Statistical methods


Sparse Representation of Signals:

A signal is said to be sparse if it can be represented in a basis or frame (Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting the obtained coefficients, sorted by their decreasing absolute values, exhibits a polynomial decay. The basis or frame is called the dictionary. Note that most natural signals and images are compressible in an appropriate dictionary. Faster is the decay, better it is, since a very good approximation of the signal can be obtained from a few coefficients. For instance, for a signal composed of a sine, the Fourier dictionary is optimal from a sparse point of view since all information is contained in a single coefficient. Wavelets have been extremely successful to represent images, most natural images present a sparse behavior in the wavelet domain, and this explains why wavelets have been chosen in the JPG2000 image compression norm. Other representations, such as curvelets, are more adequate when the data contains filaments. We have been working on several ill posed inverse problems where we have shown that sparsity is a very efficient way to regularize the problem in order to get a unique and stable solution (Starck et al, Cambridge University Press, book, 2010):