Analytic signal in many dimensions

Mikhail Tsitsvero 1 Pierre Borgnat 2 Paulo Gonçalves 1
1 DANTE - Dynamic Networks : Temporal and Structural Capture Approach
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme, IXXI - Institut Rhône-Alpin des systèmes complexes
Abstract : In this paper we extend analytic signal to the multidimensional case. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phase. Secondly we define the proper hypercomplex analytic signal as a holo-morphic hypercomplex function on the boundary of polydisk in the hypercomplex space. Next it is shown that the correct phase-shifted components can be obtained by positive frequency restriction of the Scheffers-Fourier transform based on the commutative and associative algebra generated by the set of elliptic hypercomplex numbers. Moreover we demonstrate that for d > 2 there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly. Finally the euclidean-domain construction of instantaneous amplitude is extended to manifold and manifold-like graphs and point clouds.
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Submitted on : Wednesday, September 19, 2018 - 4:51:30 PM
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Mikhail Tsitsvero, Pierre Borgnat, Paulo Gonçalves. Analytic signal in many dimensions. 2018. ⟨hal-01877403⟩

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