On the application of partial differential equations and fractional partial differential equations to images and their methods of solution

Jacobs, Byron
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This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method.