  The single-scatter approximation is fundamental in many tomographic imaging
problems including x-ray scatter imaging and optical scatter imaging for
certain media. In all cases, noisy measurements are affected by both local
scatter events and nonlocal attenuation. Prior works focus on reconstructing
one of two images: scatter density or total attenuation. However, both images
are media specific and useful for object identification.
  Nonlocal effects of the attenuation image on the data are summarized by the
broken ray transform (BRT). While analytic inversion formulas exist, poor
conditioning of the inverse problem is only exacerbated by noisy measurements
and sampling errors. This has motivated interest in the related star transforms
incorporating BRT measurements from multiple source-detector pairs. However,
all analytic methods operate on the log of the data. For media comprising
regions with no scatter a new approach is required.
  We are the first to present a joint estimation algorithm based on Poisson
data models for a single-scatter measurement geometry. Monotonic reduction of
the log-likelihood function is guaranteed for our iterative algorithm while
alternating image updates. We also present a fast algorithm for computing the
discrete BRT forward operator. Our generalized approach can incorporate both
transmission and scatter measurements from multiple source-detector pairs.
Transmission measurements resolve low-frequency ambiguity in the joint image
estimation problem, while multiple scatter measurements resolve the attenuation
image. The benefits of joint estimation, over single-image estimation, vary
with problem scaling. Our results quantify these benefits and should inform
design of future acquisition systems.
