Phase retrieval from incomplete data via weighted nuclear norm minimization

Recovering an unknown object from the magnitude of its Fourier transform is a phase retrieval problem. Here, we consider a much difficult case, where those observed intensity values are incomplete and contaminated by both salt-and-pepper and random-valued impulse noise. To take advantage of the low-rank property within the image of the object, we use a regularization term which penalizes high weighted nuclear norm values of image patch groups. For outliers (impulse noise) in the observation, the ℓ₁₋₂ metric is adopted as the data fidelity term. Then we break down the resulting optimization problem into smaller ones, for example, weighted nuclear norm proximal mapping and ℓ₁₋₂ minimization, because the nonconvex and nonsmooth subproblems have available closed-form solutions. The convergence results are also presented, and numerical experiments are provided to demonstrate the superior reconstruction quality of the proposed method.