Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl3,0

Bahri, Mawardi; Hitzer, Eckhard

doi:10.1007/s00006-006-0003-x

Cited by 70 publications

(43 citation statements)

References 6 publications

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“…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%

Hilbert Transforms in Clifford Analysis

Brackx

Knock

Schepper

2010

Geometric Algebra Computing

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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.

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Section: Resultsmentioning

confidence: 99%

Hilbert Transforms in Clifford Analysis

Brackx

Knock

Schepper

2010

Geometric Algebra Computing

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“…Together with Bülow and Sommer, Felsberg applied these CFTs to image stucture processing (key-notion: structure multivector) [50,24]. Ebling and Scheuermann [44,43] consequently applied to vector signal processing in two-and three dimensions, respectively, the following twodimensional CFT [78,63,78] to define their Clifford-Fourier transform of threedimensional multivector signals: that means, they researched the properties of F 3 {g}(ω) of (2.1) in detail when applied to full multivector signals g : R 3 → C 3,0 . This included an investigation of the uncertainty inequality for this type of CFT.…”

Section: How Clifford Algebra Square Roots Of −1 Lead To Clifford Foumentioning

confidence: 99%

“…In this context the Clifford Fourier transformations by Felsberg [50] for one-and two-dimensional signals, by Ebling and Scheuermann for twoand three-dimensional vector signal processing [44,43], and by Mawardi and Hitzer for general multivector signals in C 3,0 [78,63,78], and their respective kernels, as already reviewed in Section 2.1, should also be considered.…”

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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“…In [9,10,15], the Clifford Fourier transform (CFT) on Cl n,0 for n = 2, 3 (mod 4) has been introduced. Based on the basic concepts of Clifford algebra and its Fourier transform, we constructed Clifford algebra Cl 3,0 -valued wavelet transform 1 .…”

Section: Introductionmentioning

confidence: 99%

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

Bahri

Adji

Zhao

2010

Adv. Appl. Clifford Algebras

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Abstract. This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cln,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of R n . We express the admissibility condition in terms of the Cln,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cln,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets. Mathematics Subject Classification (2010). 15A66, 42C40, 94A12.

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Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl3,0

Cited by 70 publications

References 6 publications

Hilbert Transforms in Clifford Analysis

Hilbert Transforms in Clifford Analysis

Quaternion and Clifford Fourier Transforms and Wavelets

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

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