Exploiting Weighted Multidirectional Sparsity for Prior Enhanced Anomaly Detection in Hyperspectral Images

Abstract

Anomaly detection (AD) is an important topic in remote sensing, aiming to identify unusual or abnormal features within the data. However, most existing low-rank representation methods usually use the nuclear norm for background estimation, and do not consider the different contributions of different singular values. Besides, they overlook the spatial relationships of abnormal regions, particularly failing to fully leverage the 3D structured information of the data. Moreover, noise in practical scenarios can disrupt the low-rank structure of the background, making it challenging to separate anomaly from the background and ultimately reducing detection accuracy. To address these challenges, this paper proposes a weighted multidirectional sparsity regularized low-rank tensor representation method (WMS-LRTR) for AD. WMS-LRTR uses the weighted tensor nuclear norm for background estimation to characterize the low-rank property of the background. Considering the correlation between abnormal pixels across different dimensions, the proposed method introduces a novel weighted multidirectional sparsity (WMS) by unfolding anomaly into multimodal to better exploit the sparsity of the anomaly. In order to improve the robustness of AD, we further embed a user-friendly plug-and-play (PnP) denoising prior to optimize the background modeling under low-rank structure and facilitate the separation of sparse anomalous regions. Furthermore, an effective iterative algorithm using alternate direction method of multipliers (ADMM) is introduced, whose subproblems can be solved quickly by fast solvers or have closed-form solutions. Numerical experiments on various datasets show that WMS-LRTR outperforms state-of-the-art AD methods, demonstrating its better detection ability.

1. Introduction

With the development of remote sensing and sensor technology, hyperspectral images (HSIs) that simultaneously observe the geometric and physical characteristics of objects have received significant attention. HSIs have extensive applications in image denoising [1,2,3], anomaly detection (AD) [4,5,6], classification [7,8,9], fusion [10,11], and unmixing [12,13]. As a crucial research topic, AD aims to identify abnormal regions from HSIs that are significantly different from the background. This process usually divides the image into an abnormal part and a background part, facilitating the rapid localization of potential target objects and providing valuable data for further feature analysis and resource exploration.

Various AD methods have been proposed and studied over the past few decades. One popular method is called RX, which can be traced back to [14]. The main concept involves initially computing the mean and covariance of the background and then using the Mahalanobis distance between pixels to identify anomalies. Later, Chang et al. [15] proposed an RX method by using discriminative measures. Kwon et al. [16] introduced kernel-based RX to explore higher-order correlations between bands. Guo et al. [17] considered weighted RX by imposing different weights on anomaly and background to improve the estimation performance. Subsequently, Zhou et al. [18] developed cluster kernel RX and Ren et al. [19] suggested superpixel-based dual-window RX. Although these RX-based methods have the advantages of simple formulations and fast calculations, their AD performance will be greatly reduced in scenes with a complex background.

Another popular method is based on sparse learning [20] or low-rank representation (LRR) [21]; see, e.g., [22,23,24,25]. Xu et al. [26] introduced the low-rank and sparse representation (LRASR) method and experimentally demonstrated its advantages over RX. Recently, Zhang et al. [27] combined Mahalanobis distance with LRASR to improve detection accuracy, denoted as LSMAD. Cheng et al. [28] proposed the graph and total variation (TV) regularized LRR (GTVLRR) method to preserve geometric structures. Shen et al. [29] proposed the SaFra method, using a matrix factorization approach with framelet and saliency priors for AD. Lin et al. [30] proposed super robust principal component analysis (SuperRPCA), which integrates local neighbor information from collaborative superpixel representation detector (CSRD) superpixel computation with the global information from the RPCA framework to enhance AD. However, the above-mentioned matrix-based methods lack in-depth exploration of spectral information, which may lead to insufficient detection capabilities.

From a data representation perspective, a tensor can be a better choice [31,32,33,34,35]. Li et al. [36] introduced the prior-based tensor approximation (PTA) method, which incorporates TV regularization to enforce piecewise smoothness in the embedding space. Wang et al. [37] employed PCA for data preprocessing and utilized tensor low-rank sparse representation for AD. In order to enhance the representation of background, Sun et al. [38] introduced a low average rank with a TV regularization model for AD (LARTVAD), which leverages the characteristics of low average rank to reconstruct the background. Additionally, Feng et al. [39] introduced the adoption of tensor ring decomposition which factors in TV regularization for AD (TRDFTVAD). Ren et al. [40] proposed a unified nonconvex framework named hyperspectral AD via generalized shrinkage mappings (HADGSMs), which aims to improve the approximation of LRR-based methods. However, representation-based methods typically use the nuclear norm to separate background and anomaly in HSI [41,42]. The nuclear norm serves as a convex approximation of the rank function and fails to account for the varying contributions of different singular values, leading to errors in the background estimation of the HSI. Besides, these methods primarily focus on the relationships within the spectral dimension, overlooking the correlations in the spatial dimension. Furthermore, it is also important to point out that these methods overly emphasize the low-rank information in the spectral dimension while neglecting the fact that noise in the original HSI can disrupt the low-rank structure.

At the same time, deep learning methods are becoming increasingly prevalent in AD tasks because of their capability to extract features. The bulk of pixels are attributed to the background, with anomalies accounting for only a few pixels in the HSIs. As a result, the background typically shows small reconstruction errors, while anomalies often exhibit large reconstruction errors. To this end, Wang et al. [43] introduced the autonomous AD network (AUTO-AD), which employs fully convolutional layers to construct autoencoders (AE), improving the background reconstruction capability. In addition, Fan et al. [44] proposed robust graph AE (RGAE) to introduce graph regularization into the latent layer of AE to capture inter-pixel relationships. Besides, Xiang et al. [45] introduced a guided autoencoder (GAED) to integrate guiding modules into the network, reducing the reconstruction of anomalous targets, and consequently, enhancing the reconstruction of background features. Mu et al. [46] proposed an unsupervised framework based on a multivariate probability distribution AE (MPDA) to address AD in complex scenarios. Liu et al. [47] introduced a novel separation training strategy to suppress anomalies during background reconstruction and proposed the multiscale network (MSNet). These deep-learning-based detectors notably decrease testing time but require retraining when applied to different test scenarios, and they lack sufficient physical interpretability.

Recently, the benefits of plug-and-play (PnP) denoising have become increasingly evident in remote sensing image processing [48,49]. Zhuang et al. [50] proposed FastHyDe, a fast HSI denoising method that combines BM3D [51] and BM4D [52]. Furthermore, Fu et al. [53] proposed the denoising CNN-based AD (DeCNN-AD) method by using convolutional neural networks for noise reduction. Liu et al. [54] proposed FRCTR-PnP, which inserts BM3D denoising to reduce small tensor singular values to preserve image details. Moreover, Zhuang et al. [55] embedded the prior image extracted from FFDNet [56] into the hybrid denoising framework and proposed FastHyMix. Therefore, a natural question is whether it is possible to propose a novel method with physical interpretability that simultaneously addresses the problems of (1) inaccurate singular value contribution estimation by the nuclear norm, (2) the neglect of spatial dimensional relationships in many AD methods, and (3) the impact of noise on the background low-rank structure in the original HSI.

