Discrete tomography by convex–concave regularization and D.C. programming

• Published in: Discrete Applied Mathematics Vol. 151; no. 1; pp. 229 - 243
• Main Authors: Schüle, T., Schnörr, C., Weber, S., Hornegger, J.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Lausanne Elsevier B.V 01.10.2005; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Artificial intelligence; Calculus of variations and optimal control; Combinatorial optimization; Computer science; control theory; systems; Concave minimization; D.C. programming; Discrete tomography; Exact sciences and technology; Mathematical analysis; Mathematics; Pattern recognition. Digital image processing. Computational geometry; Sciences and techniques of general use; Theoretical computing; Discrete tomography; D.C. programming; Combinatorial optimization; Concave minimization; Parallel algorithm; Error estimation; Computer theory; Decomposition method; DC programming; Projection; Regularization method; Tomography; Convex function; Robustness; Objective function; Performance
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2005.02.028

We present a novel approach to the tomographic reconstruction of binary objects from few projection directions within a limited range of angles. A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing non-homogeneous regions, is supplemented with a concave functional enforcing binary solutions. Application of a primal-dual subgradient algorithm to a suitable decomposition of the objective functional into the difference of two convex functions leads to an algorithm which provably converges with parallel updates to binary solutions. Numerical results demonstrate robustness against local minima and excellent reconstruction performance using five projections within a range of 90 ∘ . Our approach is applicable to quite general objective functions over binary variables with constraints and thus applicable to a wide range of problems within and beyond the field of discrete tomography.