In full generality we express the classical convolution of Clifford algebra signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of Clifford algebra signals in terms of finite linear combinations of classical convolutions. Show all 6. Though the basic concepts of Clifford—Fourier transforms are well known, an implementation of analytic video sequences using multiquaternion algebras does not seem to have been realized so far.

After a short presentation of multiquaternion Clifford algebras and Clifford—Fourier transforms, a brief pedagogical review of 1D and 2D quaternion analytic signals using right quaternion Fourier transforms is given. The phase extraction procedure is fully detailed. Finally, a numerical implementation using discrete fast Fourier transforms of an analytic multiquaternion video signal is provided. Bayro-Corrochano and G. Scheuermann Eds. While in the original paper, the transform was determined for vector-valued functions only, it now will be extended to functions taking values in the entire Clifford algebra.

Next, two bases are determined under which this Fourier transform is diagonalizable. This problem will be tackled in the final section of this paper.

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform see Brackx et al. Fourier Anal.

The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained.

### Bulletin of the Belgian Mathematical Society - Simon Stevin

This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven. It is well known that Clifford geometric algebra offers a geometric interpretation for square roots of —1in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube.

Systematic research has been done [33] on the biquaternion roots of —1, abandoning the restriction to blades. This includes, e. All these roots of —1 are immediately useful in the construction of new types of geometric Clifford—Fourier transformations. We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms CFT , and study their properties from linearity to convolution. Show all 4. There have been several attempts in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra.

The first part of this chapter features certain properties of the asymptotic behaviour of the quaternion Fourier transform. In the second part we introduce the quaternion Fourier transform of a probability measure, and we establish some of its basic properties. In the final analysis, we introduce the notion of positive definite measure, and we set out to extend the classical Bochner—Minlos theorem to the framework of quaternion analysis.

Some important properties of the transform are investigated. In the last decade several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a Clifford-Fourier transform, constructed using the geometric properties of Clifford algebra.

We show the corresponding results of operational calculus, and a connection between the Fourier transform and this new transform. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the Paley-Wiener theorems are formulated. In this article, we study the quaternion ridgelet transform and curvelet transform associated to the quaternion Fourier transform QFT. We prove some properties related to such transforms, including reconstruction formulas, reproducing kernels and uncertainty principles. We introduce a new Riemannian Fourier transform for color image processing.

The construction involves spin characters and spin representations of complex Clifford algebras. Examples of applications to low-pass filtering are presented. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly demonstrate the benefit of developing problem solutions in a unified framework for algebra and geometry with the widest possible scope: from quantum computing and electromagnetism to satellite navigation, from neural computing to camera geometry, image processing, robotics and beyond.

The purpose of this chapter is to give a characterization of the class B and to give a generalization of the classical theorem of Bochner in the framework of quaternion analysis. The Hilbert transform on the real line has applications in many fields. In particular in one-dimensional signal processing, the Hilbert operator is used to extract global and instantaneous characteristics, such as frequency, amplitude, and phase, from real signals.

The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space developed within the framework of Clifford analysis.

Roughly speaking, this is a function theory of higher-dimensional holomorphic functions particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature. 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 L 1 and in the L 2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

### mathematics and statistics online

First, the basic concepts of the multivector functions, vector differential and vector derivative in geometric algebra are introduced. The aim of this paper is to define a Clifford—Fourier transform that is suitable for color image spectral analysis. There have been many attempts to define such a transformation using quaternions or Clifford algebras. We focus here on a geometric approach using group actions.

The transformation we propose is parameterized by a bivector and a quadratic form, the choice of which is related to the application to be treated. Sangwine and T. Based on the spectral representation of the Clifford Fourier transform CFT , we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. Postprocessing in computational fluid dynamics and processing of fluid flow measurements need robust methods that can deal with scalar and vector fields.

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## Clifford algebra, geometric algebra, and applications

While image processing of scalar data is a well-established discipline, there is a lack of similar methods for vector data. This paper surveys a particular approach defining convolution operators on vector fields using geometric algebra. This includes a corresponding Clifford—Fourier transform including a convolution theorem. IEEE Trans. Signal Process. The two-sided quaternionic Fourier transformation QFT was introduced in [2] for the analysis of 2D linear time-invariant partial-differential systems. In the current chapter we analyze this split further, interpret it geometrically as an orthogonal 2D planes split OPS , and generalize it to a freely steerable split of H into two orthogonal 2D analysis planes.

The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal.

## Main page of Rafal Ablamowicz

The second major result of this work is a variety of new steerable forms of the QFT, their geometric interpretation, and for each form, OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software. In this paper, we propose a new watermarking scheme resistant to geometric attacks and JPEG compression.

This method uses Fourier Clifford Transform and Harris interest points. First, we detect all circular Harris interest regions. Then, using the Delaunay-tessellation-based triangle matching method, we define robust interest region.