Motivated by the above works, this paper introduces a novel AD method named weighted multidirectional sparsity regularized low-rank tensor representation (WMS-LRTR) to address the above three limitations simultaneously. The nuclear norm is the biased estimate, which exaggerates the influence of larger singular values on the rank, leading to deviations from the true rank and affecting the background estimation. To solve this problem, we adopt the weighted tensor nuclear norm (WTNN) to impose different weights on the singular values, which effectively preserves the background tensor by adaptive shrinkage. Considering the 3D structure of HSI, there is a correlation between abnormal pixels across different dimensions. Therefore, we extend the structured sparsity to 3D space to fully leverage spatial information, which is named weighted multidirectional sparsity (WMS). WMS is introduced to extract anomaly by leveraging low-rank information across spatial and spectral dimensions, while also capturing multidirectional structured features. In addition, to avoid the influence of noise on the original HSI, a PnP denoising prior is incorporated to not only remove noise but also enhance background modeling under the low-rank assumption, thereby facilitating the separation of sparse abnormal regions. In this paper, BM4D is used as the PnP denoising prior, recognized as a 3D image denoising algorithm. Given that HSI comprises both 2D spatial information and 1D spectral information, BM4D is well-suited for HSI denoising. The overall workflow of our proposed WMS-LRTR is illustrated in Figure 1. Before detection, the HSI is preprocessed using principal component analysis (PCA) to reduce the spectral dimension. This not only decreases computation time but also improves detection performance; see Section 4.4.3. The preprocessed HSI tensor is then decomposed, with WTNN and WMS employed to construct a clean background dictionary. In addition, the PnP prior is applied to denoise the HSI, preserving its low-rank properties, and WMS is used to extract anomalies. Ultimately, the detection result is generated after the computation of WMS-LRTR.

The primary contributions of WMS-LRTR can be summarized as follows:

• We propose a novel AD method named WMS-LRTR that incorporates both the low-rank property of the background and the structured sparsity of the anomaly and enhances the robustness of AD.
• We extend the anomaly tensor to multimodal and design an adaptive dictionary construction method to generate a clean background dictionary. WTNN is employed to effectively allocate singular value contributions, while WMS leverages the correlations between abnormal pixels across different dimensions, thereby enhancing the ability to explore structured features in abnormal regions.
• We construct an efficient algorithm based on ADMM, where all subproblems are relatively easy to solve. Besides, numerical experiments on eight datasets demonstrate that the detection ability of WMS-LRTR surpasses nine benchmark AD methods.

The reminder of this paper is organized as follows. Section 2 offers a brief overview of notations and related work. Section 3 introduces a new formulation along with an optimization algorithm. Section 4 presents numerical comparisons and detailed discussions. Finally, Section 5 provides the conclusion of this paper.

2. Preliminaries

2.1. Notations

In this paper, vectors are denoted by bold lowercase letters, i.e., $\mathbf{x}$, matrices by uppercase letters, i.e., X, and tensors by upper cursive letters, i.e., $\mathcal{X}$. For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the unfolding matrix of $\mathcal{X}$ in the k-th dimension is defined as:

$$X_{\left(k\right)}=\mathtt{u}\mathtt{n}\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(\mathcal{X},k)=\left(\begin{array}{c}{X}_{\left(k\right)}^{\left(1\right)}\\ X_{\left(k\right)}^{\left(2\right)}\\ ⋮\\ X_{\left(k\right)}^{\left(n_k\right)}\end{array}\right),k=1,2,3.$$

(1)

Besides, fold is the inverse of unfold, which is defined as:

$$\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(X_{\left(k\right)},k)=\mathcal{X},k=1,2,3.$$

(2)

Next, we introduce some definitions used in the following sections.

Definition 1 

(Block circulant matrix). For a tensor $\mathcal{A}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the block circulant matrix is given by:

$$\mathtt{b}\mathtt{c}\mathtt{i}\mathtt{r}\mathtt{c}\left(\mathcal{A}\right)=\left[\begin{array}{cccc}{A}^{\left(1\right)}& A^{\left(m_3\right)}& \cdots & A^{\left(2\right)}\\ A^{\left(2\right)}& A^{\left(1\right)}& \cdots & A^{\left(3\right)}\\ ⋮& ⋮& \ddots & ⋮\\ A^{\left(m_3\right)}& A^{(m_3-1)}& \cdots & A^{\left(1\right)}\end{array}\right]\in {\mathbb{R}}^{m_1{m}_3\times m_2{m}_3},$$

(3)

where $A^{\left(i\right)}$ is the i-th frontal slice with the dimension $m_1\times m_2$.

Definition 2 

(Tensor product). For two tensors $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$ and $\mathcal{Y}\in {\mathbb{R}}^{m_2\times m_4\times m_3}$, the tensor product $\mathcal{Z}\in {\mathbb{R}}^{m_1\times m_4\times m_3}$ is defined as follows:

$$\begin{array}{c}\hfill \mathcal{Z}=\mathcal{X}\ast \mathcal{Y}=\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}\left(\mathtt{b}\mathtt{c}\mathtt{i}\mathtt{r}\mathtt{c}\right(\mathcal{X})\ast \mathtt{u}\mathtt{n}\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(\mathcal{Y},2),2).\end{array}$$

(4)

Definition 3 

(Frobenius norm). For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the Frobenius norm can be defined as:

$${\parallel \mathcal{X}\parallel }_F^2=\sum _{i=1}^{m_1}\sum _{j=1}^{m_2}\sum _{k=1}^{m_3}{\mathcal{X}}^2(i,j,k),$$

(5)

where $\mathcal{X}(i,j,k)$ represents the element located at the $(i,j,k)$ position in $\mathcal{X}$.

Definition 4 

($L_1$-norm). For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the $L_1$-norm is given by:

$${\parallel \mathcal{X}\parallel }_1=\sum _{i=1}^{m_1}\sum _{j=1}^{m_2}\sum _{k=1}^{m_3}\left|\mathcal{X}(i,j,k)\right|,$$

(6)

where $\left|\mathcal{X}\right(i,j,k\left)\right|$ is the absolute value of $\mathcal{X}(i,j,k)$.

Definition 5 

($L_{F,1}$-norm). For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the $L_{F,1}$-norm is given by:

$${\parallel \mathcal{X}\parallel }_{F,1}=\sum _{i=1}^{m_1}\sum _{j=1}^{m_2}\sqrt{\sum _{k=1}^{m_3}{\mathcal{X}}^2(i,j,k)}.$$

(7)

Definition 6 

(t-SVD [57]). For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the tensor singular value decomposition (t-SVD) is denoted as:

$$\mathcal{X}=\mathcal{U}\ast \mathcal{S}\ast {\mathcal{V}}^{\top },$$

(8)

where $\mathcal{U}\in {\mathbb{R}}^{m_1\times m_1\times m_3}$, $\mathcal{V}\in {\mathbb{R}}^{m_2\times m_2\times m_3}$ are orthogonal tensors, and $\mathcal{S}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$ is the f-diagonal tensor.

Definition 7 

(WTNN [58]). For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, WTNN is calculated as the weighted sum of all singular values of all frontal slices, that is:

$${\parallel \mathcal{X}\parallel }_{\mathit{WTNN}}=\sum _{j=1}^{m_3}\sum _{i=1}^{min(m_1,m_2)}\omega (i,i,j)\sigma (i,i,j),$$

(9)

where $\sigma (i,i,j)$ is composed of the corresponding singular values derived from t-SVD of $\mathcal{X}$ and $\omega (i,i,j)$ is the weighted tensor.

2.2. Related Work

For an HSI tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, where $m_1,m_2$ are the spatial dimensions and $m_3$ is the spectral dimension. LRR is used to represent data within a low-rank subspace. In general, the mathematical formulation can be given by:

$$\begin{array}{cc}\hfill \underset{W,E}{min}& {\parallel W\parallel }_{\ast }+\lambda {\parallel E\parallel }_1\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& X=AW+E,\hfill \end{array}$$

(10)

where $X\in {\mathbb{R}}^{m_1{m}_2\times m_3}$ is the unfolding matrix along the spectral dimension, $A\in {\mathbb{R}}^{m_1{m}_2\times n}$ represents the background dictionary, $W\in {\mathbb{R}}^{n\times m_3}$ denotes the coefficient matrix, $E\in {\mathbb{R}}^{m_1{m}_2\times m_3}$ describes the anomaly, ${\parallel W\parallel }_{\ast }$ is the nuclear norm of W, ${\parallel E\parallel }_1$ is the $L_1$-norm of E, and $\lambda >0$ is the trade-off parameter.

