Optimality in the design of overcomplete decompositions

Nick Kingsbury; H. Joel Trussell

doi:10.1117/12.825587

4 September 2009 Optimality in the design of overcomplete decompositions

Nick Kingsbury, H. Joel Trussell

Proceedings Volume 7446, Wavelets XIII; 74460R (2009) https://doi.org/10.1117/12.825587
Event: SPIE Optical Engineering + Applications, 2009, San Diego, California, United States

Abstract

We lay a philosophical framework for the design of overcomplete multidimensional signal decompositions based on the union of two or more orthonormal bases. By combining orthonormal bases in this way, tight (energy preserving) frames are automatically produced. The advantage of an overcomplete (tight) frame over a single orthonormal decomposition is that a signal is likely to have a more sparse representation among the overcomplete set than by using any single orthonormal basis. We discuss the question of the relationship between pairs of bases and the various criteria that can be used to measure the goodness of a particular pair of bases. A particular case considered is the dual-tree Hilbert-pair of wavelet bases. Several definitions of optimality are presented along with conjectures about the subjective characteristics of the ensembles where the optimality applies. We also consider relationships between sparseness and approximate representations.

Citation Download Citation

Nick Kingsbury and H. Joel Trussell "Optimality in the design of overcomplete decompositions", Proc. SPIE 7446, Wavelets XIII, 74460R (4 September 2009); https://doi.org/10.1117/12.825587

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