resolve minimization problem (5), the difference of convex algorithm (DCA) [24] which is a descent method without line search was used. Equation (5) can be decomposed into DC decomposition as F(x)=G(x)-H(x), where(6)Gx=12Ax-ϕ22+λx1,Hx=λx2.In (6), x2 is differentiable with gradient x/x2. An iterative scheme was used to solve F(x) as follows:(7)xn+1=argminx∈Rn⁡12Ax-ϕ22+λx1-x-xn,xnxn2.In each DCA iteration, there is aL1-regularized convex subproblem that needs to be solved:(8)minx∈Rn⁡12xTATAx+xTϕ+λxnxn2Tx+λx1.We use the augmented Lagrangian method and transform (8) into the following:(9)Lδx,y,u=12xTATAx+ATϕ+λxnxn2Tx+λy1+uTx-y+δ2x-y22.The subproblem is solved by minimizingLδ with respect to x, minimizing Lδ with respect to y, and updating u successively. In order to solve (9) with a fast speed of convergence, an ADMM strategy with an adaptive penalty [25] was utilized as follows:(10)xk+1=argminx⁡Lδkx,yl,ul,yk+1=argminz⁡Lδkxl+1,y,ul,uk+1=ul+δkxl+1-yl+1,δk+1=min⁡δmax,ρδk.In the above iterations, the update ofy is based on the soft-thresholding operator, where(11)Sx,ri=sgnximax⁡xi-r,0.Meanwhile, the penaltyδ was updated as an adaptive form as follows:(12)δk+1=min⁡δmax,ρδk,where δmax is an upper bound of {δk} and ρ is defined as follows:(13)ρ=ρ0,ifδyk+1-ykxk+1<ε1,otherwise,where ρ0≥1 is a constant.Algorithm1 presents the iterative process of DCA-L1-2 algorithm for FMT reconstruction. To begin with the iteration, the initial value x1 was set as L1 subproblem.Algorithm 1:DCA-L1-2 algorithm for FMT.
Initializion: set x0=0, x1≠0, εouter>0, εinner>0, and n=1 While x n - x n - 1 > ε o u t e r setz=xnxn2, x0=0, x1=xn, i=1, yi=xi, ui=0 Whilexk-xk-1>εinner xk+1=ATA+δkI-1(δkyk-z-uk) yk+1=shrinkxk+1+ukδk,λδk uk+1=uk+δk(xk+1-yk+1) δk+1=min⁡(δmax,ρδk) k=k+1 end n=n+1 xn=xk end


## 3. Experiments and Results

In this section, the simulations both on 3D digital mouse model andin vivo experiments were used to demonstrate the potential and feasibility of the DCA-L1-2 algorithm for FMT reconstruction. To investigate the performance of DCA-L1-2 algorithm, four representative regularizers, including L1/2, L1, L2, and L0, were used for a systematical comparison. More specifically, iterative reweighted least squares algorithm (IRLS-L1/2) [26], incomplete variables truncated conjugate gradient algorithm (IVTCG-L1) [27], Tikhonov regularization algorithm (Tikhonov-L2) [28], and OMP algorithm [29] were compared with the DCA-L1-2 algorithm, respectively, as the corresponding method for the above regularizers.The qualities of reconstruction results are quantitatively evaluated in terms of the absolute location error (LE) [3], reconstructed fluorescent yield (Recon. FY) [3], normalized root mean square error (NRMSE) [16], the percentage of nonzero coefficient (PNZ) [16], and time cost. The experiment codes were written in MATLAB and were performed on a desktop computer with 3.40 GHz Intel® Xeon® Processor E3-1231 and 12 G RAM.


