{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T21:28:48Z","timestamp":1781299728816,"version":"3.54.1"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T00:00:00Z","timestamp":1497916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e19060289","type":"journal-article","created":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T10:15:38Z","timestamp":1497953738000},"page":"289","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Mehler-Fock Transform in Signal Processing"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7557-4904","authenticated-orcid":false,"given":"Reiner","family":"Lenz","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Link\u00f6ping University, SE-581 83 Link\u00f6ping, Sweden"},{"name":"Department of Science and Technology, Link\u00f6ping University, SE-60174 Norrk\u00f6ping, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S1076-5670(05)38001-3","article-title":"Spectral Color Spaces: Their Structure and Transformations","volume":"138","author":"Lenz","year":"2005","journal-title":"Adv. Imaging Electron Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1017\/S0263574707003906","article-title":"Lie methods for color robot vision","volume":"26","author":"Lenz","year":"2008","journal-title":"Robotica"},{"key":"ref_3","unstructured":"Hastie, T., Tibshirani, R., and Friedman, J. (2013). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1109\/TIT.2017.2653803","article-title":"Riemannian Gaussian distributions on the space of symmetric positive definite matrices","volume":"63","author":"Said","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","unstructured":"Vilenkin, N., and Klimyk, A. (2012). Representation of Lie Groups and Special Functions: Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms, Springer."},{"key":"ref_6","unstructured":"Sneddon, I.N. (1972). 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Vis."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1364\/JOSAA.32.000381","article-title":"Reference data set for camera spectral sensitivity estimation","volume":"32","author":"Darrodi","year":"2015","journal-title":"J. Opt. Soc. Am. A Opt. Image Sci. Vis."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1137\/070712006","article-title":"Computing the Conical Function \n        \n          \n\t\t  \n             \n                 p\n\t\t\t\t \n                -\n                 1\n                 \/\n                 2\n                  +\n                  i\n                  \u03c4\n                  \n\t\t\t\t  \u03bc\n            \n\t\t\t\n                x","volume":"31","author":"Gil","year":"2009","journal-title":"SIAM J. Sci. 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NIST Handbook of Mathematical Functions Hardback and CD-ROM, Cambridge University Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/6\/289\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:39:42Z","timestamp":1760207982000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/6\/289"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,20]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2017,6]]}},"alternative-id":["e19060289"],"URL":"https:\/\/doi.org\/10.3390\/e19060289","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,6,20]]}}}