Unitary mappings and an equivalence relation between multiresolution analyses of L2(R)

Manos Papadakis

doi:10.1117/12.217599

1 September 1995 Unitary mappings and an equivalence relation between multiresolution analyses of L²(R)

Manos Papadakis

Proceedings Volume 2569, Wavelet Applications in Signal and Image Processing III; (1995) https://doi.org/10.1117/12.217599
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

This paper provides classes of unitary operations of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We also develop an equivalence relation between multiresolution analyses of L²(R). This relation called unitary equivalence is created by the action of a group of unitary operators contained in all the classes mentioned previously, in a way that the multiresolution structure and the Decomposition and Reconstruction algorithms remain invariant. A characterization of this relation in terms of the scaling functions is given. Distinct equivalence classes of multiresolution analyses are derived. Finally, we prove that B-splines give rise to non-equivalent examples.

Citation Download Citation

Manos Papadakis "Unitary mappings and an equivalence relation between multiresolution analyses of L²(R)", Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, (1 September 1995); https://doi.org/10.1117/12.217599

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