X-ray tomography is a widely used imaging technique that allows one to produce internal images in materials science or medical applications. An X-ray beam is transmitted through the sample and the projection is recorded on a detector at different angles. The most widely used reconstruction algorithm of the original image from these measurements provides satisfying results only for a large number of such projections. In order to reduce the acquisition time or the total dose absorbed by the patient, tomography users need other reconstruction algorithms that incorporate known properties of the original image. In particular, images with a discrete set of absorption values and few interfaces are frequently encountered in materials science.
Inspired by results from statistical physics of the Ising model, Lenka Zdeborova and her colleagues  have proposed a novel message-passing algorithm for discrete tomography. The algorithm works on two levels: it is iteratively sending probabilistic messages between the different light rays and different set of messages within each light ray in order to satisfy the observed values of measurements. The resulting algorithm is fast, and entirely distributed. The authors showed that for binary images an accurate reconstruction is obtained even for a very low number of measures, and its performance is robust to measurement noise. This opens the way to a new class of algorithms for practical applications of discrete tomographic reconstruction.
 Belief propagation reconstruction for discrete tomography
E. Gouillart, F. Krzakala, M. Mezard, L. Zdeborova
Inverse Problems 29,3 (2013) 035003.