The augmented lagrange multipliers method for matrix completion from corrupted samplings with application to mixed Gaussian-impulse noise removal.

Fan Meng,Chenghu Zhou,Xiaomei Yang,Xi-Nian Zuo

doi:10.1371/journal.pone.0108125

Abstract

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.

Highlights

Image denoising is highly demanded in the field of image processing, since noise is usually inevitable during the process of image acquisition and transmission, which significantly degrades the visual quality and makes subsequent high-level image analysis and understanding very difficult
Additive Gaussian noise is usually generated during image acquisition and characterized by adding each image pixel a random value from the Gaussian distribution with zero mean and standard deviation s, while impulse noise is very different in nature from Gaussian noise
We consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings (MCCS) and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the l1-norm

Summary

Introduction

Image denoising is highly demanded in the field of image processing, since noise is usually inevitable during the process of image acquisition and transmission, which significantly degrades the visual quality and makes subsequent high-level image analysis and understanding very difficult. We propose a novel denoising framework for better preserving details and lowrank features in images while removing mixed Gaussian-impulse noise, based on the theory of low-rank matrix reconstruction. The main contributions of the paper include modeling the problem of low-rank matrix recovery from incomplete and corrupted samplings of its entries, solving the convex optimization problem via the proposed ALM algorithm and creatively applying it to mixed Gaussian-impulse noise removal. Wright et al [23] have shown that under rather broad conditions the answer is affirmative: provided the error matrix E is sufficiently sparse, one can exactly recover the low-rank matrix A from D~AzE by solving the following convex optimization problem: kAkÃzlDED1, s:t:D~AzE, ð5Þ where D.D1 denotes the sum of the absolute values of matrix entries, and l is a positive weighting parameter.

Performance analysis of the proposed algorithm

Findings

Experiments and Discussion

Conclusion