Directional Uncertainty Principle for Quaternion Fourier Transform

Hitzer, Eckhard

doi:10.1007/s00006-009-0175-2

Cited by 86 publications

(48 citation statements)

References 24 publications

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“…That is, increasing the knowledge of the position decreases the knowledge of the velocity or momentum of an electron. In quaternion analysis some researches combined the uncertainty relations and the QFT [2,15,31,42]. In this section we generalize the uncertainty principle for measurable sets associated with QFT.…”

Section: Uncertainty Principlesmentioning

confidence: 99%

Uncertainty principle for measurable sets and signal recovery in quaternion domains

Kou

Yang

Zou

2017

Math Methods in App Sciences

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The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the Quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and timelimiting data.

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Section: Uncertainty Principlesmentioning

confidence: 99%

Uncertainty principle for measurable sets and signal recovery in quaternion domains

Kou

Yang

Zou

2017

Math Methods in App Sciences

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“…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. This was followed up by the establishment of a fully directional (opposed to componentwise) uncertainty principle for the QFT and the SFT [58]. Independently Mawardi et al [77] established a componentwise uncertainty principle for the QFT.…”

Section: Quaternion Fourier Transforms (Qft)mentioning

confidence: 99%

“…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. This analysis allows to compute the effect of Lorentz transformations on the spectra of these wavepackets, as well as a 4D directional spacetime uncertainty formula [58] for spacetime signals.…”

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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“…Motivated by the authors in [15][16][17], in the present paper, we propose the directional uncertainty principle related to the CQWT and then apply this uncertainty to obtain a variation on the Heisenberg type uncertainty principle and the logarithmic uncertainty principle in the context of the CQWT. The uncertainty principle describes the relation between the QFT of a quaternion function and its 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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Directional Uncertainty Principle for Quaternion Fourier Transform

Cited by 86 publications

References 24 publications

Uncertainty principle for measurable sets and signal recovery in quaternion domains

Uncertainty principle for measurable sets and signal recovery in quaternion domains

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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