Author

Talon Johnson

ORCID Identifier(s)

0000-0002-4994-215X

Graduation Semester and Year

2021

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Jianzhong Su

Second Advisor

Jianzhong Su

Third Advisor

Jimin Ren

Abstract

The evolution of technology has drastically impacted the imaging field, particularly magnetic resonance imaging (MRI). Compared to other imaging technologies, MRI offers multiple contrasting mechanisms to distinguish tissues and fat, is radiation-free, and provides anatomical and molecular information about the tissue in question. However, data acquisition times to produce those images require a patient to lie still for a relatively long time. Consequently, it may lead to the voluntary or involuntary movement of the patient due to discomfort. Combined with the underlying issue of inherent noise, MRI is often blurry and contains artifacts. Mathematically, one can describe this behavior as the convolution between the MRI and some unwanted PSF. In this thesis, we present a new approach to speed up the MR data acquisition through sparse signal reconstruction and deconvolving the unwanted convolution simultaneously. This approach is part of an ever-growing area known as compressive deconvolution. We propose a novel compressive deconvolution method for two-dimensional MRI data sets via an ℓ1 − ℓ2 regularization via ℓ1–magic and Tikonov regularization.

Keywords

ℓ1 − ℓ2 regularization, ℓ1–magic, Singular value decomposition (SVD), Tikhonov regularization, Magnetic resonance imaging (MRI)

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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