- Title
- Projection algorithms for non-separable wavelets and Clifford Fourier analysis
- Creator
- Franklin, David John
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2018
- Description
- Research Doctorate - Doctor of Philosophy (PhD)
- Description
- Fourier Analysis is a primary technique in the analysis of images, yet it has several limitations when it comes to the higher dimensional case of colour images. This thesis seeks to address some of these limitations through two main areas. First, we consider the recently developed Clifford-Fourier Transform of Brackx et al, which has the advantage over the classical Fourier Transform of combining the different channels of a colour image. We characterise the Hardy Spaces of this transform and show that functions in these Hardy spaces have monogenic extensions with bounded integral averages. We also characterise the Paley-Wiener spaces and show that functions in a Paley-Wiener space with radius R have monogenic extensions with integral averages that grow according to the radius R. Second, we consider the case of two dimensional compactly supported wavelets with orthonormal shifts and develop projection algorithms to find compactly supported, continuous wavelets with orthonormal shifts and dilations and 2 vanishing moments which are not tensor products of one dimensional wavelets. We also apply these techniques in one dimension and discover an example of an anti-symmetric, compactly supported, continuous wavelet with orthonormal shifts and dilations and 2 vanishing moments.
- Subject
- Clifford analysis; Fourier analysis; wavelets; quaternions; projections; harmonic analysis; Douglas-Rachford
- Identifier
- http://hdl.handle.net/1959.13/1395028
- Identifier
- uon:33802
- Rights
- Copyright 2018 David John Franklin
- Language
- eng
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| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | ATTACHMENT01 | Thesis | 4 MB | Adobe Acrobat PDF | View Details Download | ||
| View Details Download | ATTACHMENT02 | Abstract | 91 KB | Adobe Acrobat PDF | View Details Download | ||
| View Details Download | ATTACHMENT03 | Supporting material | 10 MB | Zip Compressed File | View Details Download |