Note that HSIs are essentially 3D, matrix representation may break the data structure. In this regard, the LRR model in (10) can be extended to tensor cases. For a tensor $\mathcal{X}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, the representative PCA-TLRSR is constructed by:

$$\begin{array}{cc}\hfill \underset{\mathcal{W},\mathcal{E}}{min}& {\parallel \mathcal{W}\parallel }_{\mathrm{WTNN}}+\lambda {\parallel \mathcal{E}\parallel }_{F,1}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathcal{X}=\mathcal{A}\ast \mathcal{W}+\mathcal{E},\hfill \end{array}$$

(11)

where $\mathcal{A}\in {\mathbb{R}}^{m_1\times n\times m_3}$ is the background dictionary tensor, $\mathcal{W}\in {\mathbb{R}}^{n\times m_2\times m_3}$ is the representation coefficient tensor, $\mathcal{E}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$ is the sparse anomaly tensor, and ${\parallel \mathcal{W}\parallel }_{\mathrm{WTNN}}$ and ${\parallel \mathcal{E}\parallel }_{F,1}$ are defined before.

It is important to point out that the $L_{F,1}$-norm in (11) does not consider the correlation between abnormal pixels across different dimensions and lacks sensitivity to spatially local information, particularly with regard to exploring high-dimensional tensor-structured data. For an image $E\in {\mathbb{R}}^{m_1{m}_2\times m_3}$, the structured sparsity [59,60] can be considered as:

$$\begin{array}{c}\hfill \Omega \left(E\right)=\sum _{j=1}^{m_3}\sum _{g\in G}{\parallel {\mathbf{e}}_g^j\parallel }_{\infty },\end{array}$$

(12)

where ${\mathbf{e}}^j\in {\mathbb{R}}^{m_1{m}_2}$ is the j-th column with $m_1{m}_2$ variables, $g\in G$ is a subset containing overlapping blocks, and $\parallel {\mathbf{e}}_g^j{\parallel }_{\infty }$ denotes the maximum number of pixels in the overlapping block g. In this paper, the block size is chosen to be $2\times 2$; see Section 4.4.4.

3. The Proposed Method

We initially propose the mathematical formulation by extending structured sparsity to the tensor case, followed by presenting the optimization algorithm in this section.

3.1. New Formulation

For the anomaly $\mathcal{E}\in {\mathbb{R}}^{m_1\times m_2\times m_3}$, it can be unfolded from two spatial dimensions and one spectral dimension as:

$$\begin{array}{c}\hfill E_{\left(i\right)}=\mathtt{u}\mathtt{n}\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(\mathcal{E},i),i=1,2,3,\end{array}$$

(13)

where $E_{\left(1\right)}\in {\mathbb{R}}^{m_2{m}_3\times m_1},E_{\left(2\right)}\in {\mathbb{R}}^{m_1{m}_3\times m_2}$, and $E_{\left(3\right)}\in {\mathbb{R}}^{m_1{m}_2\times m_3}$. Then, one can put the resulting three matrices into the structured sparsity to take into account the structured information. For each $\Omega \left(E_{\left(i\right)}\right)$, fold it back into the tensor form to get:

$$\begin{array}{c}\hfill {\mathcal{E}}_{\left(i\right)}=\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(\Omega \left(E_{\left(i\right)}\right),i),i=1,2,3,\end{array}$$

(14)

which allows us to obtain the final WMS by assigning weights to the above tensors, respectively, that is:

$$\begin{array}{c}\hfill \Omega \left(\mathcal{E}\right)={\gamma }_1{\mathcal{E}}_{\left(1\right)}+{\gamma }_2{\mathcal{E}}_{\left(2\right)}+{\gamma }_3{\mathcal{E}}_{\left(3\right)},\end{array}$$

(15)

where ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ are three weighting parameters.

In this paper, we refer to this as WMS and show the schematic diagram in Figure 2. By unfolding the anomaly tensor in three dimensions, WMS can effectively extract anomalies from horizontal, vertical, and spectral directions. Specifically, two $2\times 2$ sliding blocks of overlapping pixels are used to systematically explore the entire image, and anomalies are then identified by calculating the maximum value within each sliding block. Now, we present the WMS-LRTR model as:

$$\begin{array}{cc}\hfill \underset{\mathcal{W},\mathcal{E}}{min}& {\parallel \mathcal{W}\parallel }_{\mathrm{WTNN}}+\lambda \Omega \left(\mathcal{E}\right)+\mu \varphi \left(\mathcal{E}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathcal{X}=\mathcal{A}\ast \mathcal{W}+\mathcal{E},\hfill \end{array}$$

(16)

where $\varphi (·)$ is the PnP denoising prior [52] and $\lambda ,\mu$ are the positive trade-off parameters. Compared with PCA-TLRSR in (11), WMS-LRTR in (16) has two advantages:

• $\Omega \left(\mathcal{E}\right)$ considers the correlation between abnormal pixels across different dimensions and captures multidirectional structured features than ${\parallel \mathcal{E}\parallel }_{F,1}$, thereby preserving more spatially local anomaly characteristics.
• $\varphi \left(\mathcal{E}\right)$ complements $\Omega \left(\mathcal{E}\right)$ by filtering out noise to preserve the background low-rank structure and facilitate the separation of sparse anomalous regions, thus improving the robustness of AD.

3.2. Optimization Algorithm

This section discusses how to apply the ADMM [61] to solve (16). Two auxiliary variables $\mathcal{Z}$, $\mathcal{Y}$ are introduced first, and we reformulate (16) as:

$$\begin{array}{cc}\hfill \underset{\mathcal{Z},\mathcal{E},\mathcal{Y},\mathcal{W}}{min}& {\parallel \mathcal{Z}\parallel }_{\mathrm{WTNN}}+\lambda \Omega \left(\mathcal{E}\right)+\mu \varphi \left(\mathcal{Y}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathcal{X}=\mathcal{A}\ast \mathcal{W}+\mathcal{E},\mathcal{Z}=\mathcal{W},\mathcal{Y}=\mathcal{E},\hfill \end{array}$$

(17)

whose augmented Lagrangian function is:

$$\begin{array}{cc}\hfill L_{\beta }& (\mathcal{Z},\mathcal{E},\mathcal{Y},\mathcal{W},{\mathcal{Q}}_1,{\mathcal{Q}}_2,{\mathcal{Q}}_3)={\parallel \mathcal{Z}\parallel }_{\mathrm{WTNN}}+\lambda \Omega \left(\mathcal{E}\right)+\mu \varphi \left(\mathcal{Y}\right)\hfill \\ \hfill & +\langle {\mathcal{Q}}_1,\mathcal{X}-\mathcal{A}\ast \mathcal{W}-\mathcal{E}\rangle +{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{X}-\mathcal{A}\ast \mathcal{W}-\mathcal{E}\parallel }_F^2\hfill \\ \hfill & +\langle {\mathcal{Q}}_2,\mathcal{Z}-\mathcal{W}\rangle +{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{Z}-\mathcal{W}\parallel }_F^2\hfill \\ \hfill & +\langle {\mathcal{Q}}_3,\mathcal{Y}-\mathcal{E}\rangle +{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{Y}-\mathcal{E}\parallel }_F^2,\hfill \end{array}$$

(18)

where ${\mathcal{Q}}_1$, ${\mathcal{Q}}_2$, ${\mathcal{Q}}_3$ represent the Lagrange multipliers and $\beta >0$ is the penalty parameter. According to the ADMM, Equation (17) can be solved by iteratively updating one variable while keeping the others fixed.

