  Ptychography has risen as a reference X-ray imaging technique: it achieves
resolutions of one billionth of a meter, macroscopic field of view, or the
capability to retrieve chemical or magnetic contrast, among other features. A
ptychographyic reconstruction is normally formulated as a blind phase retrieval
problem, where both the image (sample) and the probe (illumination) have to be
recovered from phaseless measured data. In this article we address a nonlinear
least squares model for the blind ptychography problem with constraints on the
image and the probe by maximum likelihood estimation of the Poisson noise
model. We formulate a variant model that incorporates the information of
phaseless measurements of the probe to eliminate possible artifacts. Next, we
propose a generalized alternating direction method of multipliers designed for
the proposed nonconvex models with convergence guarantee under mild conditions,
where their subproblems can be solved by fast element-wise operations.
Numerically, the proposed algorithm outperforms state-of-the-art algorithms in
both speed and image quality.
