Optimized signal expansions for sparse representation

Traditional signal decompositions such as transforms, filterbanks, and wave lets generate signal expansions using the analysis-synthesis setting: The e xpansion coefficients are found by taking the inner product of the signal,v ith the corresponding analysis vector. In this paper, we try to free oursel ves from the analysis-synthesis paradigm by concentrating on the synthesis or reconstruction part of the signal expansion. Ignoring the analysis issue completely, we construct sets of synthesis vectors, which are denoted wave form dictionaries, for efficient signal representation, Within this framewo rk, we present an algorithm for designing waveform dictionaries that allow sparse representations: The objective is to approximate a training signal u sing a small number of dictionary vectors. Our algorithm optimizes the dict ionary vectors with respect to the average nonlinear approximation error, i .e,, the error resulting when keeping a fixed number n of expansion coeffic ients but not necessarily the first n coefficients. Using signals from a Ga ussian, autoregressive process with correlation factor 0.95, it is demonstr ated that for established signal expansions like the Karhunen-Loeve transfo rm, the lapped orthogonal transform, and the biorthogonal 7/9 wavelet, it i s possible to improve the approximation capabilities by up to 30% by fine t uning of the expansion vectors.