• The $\mathcal{Z}$-subproblem can be simplified to:

  $$\begin{array}{cc}\hfill \underset{\mathcal{Z}}{min}& {\parallel \mathcal{Z}\parallel }_{\mathrm{WTNN}}+{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{Z}-{\mathcal{W}}^k+{\displaystyle \frac{{\mathcal{Q}}_2^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (19)

  where (19) can be addressed using the weighted tensor singular value thresholding according to [58].
• The $\mathcal{E}$-subproblem can be solved by:

  $$\begin{array}{cc}\hfill \underset{\mathcal{E}}{min}\lambda \Omega \left(\mathcal{E}\right)& +{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{X}-\mathcal{A}\ast {\mathcal{W}}^k-\mathcal{E}+{\displaystyle \frac{{\mathcal{Q}}_1^k}{\beta }}\parallel }_F^2\hfill \\ \hfill & +{\displaystyle \frac{\beta }{2}}{\parallel {\mathcal{Y}}^k-\mathcal{E}+{\displaystyle \frac{{\mathcal{Q}}_3^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (20)

  which is equivalent to:

  $$\begin{array}{cc}\hfill \underset{\mathcal{E}}{min}& \lambda \Omega \left(\mathcal{E}\right)+\beta \parallel \mathcal{E}-{\displaystyle \frac{{\mathcal{T}}^k}{2}}{\parallel }_F^2,\hfill \end{array}$$

  (21)

  where ${\mathcal{T}}^k=\mathcal{X}-\mathcal{A}\ast {\mathcal{W}}^k+{\displaystyle \frac{{\mathcal{Q}}_1^k}{\beta }}+{\mathcal{Y}}^k+{\displaystyle \frac{{\mathcal{Q}}_3^k}{\beta }}$. Then, Equation (21) can be written as:

  $$\begin{array}{c}\hfill \underset{E_{\left(i\right)}}{min}\lambda \Omega \left(E_{\left(i\right)}\right)+\beta {\parallel E_{\left(i\right)}-\frac{T_{\left(i\right)}^k}{2}\parallel }_F^2,i=1,2,3,\end{array}$$

  (22)

  where $E_{\left(i\right)}=\mathtt{u}\mathtt{n}\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(\mathcal{E},i),i=1,2,3.$ According to [59], Equation (22) can be solved through the proximal operator. It then follows from the quadratic min-cost flow technique that:

  $$\begin{array}{c}\hfill E_{\left(i\right)}^{k+1}=\mathtt{P}\mathtt{r}\mathtt{o}\mathtt{x}({\displaystyle \frac{T_{\left(i\right)}^k}{2}},{\displaystyle \frac{2\lambda }{\beta }}),i=1,2,3,\end{array}$$

  (23)

  then the solution can be obtained by folding $E_{\left(i\right)}$ and assigning weights to the above tensors, respectively, that is:

  $$\begin{array}{cc}\hfill {\mathcal{E}}^{k+1}& ={\gamma }_1\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(E_{\left(1\right)}^{k+1},1)\hfill \\ & +{\gamma }_2\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(E_{\left(2\right)}^{k+1},2)+{\gamma }_3\mathtt{f}\mathtt{o}\mathtt{l}\mathtt{d}(E_{\left(3\right)}^{k+1},3).\hfill \end{array}$$

  (24)
• The $\mathcal{Y}$-subproblem can be rewritten as:

  $$\begin{array}{cc}\hfill \underset{\mathcal{Y}}{min}& \mu \varphi \left(\mathcal{Y}\right)+{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{Y}-{\mathcal{E}}^{k+1}+{\displaystyle \frac{{\mathcal{Q}}_3^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (25)

  where $\varphi \left(\mathcal{Y}\right)$ is the PnP information, serving as a method for prior denoising supported by BM4D. For convenience, the solution of (25) is given by:

  $$\begin{array}{c}\hfill {\mathcal{Y}}^{k+1}=\mathtt{D}\mathtt{e}\mathtt{n}\mathtt{o}\mathtt{i}\mathtt{s}\mathtt{e}\mathtt{r}({\mathcal{E}}^k-{\displaystyle \frac{{\mathcal{Q}}_3^k}{\beta }},\sqrt{{\displaystyle \frac{\mu }{\beta }}}),\end{array}$$

  (26)

  where $\sqrt{{\displaystyle \frac{\mu }{\beta }}}$ is the noise level.
• The $\mathcal{W}$-subproblem can be transformed to:

  $$\begin{array}{cc}\hfill \underset{\mathcal{W}}{min}& {\parallel \mathcal{X}-\mathcal{A}\ast \mathcal{W}-{\mathcal{E}}^{k+1}+{\displaystyle \frac{{\mathcal{Q}}_1^k}{\beta }}\parallel }_F^2+{\parallel {\mathcal{Z}}^{k+1}-\mathcal{W}+{\displaystyle \frac{{\mathcal{Q}}_2^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (27)

  which can be solved by:

  $$\begin{array}{c}\hfill {\mathcal{W}}^{k+1}={({\mathcal{A}}^{†}\ast \mathcal{A}+\mathcal{I})}^{-1}({\mathcal{A}}^{†}\ast (\mathcal{X}-{\mathcal{E}}^{k+1}+{\displaystyle \frac{{\mathcal{Q}}_1^k}{\beta }})+{\mathcal{Z}}^{k+1}+{\displaystyle \frac{{\mathcal{Q}}_2^k}{\beta }}),\end{array}$$

  (28)

  where ${\mathcal{A}}^{†}$ is the transpose of $\mathcal{A}$.
• The Lagrange multipliers ${\mathcal{Q}}_1,{\mathcal{Q}}_2,{\mathcal{Q}}_3$ are updated by:

  $$\left\{\begin{array}{c}{\mathcal{Q}}_1^{k+1}={\mathcal{Q}}_1^k+\beta (\mathcal{X}-\mathcal{A}\ast {\mathcal{W}}^{k+1}-{\mathcal{E}}^{k+1}),\hfill \\ {\mathcal{Q}}_2^{k+1}={\mathcal{Q}}_2^k+\beta ({\mathcal{Z}}^{k+1}-{\mathcal{W}}^{k+1}),\hfill \\ {\mathcal{Q}}_3^{k+1}={\mathcal{Q}}_3^k+\beta ({\mathcal{Y}}^{k+1}-{\mathcal{E}}^{k+1}),\hfill \\ \beta =\min \{\kappa \beta ,{\beta }_{max}\}.\hfill \end{array}\right.$$

  (29)

To sum up, the whole iterative framework is presented in Algorithm 1.

Algorithm 1 Optimization framework

Input: Given data $\mathcal{X}$, dictionary $\mathcal{A}$, parameters $\lambda ,\mu ,\beta ,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\beta }_{max}={10}^8,\kappa =1.1,ϵ={10}^{-6}$ Initialize: ${\mathcal{Z}}^0,{\mathcal{L}}^0,{\mathcal{Y}}^0,{\mathcal{W}}^0,{\mathcal{E}}^0,{\mathcal{Q}}_1^0,{\mathcal{Q}}_2^0,{\mathcal{Q}}_3^0$ initialized to 0, set iteration number $k=0$ While not converged do    1: Update ${\mathcal{Z}}^{k+1}$ by weighted tensor singular value thresholding [58]    2: Update ${\mathcal{E}}^{k+1}$ by (24)    3: Update ${\mathcal{Y}}^{k+1}$ by (26)    4: Update ${\mathcal{W}}^{k+1}$ by (28)    5: Update ${\mathcal{Q}}_1^{k+1},{\mathcal{Q}}_2^{k+1},{\mathcal{Q}}_3^{k+1}$ by (29)    6: check convergence: $$\begin{gathered} \max \left\{\begin{array}{c}\parallel {\mathcal{Z}}^{k+1}-{\mathcal{Z}}^k{\parallel }_{\infty }, \\ \parallel {\mathcal{E}}^{k+1}-{\mathcal{E}}^k{\parallel }_{\infty }, \\ \parallel {\mathcal{Y}}^{k+1}-{\mathcal{Y}}^k{\parallel }_{\infty }, \\ {\parallel {\mathcal{W}}^{k+1}-{\mathcal{W}}^k\parallel }_{\infty }\end{array}\right\}\le ϵ \end{gathered}$$    7: $k=k+1$ End While Output: $\mathcal{E}$

