Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints.

The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Fréchet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp-Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.