Discrete Tomography by Continuous Multilabeling Subject to Projection Constraints

A Novel Convex Relaxation for Non-binary Discrete Tomography

“…Comparison to previously used relaxation. It can be shown that the algorithms in [9,19,17,3] use the following relaxation.…”

“…Several algorithms have been proposed to solve the discrete tomography problem. Among them are (i) linear programming-based algorithms [9,17], (ii) belief propagation [8], (iii) network flow techniques [3], (iv) convex-convave programming [19,14] (v) evolutionary algorithms [2] and other heuristic algorithms [4,6,11,12]. Not all approaches are applicable to the general discrete tomography problem we treat here: algorithms [9,17,8,6,11,12] only support binary labels, while [4,3] solves only the feasiblity problem and does not permit any energy.…”

“…Not all approaches are applicable to the general discrete tomography problem we treat here: algorithms [9,17,8,6,11,12] only support binary labels, while [4,3] solves only the feasiblity problem and does not permit any energy. Algorithms [19,14] are applicable to the setting we propose but are purely primal algorithms and do not output dual lower bounds (which we do). Hence, one cannot judge proximity of solutions computed by [19,14] to global optimal ones, prohibiting its use in branch and bound.…”

“…Hence, one cannot judge proximity of solutions computed by [19,14] to global optimal ones, prohibiting its use in branch and bound. In case convex relaxations were considered [9,19,17,3], they were less tight than the one we propose, leading to inferior dual bounds.…”

“…Algorithms [19,14] are applicable to the setting we propose but are purely primal algorithms and do not output dual lower bounds (which we do). Hence, one cannot judge proximity of solutions computed by [19,14] to global optimal ones, prohibiting its use in branch and bound. In case convex relaxations were considered [9,19,17,3], they were less tight than the one we propose, leading to inferior dual bounds.…”