On the Application of Partial Differential Equations and Fractional Partial Differential Equations to Images and Their Methods of Solution.

Abstract: 

This body of work examines the plausibility of applying partial differential equations and time-fractional partial differential equations to images. The standard diffusion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with diffusive properties and a binarizing effect due to the source term. We examine the eff ects of applying this model to a class of images known as document images; images that largely comprise text. The effects 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 affords 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: finite-differences 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.