### 3.1. Numerical Simulation Experiments

A 33 mm height torso extracted from a 3D mouse atlas was utilized to simulate the heterogeneity of biological tissues [30]. Figure 1(a) shows the mouse model with six organs. Table 1 lists the specific optical parameters. A cylinder with a radius of 0.8 mm and a height of 1.6 mm was positioned at 17.8, 6.6, and 16.4 mm to mimic the fluorescent target. The actual fluorescent yield was set to be 0.05 mm−1. For excitation, we used 18 excitation sources being located on the plane of Z=16.4 mm as shown in Figure 2(a). The surface data on the opposite side with a 120° field of view (FOV) were measured for each excitation source. A total of 18 datasets were assembled for the subsequent reconstruction process.Table 1
Optical parameters for the heterogeneous model.
Organs μ a , e x (mm−1) μ s , e x ′ (mm−1) μ a , e m (mm−1) μ s , e m ′ (mm−1) Muscle 0.0052 1.08 0.0068 1.03 Heart 0.0083 1.01 0.0104 0.99 Lungs 0.0133 1.97 0.0203 1.95 Liver 0.0329 0.70 0.0176 0.65 Kidneys 0.0660 2.25 0.0380 2.02 Stomach 0.0114 1.74 0.0070 1.36Figure 1
The mouse model and the 3D views of the reconstructed results. (a) The mouse model with single target. (b–f) The 3D views of the reconstructed results where the red cylinder is the real fluorescent target.
(a)
3D mouse with single target (b)
L 1 - 2 (c)
L 1 / 2 (d)
L 1 (e)
L 2 (f)
L 0Figure 2
The plane of 18 excitation sources and the 2D views (Z=16.4 mm) of the reconstructed results. (a) The plane of 18 excitation sources with 120° FOV. (b–f) The 2D views (Z=16.4 mm) of the reconstructed results by six comparative methods. The black circle denotes real position of fluorescent target.
(a)
The plane of excitation source ofz=16.4 (b)
L 1 - 2 (c)
L 1 / 2 (d)
L 1 (e)
L 2 (f)
L 0The forward FEM mesh was discretized into 24231 nodes and 128300 tetrahedral elements. Meanwhile, the FEM mesh for inverse reconstruction was discretized into 2601 nodes and 12752 tetrahedral elements. The mutual coherence of the system matrix for inverse reconstruction was 99.96%.Figures1(b)–1(f) present a comparison of reconstruction results with 3D views for single fluorescent target. The corresponding 2D section-views on the excitation plane are demonstrated in Figures 2(b)–2(f). Table 2 gives the quantitative results of the five regularization methods.Table 2
Quantitative results in single target reconstruction experiment.
Methods LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) DCA-L1-2 0.436 0.039 19 1.15 1.32 IRLS-L1/2 0.668 0.032 29 1.86 1.67 IVTCG-L1 1.279 0.005 67 2.47 31.74 Tikhonov-L2 2.169 0.004 79 11.96 21.96 OMP 1.135 0.004 63 9.83 1.21As shown in Figures1(b)–1(f), Figures 2(b)–2(f), and Table 2, reconstruction results of L1-2 and L1/2 were remarkable. Compared to the other three methods, results of DCA-L1-2 and IRLS-L1/2 have fewer artifacts, lower LE, lower NRMSE, and lower PNZ. Meanwhile, the Recon. FY by DCA-L1-2 and IRLS-L1/2 were closer to 0.05 mm−1. The proposed DCA-L1-2 completely outperforms the other methods, except for a slightly larger time consumption compared to OMP.Generally speaking, the quality of FMT reconstruction is influenced by the number of excitation sources. To test the stability of the algorithm, different numbers of excitation sources were used for reconstruction. Table3 gives the corresponding reconstructed results with 18, 12, 8, and 4. Obviously, the decreased number of excitation sources leads to significant reduction of measurements. Nevertheless, the results of DCA-L1-2 are generally satisfactory even in the case of 4 excitation sources.Table 3
The results of DCA-L1-2 with different excitation sources.
The number of excitation sources LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) 18 0.436 0.039 19 1.15 1.67 12 0.497 0.034 21 1.28 1.32 8 0.518 0.029 23 2.34 1.23 4 0.614 0.014 36 3.73 0.89The quality of reconstructed results is sensitive to measurement noise because of the severe ill-condition of system matrix. For stability test, four different levels of Gaussian noise (5%, 15%, 25%, and 35%) were added to the synthetic measurements. Table4 shows the reconstruction results under 4 different noise levels. It shows that the DCA-L1-2 algorithm is quite resilient with Gaussian noise.Table 4
Impact of Gaussian noise on DCA-L1-2.
Noise level (%) LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) 5 0.436 0.039 19 1.15 1.67 15 0.437 0.039 19 1.15 1.67 25 0.437 0.038 19 1.15 1.72 35 0.514 0.038 20 1.33 1.78


