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In imaging modalities recording diffraction data, the original image can bereconstructed assuming known phases. When phases are unknown, oversampling anda constraint on the support region in the original object can be used to solvea non-convex optimization problem.  Such schemes are ill-suited to find the optimum solution for sparse data,since the recorded image does not correspond exactly to the original wavefunction. We construct a convex optimization problem using a relaxed supportconstraint and a maximum-likelihood treatment of the recorded data as a samplefrom the underlying wave function. We also stress the need to use relevantwindowing techniques to account for the sampled pattern being finite.  On simulated data, we demonstrate the benefits of our approach in terms ofvisual quality and an improvement in the crystallographic R-factor from .4 to.1 for highly noisy data.

Using Convex Optimization of Autocorrelation with Constrained Support and Windowing for Improved Phase Retrieval Accuracy