A Convex Formulation for Binary Tomography

TL;DR

This paper introduces a novel convex formulation for binary tomography using the Lagrange dual, enabling efficient reconstruction from limited and noisy projection data, and demonstrating promising results compared to existing methods.

Contribution

It proposes a convex relaxation approach based on the Lagrange dual for binary tomography, improving solution feasibility and accuracy over traditional non-convex methods.

Findings

Successfully reconstructs binary images with unique solutions

Finds commonalities in multiple solutions

Outperforms Total Variation and DART methods in experiments

Abstract

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 non-convex, 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 LASSO 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 (2x2, 3x3, 4x4) we find, however, that if the problem has a unique solution, our dual approach finds it. In case of multiple solutions, our approach finds the commonalities between the solutions. Further experiments on realistic numerical phantoms and an experiment on X-ray dataset show that our method compares favourably to Total Variation and DART.