### 3.2.In Vivo Evaluation with Implanted Fluorophore

The performance of DCA-L1-2 and IRLS-L1/2 was remarkably compared to the other three methods in the simulation experiments. In this section, we further evaluated the performance of the proposed algorithm within vivo experimental data [8].In this experiment, an adult BALB/C mouse with a glass tube implanted into its abdomen was used. The experimental data was acquired by a hybrid FMT/Micro-CT system [8]. The glass tube (0.6 mm and the height of 2.8 mm) was filled with Cy5.5 solution (with the extinction coefficient of about 0.019 mm−1 μM-1 and quantum efficiency of 0.23 at the peak excitation wavelength of 671 nm) [31]. The center of the target was determined at 21.1, 27.8, and 7.4 mm by the Micro-CT. The fluorescent yield of Cy5.5 was 0.0402 mm−1 [32]. For reconstruction, the CT data was segmented into five major anatomical components, including muscle, heart, lungs, liver, and kidneys. Table 5 shows the optical properties of different organs [33].Table 5
Optical parameters of the mouse model at 670 nm and 710 nm.
Organs 670 nm 710 nm μ a , e x (mm−1) μ s , e x ′ (mm−1) μ a , e m (mm−1) μ s , e m ′ (mm−1) Muscle 0.075 0.412 0.043 0.350 Heart 0.051 0.944 0.030 0.870 Lungs 0.170 2.157 0.097 2.093 Liver 0.304 0.668 0.176 0.629 Kidneys 0.058 2.204 0.034 2.021For inverse reconstruction, the segmented mouse torso data was discretized into a mesh with 3049 nodes and 14932 tetrahedral elements. Mutual coherence of the system matrix for inverse reconstruction was 99.87%. Comparison results between DCA-L1-2 and IRLS-L1/2 are shown in Table 6 and Figure 3. The 3D views of reconstructed results forin vivo experiments via DCA-L1-2 are shown in Figure 4.Table 6
Quantitative results ofin vivo experiments.
Method LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) DCA-L1-2 1.426 0.034 27 0.13 0.53 IRLS-L1/2 1.705 0.019 45 0.26 1.90Figure 3
Reconstructed resultsin vivo experiment on adult BALB/C mouse. (a) The 3D view of the reconstructed results in which the red glass tube is the implanted fluorescent target and the green target denotes reconstructed results. (b) The 2D views (Z=7.4 mm) of the reconstructed results. The black circle denotes the real positions of fluorescent target.
(a) (b)Figure 4
The reconstructed results ofin vivo experiments via DCA-L1-2. (a) 3D rendering of the mouse. (b) The reconstructed result in 3D view. (c) The photon density distribution of the reconstructed result in 3D view. (d–f) The 2D views of the overlapped result with corresponding CT slices.


## 3.1. Numerical Simulation Experiments

A 33 mm height torso extracted from a 3D mouse atlas was utilized to simulate the heterogeneity of biological tissues [30]. Figure 1(a) shows the mouse model with six organs. Table 1 lists the specific optical parameters. A cylinder with a radius of 0.8 mm and a height of 1.6 mm was positioned at 17.8, 6.6, and 16.4 mm to mimic the fluorescent target. The actual fluorescent yield was set to be 0.05 mm−1. For excitation, we used 18 excitation sources being located on the plane of Z=16.4 mm as shown in Figure 2(a). The surface data on the opposite side with a 120° field of view (FOV) were measured for each excitation source. A total of 18 datasets were assembled for the subsequent reconstruction process.Table 1
Optical parameters for the heterogeneous model.
Organs μ a , e x (mm−1) μ s , e x ′ (mm−1) μ a , e m (mm−1) μ s , e m ′ (mm−1) Muscle 0.0052 1.08 0.0068 1.03 Heart 0.0083 1.01 0.0104 0.99 Lungs 0.0133 1.97 0.0203 1.95 Liver 0.0329 0.70 0.0176 0.65 Kidneys 0.0660 2.25 0.0380 2.02 Stomach 0.0114 1.74 0.0070 1.36Figure 1
The mouse model and the 3D views of the reconstructed results. (a) The mouse model with single target. (b–f) The 3D views of the reconstructed results where the red cylinder is the real fluorescent target.
(a)
3D mouse with single target (b)
L 1 - 2 (c)
L 1 / 2 (d)
L 1 (e)
L 2 (f)
L 0Figure 2
The plane of 18 excitation sources and the 2D views (Z=16.4 mm) of the reconstructed results. (a) The plane of 18 excitation sources with 120° FOV. (b–f) The 2D views (Z=16.4 mm) of the reconstructed results by six comparative methods. The black circle denotes real position of fluorescent target.
(a)
The plane of excitation source ofz=16.4 (b)
L 1 - 2 (c)
L 1 / 2 (d)
L 1 (e)
L 2 (f)
L 0The forward FEM mesh was discretized into 24231 nodes and 128300 tetrahedral elements. Meanwhile, the FEM mesh for inverse reconstruction was discretized into 2601 nodes and 12752 tetrahedral elements. The mutual coherence of the system matrix for inverse reconstruction was 99.96%.Figures1(b)–1(f) present a comparison of reconstruction results with 3D views for single fluorescent target. The corresponding 2D section-views on the excitation plane are demonstrated in Figures 2(b)–2(f). Table 2 gives the quantitative results of the five regularization methods.Table 2
Quantitative results in single target reconstruction experiment.
Methods LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) DCA-L1-2 0.436 0.039 19 1.15 1.32 IRLS-L1/2 0.668 0.032 29 1.86 1.67 IVTCG-L1 1.279 0.005 67 2.47 31.74 Tikhonov-L2 2.169 0.004 79 11.96 21.96 OMP 1.135 0.004 63 9.83 1.21As shown in Figures1(b)–1(f), Figures 2(b)–2(f), and Table 2, reconstruction results of L1-2 and L1/2 were remarkable. Compared to the other three methods, results of DCA-L1-2 and IRLS-L1/2 have fewer artifacts, lower LE, lower NRMSE, and lower PNZ. Meanwhile, the Recon. FY by DCA-L1-2 and IRLS-L1/2 were closer to 0.05 mm−1. The proposed DCA-L1-2 completely outperforms the other methods, except for a slightly larger time consumption compared to OMP.Generally speaking, the quality of FMT reconstruction is influenced by the number of excitation sources. To test the stability of the algorithm, different numbers of excitation sources were used for reconstruction. Table3 gives the corresponding reconstructed results with 18, 12, 8, and 4. Obviously, the decreased number of excitation sources leads to significant reduction of measurements. Nevertheless, the results of DCA-L1-2 are generally satisfactory even in the case of 4 excitation sources.Table 3
The results of DCA-L1-2 with different excitation sources.
The number of excitation sources LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) 18 0.436 0.039 19 1.15 1.67 12 0.497 0.034 21 1.28 1.32 8 0.518 0.029 23 2.34 1.23 4 0.614 0.014 36 3.73 0.89The quality of reconstructed results is sensitive to measurement noise because of the severe ill-condition of system matrix. For stability test, four different levels of Gaussian noise (5%, 15%, 25%, and 35%) were added to the synthetic measurements. Table4 shows the reconstruction results under 4 different noise levels. It shows that the DCA-L1-2 algorithm is quite resilient with Gaussian noise.Table 4
Impact of Gaussian noise on DCA-L1-2.
Noise level (%) LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) 5 0.436 0.039 19 1.15 1.67 15 0.437 0.039 19 1.15 1.67 25 0.437 0.038 19 1.15 1.72 35 0.514 0.038 20 1.33 1.78


