Wavefront Reconstruction from Noisy Fringe Observations via Sparse Coding

In this paper, we use sparse modeling for processing phase-shifting interferometry measurements. The proposed approach takes into full consideration the Poissonian (photon counting) measurements. In this way we are targeting at optimal sparse reconstruction both phase and magnitude taking into consideration all details of the observation formation. Many images (and signals) admit sparse representations in the sense that they are well approximated by linear combinations of a small number of functions taken from a know set. The topic of sparse and redundant representations has attracted tremendous interest from the research community in the last ten years. This interest stems from the role that the low dimensional models play in many signal and image areas such as compression, restoration, classification, and design of priors and regularizers, just to name a few [1]. Let n R ∈ c denote a vector representing an image, or a patch of it, and let us assume that c admits a sparse representation or sparse coding with respect to the columns of a given matrix m n× ∈R Ψ ; i.e., it is possible to write θ Ψ c = , where m R ∈ θ is a vector containing only a few non-zero components. The matrix Ψ is termed a synthesis operator (or dictionary) because in the writing