Random Models for Sparse Signals Expansion on Unions of Bases With Application to Audio Signals

A new approach for signal expansion with respect to hybrid dictionaries, based upon probabilistic modeling is proposed and studied. The signal is modeled as a sparse linear combination of waveforms, taken from the union of two orthonormal bases, with random coefficients. The behavior of the analysis coefficients, namely inner products of the signal with all basis functions, is studied in details, which shows that these coefficients may generally be classified in two categories: significant coefficients versus insignificant coefficients. Conditions ensuring the feasibility of such a classification are given. When the classification is possible, it leads to efficient estimation algorithms, that may in turn be used for denoising or coding purposes. The proposed approach is illustrated by numerical experiments on audio signals, using MDCT bases. However, it is general enough to be applied without much modifications in different contexts, for example in image processing.