## 3.2.In Vivo Evaluation with Implanted Fluorophore

The performance of DCA-L1-2 and IRLS-L1/2 was remarkably compared to the other three methods in the simulation experiments. In this section, we further evaluated the performance of the proposed algorithm within vivo experimental data [8].In this experiment, an adult BALB/C mouse with a glass tube implanted into its abdomen was used. The experimental data was acquired by a hybrid FMT/Micro-CT system [8]. The glass tube (0.6 mm and the height of 2.8 mm) was filled with Cy5.5 solution (with the extinction coefficient of about 0.019 mm−1 μM-1 and quantum efficiency of 0.23 at the peak excitation wavelength of 671 nm) [31]. The center of the target was determined at 21.1, 27.8, and 7.4 mm by the Micro-CT. The fluorescent yield of Cy5.5 was 0.0402 mm−1 [32]. For reconstruction, the CT data was segmented into five major anatomical components, including muscle, heart, lungs, liver, and kidneys. Table 5 shows the optical properties of different organs [33].Table 5
Optical parameters of the mouse model at 670 nm and 710 nm.
Organs 670 nm 710 nm μ a , e x (mm−1) μ s , e x ′ (mm−1) μ a , e m (mm−1) μ s , e m ′ (mm−1) Muscle 0.075 0.412 0.043 0.350 Heart 0.051 0.944 0.030 0.870 Lungs 0.170 2.157 0.097 2.093 Liver 0.304 0.668 0.176 0.629 Kidneys 0.058 2.204 0.034 2.021For inverse reconstruction, the segmented mouse torso data was discretized into a mesh with 3049 nodes and 14932 tetrahedral elements. Mutual coherence of the system matrix for inverse reconstruction was 99.87%. Comparison results between DCA-L1-2 and IRLS-L1/2 are shown in Table 6 and Figure 3. The 3D views of reconstructed results forin vivo experiments via DCA-L1-2 are shown in Figure 4.Table 6
Quantitative results ofin vivo experiments.
Method LE (mm) Recon. FY (mm−1) NRMSE (%) PNZ (%) Time (s) DCA-L1-2 1.426 0.034 27 0.13 0.53 IRLS-L1/2 1.705 0.019 45 0.26 1.90Figure 3
Reconstructed resultsin vivo experiment on adult BALB/C mouse. (a) The 3D view of the reconstructed results in which the red glass tube is the implanted fluorescent target and the green target denotes reconstructed results. (b) The 2D views (Z=7.4 mm) of the reconstructed results. The black circle denotes the real positions of fluorescent target.
(a) (b)Figure 4
The reconstructed results ofin vivo experiments via DCA-L1-2. (a) 3D rendering of the mouse. (b) The reconstructed result in 3D view. (c) The photon density distribution of the reconstructed result in 3D view. (d–f) The 2D views of the overlapped result with corresponding CT slices.


## 4. Discussion and Conclusion

In this paper, novelL1-2 norm regularization was proposed to solve the inverse problem of FMT with highly coherent system matrix. To accurately recover the small fluorescent target, an iterative method based on DCA algorithm was presented to solve the nonconvex L1-2 minimization problem. And the ADMM method with an adaptive penalty was used to get fast convergence for the subproblem.Simulated data on a 3D heterogeneous mouse model andin vivo experimental data acquired by a hybrid FMT/Micro-CT system were used to demonstrate the feasibility of the DCA-L1-2 algorithm for FMT. The comparative results of single target show that the DCA-L1-2 algorithm has better performance compared to other typical algorithms based on L1/2, L1, L2, and L0 norm regularizer. The robustness tests further illustrate that the DCA-L1-2 algorithm is stable and robust to measurement noise. In addition, decreasing the number of excitation sources from 18 to 4, DCA-L1-2 still yields satisfactory results.However, the reconstructed fluorescent yield of the proposed method was still smaller than the true value. So new strategies that may further improve fluorescent yield will be our future research focuses. Moreover, we will also focus on investigating the multitargets resolution and new application ofL1-2 norm regularizer in other imaging modalities in the near future.In conclusion, both numerical experiments andin vivo experiments validated the good performance of L1-2 regularizer for FMT. Moreover, comparative experiments indicate that L1-2 outperforms the iterative reweighted strategies for lp with p=1/2 when system matrix is highly coherent.



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*Source: 5065217-2016-12-06.xml*