An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems

TL;DR

This paper introduces a fast augmented Lagrangian algorithm for constrained linear inverse problems in imaging, effectively handling large-scale, non-smooth regularization tasks like deconvolution and MRI reconstruction.

Contribution

It develops an efficient augmented Lagrangian-based method tailored for basis pursuit denoising in imaging, with proven convergence and competitive performance.

Findings

Effective in large-scale image reconstruction tasks

Handles non-smooth regularizers like total variation and wavelets

Achieves competitive results compared to state-of-the-art methods

Abstract

We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called "alternating direction method of multipliers", for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.