An Extended Primal-Dual Algorithm Framework for Nonconvex Problems with Application to Nonlinear Imaging

TL;DR

This paper introduces an extended primal-dual algorithm framework tailored for nonconvex optimization problems, particularly effective in nonlinear imaging applications like spectral computed tomography, demonstrating strong convergence and reconstruction performance.

Contribution

The work develops a novel extended primal-dual framework with six specific iterative schemes for nonconvex problems, including convergence analysis and application to nonlinear imaging.

Findings

Effective image reconstruction in nonlinear imaging scenarios.

Convergence of schemes under various dual variable conditions.

Numerical results show good performance with non-standard data.

Abstract

We propose an extended primal-dual algorithm framework for solving a general nonconvex optimization model. This work is motivated by image reconstruction problems in a class of nonlinear imaging, where the forward operator can be formulated as a nonlinear convex function with respect to the reconstructed image. Using the proposed framework, we put forward six specific iterative schemes, and present their detailed mathematical explanation. We also establish the relationship to existing algorithms. Moreover, under proper assumptions, we analyze the convergence of the schemes for the general model when the optimal dual variable regarding the nonlinear operator is non-vanishing. As a representative, the image reconstruction for spectral computed tomography is used to demonstrate the effectiveness of the proposed algorithm framework. By special properties of the concrete problem, we further prove the convergence of these customized schemes when the optimal dual variable regarding the nonlinear operator is vanishing. Finally, the numerical experiments show that the proposed algorithm has good performance on image reconstruction for various data with non-standard scanning configuration.