  Spectroscopic ptychography is a powerful technique to determine the chemical
composition of a sample with high spatial resolution. In spectro-ptychography,
a sample is rastered through a focused x-ray beam with varying photon energy so
that a series of phaseless diffraction data are recorded. Each chemical
component in the material under investigation has a characteristic absorption
and phase contrast as a function of photon energy. Using a dictionary formed by
the set of contrast functions of each energy for each chemical component, it is
possible to obtain the chemical composition of the material from high
resolution multi-spectral images. This paper presents SPA (Spectroscopic
Ptychography with ADMM), a novel algorithm to iteratively solve the
spectroscopic blind ptychography problem. We design first a nonlinear
spectro-ptychography model based on Poisson maximum likelihood, and construct
then the proposed method based on fast iterative splitting operators. SPA can
be used to retrieve spectral contrast when considering both a known or an
incomplete (partially known) dictionary of reference spectra. By coupling the
redundancy across different spectral measurements, the proposed algorithm can
achieve higher reconstruction quality when compared to standard
state-of-the-art two-step methods. We demonstrate how SPA can recover accurate
chemical maps from Poisson-noised measurements, and also show its enhanced
robustness when reconstructing reduced redundancy ptychography data using large
scanning stepsizes.
