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.

Date Published: 4 September 2009
PDF: 13 pages
Proc. SPIE 7446, Wavelets XIII, 74460R (4 September 2009); doi: 10.1117/12.825587

Published in SPIE Proceedings Vol. 7446:
Wavelets XIII
Vivek K. Goyal; Manos Papadakis; Dimitri Van De Ville, Editor(s)