3.3. Dictionary Construction

In LRTR-based AD methods, the quality of the background dictionary is important for enhancing AD accuracy. Some existing methods regard the entire HSI as a dictionary, and this construction method usually has two drawbacks [62,63]. First, the dictionary contains too many elements, resulting in significant computational overhead. Second, HSI is inevitably affected by noise during the acquisition and transmission. Therefore, we perform PCA preprocessing on the HSI tensor before dictionary construction to reduce both the computational burden and the impact of noisy bands. Tensor robust principal component analysis (TRPCA) [57] utilizes the tensor nuclear norm and and $L_1$-norm to decompose the input data tensor into low-rank and sparse tensors. Inspired by TRPCA, our dictionary construction strategy employs WTNN appropriately allocate the singular value weights to preserve the low-rank characteristics of the background, while $\Omega \left(\mathcal{S}\right)$ is used to merge the abnormal pixels of each band from spatial and spectral dimensions, enhancing anomaly sparsity and resulting in a cleaner background:

$$\begin{array}{cc}\hfill \underset{\mathcal{L},\mathcal{S}}{min}& {\parallel \mathcal{L}\parallel }_{\mathrm{WTNN}}+\alpha \Omega \left(\mathcal{S}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathcal{X}=\mathcal{L}+\mathcal{S},\hfill \end{array}$$

(30)

where $\alpha$ is the positive trade-off parameter, $\mathcal{L}$ is the low-rank background, and $\mathcal{S}$ is the sparse anomaly. Equation (30) is solved using ADMM with the augmented Lagrangian function given by:

$$\begin{array}{cc}\hfill L_{\beta }& (\mathcal{L},\mathcal{S},{\mathcal{Q}}_4)={\parallel \mathcal{L}\parallel }_{\mathrm{WTNN}}+\alpha \Omega \left(\mathcal{S}\right)\hfill \\ \hfill & +\langle {\mathcal{Q}}_4,\mathcal{X}-\mathcal{L}-\mathcal{S}\rangle +{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{X}-\mathcal{L}-\mathcal{S}\parallel }_F^2,\hfill \end{array}$$

(31)

where ${\mathcal{Q}}_4$ represents the Lagrange multiplier and $\beta >0$ is the penalty parameter. Equation (31) can be solved in a similar way as in Section 3.2.

• The $\mathcal{L}$-subproblem can be simplified to:

  $$\begin{array}{cc}\hfill \underset{\mathcal{L}}{min}& {\parallel \mathcal{L}\parallel }_{\mathrm{WTNN}}+{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{X}-\mathcal{L}-{\mathcal{S}}^k+{\displaystyle \frac{{\mathcal{Q}}_4^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (32)

  where the solution of (32) is similar to (19).
• The $\mathcal{S}$-subproblem can be simplified to:

  $$\begin{array}{cc}\hfill \underset{\mathcal{S}}{min}& \alpha \Omega \left(\mathcal{S}\right)+{\displaystyle \frac{\beta }{2}}{\parallel \mathcal{X}-{\mathcal{L}}^k-\mathcal{S}+{\displaystyle \frac{{\mathcal{Q}}_4^k}{\beta }}\parallel }_F^2,\hfill \end{array}$$

  (33)

  which has a closed-form solution similar to (22).
• The Lagrange multipliers ${\mathcal{Q}}_4$ is updated by:

  $$\left\{\begin{array}{c}{\mathcal{Q}}_4^{k+1}={\mathcal{Q}}_4^k+\beta (\mathcal{X}-{\mathcal{L}}^{k+1}-{\mathcal{S}}^{k+1}),\hfill \\ \beta =\min \{\kappa \beta ,{\beta }_{max}\}.\hfill \end{array}\right.$$

  (34)

The resulting background component $\mathcal{L}$ is then used as the constructed dictionary $\mathcal{A}$, i.e., $\mathcal{A}=\mathcal{L}$. This construction strategy is based on LRTR, and the resulting tensor dictionary $\mathcal{A}$ preserves the inherent spatial–spectral structure.

4. Experiments and Discussions

In this section, WMS-LRTR is compared with nine benchmark methods, including RX [14], LRASR [26], GTVLRR [28], AUTO-AD [43], RGAE [44], DeCNN-AD [53], PTA [36], PCA-TLRSR [37], and LARTVAD [38]. Among them, RX, LRASR, and GTVLRR are three classical methods, and others are proposed in recent years. All experiments in this paper are performed on Matlab R2018a and Pytorch 1.9.0 on a computer with Intel i5-10210U CPU@1.60 GHz and 12 GB RAM in Windows 10.

4.1. Dataset Description

To evaluate the effectiveness of WMS-LRTR, thirteen datasets from Airport–Beach–Urban (ABU) database are considered. ABU was captured by the airborne visible/infrared imaging spectrometer (AVIRIS) sensor and the reflective optics system imaging spectrometer (ROSIS) sensor and contains three subscenes in total: airport, beach, and urban, see Figure 3. The size is $100\times 100\times N$ or $150\times 150\times N$, where N is in the range from 102 to 207. The details of these datasets are listed in

Table 1. The details of the thirteen datasets.

Dataset	Sensor	Location	Size	Bands	Spatial Resolution (m)	Spectral Resolution (nm)
Airport-1	AVIRIS	Los Angeles	100 × 100	205	7.1	10.0
Airport-2	AVIRIS	Los Angeles	100 × 100	205	7.1	10.0
Airport-3	AVIRIS	Los Angeles	100 × 100	205	7.1	10.0
Airport-4	AVIRIS	Gulfport	100 × 100	191	3.4	10.0
Beach-1	AVIRIS	Cat Island	150 × 150	188	17.2	10.0
Beach-2	AVIRIS	San Diego	100 × 100	193	7.5	10.0
Beach-3	AVIRIS	Bay Champagne	100 × 100	188	4.4	10.0
Beach-4	ROSIS	Pavia	150 × 150	102	1.3	10.0
Urabn-1	AVIRIS	Texas Coast	100 × 100	204	17.2	10.0
Urabn-2	AVIRIS	Texas Coast	100 × 100	207	17.2	10.0
Urabn-3	AVIRIS	Gainesville	100 × 100	191	3.5	10.0
Urabn-4	AVIRIS	Los Angeles	100 × 100	205	7.1	10.0
Urabn-5	AVIRIS	Los Angeles	100 × 100	205	7.1	10.0

.

