# L0TV: A Sparse Optimization Method for Impulse Noise Image Restoration

## TL;DR

This paper introduces a novel sparse optimization method, TV-PADMM, for impulse noise image restoration that outperforms existing methods by effectively solving a non-convex -norm based TV restoration problem.

## Contribution

It proposes a new TV-PADMM algorithm that addresses non-convex -norm TV restoration, with proven convergence, for improved impulse noise removal in images.

## Key findings

- TV-PADMM outperforms state-of-the-art methods in impulse noise removal.
- The method effectively handles non-convex -norm optimization problems.
- Experimental results show superior denoising and deblurring performance.

## Abstract

Total Variation (TV) is an effective and popular prior model in the field of regularization-based image processing. This paper focuses on total variation for removing impulse noise in image restoration. This type of noise frequently arises in data acquisition and transmission due to many reasons, e.g. a faulty sensor or analog-to-digital converter errors. Removing this noise is an important task in image restoration. State-of-the-art methods such as Adaptive Outlier Pursuit(AOP) \cite{yan2013restoration}, which is based on TV with $\ell_{02}$-norm data fidelity, only give sub-optimal performance. In this paper, we propose a new sparse optimization method, called $\ell_0TV$-PADMM, which solves the TV-based restoration problem with $\ell_0$-norm data fidelity. To effectively deal with the resulting non-convex non-smooth optimization problem, we first reformulate it as an equivalent biconvex Mathematical Program with Equilibrium Constraints (MPEC), and then solve it using a proximal Alternating Direction Method of Multipliers (PADMM). Our $\ell_0TV$-PADMM method finds a desirable solution to the original $\ell_0$-norm optimization problem and is proven to be convergent under mild conditions. We apply $\ell_0TV$-PADMM to the problems of image denoising and deblurring in the presence of impulse noise. Our extensive experiments demonstrate that $\ell_0TV$-PADMM outperforms state-of-the-art image restoration methods.