﻿ 基于电阻抗成像问题的空腔重构算法 Cavity Reconstruction Algorithm Based on Electrical Impedance Tomography

Vol.04 No.02(2015), Article ID:15309,7 pages
10.12677/AAM.2015.42024

Cavity Reconstruction Algorithm Based on Electrical Impedance Tomography

Tianhong Feng

School of Mathematics, Dongbei University of Finance & Economics, Dalian Liaoning

Email: fength1212@163.com

Received: May 7th, 2015; accepted: May 22nd, 2015; published: May 27th, 2015

ABSTRACT

Electrical impedance tomography problem refers to the imaging of electrical parameters inside the object by measuring the current and voltage value of object surfaces [1] -[3] . An algorithm is proposed aiming at the reconstruction of homogeneous medium in the electrical impedance tomography with cavity. The basic idea of the algorithm is using analytic continuation to transfer the original problem to the Cauchy problem of circle domain; Newton-type iterative method is used to solve the nonlinear equations, getting the assemblage whose normal derivative is zero satisfying the solution of Cauchy problem, and then the boundary of the cavity is gotten. At the same time, numerical examples of several kinds of special shaped cavity reconstruction are presented to demonstrate the feasibility of this algorithm.

Keywords:Electrical Impedance Tomography, Cauchy Problem, Newton-Type Iterative Method

Email: fength1212@163.com

1. 问题的描述

(1.1)

(1.2)

(1.3)

2. 数学模型的建立

，于是(1.1)和(1.2)式转化为极坐标系下的Cauchy问题：

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.8)和(2.9)式构成特征值问题，显然其特征值和特征函数分别为

(2.10)

(2.11)

(2.12)

，有，从而，进而可得

(2.13)

(2.14)

3. 重构算法

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

4. 数值实验

(a)(b)

Figure 1. (a) Reconstruction of kite shape cavity; (b) Reconstruction with noise data

(a)(b)

Figure 2. (a) Reconstruction of peanut shape cavity; (b) Reconstruction with noise data

(a)(b)

Figure 3. (a) Reconstruction of piecewise smooth cavity; (b) Reconstruction with noise data

5. 总结

Cavity Reconstruction Algorithm Based on Electrical Impedance Tomography. 应用数学进展,02,189-196. doi: 10.12677/AAM.2015.42024

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