A Fast Algorithm for the Constrained Formulation of Compressive Image Reconstruction and Other Linear Inverse Problems

TL;DR

This paper introduces a fast, efficient augmented Lagrangian algorithm specifically designed for constrained linear inverse problems in image reconstruction, including compressive sensing and deconvolution, addressing challenges of high dimensionality and non-smoothness.

Contribution

The paper presents a novel augmented Lagrangian-based algorithm tailored for basis pursuit denoising in image recovery, improving computational efficiency over existing methods.

Findings

Algorithm effectively handles large-scale problems

Demonstrates superior speed and accuracy in experiments

Applicable to various imaging inverse problems

Abstract

Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard approach to deal with ILIP uses a constrained optimization problem, where a regularization function is minimized under the constraint that the solution explains the observations sufficiently well. The regularizer and constraint are usually convex; however, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated much research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the category of augmented Lagrangian methods, can be used to deal with a variety of imaging ILIP, including deconvolution and reconstruction from compressive observations (such as MRI). Experiments testify for the effectiveness of the proposed method.