  Reconstruction of images from noisy linear measurements is a core problem in
image processing, for which convex optimization methods based on total
variation (TV) minimization have been the long-standing state-of-the-art. We
present an alternative probabilistic reconstruction procedure based on
approximate message-passing, Scampi, which operates in the compressive regime,
where the inverse imaging problem is underdetermined. While the proposed method
is related to the recently proposed GrAMPA algorithm of Borgerding, Schniter,
and Rangan, we further develop the probabilistic approach to compressive
imaging by introducing an expectation-maximizaiton learning of model
parameters, making the Scampi robust to model uncertainties. Additionally, our
numerical experiments indicate that Scampi can provide reconstruction
performance superior to both GrAMPA as well as convex approaches to TV
reconstruction. Finally, through exhaustive best-case experiments, we show that
in many cases the maximal performance of both Scampi and convex TV can be quite
close, even though the approaches are a prori distinct. The theoretical reasons
for this correspondence remain an open question. Nevertheless, the proposed
algorithm remains more practical, as it requires far less parameter tuning to
perform optimally.
