Best Basis Compressive Sensing

This paper proposes an extension of compressive sensing that allows to express the sparsity prior in a dictionary of bases. This enables the use of a fixed set of non-adaptive linear measurements and an adaptive recovery process. This reconstruction optimizes the basis to the structure of the sensed signal. An iterative thresholding algorithm is used to perform both the recovery and the estimation of the best basis. Numerical experiments on sounds and geometrical images show that adaptivity is indeed crucial to capture the regularity of complex natural signals.