A splitting primal-dual proximity algorithm for solving composite optimization problems

TL;DR

This paper introduces a splitting primal-dual proximity algorithm for composite convex optimization, with applications to image reconstruction, demonstrating effective image quality with sparse data.

Contribution

It proposes a novel splitting primal-dual proximity algorithm and a preconditioned variant that do not require prior knowledge of operator norms, with proven convergence.

Findings

Effective image reconstruction with sparse projections

Ability to combine different loss functions in regularization

Converges under theoretical guarantees

Abstract

Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Further, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.