4.2. Implementation Details

4.2.1. Performance Indicators

In this paper, we employ three commonly used evaluation metrics, namely, AD map, 3D receiver operating characteristic (ROC) curve [64], and area under the ROC curve (AUC). The interplay among detection probability ${\mathrm{P}}_{\mathrm{D}}$, false alarm probability ${\mathrm{P}}_{\mathrm{F}}$, and detection threshold $\tau$ is depicted by the 3D ROC curve. Furthermore, three 2D ROC curves representing (${\mathrm{P}}_{\mathrm{D}}$, ${\mathrm{P}}_{\mathrm{F}}$), (${\mathrm{P}}_{\mathrm{D}}$, $\tau$), and (${\mathrm{P}}_{\mathrm{F}}$, $\tau$) can be obtained from the 3D ROC curve. The ROC curve of (${\mathrm{P}}_{\mathrm{D}}$, ${\mathrm{P}}_{\mathrm{F}}$) is near the upper left corner and (${\mathrm{P}}_{\mathrm{D}}$, $\tau$) is near the upper right corner, indicating better detection performance. On the contrary, the ROC curve of (${\mathrm{P}}_{\mathrm{F}}$, $\tau$) is close to the lower left corner, indicating a stronger ability to suppress the background. Consequently, we can derive ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$, ${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}$, and ${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}$ from these curves. Integrating these evaluation indicators with each other, the following four comprehensive evaluation indicators can be obtained, which are, respectively, called ${\mathrm{AUC}}_{\mathrm{TD}}$, ${\mathrm{AUC}}_{\mathrm{BS}}$, ${\mathrm{AUC}}_{\mathrm{TDBS}}$, and ${\mathrm{AUC}}_{\mathrm{ODP}}$. Typically, ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ and ${\mathrm{AUC}}_{\mathrm{TDBS}}$ are used to assess the efficiency of the detector, while ${\mathrm{AUC}}_{\mathrm{ODP}}$ assesses the overall detection performance. These three indicators are considered the most important. Moreover, ${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}$ and ${\mathrm{AUC}}_{\mathrm{TD}}$ evaluate the target detection capability. ${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}$ and ${\mathrm{AUC}}_{\mathrm{BS}}$ evaluate the background suppression capability.

4.2.2. Parameter Selection

In WMS-LRTR, there exist three parameters, i.e., $\lambda$, $\mu$, and $\beta$. Their variation range is set from ${10}^{-6}$ to ${10}^4$. The ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ changes under different parameter settings on Urban-3, Urban-4, and Urban-5 are shown in Figure 4a–f, respectively. Then, we discuss the selection of spectral dimensions for PCA. Figure 5 illustrates the ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ values vary with the spectral dimension. The best spectral dimensions for the thirteen datasets are 4, 5, 12, 5, 12, 12, 11, 12, 8, 8, 4, 6, and 8, respectively. In all experiments, when {$\lambda$, $\mu$, $\beta$} are set to {${10}^{-2}$, ${10}^{-2}$, ${10}^{-1}$}, WMS-LRTR exhibits the superior performance. For convenience, ${\gamma }_1,{\gamma }_2,{\gamma }_3$ are set to $1/3$. The parameters of the compared methods are set to the optimal values recommended in the respective literature. For LRASR, the number of clusters K is set to 15, with each cluster containing 20 atoms. The parameters $\lambda$ and $\beta$ are varied within the range of 0.01 to 1. For GTVLRR, the parameters $\lambda$, $\beta$, and $\gamma$ are set to 0.05, 0.2, and 0.02, respectively. The trade-off parameter $\lambda$, the number of superpixels S, and the dimension of the hidden layer $n\_hid$ in RGAE are set to 0.01, 150, and 100, respectively. The parameters of DeCNN-AD are K, $\beta$, and $\lambda$ and we set them to 5, 0.1, and 0.1. For PTA, the truncated low-rank r, $\alpha$, and $\tau$ are set to 1, 1, and 0.01, respectively, while $\mu$ is chosen from the range 0.1 to 0.5. The parameters in PCA-TLRSR are set based on the recommendations provided in [37]. For the parameters K, $\beta$, and $\lambda$ in LARTVAD, we set them to 80, 0.1, and 1, respectively.

4.3. Numerical Results

4.3.1. Experiments on the Airport Scenes

Figure 3 illustrates the AD maps for all the compared methods on the Airport scenes. Among all the methods, RX, LRASR, and GTVLRR detect fewer anomalies. AUTO-AD and RGAE demonstrate strong background suppression capabilities, but their detection performance is not excellent. PTA shows better detection ability but tends to generate redundant background components in the AD map. Among the remaining methods, the proposed WMS-LRTR successfully detects more anomalies while effectively suppressing the background.

In addition, the ROC curves for Airport-1 and Airport-2 are shown in Figure 6 and Figure 7. As can be seen in Figure 6a and Figure 7a, WMS-LRTR surpasses all methods to achieve the best performance. In Figure 6b and Figure 7b, WMS-LRTR shows slightly inferior to PTA. However, in Figure 6c and Figure 7c, PTA performs significantly worse than WMS-LRTR. Therefore, as demonstrated in Figure 6d and Figure 7d, WMS-LRTR exhibits better overall performance compared to all other methods. Additionally,

Table 2. The AUC values on the Airport scenes.

Dataset	AUC	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Airport-1	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.8221	0.7284	0.8997	0.6941	0.6387	0.8662	0.9109	0.9420	0.9202	0.9435
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.0987	0.1711	0.2665	0.1595	0.0506	0.1562	0.3471	0.3088	0.2540	0.3284
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0424	0.1209	0.1153	0.0991	0.0255	0.0689	0.1191	0.0918	0.0816	0.1001
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	0.9208	0.8996	1.1647	0.8536	0.6889	1.0224	1.2580	1.2508	1.1742	1.2718
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.7797	0.6075	0.7844	0.5950	0.6128	0.7974	0.7918	0.8502	0.8386	0.8433
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0563	0.0502	0.1496	0.0603	0.0252	0.0873	0.2279	0.2170	0.1724	0.2282
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	0.8784	0.7786	1.0493	0.7544	0.6635	0.9536	1.1388	1.1590	1.0927	1.1717
Airport-2	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.8403	0.8707	0.8670	0.6764	0.7470	0.9656	0.9411	0.9543	0.9387	0.9704
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1841	0.3156	0.3175	0.1976	0.0770	0.3257	0.4334	0.3705	0.2845	0.3807
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0516	0.1613	0.1379	0.0862	0.0196	0.0476	0.1292	0.0753	0.0692	0.0652
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0245	1.1863	1.1845	0.8439	0.8239	1.2913	1.3745	1.3248	1.2233	1.3511
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.7888	0.7094	0.7291	0.5902	0.7274	0.9180	0.8119	0.8790	0.8696	0.9052
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1325	0.1542	0.1797	0.0814	0.0574	0.2781	0.3042	0.2952	0.2154	0.3155
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	0.9709	1.0249	1.0467	0.7578	0.8044	1.2437	1.2453	1.2495	1.1541	1.2859
Airport-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9288	0.9234	0.9231	0.9210	0.8873	0.9235	0.9247	0.9540	0.8877	0.9579
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.0660	0.0562	0.0695	0.1278	0.0511	0.0676	0.1665	0.1398	0.1203	0.1333
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0145	0.0126	0.0155	0.0395	0.0057	0.0123	0.0416	0.0279	0.0326	0.0194
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	0.9948	0.9796	0.9927	1.0488	0.9384	0.9916	1.0911	1.0945	1.0080	1.0912
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9144	0.9108	0.9077	0.8815	0.8816	0.9117	0.8831	0.9268	0.8551	0.9385
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0516	0.0436	0.0540	0.0883	0.0454	0.0553	0.1249	0.1119	0.0877	0.1139
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	0.9804	0.9670	0.9772	1.0094	0.9327	0.9793	1.0496	1.0666	0.9754	1.0718
Airport-4	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9526	0.9566	0.9836	0.9840	0.7508	0.9239	0.9841	0.9933	0.9173	0.9961
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.0736	0.3747	0.4437	0.4071	0.1172	0.4229	0.6476	0.4350	0.0931	0.5110
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0248	0.1053	0.0942	0.0267	0.0749	0.1646	0.1044	0.0924	0.0311	0.0427
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0262	1.3313	1.4273	1.3911	0.8679	1.3467	1.6317	1.4283	1.0104	1.5071
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9278	0.8513	0.8894	0.9573	0.6759	0.7593	0.8798	0.9008	0.8862	0.9534
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0489	0.2693	0.3496	0.3804	0.0423	0.2583	0.5432	0.3426	0.0620	0.4684
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0015	1.2260	1.3332	1.3644	0.7930	1.1822	1.5273	1.3358	0.9793	1.4645

↑ indicates a higher result is better, ↓ indicates a lower result is better, bold indicates the best result, and underline indicates the second-best result.

shows the AUC values for all methods on the Airport scenes. As can be seen from Table 2, for ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$, which evaluates detector efficiency, WMS-LRTR outperforms all other methods, achieving the highest value across the four datasets. Besides, for ${\mathrm{AUC}}_{\mathrm{ODP}}$, which assesses overall performance, WMS-LRTR achieves either the best or the second-best value across the datasets, further demonstrating its effectiveness.

