Multi-dimensional continuous wavelet transforms and generalized fourier transforms in Clifford analysis

This work covers three mathematical analysis domains: Continuous Wavelet Transform, Fourier Transform and Clifford Analysis, and consists of three parts in which these domains interact. The one-dimensional continuous wavelet transform (CWT) is a successful tool for signal and image analysis, with applications in mathematics, physics and engineering. Higher dimensional CWTs typically originate as tensor products of one-dimensional phenomena. Clifford analysis offers a natural generalization to higher dimension of the theory of holomorphic functions in the complex plane. The generalized holomorphic functions, known as monogenic functions, are null-solutions of the so-called Dirac operator, a first order rotationally invariant differential operator factorizing the Laplacian in higher dimensions. This factorization of the Laplace operator establishes a special relationship between monogenic functions and harmonic functions of several variables, in that the properties of monogenic functions constitute a refinement of those of harmonic functions. An intrinsic feature of Clifford analysis is that it encompasses all dimensions at once, as opposed to the usual tensorial approaches. This true multi-dimensional nature allows for a very specific construction of higher dimensional wavelets and the development of the corresponding CWT-theory, based on generalizations to higher dimension of classical orthogonal polynomials on the real line. In Part I this wavelet construction procedure is presented within the usual, orthogonal Clifford analysis framework, while in Part III we generalize it to the metric dependent setting of Clifford analysis. The latter gives rise to so-called anisotropic Clifford-wavelets which are adaptable to preferential, not necessarily orthogonal, directions in the signals or textures to be analysed. The central topic of Part II is the development of a new multi-dimensional Fourier transform in the framework of Clifford analysis, the so-called Clifford-Fourier transform. It arises as a theoretical construct quite naturally in the spirit of the above mentioned refinement of harmonic functions by monogenic ones.

Author dc:creator

• De Schepper, Nele

Contributors dc:contributor

• Brackx, F
• Sommen, F

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

und

Identifier

https://biblio.ugent.be/publication/470686
http://doi.org/1854/8873
https://biblio.ugent.be/publication/470686/file/1878971

OAI identifier oai:identifier

oai:archive.ugent.be:470686