Robust Low-Rank Analysis with Adaptive Weighted Tensor for Image Denoising

Highlights

• The adaptive weight tensor can effectively retain useful singular values and better preserve the low-rank properties of the unfolding matrix.

• The proposed algorithm considers the spatial information and spectral information at the same time.

Abstract

In order to obtain better denoising results, this paper proposes the Robust Low-Rank Analysis with Adaptive Weighted Tensor (AWTD) method for image denoising tasks. On one hand, it uses the latest adaptive weight tensor, which obtains the low-rank approximation of the tensor by adding adaptive weights to the unfolding matrix of the tensor. The adaptive weight tensor can effectively retain useful singular values and better preserve the low-rank properties of the unfolding matrix. On the other hand, the proposed algorithm considers the spatial information and spectral information at the same time: for the RGB images, it retains the structural information inside the image patch and the connection between different channels (the spatial information of the image); for the hyperspectral images, it also retains the spectral information of the hyperspectral images. The experimental results show that the proposed method is superior to other test methods.

Introduction

The purpose of image denoising is to improve the quality of a given image due to noise. Various denoising technologies can effectively improve the image quality and increase the signal-to-noise ratio, to better reflect the original information of the image. With the development of computer technology and digital image processing technology, image denoising has been widely used in many fields such as medicine, public security, and military. Existing image denoising algorithms can be divided into traditional two-dimensional denoising methods and tensor-based multi-linear models.

There are many existing two-dimensional denoising algorithms, and they include local and non-local methods. The local methods [1], [2] use a designed filter to perform a convolution operation on the entire image, and they will lose the global structural information of the image. Correspondingly, there are many methods based on non-local algorithms [3], [4], [5], [6]. These algorithms use non-local means for image denoising and use the self-similarity of the image to remove noise. In addition, the method of rank minimization is also a popular research direction [7], [8], [9], [10], [11], [12]. The rank minimization problem is NP-hard, but in some cases, nuclear norm minimization can be used to approximate the original problem [13], [14]. Although the above methods have achieved good results in image denoising, they are only suitable for gray-scale image denoising tasks. Because they only consider the structural information inside the image patch and ignore the relationship between the different color channels in the RGB image, which limits their usefulness for extracting information from a multi-dimensional perspective.

The tensor-based multi-linear model can better deal with high-dimensional denoising tasks and can use the multi-linear structure to provide better understanding and higher accuracy. There are currently two popular directions of the tensor model: tensor decomposition and tensor low-rank. The tensor decomposition is usually divided into CANDECOMP/PARAFAC (CP) decomposition [15], [16] and Tucker decomposition (High-order SVD) [17], [18], as well as the newly proposed Ring decomposition model [19] and the special form of Tucker decomposition [20], [21] in recent years. The tensor decomposition is very similar to the Principal Component Decomposition (PCA) [22], [23] of the matrix, and PCA is sensitive to outliers and serious pollution (non-Gaussian noise), so the CP and Tucker decomposition also cannot successfully deal with these problems. Someone has proposed some algorithms to strengthen tensor decomposition to solve the problem of outliers and missing data [24], [25], [26]. Unfortunately, they lack global optimality guarantees.

In practice, whether it is caused by outliers or missing data, tensor data is often low-rank. This means that the main part of the data in practice is usually affected by a small number of potential factors. Therefore, the nuclear norm corresponding to the matrix is minimized, and the image denoising task can be accomplished by minimizing the tensor nuclear norm [27], [28], [29]. Although these methods have been successfully applied to color image denoising, they have not considered the adaptive weight of the tensor nuclear norm and therefore lack flexibility.

In summary, it can be concluded that on the one hand, the traditional two-dimensional methods are matrix-based and only consider the local information of the image patches or the structure information of the entire image patch. This ignores the relationship within the image block or between the entire image block and different channels (spatial information), which limits the scope of application of the algorithm in practical applications. On the other hand, in tensor-based multi-linear models, tensor decomposition cannot well overcome the problems of outliers and missing data. In the tensor low-rank models, the tensor nuclear norm minimization strategy of low-rank approximation does not take into account the adaptive weight of the nuclear norm, so it lacks flexibility.

In recent years, Kwok-Po Ng [30] proposed an adaptive weight tensor complement method for hyperspectral remote sensing image denoising (AWTC). By considering the contribution of spatial, spectral, and temporal information in each dimension to adaptively determine the weights, a new weighted tensor low-rank regularization model is constructed. In this paper, we apply the adaptive weight tensor proposed in the AWTC model to the High-order RPCA (HORPCA) [29] model in the tensor nuclear norm minimization strategy, and propose Robust Low-Rank Analysis with Adaptive Weighted Tensor for Image Denoising (AWTD). AWTD not only overcomes the problem of losing spatial information in the traditional two-dimensional method but also introduces an adaptive weight tensor, which is more flexible than the High-order RPCA (HORPCA) model.

In this paper, we make three following main contributions:

• By introducing the adaptive weight tensor, a more flexible Robust Low-Rank Analysis with Adaptive Weighted Tensor for Image Denoising (AWTD) model is proposed to better complete the low-rank regularization of the tensor.

• The results of different types of denoising experiments are shown, including RGB images with special and complex structures and hyperspectral remote sensing images.

• The distribution of the weights of different unfolding matrices on different noise levels and different types of images is discussed. The sensitivity analysis of the parameters and the convergence analysis of the algorithm are given.

The rest of this paper is organized as follows. In Section 2, we will give an introduction to mathematical symbols and review some definitions of matrices and tensors. We will first introduce the adaptive weight in Section 3, then we will propose our model: Robust Low-Rank Analysis with Adaptive Weighted Tensor for Image Denoising (AWTD). The experimental results and the sensitivity analysis of the parameters will be shown in Section 4. Finally, this paper will end with a conclusion in Section 5.