On the Clifford-Fourier Transform

Bie, Hendrik De; Xu, Yuan

doi:10.1093/imrn/rnq288

Cited by 37 publications

(93 citation statements)

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“…So far, we have applied these ideas in 3 different directions of hypercomplex FTs, namely -k-vector Fourier transforms ( [14]) -radially deformed Fourier transforms ( [15,16]) -Clifford-Fourier transforms ( [17,13]). …”

Section: Overview Of Recent Resultsmentioning

confidence: 99%

“…In recent work (see [13][14][15][16][17]) we have developed a different methodology: we start from a list of properties or general mathematical principles we want a hypercomplex Fourier transform to have, and then determine all kernels that satisfy these properties.…”

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confidence: 99%

“…In particular we devote our attention to the so-called Clifford-Fourier transform (as studied in [17,13]). Subsequently, we introduce a new fractional version of the Clifford-Fourier transform (CFT).…”

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confidence: 99%

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Fractional Fourier transforms of hypercomplex signals

Bie

Schepper

2012

SIViP

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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.

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Section: Overview Of Recent Resultsmentioning

confidence: 99%

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confidence: 99%

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Fractional Fourier transforms of hypercomplex signals

Bie

Schepper

2012

SIViP

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“…The question whether F H ± can be written as an integral transform is answered positively in the case of even dimension by De Bie and Xu in [39]. The integral kernel of this transform is not easy to obtain and looks quite complicated.…”

Section: The Clifford Fourier Transform In the Light Of Clifford Analmentioning

confidence: 99%

Quaternion and Clifford Fourier Transforms and Wavelets

Hitzer¹,

Sangwine²

2013

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Abstract. We survey the historical development of quaternion and Clifford Fourier transforms and wavelets. Keywords. Quaternions, Clifford Algebra, Fourier Transforms, Wavelet Transforms.The development of hypercomplex Fourier transforms and wavelets has taken place in several different threads, reflected in the overview of the subject presented in this chapter. We present in Section 1 an overview of the development of quaternion Fourier transforms, then in Section 2 the development of Clifford Fourier transforms. Finally, since wavelets are a more recent development, and the distinction between their quaternion and Clifford algebra approach has been much less pronounced than in the case of Fourier transforms, Section 3 reviews the history of both quaternion and Clifford wavelets.We recognise that the history we present here may be incomplete, and that work by some authors may have been overlooked, for which we can only offer our humble apologies. [90,91] on Clifford Fourier and Laplace transforms further explained in Section 2.2. Ernst and Delsuc's quaternion transforms were two-dimensional (that is they had two independent variables) and proposed for application to nuclear magnetic resonance (NMR) imaging. Written in terms of two independent time variables 1 t 1 and t 2 , the forward transforms 1 The two independent time variables arise naturally from the formulation of twodimensional NMR spectroscopy. Quaternion Fourier Transforms (QFT)1

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“…It has received increasing attention since the seminal work of S. Sangwine and co-authors whose original idea was to embed the acquisition space of RGB color channels into the 3-dimensional vector space of imaginary quaternions ( [17], [29]). Lots of works are nowadays devoted to the more general question of defining Fourier transforms in the framework of Clifford algebras (see for instance [4], [6], [7], [8] [9], [10], [11], [14], [15], [16], [18], [19]). In this contribution, we adress this issue from the viewpoint of group actions and representations which are the basic notions of the abstract Fourier transform theory ( [32]).…”

Section: Introductionmentioning

confidence: 99%

Spin Geometry and Image Processing

Berthier

2013

Adv. Appl. Clifford Algebras

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Abstract. We give a survey of applications of spin geometry to image processing. We mainly focus on the problem of defining geometric Fourier transforms for both grey-level and color images. The definitions we propose rely on a spin generalization of the usual notion of character. We consider three possibilities for the actions of these spin characters: by using the spinor representation of grey-level image surfaces; by considering grey-level and color images as sections of associated bundles built first with standard representations and then with spin representations. Examples of applications to low-pass filtering are presented.Mathematics Subject Classification (2010). Primary 68U10, 53C27; Secondary 53A05, 43A32.

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On the Clifford-Fourier Transform

Cited by 37 publications

References 20 publications

Fractional Fourier transforms of hypercomplex signals

Fractional Fourier transforms of hypercomplex signals

Quaternion and Clifford Fourier Transforms and Wavelets

Spin Geometry and Image Processing

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