Keywords

Anisotropic Conductivity, Inverse Problems, Beltrami Equation, Reconstruction

Abstract

Some single-physics medical imaging methods fail to reliably discriminate benign from malignant biological tissue. In partial response, a new class of inverse problems employ multi-physics phenomena, coupling methods of high resolution with methods of high contrast. In this thesis we are concerned with one such problem in which an anisotropic electrical conductivity is to be recovered from some internal knowledge of the current density field generated by maintaining a fixed boundary voltage. This internal data is obtained by the method of Current Density Impedance Imaging (CDII), which combines Maxwell's equations with Magnetic Resonance Imaging (MRI) measurements. The anisotropy class is also known from MRI measurements of the diffusion of water molecules. The mathematical problem, originally formulated by Hoell, Moradifam and Nachman in \cite{hoell14}, asked for the determination of the conformal factor from internal knowledge of the magnitude of the current density field in the metric determined by the conformal class. In their original work they proved unique determination, but no reconstruction has been proposed.

In here, we develop a reconstruction method for the two-dimensional problem. The proposed method makes use of an isothermal diffeomorphism which reduces the anisotropic problem to the isotropic one. Then, the pullback of a reconstruction method from the isotropic case fully recovers the anisotropic conductivity. In addition, we propose an approach to recovering the isothermal diffeomorphism via elliptic PDE theory. The method is implemented and tested in numerical experiments.

Completion Date

2026

Semester

Spring

Committee Chair

Dr. Alexandru Tamasan

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

School of Data, Mathematical, and Statistical Sciences

Format

PDF

Document Type

Dissertation

Identifier

DP0053305

Download

Share

COinS
 

Accessibility Statement

This item was created or digitized prior to April 24, 2027, or is a reproduction of legacy media created before that date. It is preserved in its original, unmodified state specifically for research, reference, or historical recordkeeping. In accordance with the ADA Title II Final Rule, the University Libraries provides accessible versions of archival materials upon request. To request an accommodation for this item, please submit an accessibility request form.