BioMed Research International

Volume 2016, Article ID 5065217, 9 pages

http://dx.doi.org/10.1155/2016/5065217

## Fast and Robust Reconstruction for Fluorescence Molecular Tomography via Regularization

School of Information Sciences and Technology, Northwest University, Xi’an, Shaanxi 710027, China

Received 19 June 2016; Revised 10 September 2016; Accepted 25 September 2016

Academic Editor: Jun Zhang

Copyright © 2016 Haibo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 means minimization may not produce the sparsest solution. In this paper, we propose a novel reconstruction method by minimization of the difference of and norms. To solve the nonconvex 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 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 and* in vivo* experimental data. The results demonstrate that the DCA for minimization outperforms the representative algorithms for , , , and when the system matrix is highly coherent.

#### 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, some* a 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, norm regularizer is a common penalty term in spite of its over-smoothness and results with lower resolution [8]. Another common regularizer is 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 norm regularizer as the convex relaxation of has become a widely used sparsity-inducing norm for FMT reconstruction [10–13]. However, norm regularizer is not always providing the sparsest solution for the inverse problem of FMT [14]. This gives way to nonconvex () norm regularizer, which has been applied to optical tomography and was found to have better results than does [15]. Some comparative studies show that nonconvex () norm regularizer with near 1/2 performs the best result among regularizers of () norm [16].

Recently, a new nonconvex regularizer named has been proposed and produced better solution than () 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 norm regularizer outperforms and 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, new norm regularization was proposed to improve the FMT imaging. In our method, a difference of convex algorithm (DCA) was presented to solve the nonconvex 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 and* in vivo* experimental data.

The outline of this paper is as follows. Section 2 elaborates the forward model and norm regularization algorithm. Section 3 demonstrates the feasibility and effectiveness of the method with both simulated data and* in 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:where subscript and denote excitation light and emission light, respectively. is the domain under consideration. and are diffusion coefficients with , as absorption coefficients for excitation and emission wavelengths, is the anisotropy parameter, and , are the reduced scattering coefficients. and denote the photon density. is the unknown fluorescent yield to be reconstructed. Using the finite elements method (FEM), (1) can be linearly discretized as follows:where and denote the system matrix at excitation and emission wavelengths, respectively. The symmetric matrix is obtained by discretizing the unknown fluorescent yield distribution. The final linear relationship between the unknown fluorescence yield and the measured surface data can be obtained as follows:where is linear system matrix which is large-sized and ill-posed.

##### 2.2. Inverse Reconstruction of FMT by DCA- 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 aswhere and are different columns of . The mutual coherence of system matrix derived by FEM method is always as high as above 90% [19]. In the highly coherent regime of CS, () and norm regularizers are expected to yield the sparest solution that regularization always fails to [17, 18].

Recently, a DCA- algorithm was proposed and theoretical properties of minimization have been proved in papers [17, 23]. Considering the advantages of minimization, we converted linear matrix equation (3) into the following unconstrained optimization problem:where is a regularization parameter which is usually empirically selected and denotes the 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 , where

In (6), is differentiable with gradient . An iterative scheme was used to solve as follows:

In each DCA iteration, there is a -regularized convex subproblem that needs to be solved:

We use the augmented Lagrangian method and transform (8) into the following:

The subproblem is solved by minimizing with respect to , minimizing with respect to , and updating successively. In order to solve (9) with a fast speed of convergence, an ADMM strategy with an adaptive penalty [25] was utilized as follows:

In the above iterations, the update of is based on the soft-thresholding operator, where

Meanwhile, the penalty was updated as an adaptive form as follows:where is an upper bound of and is defined as follows:where is a constant.

Algorithm 1 presents the iterative process of DCA- algorithm for FMT reconstruction. To begin with the iteration, the initial value was set as subproblem.