Fast Image Recovery Using Variable Splitting and Constrained Optimization

TL;DR

This paper introduces a fast, convergent algorithm for image restoration that efficiently handles regularization terms like wavelet-based and total variation, outperforming existing methods in speed.

Contribution

It presents a novel algorithm based on variable splitting and augmented Lagrangian methods for faster image recovery with proven convergence.

Findings

Faster than current state-of-the-art methods.

Applicable to wavelet-based and total variation regularization.

Proven convergence of the proposed algorithm.

Abstract

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an $\ell_2$ data-fidelity term and a non-smooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called "alternating direction method of multipliers", for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.