Nele De Schepper

The Two-Dimensional Clifford-Fourier Transform

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L1 and in the L2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

Chapter 2 The Fourier Transform in Clifford Analysis

Publisher Summary This chapter focuses on the Fourier transform in Clifford analysis. This chapter includes an introductory section on Clifford analysis, and each section starts with an introductory situation. This chapter presents the new Clifford–Fourier transform is given in terms of an operator exponential, or alternatively, by a series representation. Particular attention is directed to the two-dimensional (2D) case since then the Clifford–Fourier kernel can be written in a closed form. This chapter also discusses the fractional Fourier transform wherein, it is shown that the traditional and the Clifford analysis approach coincide. This chapter develops the theory for the Clifford–Hermite and Clifford–Gabor filters for early vision. This chapter faced with the following situation: In dimension greater than two, we have a first Clifford–Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension, two both transforms coincide.

The Clifford–Gegenbauer polynomials and the associated continuous wavelet transform

In the framework of Clifford analysis, Clifford–Gegenbauer and generalized Clifford–Gegenbauer polynomials are constructed. Orthogonality relations and Rodrigues formulae are established. It is shown that they are the appropriate building blocks for new specific wavelet kernel functions for a higher dimensional continuous wavelet transform. E-mail: fs@cage.ugent.be E-mail: nds@cage.ugent.be

Convolution Products for Hypercomplex Fourier Transforms

Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.

Fractional Fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given.

The Fractional Clifford-Fourier Transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.

Clifford-Jacobi Polynomials and the Associated Continuous Wavelet Transform in Euclidean Space

Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are presented within the framework of Clifford analysis. These multi-dimensional wavelets are constructed by taking the Clifford-monogenic extension to ℝ m+1 of specific functions in ℝm generalizing the traditional Jacobi weights. The notion of Clifford-monogenic function is a direct higher dimensional generalization of that of holomorphic function in the complex plane. Moreover, crucial to this construction is the orthogonal decomposition of the space of square integrable functions into the Hardy space H 2 (ℝm) and its orthogonal complement. In this way a nice relationship is established between the theory of the Clifford Continuous Wavelet Transform on the one hand, and the theory of Hardy spaces on the other hand. Furthermore, also new multi-dimensional polynomials, the so-called Clifford-Jacobi polynomials, are obtained.

The higher dimensional Hermite transform: a new approach

In this article we expand the filter functions of the classical Hermite transform into the Clifford-Hermite polynomials. Furthermore, we construct a new higher dimensional Hermite transform within the framework of Clifford analysis using the radial and generalized Clifford-Hermite polynomials. Finally we compare this newly introduced Clifford-Hermite transform with the Clifford-Hermite Continuous Wavelet transform.

The Cylindrical Fourier Transform

The aim of this paper is to show the application potential of the cylindrical Fourier transform, which was recently devised as a new integral transform within the context of Clifford analysis. Next to the approximation approach where, using density arguments, the spectrum of various types of functions and distributions may be calculated starting from the cylindrical Fourier images of the L 2-basis functions in ℝ m , direct computation methods are introduced for specific distributions supported on the unit sphere, and an illustrative example is worked out.

Metric Dependent Clifford Analysis with Applications to Wavelet Analysis

In earlier research multi-dimensional wavelets have been constructed in the framework of Clifford analysis. Clifford analysis, centered around the notion of monogenic functions, may be regarded as a direct and elegant generalization to higher dimension of the theory of the holomorphic functions in the complex plane. This Clifford wavelet theory might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one.