# Multi-band Weighted $l_p$ Norm Minimization for Image Denoising

## TL;DR

This paper introduces a multi-band weighted $l_p$ norm minimization model for image denoising that more accurately approximates low rank matrices by considering the importance of different components, outperforming existing methods.

## Contribution

The paper proposes the MBWPNM model, which improves low rank matrix approximation by incorporating weighted $l_p$ norms and prior knowledge of component importance, with an efficient solution method.

## Key findings

- MBWPNM achieves superior denoising performance on additive white Gaussian noise.
- The method outperforms several state-of-the-art algorithms in experiments.
- Global optimum can be efficiently obtained via generalized soft-thresholding.

## Abstract

Low rank matrix approximation (LRMA) has drawn increasing attention in recent years, due to its wide range of applications in computer vision and machine learning. However, LRMA, achieved by nuclear norm minimization (NNM), tends to over-shrink the rank components with the same threshold and ignore the differences between rank components. To address this problem, we propose a flexible and precise model named multi-band weighted $l_p$ norm minimization (MBWPNM). The proposed MBWPNM not only gives more accurate approximation with a Schatten $p$-norm, but also considers the prior knowledge where different rank components have different importance. We analyze the solution of MBWPNM and prove that MBWPNM is equivalent to a non-convex $l_p$ norm subproblems under certain weight condition, whose global optimum can be solved by a generalized soft-thresholding algorithm. We then adopt the MBWPNM algorithm to color and multispectral image denoising. Extensive experiments on additive white Gaussian noise removal and realistic noise removal demonstrate that the proposed MBWPNM achieves a better performance than several state-of-art algorithms.