Analysis-form sparse phase retrieval using variable-splitting

Analysis-form sparsity, like total variation regularization, is incorporated into a variable splitting framework for the phase retrieval image reconstruction problem. This framework effectively applies variable splitting and the alternating direction method of multipliers to recover the Fourier transform coefficients of an image that are consistent with the squared-magnitude measurements of those Fourier coefficients and have a compressible or sparse isotropic total variation image-domain transform representation. The robustness to additive noise and outliers is demonstrated on a moderately sized Shepp-Logan phantom image showing good quality reconstructions for 60 dB additive white Gaussian noise and anywhere from zero to 1% (41) outliers.