  In this paper, we report the development of the generalized proximal
smoothing (GPS) algorithm for phase retrieval of noisy data. GPS is a
optimization-based algorithm, in which we relax both the Fourier magnitudes and
object constraints. We relax the object constraint by introducing the
generalized Moreau-Yosida regularization and heat kernel smoothing. We are able
to readily handle the associated proximal mapping in the dual variable by using
an infimal convolution. We also relax the magnitude constraint into a least
squares fidelity term, whose proximal mapping is available. GPS alternatively
iterates between the two proximal mappings in primal and dual spaces,
respectively. Using both numerical simulation and experimental data, we show
that GPS algorithm consistently outperforms the classical phase retrieval
algorithms such as hybrid input-output (HIO) and oversampling smoothness (OSS),
in terms of the convergence speed, consistency of the phase retrieval, and
robustness to noise.
