Robust phase retrieval with sparsity under nonnegativity constraints

High-resolution phase retrieval is challenging due to the low signal-to-noise ratio of measurements. This work utilizes variable splitting and alternating minimization to simultaneously enforce a 1-norm data fit penalty, an analysis-form sparse regularizer, and nonnegativity or real-valued image constraints to resolve an image from squared-magnitude measurements. The reconstruction algorithm incorporates real-valued and nonnegativity constraints via enforcing Hermitian symmetry on Fourier transform coefficients and projecting the reconstructed image onto the nonnegative orthant. The proposed method preserves image quality more robustly than unconstrained analysis-form sparse phase retrieval on the Shepp-Logan phantom, for a range of noise levels and outlier measurements.