Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Electrical, Computer & Energy Engineering

First Advisor

Shannon M. Hughes

Second Advisor

Youjian Liu

Third Advisor

Lijun Chen

Fourth Advisor

Jem Corcoran

Fifth Advisor

Elizabeth Bradley

Abstract

While the field of image processing has been around for some time, new applications across many diverse areas, such as medical imaging, remote sensing, astrophysics, cellular imaging, computer vision, and many others, continue to demand more and more sophisticated image processing techniques. These areas inherently rely on the development of novel methods and algorithms for their success. Many important cases in these applications can be posed as problems of reversing the action of certain linear operators. Recently, patch-based methods for image reconstruction have been shown to work exceptionally well in addressing these inverse problems, establishing new state-of-the-art benchmarks for many of them, and even approaching estimated theoretical limits of performance.

However, there is still space and need for improvement, particularly in highly specialized domains. The purpose of this thesis will be to improve upon these prior patch-based image processing methods by developing a computationally efficient way to model the underlying set of patches as arising from a low-dimensional manifold. In contrast to other works that have attempted to use a manifold model for patches, ours will rely on the machinery of kernel methods to efficiently approximate the solution. This will make our approach much more suitable for practical use than those of our predecessors. We will show experimental results paralleling or exceeding those of modern state-of-the-art image processing algorithms for several inverse problems. Additionally, near the end of the thesis, we will revisit the problem of learning a representation for the manifold from its samples and develop an improved approach for it. In contrast to prior methods for manifold learning, our kernel-based strategy will be robust to issues of learning from very few or noisy samples, and it will readily allow for interpolation along or projection onto the manifold.

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