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Please use this identifier to cite or link to this item: http://hdl.handle.net/1974/5986

Title: Sobolev Gradient Flows and Image Processing
Authors: Calder, Jeffrey

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Keywords: Partial Differential Equations
Image Diffusion
Gradient Flows
Gradient Descent
Gradient Ascent
Sobolev Spaces
Image Sharpening
Anisotropic Diffusion
Perona-Malik Paradox
Issue Date: 2010
Series/Report no.: Canadian theses
Abstract: In this thesis we study Sobolev gradient flows for Perona-Malik style energy functionals and generalizations thereof. We begin with first order isotropic flows which are shown to be regularizations of the heat equation. We show that these flows are well-posed in the forward and reverse directions which yields an effective linear sharpening algorithm. We furthermore establish a number of maximum principles for the forward flow and show that edges are preserved for a finite period of time. We then go on to study isotropic Sobolev gradient flows with respect to higher order Sobolev metrics. As the Sobolev order is increased, we observe an increasing reluctance to destroy fine details and texture. We then consider Sobolev gradient flows for non-linear anisotropic diffusion functionals of arbitrary order. We establish existence, uniqueness and continuous dependence on initial data for a broad class of such equations. The well-posedness of these new anisotropic gradient flows opens the door to a wide variety of sharpening and diffusion techniques which were previously impossible under L2 gradient descent. We show how one can easily use this framework to design an anisotropic sharpening algorithm which can sharpen image features while suppressing noise. We compare our sharpening algorithm to the well-known shock filter and show that Sobolev sharpening produces natural looking images without the "staircasing" artifacts that plague the shock filter.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-08-25 10:44:12.23
URI: http://hdl.handle.net/1974/5986
Appears in Collections:Queen's Graduate Theses and Dissertations
Department of Mathematics and Statistics Graduate Theses

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