Adaptive Time - Frequency Decompositions with Matching

Computing the optimal expansion of a signal in a redundant dictionary of waveforms is an NP-hard problem. We introduce a greedy algorithm, called a matching pursuit, which computes a sub-optimal expansion. The dictionary waveforms which best match a signal's structures are chosen iteratively. An orthogonalized version of the matching pursuit is also developed. Matching pursuits are general procedures for computing adaptive signal representations. With a dictionary of Gabor functions, a matching pursuit deenes an adaptive time-frequency transform. We derive a signal energy distribution in the time-frequency plane which does not contain interference terms, unlike the Wigner and Cohen class distributions. Matching pursuits are chaotic maps whose attractors deene a generic noise with respect to the dictionary. We derive an algorithm that isolates the coherent structures of a signal and describe an application to pattern extraction from noisy signals.