Book chapter
Scaling, Wavelets, Image Compression, and Encoding
Analysis for Science, Engineering and Beyond, pp.215-252
Springer Proceedings in Mathematics, Springer Berlin Heidelberg
05/18/2011
DOI: 10.1007/978-3-642-20236-0_8
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 ON, 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 ON, and tensor products is that we get easy formulas for generating all the choices of matrices going into our algorithms.
Details
- Title: Subtitle
- Scaling, Wavelets, Image Compression, and Encoding
- Creators
- Palle E. T Jorgensen - University of IowaMyung-Sin Song - Southern Illinois University Edwardsville
- Resource Type
- Book chapter
- Publication Details
- Analysis for Science, Engineering and Beyond, pp.215-252
- Publisher
- Springer Berlin Heidelberg; Berlin, Heidelberg
- Series
- Springer Proceedings in Mathematics
- DOI
- 10.1007/978-3-642-20236-0_8
- eISSN
- 2190-5622
- ISSN
- 2190-5614
- Language
- English
- Date published
- 05/18/2011
- Academic Unit
- Mathematics
- Record Identifier
- 9984241038902771
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