扩散光学层析成像的线性抽样型算法研究

论文总字数：24241字

摘 要

扩散光学层析成像又称漫射光学层析成像（Diffuse Optical Tomography, DOT）,是一种利用近红外光（600-900nm）照射生物组织获得成像的方法，已被广泛应用于生物医学检测中。成像原理是，用激发源照射生物组织体，激发源放射的光子在组织体内被散射和吸收后，通过放置在生物组织体表面的接收器测量得到生物组织体表面的光强分布信息，光强分布信息中包含了生物组织体的光学性质，如散射系数和折射系数等。成像问题的数学模型依赖于生物组织体的空间尺度，常用的模型包括麦克斯韦方程组、辐射输运方程和扩散方程等。本课题采用扩散方程模型，对应的成像问题归结为扩散方程反问题。本课题研究由边界测量数据重构生物组织体内部异常区域的位置、大小和形状等几何信息。对于正问题，本文利用边界积分方程方法，用单层位势表示光强分布，由边界条件得到关于密度函数的边界积分方程组。对该边界积分方程组，提出一种离散化方法，求解得到光强分布的数值解。对于反问题，本文采用非迭代的数值方法（指示函数法）求解。最后，进行了数值实验，数值结果表明本文采用的方法是有效可行的。

关键词：扩散光学层析成像；扩散方程；边界积分方程方法；正问题；反问题；

Linear Sampling Algorithms for Diffuse Optical Tomography

Abstract

Diffuse optical tomography is also called diffuse optical tomography (DOT). It is an imaging method by using near-infrared light (600-900 nm). It has been widely used in biomedical testing. The imaging principle is that the biological tissue is irradiated with an excitation source. The excitation source emitted photons, which are scattered and absorbed in the tissue. Then the light intensity distribution information on the surface of the biological tissue is measured by a receiver placed on the surface of the biological tissue. The distribution information includes the optical properties of the biological tissue, such as the scattering coefficient and the refractive coefficient. The mathematical model of the imaging problem depends on the spatial scale of the biological tissue. Commonly used models include Maxwell's equations, radiation transport equations and diffusion equations. The diffusion equation model is used in this paper, and the corresponding imaging problem is reduced to the inverse problem of the diffusion equation. This paper studies the problem of reconstructing the geometrical information such as the position, size and shape of the abnormal region inside the biological tissue from the boundary measurement data. For the forward problem, the boundary integral equation method is used in this paper. The light intensity distribution represent by a single layer potential. Then the boundary integral equations for the density function can be obtained. A discrete method is proposed for the boundary integral equations and numerical solution of the intensity distribution can be obtained. For the inverse problem, a non-iterative numerical method (indicattor function method) is used in this paper. Finally, numerical experiments are carried out to show that the method adopted in this paper is effective and feasible.

Keywords: diffuse optical tomography; diffusion equation; boundary integral equation method; forward problem; inverse problem.

目 录

摘要 ……………………………………………………………………………………………………Ⅰ

Abstract …………………………………………………………………………………………… Ⅱ

1. 引言…………………………………………………………………………………1

1.1 问题模型及概述…………………………………………………………………1

1.2 本文的主要工作…………………………………………………………………2

1. 基本工具与理论 ………………………………………………………………4

2.1 关于扩散方程正问题的基本工具和理论………………………………………4

2.2 关于扩散方程反问题的基本工具和理论………………………………………4

1. 正问题的数值求解 ………………………………………………………………6

3.1 正问题求解………………………………………………………………………6

3.1.1 位势理论得到的积分方程组 …………………………………………6

3.1.2 离散积分方程组 ……………………………………………………7

3.1.3 u的计算 ……………………………………………………………9

3.2 正问题算例 …………………………………………………………………10

1. 反问题的数值求解 ………………………………………………………………17

4.1 反问题求解 ……………………………………………………………………17

4.2 反问题算例 ……………………………………………………………………19

致谢 ………………………………………………………………………………………22

参考文献（References） ………………………………………………………………23

第一章 引言

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