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Entropy 2017, 19(6), 289; doi:10.3390/e19060289

The Mehler-Fock Transform in Signal Processing

1
Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden
2
Department of Science and Technology, Linköping University, SE-60174 Norrköping, Sweden
Received: 25 April 2017 / Revised: 13 June 2017 / Accepted: 15 June 2017 / Published: 20 June 2017
(This article belongs to the Special Issue Information Geometry II)
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Abstract

Many signals can be described as functions on the unit disk (ball). In the framework of group representations it is well-known how to construct Hilbert-spaces containing these functions that have the groups SU(1,N) as their symmetry groups. One illustration of this construction is three-dimensional color spaces in which chroma properties are described by points on the unit disk. A combination of principal component analysis and the Perron-Frobenius theorem can be used to show that perspective projections map positive signals (i.e., functions with positive values) to a product of the positive half-axis and the unit ball. The representation theory (harmonic analysis) of the group SU(1,1) leads to an integral transform, the Mehler-Fock-transform (MFT), that decomposes functions, depending on the radial coordinate only, into combinations of associated Legendre functions. This transformation is applied to kernel density estimators of probability distributions on the unit disk. It is shown that the transform separates the influence of the data and the measured data. The application of the transform is illustrated by studying the statistical distribution of RGB vectors obtained from a common set of object points under different illuminants. View Full-Text
Keywords: Mehler-Fock transform; kernel density estimator; signal processing; harmonic analysis; SU(1,1); positive signals Mehler-Fock transform; kernel density estimator; signal processing; harmonic analysis; SU(1,1); positive signals
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Lenz, R. The Mehler-Fock Transform in Signal Processing. Entropy 2017, 19, 289.

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