  In imaging modalities recording diffraction data, the original image can be
reconstructed assuming known phases. When phases are unknown, oversampling and
a constraint on the support region in the original object can be used to solve
a 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 wave
function. We construct a convex optimization problem using a relaxed support
constraint and a maximum-likelihood treatment of the recorded data as a sample
from the underlying wave function. We also stress the need to use relevant
windowing techniques to account for the sampled pattern being finite.
  On simulated data, we demonstrate the benefits of our approach in terms of
visual quality and an improvement in the crystallographic R-factor from .4 to
.1 for highly noisy data.
