Quaternion Fourier Transform on Quaternion Fields and Generalizations

Hitzer, Eckhard

doi:10.1007/s00006-007-0037-8

Cited by 238 publications

(163 citation statements)

References 23 publications

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“…It is shown that many WFT properties still hold but others have to be modified. Another generalization using the kernel of the (right-sided) quaternion Fourier transform [11,12] was introduced in [13]. In the present paper, we continue this generalization to real Clifford algebra Cl 0,n .…”

Section: Introductionmentioning

confidence: 84%

A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

Bahri

2013

Quaternion and Clifford Fourier Transforms and Wavelets

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Section: Introductionmentioning

confidence: 84%

A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

Bahri

2013

Quaternion and Clifford Fourier Transforms and Wavelets

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“…This made possible a quaternion Fourier transform of a one-dimensional signal: [24,25], Hitzer thoroughly studied the quaternion Fourier transform (QFT) applied to quaternion-valued functions in [54]. As part of this work a quaternion split 5) was devised and applied, which led to a better understanding of GL(R 2 ) transformation properties of the QFT spectrum of two-dimensional images, including colour images, and opened the way to a generalization of the QFT concept to a full spacetime Fourier transformation (SFT) for spacetime algebra C 3,1 -valued signals.…”

Section: Quaternion Fourier Transforms (Qft)mentioning

confidence: 99%

“…So the relative order of the factors in F n {f }(ω) becomes important, see [66] for a systematic investigation and comparison. In the context of generalizing quaternion Fourier transforms (QFT) via algebra isomorphisms to higher dimensional Clifford algebras, Hitzer [54] constructed a spacetime Fourier transform (SFT) in the full algebra of spacetime C 3,1 , which includes the CFT (2.1) as a partial transform of space. Implemented analogous (isomorphic) to the orthogonal 2D planes split of quaternions, the SFT permits a natural spacetime split, which algebraically splits the SFT into right-and left propagating multivector wave packets.…”

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

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 plays an important role in quaternionic signal processing. Based on the Heisenberg type uncertainty principle for the quaternion Fourier transform (QFT) [13,14], the authors in [7] proposed a component-wise uncertainty principle associated with the CQWT.…”

Section: Introductionmentioning

confidence: 99%

A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

Bahri

Ashino

2017

Abstract and Applied Analysis

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The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

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Quaternion Fourier Transform on Quaternion Fields and Generalizations

Cited by 238 publications

References 23 publications

A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

A Generalized Windowed Fourier Transform in Real Clifford Algebra Cl 0,n

Quaternion and Clifford Fourier Transforms and Wavelets

A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

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