4.3.2. Experiments on the Beach Scenes

Figure 3 depicts the visualization maps of compared methods on the Beach scenes. For Beach-1 and Beach-2, AUTO-AD and RGAE perform poorly, while other methods are able to identify the majority of the anomalies. Among these methods, WMS-LRTR exhibits better background suppression ability and detection efficiency. For Beach-3 and Beach-4, AUTO-AD based on deep learning shows excellent background suppression ability. Among the other methods, anomalies detected by WMS-LRTR are obvious, although it should be acknowledged that its background suppression ability is slightly weaker than AUTO-AD.

Figure 8 and Figure 9 give the related ROC curves on Beach-2 and Beach-3. In Figure 8a,b and Figure 9a,b, the ROC curve of WMS-LRTR is closest to the upper left and right corners, respectively. In Figure 8c and Figure 9c, WMS-LRTR is slightly inferior to AUTO-AD and RGAE. Besides, the AUC values on the Beach scenes are provided in

Table 3. The AUC values on the Beach scenes.

Dataset	AUC	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Beach-1	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9804	0.9155	0.9720	0.9510	0.9395	0.9635	0.9742	0.9673	0.9606	0.9883
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.2496	0.2161	0.3081	0.1288	0.1055	0.2778	0.2711	0.3298	0.2265	0.3715
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0065	0.0460	0.0237	0.0183	0.0142	0.0181	0.0177	0.0235	0.0145	0.0111
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.2304	1.1316	1.2801	1.0798	1.0450	1.1913	1.2452	1.2971	1.1871	1.3598
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9742	0.8694	0.9483	0.9327	0.9253	0.9454	0.9565	0.9437	0.9460	0.9772
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.2431	0.1701	0.2845	0.1105	0.0913	0.2096	0.2534	0.3063	0.2119	0.3604
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.2238	1.0856	1.2565	1.0615	1.0308	1.1732	1.2275	1.2735	1.1725	1.3487
Beach-2	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9106	0.6357	0.9274	0.8803	0.9020	0.9067	0.9167	0.9273	0.9230	0.9604
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1530	0.1335	0.2236	0.0472	0.0200	0.1774	0.1701	0.2454	0.1154	0.2820
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0488	0.0907	0.0469	0.0173	0.0180	0.0320	0.0536	0.0477	0.0253	0.0398
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0636	0.7692	1.1510	0.9275	0.9220	1.0841	1.0868	1.1727	1.0384	1.2424
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.8618	0.5450	0.8805	0.8630	0.8840	0.8747	0.8631	0.8796	0.8977	0.9206
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1042	0.0428	0.1767	0.0299	0.0020	0.1454	0.1165	0.1977	0.0901	0.2422
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0148	0.6786	1.1041	0.9101	0.9040	1.0522	1.0333	1.1250	1.0130	1.2026
Beach-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9998	0.9953	0.9923	0.9991	0.8668	0.9985	0.9989	0.9985	0.9939	0.9993
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.5314	0.5578	0.6527	0.3724	0.3569	0.5439	0.5679	0.6387	0.5640	0.6765
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0259	0.0796	0.1430	0.0029	0.0397	0.0428	0.0459	0.0629	0.0490	0.0639
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.5312	1.5531	1.6450	1.3715	1.2237	1.5424	1.5668	1.6372	1.5579	1.6758
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9739	0.9157	0.8493	0.9962	0.8271	0.9557	0.9530	0.9356	0.9449	0.9354
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.5055	0.4782	0.5097	0.3695	0.3172	0.5011	0.5220	0.5758	0.5150	0.6126
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.5053	1.4736	1.5020	1.3685	1.1840	1.4996	1.5209	1.5743	1.5089	1.6118
Beach-4	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9538	0.9216	0.9796	0.9838	0.9041	0.9680	0.9701	0.9463	0.9578	0.9724
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1343	0.1949	0.2420	0.1047	0.2210	0.1882	0.3579	0.2991	0.3090	0.3287
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0233	0.0510	0.0236	0.0012	0.0377	0.0081	0.0353	0.0807	0.0717	0.0694
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0881	1.1167	1.2217	1.0885	1.1252	1.1562	1.3280	1.2454	1.2668	1.3010
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9305	0.8708	0.9560	0.9826	0.8664	0.9599	0.9349	0.8656	0.8861	0.9030
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1110	0.1440	0.2185	0.1036	0.1834	0.1801	0.3226	0.2184	0.2373	0.2593
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0648	1.0657	1.1981	1.0873	1.0875	1.1481	1.2927	1.1647	1.1951	1.2317

↑ indicates a higher result is better, ↓ indicates a lower result is better, bold indicates the best result, and underline indicates the second-best result.

. As shown in Table 3, both ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ and ${\mathrm{AUC}}_{\mathrm{ODP}}$ achieve either the best or suboptimal performance across the four datasets, further demonstrating the better efficiency and overall detection capability of WMS-LRTR.

4.3.3. Experiments on the Urban Scenes

Figure 3 shows the detection visual results for the Urban scenes. AUTO-AD and RGAE exhibit good background suppression ability on these five datasets, but they fail to detect all anomalies or mistakenly classify background pixels as anomalies. LRASR, GTVLRR, and DeCNN-AD mistakenly identify more background as abnormal targets, resulting in a poorer visual effect. RX and LARTVAD do not perform well in AD, as many anomalies are not detected. PTA, PCA-TLRSR, and WMS-LRTR are able to detect all anomalies Compared with the previous two methods, WMS-LRTR exhibits better detection effectiveness and identifies fewer background pixels as anomalies.

The ROC curves of the Urban scenes are shown in Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. From the ROC curves of (${\mathrm{P}}_{\mathrm{D}}$, ${\mathrm{P}}_{\mathrm{F}}$) on these datasets, WMS-LRTR is consistently in the upper left corner, indicating its significantly superior performance compared to other methods. Besides, there is a crossover between WMS-LRTR and other methods for the ROC curves of (${\mathrm{P}}_{\mathrm{D}}$, $\tau$), but WMS-LRTR is the closest to the upper right corner. WMS-LRTR performs slightly worse than AUTO-AD and RGAE for the ROC curves of (${\mathrm{P}}_{\mathrm{F}}$, $\tau$), but these two methods show poorer performance on the other ROC curves. From Figure 10c, Figure 11c, Figure 12c, Figure 13c and Figure 14c, it is evident that the ROC curve of WMS-LRTR consistently appears in the lower left corner across these five datasets, which demonstrates that WMS-LRTR maintains a low false alarm rate across different threshold $\tau$. In addition, all AUC values are presented in

Table 4. The AUC values on the Urban scenes.

