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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/15144 |
| Date | 11 August 2014 |
| Creators | Jacobs, Byron |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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