A Convex Formulation for Binary Tomography

Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a constrained least-squares problem. As the constraints are nonconvex, many approaches for solving it rely on either relaxing the constraints or heuristics. In this paper, we propose a novel convex formulation, based on the Lagrange dual of the constrained least-squares problem. The resulting problem is a generalized least absolute shrinkage and selection operator problem, which can be solved efficiently. It is a relaxation in the sense that it can only be guaranteed to give a feasible solution, not necessarily the optimal one. In exhaustive experiments on small images ( $2\times 2$ , $3\times 3$ , $4\times 4$ ), we find, however, that if the problem has a unique solution, our dual approach finds it. In the case of multiple solutions, our approach finds the commonalities between the solutions. Further experiments on realistic numerical phantoms and an experiment on the X-ray dataset show that our method compares favorably to Total Variation and DART.