  We consider X-ray coherent scatter imaging, where the goal is to reconstruct
momentum transfer profiles (spectral distributions) at each spatial location
from multiplexed measurements of scatter. Each material is characterized by a
unique momentum transfer profile (MTP) which can be used to discriminate
between different materials. We propose an iterative image reconstruction
algorithm based on a Poisson noise model that can account for photon-limited
measurements as well as various second order statistics of the data. To improve
image quality, previous approaches use edge-preserving regularizers to promote
piecewise constancy of the image in the spatial domain while treating each
spectral bin separately. Instead, we propose spectrally grouped regularization
that promotes piecewise constant images along the spatial directions but also
ensures that the MTPs of neighboring spatial bins are similar, if they contain
the same material. We demonstrate that this group regularization results in
improvement of both spectral and spatial image quality. We pursue an
optimization transfer approach where convex decompositions are used to lift the
problem such that all hyper-voxels can be updated in parallel and in
closed-form. The group penalty introduces a challenge since it is not directly
amendable to these decompositions. We use the alternating directions method of
multipliers (ADMM) to replace the original problem with an equivalent sequence
of sub-problems that are amendable to convex decompositions, leading to a
highly parallel algorithm. We demonstrate the performance on real data.
