Weighted Schatten $p$-Norm Minimization for Image Denoising and Background Subtraction

TL;DR

This paper introduces Weighted Schatten p-Norm Minimization (WSNM), a flexible low-rank approximation method that improves image denoising and background subtraction by better modeling the importance of singular values.

Contribution

The paper proposes WSNM, a novel low-rank approximation model that generalizes nuclear norm minimization with weighted Schatten p-norm, offering better accuracy and flexibility.

Findings

WSNM outperforms state-of-the-art methods in denoising quality.

WSNM effectively models complex and dynamic scenes.

The solution can be efficiently obtained via generalized iterated shrinkage.

Abstract

Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten $p$-Norm Minimization (WSNM), to generalize the NNM to the Schatten $p$-norm minimization with weights assigned to different singular values. The proposed WSNM not only gives better approximation to the original low-rank assumption, but also considers the importance of different rank components. We analyze the solution of WSNM and prove that, under certain weights permutation, WSNM can be equivalently transformed into independent non-convex $l_p$-norm subproblems, whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g., image denoising and background subtraction. Extensive experimental results show, both qualitatively and quantitatively, that the proposed WSNM can more effectively remove noise, and model complex and dynamic scenes compared with state-of-the-art methods.