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Fractal Imaging Theory and Applications beyond Compression

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dc.contributor.advisor Kunze, Herb
dc.contributor.advisor La Torre, Davide Demers, Matthew 2012-05-07 2012-05-14T18:36:51Z 2012-05-14T18:36:51Z 2012-05-14
dc.description.abstract The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favourably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing. en_US
dc.description.sponsorship The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research. en_US
dc.language.iso en en_US
dc.rights.uri *
dc.subject Applied Analysis en_US
dc.subject Fractals en_US
dc.subject Imaging en_US
dc.subject Edge Detection en_US
dc.subject Iterated Function Systems en_US
dc.subject Texturing en_US
dc.subject Cryptography en_US
dc.subject Inverse Problems en_US
dc.title Fractal Imaging Theory and Applications beyond Compression en_US
dc.type Thesis en_US Mathematics and Statistics en_US Doctor of Philosophy en_US Department of Mathematics and Statistics en_US

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