Belief Propagation Reconstruction for Discrete Tomography

Emmanuelle Gouillart (1) , Florent Krzakala (2) , Marc Mezard (3) , Lenka Zdeborová (4)

1 SVI - Surface du Verre et Interfaces
2 LPCT - Laboratoire de Physico-Chimie Théorique
3 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
4 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We consider the reconstruction of a two-dimensional discrete image from a set of tomographic measurements corresponding to the Radon projection. Assuming that the image has a structure where neighbouring pixels have a larger probability to take the same value, we follow a Bayesian approach and introduce a fast message-passing reconstruction algorithm based on belief propagation. For numerical results, we specialize to the case of binary tomography. We test the algorithm on binary synthetic images with different length scales and compare our results against a more usual convex optimization approach. We investigate the reconstruction error as a function of the number of tomographic measurements, corresponding to the number of projection angles. The belief propagation algorithm turns out to be more efficient than the convex-optimization algorithm, both in terms of recovery bounds for noise-free projections, and in terms of reconstruction quality when moderate Gaussian noise is added to the projections.

Mots clés: tomography reconstruction; discrete tomography; belief propagation; approximate message passing

Domaines: Analyse numérique [cs.NA]

Emmanuelle Gouillart, Florent Krzakala, Marc Mezard, Lenka Zdeborová. Belief Propagation Reconstruction for Discrete Tomography. Inverse Problems, 2013, 29 (3), pp.035003. ⟨10.1088/0266-5611/29/3/035003⟩. ⟨hal-00750511v2⟩