  This paper proposes a novel algorithm for image phase retrieval, i.e., for
recovering complex-valued images from the amplitudes of noisy linear
combinations (often the Fourier transform) of the sought complex images. The
algorithm is developed using the alternating projection framework and is aimed
to obtain high performance for heavily noisy (Poissonian or Gaussian)
observations. The estimation of the target images is reformulated as a sparse
regression, often termed sparse coding, in the complex domain. This is
accomplished by learning a complex domain dictionary from the data it
represents via matrix factorization with sparsity constraints on the code
(i.e., the regression coefficients). Our algorithm, termed dictionary learning
phase retrieval (DLPR), jointly learns the referred to dictionary and
reconstructs the unknown target image. The effectiveness of DLPR is illustrated
through experiments conducted on complex images, simulated and real, where it
shows noticeable advantages over the state-of-the-art competitors.
