  A variational approach to reconstruction of phase and amplitude of a
complex-valued object from Poissonian intensity observations is developed. The
observation model corresponds to the typical optical setups with a phase
modulation of wavefronts. The transform domain sparsity is applied for the
amplitude and phase modeling. It is demonstrated that this modeling results in
the essential advantage of the developed algorithm for heavily noisy
observations corresponding to a short exposure time in optical experiments. We
consider also two simplified versions of this algorithm where the sparsity
modeling of phase and amplitude is omitted. In the simulation study we compare
the developed algorithms versus the Gerchberg-Saxton and truncation Wirtinger
flow algorithms. The latter algorithm being the maximum likelihood based is the
state-of-the-art for the phase retrieval from Poissonian observations. For
noisy and very noisy observations the proposed algorithm demonstrates a
valuable advantage.
