Author ORCID Identifier

https://orcid.org/0000-0001-6184-0183

Date of Award

Spring 6-2024

Document Type

Thesis (Ph.D.)

Department or Program

Mathematics

First Advisor

Anne Gelb

Second Advisor

Geoffrey Luke

Abstract

Inverse problems are prevalent in many fields of science and engineering, such as signal processing and medical imaging. In such problems, indirect data are used to recover information regarding some unknown parameters of interest. When these problems fail to be well-posed, the original problems must be modified to include additional constraints or optimization terms, giving rise to so-called regularization techniques. Classical methods for solving inverse problems are often deterministic and focus on finding point estimates for the unknowns. Some newer methods approach the solving of inverse problems by instead casting them in a statistical framework, allowing for novel point estimate approaches and for the recovery of uncertainty information. In this dissertation, we first use a deterministic approach in the context of a medical imaging application to reconstruct volumetric images of blood vessels while enforcing sparsity in the edge domain. We then propose and investigate methods for the statistical inference of complex-valued signals as well as techniques for volumetric reconstruction using complex-valued synthetic aperture radar data.

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