Fast Proximal Gradient Methods for Nonsmooth Convex Optimization for Tomographic Image Reconstruction

TL;DR

This paper introduces accelerated proximal gradient methods tailored for nonsmooth convex optimization in tomographic image reconstruction, providing theoretical convergence guarantees and demonstrating practical efficiency through numerical experiments.

Contribution

It presents the Fast Proximal Gradient Method and Monotone FPGM with proven convergence rates, expanding understanding of their behavior in tomographic reconstruction.

Findings

Methods are competitive in practical tomographic scenarios.

Convergence rate of $O(1/k^{2})$ established for the algorithms.

Experimental comparison reveals new insights into accelerated proximal gradient algorithms.

Abstract

The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a $O\left(1/k^{2}\right)$ non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.