International audienceWe propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains

In this article, we discuss the evaluation of Blind Audio Source Separation (BASS) algorithms. Depending on the exact application, different distortions can be allowed between an estimated source and the wanted true source. We consider four different sets of such allowed distortions, from time-invariant gains to time-varying filters. In each case we decompose the estimated source into a true source part plus error terms corresponding to interferences, additive noise and algorithmic artifacts. Then we derive a global performance measure using an energy ratio, plus a separate performance measure for each error term. These measures are computed and discussed on the results of several BASS problems with various difficulty levels.

International audienceThe purpose of this correspondence is to generalize a result by Donoho and Huo and Elad and Bruckstein on sparse representations of signals in a union of two orthonormal bases for R^N. We consider general (redundant) dictionaries for R^N, and derive sufficient conditions for having unique sparse representations of signals in such dictionaries. The special case where the dictionary is given by the union of L \ge 2 orthonormal bases for R^N is studied in more detail. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of the \ell^0 optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant