Design of signal expansions for sparse representation

Traditional signal decompositions generate signal expansions using the analysis-synthesis setting: the expansion coefficients are found by taking the inner product of the signal with the corresponding analysis vector. In this paper we try to free ourselves 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, denoted waveform dictionaries, for sparse signal representation. The objective is to approximate a training signal using a small number of dictionary vectors. Our algorithm optimize the dictionary vectors with respect to the average non-linear approximation error. Using signals from a Gaussian, autoregressive process with correlation factor 0.95, it is demonstrated that for established signal expansions like the Karhunen-Loeve transform, the lapped orthogonal transform, and the biorthogonal 7/9 wavelet, it is possible to improve the approximation capabilities by up to 30% by optimizing the expansion vectors