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Sobolev gradient flows and image pro...

Calder, Jeffrey William.

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Sobolev gradient flows and image processing.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Sobolev gradient flows and image processing./
Author:	Calder, Jeffrey William.
Description:	140 p.
Notes:	Source: Masters Abstracts International, Volume: 49-04, page: 2515.
Contained By:	Masters Abstracts International49-04.
Subject:	Applied Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=MR70245
ISBN:	9780494702451

Sobolev gradient flows and image processing.
Calder, Jeffrey William.

Sobolev gradient flows and image processing. - 140 p.

Source: Masters Abstracts International, Volume: 49-04, page: 2515.

Thesis (M.Sc.)--Queen's University (Canada), 2010.

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.

ISBN: 9780494702451Subjects--Topical Terms:

1669109
Applied Mathematics.

Sobolev gradient flows and image processing.
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020 $a 9780494702451
035 $a (UMI)AAIMR70245
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040 $a UMI $c UMI
100 1 $a Calder, Jeffrey William. $3 1679373
245 1 0 $a Sobolev gradient flows and image processing.
300 $a 140 p.
500 $a Source: Masters Abstracts International, Volume: 49-04, page: 2515.
502 $a Thesis (M.Sc.)--Queen's University (Canada), 2010.
520 $a 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.
590 $a School code: 0283.
650 4 $a Applied Mathematics. $3 1669109
650 4 $a Mathematics. $3 515831
690 $a 0364
690 $a 0405
710 2 $a Queen's University (Canada). $3 1017786
773 0 $t Masters Abstracts International $g 49-04.
790 $a 0283
791 $a M.Sc.
792 $a 2010
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=MR70245