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Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

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Abstract

The purpose of this paper is to show that algorithms in a diverse set of applications may be cast in the context of relations on a finite set of operators in Hilbert space. The Cuntz relations for a finite set of isometries form a prototype of these relations. Such applications as entropy encoding, analysis of correlation matrices (Karhunen-Loève), fractional Brownian motion, and fractals more generally, admit multi-scales. In signal/image processing, this may be implemented with recursive algorithms using subdivisions of frequency-bands; and in fractals with scale similarity. In Karhunen-Loève analysis, we introduce a diagionalization procedure; and we show how the Hilbert space formulation offers a unifying approach; as well as suggesting new results.

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Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, USA

    Palle E. T. Jorgensen

  2. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026, USA

    Myung-Sin Song

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  1. Palle E. T. Jorgensen
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Jorgensen, P.E.T., Song, MS. Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms. Acta Appl Math 108, 489–508 (2009). https://doi.org/10.1007/s10440-009-9529-y

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