DC-Programming versus ℓ₀-Superiorization for Discrete Tomography

In this paper we focus on the reconstruction of sparse solutions to underdetermined systems of linear equations with variable bounds. The problem is motivated by sparse and gradient-sparse reconstruction in binary and discrete tomography from limited data. To address the ℓ₀-minimization problem we consider two approaches: DC-programming and ℓ₀-superiorization. We show that ℓ₀-minimization over bounded polyhedra can be equivalently formulated as a DC program. Unfortunately, standard DC algorithms based on convex programming often get trapped in local minima. On the other hand, ℓ₀-superiorization yields comparable results at significantly lower costs.