The Clifford-Fourier Transform

2012

SIViP

Self Cite

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.

Section: Overview Of Recent Resultsmentioning

confidence: 99%

Fractional Fourier transforms of hypercomplex signals

Bie

2012

SIViP

Self Cite

“…It becomes apparent, also from the given examples, that the Clifford analysis framework is most appropriate to develop these multidimensional Hilbert transforms. That Clifford analysis could be a powerful tool in multidimensional signal analysis became already clear during the last decade from the several constructions of multidimensional Fourier transforms with quaternionic or Clifford algebra valued kernels with direct applications in signal analysis and pattern recognition, see [20,21,24,[32][33][34]39] and also the review paper [23] wherein the relations between the different approaches are established. In view of the fact that in the underly-ing paper the interaction of the Clifford-Hilbert transforms with only the standard Fourier transform was considered, their interplay with the various Clifford-Fourier transforms remains an intriguing and promising topic for further research.…”

Section: Resultsmentioning

confidence: 99%

“…As each function in the Hardy space H 2 (R m+1 ± ), (20), possesses a non-tangential L 2 boundary limit for x 0 → 0±, one is lead to the introduction of the Hardy space…”

Section: Definition and Propertiesmentioning

confidence: 99%

Hilbert Transforms in Clifford Analysis

Brackx

Knock

Geometric Algebra Computing

2010

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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 as well as 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, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature.

“…Since moreover it holds that tθ ± = −iθ ± |t, it can be readily verified that the functions f ( x, t + x 0 |t|) θ ± are monogenic. This is the way in which monogenic Fourier kernels were constructed by T. Qian and A. Mc Intosh ([Li et al, 1994]), see also [Sommen, 1988] and [Brackx et al, 2005].…”

Section: The Cauchy-kovalevskaya Extensionmentioning

confidence: 99%

Introductory Clifford Analysis

Sommen

2014

Operator Theory

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In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy-Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered both as a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated to the vectorial part of the Cauchy-Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter's theorem on the one hand and the Cauchy-Kovalevskaya extension theorem on the other. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert and Radon transforms, which are important both within the theoretical framework and in view of possible applications.