Optimized signal expansions for sparse representation

Traditional signal decompositions such as transforms, filterbanks, and wavelets 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, which are denoted waveform dictionaries, for efficient signal representation. Within this framework, we present an algorithm for designing waveform dictionaries that allow sparse representations: The objective is to approximate a training signal using a small number of dictionary vectors. Our algorithm optimizes the dictionary vectors with respect to the average nonlinear approximation error, i.e., the error resulting when keeping a fixed number of expansion coefficients but not necessarily the first coefficients. Using signals from a Gaussian, autoregressive process with correlation factor 0.95, it is demonstrated that for established signal expansions like the Karhunen–Loéve transform, the lapped orthogonal transform, and the biorthogonal 7/9 wavelet, it is possible to improve the approximation capabilities by up to 30% by fine tuning of the expansion vectors.