The unified theory of n-dimensional complex and hypercomplex analytic signals

Hahn, Stefan L.; Snopek, Kajetana M.

doi:10.2478/v10175-011-0021-2

Cited by 26 publications

(26 citation statements)

References 20 publications

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“…These generalizations were mainly based on non-commutative algebras like quaternions, Caley-Dickson and Clifford algebras, see e.g. [10,26,27,28,11,3,29,12,30,31,32,33,34]. In this paper we show that a commutative and associative hypercomplex algebra yields a nice approach to define multidimensional analytic signal.…”

Section: Scheffers Algebra-valued Fourier Transformmentioning

confidence: 76%

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Multidimensional Analytic Signal with Application on Graphs

Tsitsvero

Borgnat

Gonçalvès

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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Section: Scheffers Algebra-valued Fourier Transformmentioning

confidence: 76%

“…Appropriate generalizations of analytic signals to two or more dimensions and its connection with complex and hypercomplex analysis were studied in the recent decades. This was motivated by the development of signal processing methods for image analysis [3,4,5,6] and analysis of multivariate signals [7,8,9,10,11,12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Analytic Signal with Application on Graphs

Tsitsvero

Borgnat

Gonçalvès

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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“…In the last few years some generalizations of the Fourier transform (defined as in (1.1)) to the octonion and higher-order algebras appeared in the literature [17,29,30,31,32]. They are defined on the basis of the Cayley-Dickson algebras and called the Cayley-Dickson Fourier transforms.…”

Section: Introductionmentioning

confidence: 99%

A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

Błaszczyk

2020

Multidim Syst Sign Process

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The paper is devoted to the development of the octonion Fourier transform (OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a continuation and generalization of earlier work by Błaszczyk and Snopek, where they proved few essential properties of the OFT of real-valued functions, e.g. symmetry properties. The results of this article focus on proving that the OFT is well-defined for octonion-valued functions and almost all well-known properties of classical (complex) Fourier transform (e.g. argument scaling, modulation and shift theorems) have their direct equivalents in octonion setup. Those theorems, illustrated with some examples, lead to the generalization of another result presented in earlier work, i.e. Parseval and Plancherel Theorems, important from the signal and system processing point of view. Moreover, results presented in this paper associate the OFT with 3-D LTI systems of linear PDEs with constant coefficients. Properties of the OFT in context of signal-domain operations such as derivation and convolution of Rvalued functions will be stated. There are known results for the QFT, but they use the notion of other hypercomplex algebra, i.e. double-complex numbers. Considerations presented here require defining other higher-order hypercomplex structure, i.e. quadruple-complex numbers. This hypercomplex generalization of the Fourier transformation provides an excellent tool for the analysis of 3-D LTI systems.

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“…Octonion has been widely studied by Baez [11]. Recently, Many experts and scholars are dedicated to octonionic analysis and obtain some results such as Cauchy integral formula for regular function, Hardy space, Bergman space [12][13][14][15]. H. Y. Wang and his collaborators studied the right inverse of Dirac in octonion space and generalized octonionic analysis to octonionic analysis of several variables [16][17].…”

Section: Introductionmentioning

confidence: 99%