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Scaling, Wavelets, Image Compression, and Encoding

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Analysis for Science, Engineering and Beyond

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 6))

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Abstract

In this paper we develop a family of multi-scale algorithms with the use of filter functions in higher dimension.

While our primary application is to images, i.e., processes in two dimensions, we prove our theorems in a more general context, allowing dimension 3 and higher.

The key tool for our algorithms is the use of tensor products of representations of certain algebras, the Cuntz algebras O N , from the theory of algebras of operators in Hilbert space. Our main result offers a matrix algorithm for computing coefficients for images or signals in specific resolution subspaces. A special feature with our matrix operations is that they involve products and iteration of slanted matrices. Slanted matrices, while large, have many zeros, i.e., are sparse. We prove that as the operations increase the degree of sparseness of the matrices increase. As a result, only a few terms in the expansions will be needed for achieving a good approximation to the image which is being processed. Our expansions are local in a strong sense.

An additional advantage with the use of representations of the algebras O N , and tensor products is that we get easy formulas for generating all the choices of matrices going into our algorithms.

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Acknowledgements

The first named author was supported in part by a grant from the US NSF. Also first named author acknowledges partial support of the Swedish Foundation for International Cooperation in Research and Higher education (STINT) during his visits to Lund University.

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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

Authors
  1. Palle E. T. Jorgensen
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  2. Myung-Sin Song
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Corresponding author

Correspondence to Palle E. T. Jorgensen .

Editor information

Editors and Affiliations

  1. Centre for Mathematical Sciences, Lund University, Sölvegatan 18, Lund, 221 00, Sweden

    Kalle Åström

  2. Dept. Mathematics, Luleå University of Technology, Luleå, 971 87, Sweden

    Lars-Erik Persson

  3. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Sergei D. Silvestrov

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Jorgensen, P.E.T., Song, MS. (2012). Scaling, Wavelets, Image Compression, and Encoding. In: Åström, K., Persson, LE., Silvestrov, S. (eds) Analysis for Science, Engineering and Beyond. Springer Proceedings in Mathematics, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20236-0_8

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