Linear Total Variation Approximate Regularized Nuclear Norm Optimization for Matrix Completion

Xu Han,Huazhong Shu,Yang Chen,Lu Wang,Lotfi Senhadji,Jiasong Wu

doi:10.1155/2014/765782

Xu Han, Huazhong Shu + Show 4 more

Open Access

https://doi.org/10.1155/2014/765782

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Abstract

Matrix completion that estimates missing values in visual data is an important topic in computer vision. Most of the recent studies focused on the low rank matrix approximation via the nuclear norm. However, the visual data, such as images, is rich in texture which may not be well approximated by low rank constraint. In this paper, we propose a novel matrix completion method, which combines the nuclear norm with the local geometric regularizer to solve the problem of matrix completion for redundant texture images. And in this paper we mainly consider one of the most commonly graph regularized parameters: the total variation norm which is a widely used measure for enforcing intensity continuity and recovering a piecewise smooth image. The experimental results show that the encouraging results can be obtained by the proposed method on real texture images compared to the state-of-the-art methods.

Highlights

The problem of matrix completion, which can be seen as the extension of recently developed compressed sensing (CS) theory [1,2,3], plays an important role in the field of signal and image processing [4,5,6,7,8,9,10,11]
The key point of the proposed approach is the combination of the nuclear norm and the linear total variation approximate norm; the optimization problem is described as min γ‖X‖LTVA
It can be observed from these two figures that the best result is obtained for the value of γ near to 0.5, which corresponds to the case where the two norms are equivalently used in (9)

Summary

Introduction

The problem of matrix completion, which can be seen as the extension of recently developed compressed sensing (CS) theory [1,2,3], plays an important role in the field of signal and image processing [4,5,6,7,8,9,10,11]. The total variation (TV) norm has demonstrated its usefulness as a graph regularizer in the field of image processing, so we propose here a method that combines the nuclear norm with the linear TV approximate norm to solve the problem of matrix completion. We call it the linear total variation approximate regularized nuclear norm (LTVNN) minimization problem. This combination optimization problem will be solved by simple and efficient optimization scheme based on the alternating direction method of multipliers (ADMM) model [20, 21].

Proposed Method

Convergence Analysis

Experiments

Conclusion