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Phase retrieval for Fresnel measurements using a shearlet sparsity constraint

We consider the problem of phase retrieval in the Fresnel regime. In recent years, several techniques have been used to solve this problem applying different a priori assumptions on the two-dimensional object in space, such as positivity, finite support and amplitude constraints. In this paper, we propose a new constraint, namely the assumption that the object possesses a sparse representation in a shearlet frame. We show how a shearlet soft-thresholding procedure can be used for phase reconstruction with Fresnel data. As it turns out, the shearlet sparsity constraint yields reconstruction results that are far superior to the support constraint, and comparable to the support plus positivity constraint.