Date of Award

December 2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Engineering

First Advisor

Roshan M D'Souza

Committee Members

Krishna M Pillai, Zeyun Yu, Mohammad H Rahman, Ahmadreza Baghaie, Ahmad P Tafti

Keywords

4D-Flow MRI, Compressed sensing, Denoising, DMD, Dynamic, Kalman filter

Abstract

Recent advancements in data collection methods and equipment have resulted in a huge increase in the amount of data collected by observing various types of physical phenomena. Regardless of the amount of data collected, it is well known for many physical systems, the so-called information rank of the collected data is much lower than the rank of the data itself. This usually means the data may be represented sparsely in terms of a properly-chosen basis. This realization has led to methods for storing large amounts of data through compression by sacrificing negligible data quality. More importantly, with the advent of compressed sensing techniques, using an appropriate representation basis and sampling technique, it is now possible to sample data far below the Shannon-Nyquist limit thus speeding up data acquisition and also reducing the complexity of data-acquisition hardware. In this research, we explore the application of various modern data analysis techniques such as proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), compressed sensing, and Kalman filter and smoother in the data-driven analysis of dynamic systems with many degrees of freedom. This research has resulted in four novel methods. The first method is developed for denoising and spatial resolution enhancement of 4D-Flow MRI data based on POD and sparse reconstruction. The second method combines DMD and compressed sensing and takes discrete cosine transform (DCT) as the representation basis for dynamic denoising and gappy data reconstruction in 2D. The third method is a fast and parameter-free dynamic denoising method which combines a reduced-order model (ROM), a Kalman filter and smoother, and a DMD-based forward model. The fourth method is developed for reconstructing a 2D incompressible flow field by taking sparse measurements from the Fourier domain. As the reconstruction basis, a custom divergence-free set of basis vectors are derived and implemented.

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