Mathematical Problems in Engineering

Volume 2015, Article ID 412315, 9 pages

http://dx.doi.org/10.1155/2015/412315

## Image Reconstruction of Two-Dimensional Highly Scattering Inhomogeneous Medium Using MAP-Based Estimation

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Received 5 July 2014; Revised 11 November 2014; Accepted 15 November 2014

Academic Editor: Subhash Chandra Mishra

Copyright © 2015 Hong Qi 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

A maximum a posteriori (MAP) estimation based on Bayesian framework is applied to image reconstruction of two-dimensional highly scattering inhomogeneous medium. The finite difference method (FDM) and conjugate gradient (CG) algorithm serve as the forward and inverse solving models, respectively. The generalized Gaussian Markov random field model (GGMRF) is treated as the regularization, and finally the influence of the measurement errors and initial distributions is investigated. Through the test cases, the MAP estimate algorithm is demonstrated to greatly improve the reconstruction results of the optical coefficients.

#### 1. Introduction

The optical information reconstruction in nonhomogeneous medium exposed to collimated short-pulse irradiation has been a research focus in transient radiative transfer recently. It is widely used in many fields, such as nondestructive testing, optical tomography, infrared remote sensing, information processing, combustion diagnosis, biology, and medicine [1–7]. Among these fields, the diffuse optical tomography (DOT) is an emerging technology that can reconstruct the optical properties of internal biological tissues from the measured transient transmittance and reflectance signals at the boundary of tissues. The DOT is characterized as being noninvasive, portable, and producing real-time images of clinically relevant parameters. Commonly, the near-infrared laser pulse is used as the light source in DOT, which does no harm to the detected biological issues. The photon propagation in biological medium which is highly scattering always obeys diffusion equation. It offers the significant advantages of both flexibility to geometry and heterogeneity of tissue and is easy to be solved using numerical methods [8].

Using the information carried by scattering photon, combined with the photon transmission model and the optimization algorithm, the information reconstruction can be converted to an inverse problem. The difficulties are how to choose radiative transfer models, detect transmission and reflected signals, and apply inverse optimization algorithms. Many research groups have done a lot of work in this field. Yamada and Hasegawa applied the finite element method (FEM) to solve the parabolic diffusion approximation equation and the transient transmission of pulse in 2D and 3D cylindrical media [9]. Schweiger et al. used the FEM to solve the diffusion approximation equation and the problems of light spread in biological tissue under different boundary conditions and light source [10, 11]. Guven et al. provided a computationally efficient iterative algorithm to simultaneously estimate the optical image and the unknown a priori model parameters [12]. More recently, Wang and Zuo proposed a variational model based on the maximum posterior probability (MAP) to reconstruct images degraded by Rician noise [13]. Shi et al. developed an efficient regularization-based reconstruction algorithm to increase the computational speed with low memory consumption [14]. Zhang et al. proposed a new method to resolve the fluorescence molecular tomography inverse problem based on MAP estimation with structural priors in a Bayesian framework [15].

However, to the authors’ best knowledge, there are few reports concerning the image reconstruction of biological tissues with the MAP estimation in the field of transient radiative transfer problems [12, 16]. In this work, based on the time-domain diffusion equation, the diffusion coefficient distribution of the biological tissues is reconstructed under the condition of uniform absorption coefficient firstly, and then the absorption coefficient and reduced scattering coefficient distribution are reconstructed simultaneously considering inhomogeneous absorption coefficient. The edges of the reconstruction image become clearer with the MAP estimation. Compared with the unconstrained method, the MAP estimation algorithm greatly improves the reconstruction quality and reduces the normalized root-mean square error (NRMSE).

#### 2. Materials and Methods

##### 2.1. The Principle of the Forward Problem

To reconstruct the optical parameters of semitransparent media, a forward model is needed to simulate the photon propagation in semitransparent media and calculate transmission and reflected signals. The photon propagation in highly scattering medium obeys diffusion equation [12, 16]. The two-dimensional time-resolved diffusion equation is given as [10]where denotes the location and denotes the time. indicates the medium. is the refractive index. is the speed of light in vacuum. is the incident radiation. is the radiative source term. is the absorption coefficient. is the diffusion coefficient, which is determined by where is called reduced scattering coefficient and is the scattering coefficient. is the scattering asymmetry parameter.

The initial and boundary conditions are expressed as follows.

Initial condition

Considering that the refractive indexes in the two sides of boundary are inconsistent, we employ the partial current boundary condition [17] in the four boundaries of two-dimensional media:where indicates the boundary. is the outward normal unit vector of the border. can be obtained bywhere and the critical angle .

By replacing the spatial and temporal derivatives with the fully implicit scheme of finite difference approximations, (1) can be expressed as follows:

For square grids (), the diffusion equation can be transformed into linear equations:

Equation (7) can be abbreviated as

Equation (8) is solved by a stabilized biconjugate gradient method (Bi-CGSTAB) [18] in the present study.

##### 2.2. The Principle of the Inverse Problem

The reconstruction of unknown optical parameters is an ill-posed inverse problem. So the regularization technique should be used to make the solution well-behaved. For maximum a posteriori (MAP) estimation deduced from Bayesian theory can lead to Tikhonov regularization, the inversion of the distribution of the optical parameters is converted to the Bayesian estimation problems for image reconstruction by means of probability distribution models [15, 19]. The MAP estimation of the unknown image based on Bayesian theorem can be written as [20]where denotes the retrieval optical parameters. denotes the measured value. is the conditional probability density and is the prior probability density of the image. The first term on the right is the agreement term and the second is the regularization term. In the inverse transient radiative problem, the objective function is usually given bywhere denotes the retrieval optical parameters. is the estimated value of the th detector at the th moment when the th light source is working. is the measured value of the th detector at the th moment when the th light source is working. is the regularization term.

To preserve the edges of the image, the generalized Gaussian Markov random field (GGMRF) model is used for image reconstruction. The regularization term can be defined as [21]where is the image sharpness parameter. is the scale parameter. is the set of all neighboring pixel pairs. is the neighboring position of . is the weight coefficient.

The reconstruction of optical image is equal to minimization of the objective function, which needs to obtain the gradient of objective function with respect to the optical parameters. The adjoint differentiation (AD) method is an efficient method to calculate the gradient, which can reduce the consumption of computing time. The gradient of objective function with respect to the optical parameters can be expressed aswhere is the sensitivity of with respect to and can be obtained by differentiating (8) with respect to , which yieldsThe intensity and matrix can be calculated by the forward model. The solution of (18) is similar to the solution forward model using the Bi-CGSTAB. is the derivative of objective function with respect to intensity and can be obtained recursively using the chain rulewithwhere is the sampling time which means the time consumed in the measurement of transmission and reflected signals. is obtained by partially differentiating objective function with respect to :

Differentiating (8) with respect to , we can obtainwith

The gradient of the objective function with respect to the optical parameters can be calculated by (12)–(18). After calculation of the gradient, CG method can be employed to solve inverse problem [12].

The flowchart of the image reconstruction is shown in Figure 1.