Desenvolvimento de um algoritmo para reconstrução de imagens utilizando a técnica de tomografia por impedância elétrica

Abstract

This work makes a mathematical study aiming to solve the problem of image's reconstruction in Electrical Impedance Tomography. In this image technique, electrodes are positioned on the boundary/border of a volume to be studied. In two of them, patterns of currents are "injected" and in the remaining electrodes electric potentials are measured. Through these data it is possible to estimate the electrical conductivity or resistivity within the region assessed, thus forming an image of it using its electrical properties. In order to establish this estimate, it is necessary to solve two problems: the forward and the inverse problem. The forward problem consists in solving the generalized Laplace equation, which governs the potential within the studied region. To accomplish that, numerical methods are used, such as the Finite Element Method, the Boundary Element Method or the Finite Difference Method which was the method used in this work. By solving the forward problem and the measurements of the potential contour the inverse problem is solved. In this process, the potential is calculated and measured values of potential are placed in an error functional and the distribution of conductivity that minimizes the value of this functional is searched. A Minimization procedure known as simulated annealing applied to the functional can to resolve the Electrical Impedance Tomography's inverse problem.