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

摘要

This body of work examines the plausibility of applying partial di erential equations andudtime-fractional partial di erential equations to images. The standard di usion equationudis coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain audmodel with di usive properties and a binarizing e ect due to the source term. We examineudthe e ects of applying this model to a class of images known as document images;udimages that largely comprise text. The e ects of this model result in a binarization processudthat is competitive with the state-of-the-art techniques. Further to this application,udwe provide a stability analysis of the method as well as high-performance implementationudon general purpose graphical processing units. The model is extended to includeudtime derivatives to a fractional order which a ords us another degree of control over thisudprocess and the nature of the fractionality is discussed indicating the change in dynamicsudbrought about by this generalization. We apply a semi-discrete method derived byudhybridizing the Laplace transform and two discretization methods: nite-di erences andudChebyshev collocation. These hybrid techniques are coupled with a quasi-linearizationudprocess to allow for the application of the Laplace transform, a linear operator, to audnonlinear equation of fractional order in the temporal domain. A thorough analysisudof these methods is provided giving rise to conditions for solvability. The merits anduddemerits of the methods are discussed indicating the appropriateness of each method.