Sparse and Structured Decompositions of Audio Signals in Overcomplete Spaces
We investigate the notion of “sparse decompositions” of audio signals in overcomplete spaces, ie when the number of basis functions is greater than the number of signal samples. We show that, with a low degree of overcompleteness (typically 2 or 3 times), it is possible to get good approximation of the signal that are sparse, provided that some “structural” information is taken into account, ie the localization of significant coefficients that appears to form clusters. This is illustrated with decompositions on a union of local cosines (MDCT) and discrete wavelets (DWT), that are shown to perform well on percussive signals, a class of signals that is difficult to sparsely represent on pure (local) Fourier bases. Finally, the obtained clusters of individuals atoms are shown to carry higher levels of information, such as a parametrization of partials or attacks, and this is potentially useful in an information retrieval context.