Keywords
Current density impedance imaging, 1-laplacian, inverse boundary value problems, complete electrode model, electrical impedance tomography
Abstract
We study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity from the magnitude of one current density field inside. The method now known as Current Density Impedance Imaging is based on solving boundary value problems for the 1-Laplacian in an appropriate Riemann metric space. We consider two types of methods: the ones based on level sets and a variational approach, which aim to solve specific boundary value problem associated with the 1-Laplacian. We will address the Cauchy and Dirichlet problems with full and partial data, and also the Complete Electrode Model (CEM). The latter model is known to describe most accurately the voltage potential distribution in a conductive body, while taking into account the transition of current from the electrode to the body. For the CEM the problem is non-unique. We characterize the non-uniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.
Notes
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Graduation Date
2014
Semester
Summer
Advisor
Tamasan, Alexandru
Degree
Doctor of Philosophy (Ph.D.)
College
College of Sciences
Department
Mathematics
Degree Program
Mathematics
Format
application/pdf
Identifier
CFE0005437
URL
http://purl.fcla.edu/fcla/etd/CFE0005437
Language
English
Release Date
August 2014
Length of Campus-only Access
None
Access Status
Doctoral Dissertation (Open Access)
Subjects
Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic
STARS Citation
Veras, Johann, "Electrical Conductivity Imaging via Boundary Value Problems for the 1-Laplacian" (2014). Electronic Theses and Dissertations. 4565.
https://stars.library.ucf.edu/etd/4565