Utilising duality in discrete tomography problems

Abstract

We consider the problem of reconstructing binary images given the line sums (projections) of the cell values along a few prescribed directions. This problem is computationally hard, and the projection data can be noisy, so in many cases finding an exact reconstruction is not feasible. For this reason we work with a least-squares formulation of the problem, where the aim is to find a binary image of which the vector of its line sums is close to the given projection data. In this paper, we will mostly focus on preprocessing algorithms that can partially reconstruct the image, assigning values to some of the cells but leaving others undetermined.

The two main approaches we investigated are both based on some form of duality. The first one is based on Langrangian duality: in a recent paper, A. Kadu and T. van Leeuwen introduced a dual formulation of the discrete tomography problem and used the optimal solution of the dual problem to partially reconstruct the binary image. We will further analyse this approach. In particular, we will show that a certain relaxation gives raise to an equivalent dual problem, which opens up ways to compute the dual optimal through different methods. Additionally, we give sufficient conditions under which a non-optimal dual solution enforces the same cell value assignment as the optimal solution.

The approach is based on roof duality, a concept introduced in 1984 as a way to find lower bounds on real, quadratic functions with a binary domain. Additionally, methods that compute the roof duality lower bound can also identify some variable assignments that must hold for any minimiser of the function. While algorithms based on roof duality have become more sophisticated over the years, to our knowledge it has not been applied in the field of discrete tomography so far. We will show that in many cases roof duality based methods will not be able find good partial reconstructions.