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The Mehler-Fock Transform in Signal Processing
Linköping University, Department of Science and Technology, Media and Information Technology. Linköping University, Department of Electrical Engineering. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-7557-4904
2017 (English)In: Entropy, ISSN 1099-4300, E-ISSN 1099-4300, Vol. 19, no 6, 289Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
MDPI AG , 2017. Vol. 19, no 6, 289
Keyword [en]
Mehler-Fock transform; kernel density estimator; signal processing; harmonic analysis; SU(1, 1); positive signals
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-139401DOI: 10.3390/e19060289ISI: 000404454500051OAI: oai:DiVA.org:liu-139401DiVA: diva2:1129889
Note

Funding Agencies|Swedish Research Council [2014-6227]

Available from: 2017-08-07 Created: 2017-08-07 Last updated: 2017-11-29

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Lenz, Reiner
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