Dataset	AUC	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Urban-1	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9907	0.9452	0.8278	0.9886	0.9821	0.9238	0.9808	0.9810	0.9773	0.9942
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.3143	0.4198	0.3212	0.2245	0.3749	0.4661	0.5001	0.4977	0.5206	0.5271
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0556	0.1707	0.1681	0.0050	0.0168	0.1499	0.0882	0.0621	0.0657	0.0682
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.3050	1.3650	1.1490	1.2131	1.3570	1.3899	1.4809	1.4787	1.4979	1.5213
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9351	0.7745	0.6597	0.9836	0.9653	0.7739	0.8926	0.9189	0.9116	0.9260
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.2587	0.2491	0.1531	0.2195	0.3581	0.3162	0.4119	0.4356	0.4549	0.4589
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.2494	1.1942	0.9810	1.2081	1.3402	1.2400	1.3927	1.4166	1.4322	1.4531
Urban-2	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9946	0.8640	0.8499	0.9893	0.9871	0.9340	0.9592	0.9854	0.9597	0.9959
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1178	0.0636	0.1324	0.0812	0.1101	0.2047	0.2001	0.1870	0.1320	0.2144
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0135	0.0186	0.0424	0.0014	0.0053	0.0392	0.0130	0.0228	0.0142	0.0243
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.1124	0.9276	0.9823	1.0705	1.0973	1.1387	1.1593	1.1724	1.0917	1.2103
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9811	0.8454	0.8075	0.9879	0.9819	0.8948	0.9462	0.9626	0.9455	0.9716
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1043	0.0450	0.0900	0.0798	0.1049	0.1655	0.1871	0.1642	0.1178	0.1901
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0989	0.9090	0.9399	1.0690	1.0920	1.0995	1.1464	1.1496	1.0776	1.1861
Urban-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9513	0.9521	0.9430	0.9891	0.8223	0.9635	0.9684	0.9906	0.9679	0.9936
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.0963	0.3686	0.4351	0.2654	0.1018	0.4163	0.3170	0.4459	0.3643	0.4608
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0351	0.1135	0.1106	0.0089	0.0376	0.0601	0.0562	0.0693	0.0672	0.0346
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0476	1.3207	1.3781	1.2545	0.9241	1.3779	1.2854	1.4365	1.3322	1.4544
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9162	0.8386	0.8324	0.9802	0.7847	0.9035	0.9122	0.9213	0.9007	0.9590
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0612	0.2551	0.3245	0.2565	0.0642	0.3563	0.2608	0.3766	0.2971	0.4262
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0125	1.2072	1.2676	1.2457	0.8865	1.3198	1.2292	1.3672	1.2651	1.4198
Urban-4	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9887	0.5991	0.9326	0.9867	0.9862	0.9744	0.9632	0.9313	0.9796	0.9887
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.0891	0.0562	0.0682	0.0379	0.0372	0.0993	0.1034	0.1113	0.0593	0.1130
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0114	0.0191	0.0089	0.0008	0.0014	0.0183	0.0054	0.0162	0.0074	0.0146
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.0778	0.8188	1.0085	1.0246	1.0234	1.0737	1.0666	1.0426	1.0389	1.1017
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9773	0.7767	0.9283	0.9859	0.9848	0.9561	0.9578	0.9151	0.9722	0.9741
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0777	0.0291	0.0622	0.0371	0.0358	0.0810	0.0980	0.0951	0.0519	0.0984
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0664	0.6362	0.9919	1.0238	1.0220	1.0554	1.0612	1.0263	1.0316	1.0871
Urban-5	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9692	0.9076	0.9304	0.8728	0.9569	0.8339	0.9136	0.9691	0.9043	0.9804
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1461	0.2138	0.2611	0.0707	0.2533	0.3065	0.2852	0.2853	0.1721	0.3158
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0437	0.0899	0.0844	0.0062	0.0179	0.1742	0.0417	0.0688	0.0660	0.0627
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.1153	1.1214	1.1915	0.9435	1.2102	1.1404	1.1988	1.2544	1.0764	1.2962
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9255	0.8177	0.8460	0.8666	0.9390	0.6597	0.8719	0.9003	0.8383	0.9177
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1024	0.1239	0.1767	0.0645	0.2354	0.1323	0.2435	0.2165	0.1061	0.2531
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0716	1.0314	1.1072	0.9372	1.1922	0.9662	1.1570	1.1856	1.0104	1.2335

↑ indicates a higher result is better, ↓ indicates a lower result is better, bold indicates the best result, and underline indicates the second-best result.

. For ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$, which evaluates the effectiveness of detectors, WMS-LRTR obtains the optimal value on the five datasets. For ${\mathrm{AUC}}_{\mathrm{ODP}}$, which evaluates the overall detection performance, WMS-LRTR also achieves the optimal value on all datasets. Finally,

Table 5. The average AUC values on the thirteen datasets.

Dataset	AUC	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Average	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9448	0.8627	0.9253	0.9166	0.8747	0.9343	0.9543	0.9646	0.9452	0.9801
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.1734	0.2417	0.2878	0.1711	0.1444	0.2810	0.3360	0.3303	0.2473	0.3572
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.0305	0.0830	0.0780	0.0241	0.0242	0.0643	0.0578	0.0570	0.0458	0.0474
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.1183	1.1170	1.2136	1.0855	1.0190	1.2113	1.2902	1.2950	1.1926	1.3372
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.9143	0.7948	0.8476	0.8925	0.8505	0.8700	0.8965	0.9077	0.8994	0.9327
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.1429	0.1580	0.2099	0.1447	0.1202	0.2128	0.2782	0.2733	0.2015	0.3098
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.0876	1.0214	1.1350	1.0613	0.9948	1.1471	1.2325	1.2380	1.1468	1.2899

↑ indicates a higher result is better, ↓ indicates a lower result is better, bold indicates the best result, and underline indicates the second-best result.

presents the average AUC values across the thirteen datasets. Except for ${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}$, all other indicators show that WMS-LRTR achieved the best values. Furthermore, the ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ is increased by about 1.6% compared to PCA-TLRSR. These quantitative experimental results clearly demonstrate the superiority of WMS-LRTR over other competing methods.

4.4. Discussion

4.4.1. Statistical Separability Analysis

In this section, we utilize boxplots to visualize ability of WMS-LRTR to distinguish anomalies from the background. The resulting boxplots are shown in Figure 15, where the green box illustrates the distribution of the background, and the red box represents the distribution of the anomalies. The distance between the two boxes serves as an intuitive indicator of the ability to separate anomalies from the background. A larger distance between the two boxes indicates better AD ability. For Urban-1, Urban-3, and Urban-5, WMS-LRTR has the best ability to separate anomaly and background. For Airport-1 and Beach-2, all methods exhibit poor separation, but WMS-LRTR still achieves the best performance. However, for Beach-3, Urban-2, and Urban-4, the separation capability of WMS-LRTR is slightly lower than AUTO-AD and RGAE, resulting in a suboptimal result. In summary, the proposed WMS-LRTR method demonstrates a better capability for separating anomalies from the background.

4.4.2. Noise Resistance Analysis

This section discusses the noise resistance ability of WMS-LRTR. In the experiment, we apply Gaussian noise with a density of 0.2 to each band and randomly select 20 bands with a density of 0.2 salt and pepper noise to generate two synthetic datasets, called Noisy Beach-3 and Noisy Urban-3. The visualization results are depicted in Figure 16, with the corresponding AUC values provided in

Table 6. The comparison of AUC values under noise environments.

Dataset	AUC	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Noisy Beach-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.9267	0.7543	0.6208	0.7905	0.8067	0.7806	0.9740	0.8109	0.8602	0.9755
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.5386	0.5523	0.5122	0.2924	0.3427	0.5665	0.5110	0.5785	0.7330	0.4833
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.2453	0.3667	0.4267	0.0607	0.0630	0.3506	0.1056	0.3105	0.5712	0.0491
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.4652	1.3066	1.1330	1.0829	1.1494	1.3470	1.4850	1.3894	1.5932	1.4589
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.6814	0.3876	0.1941	0.7297	0.7437	0.4300	0.8684	0.5004	0.2889	0.9264
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.2933	0.1856	0.0854	0.2316	0.2797	0.2158	0.4054	0.2680	0.1618	0.4342
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	1.2200	0.9400	0.7063	1.0221	1.0864	0.9964	1.3794	1.0790	1.0220	1.4097
Noisy Urban-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}\uparrow$	0.6873	0.5919	0.5124	0.6794	0.4518	0.5492	0.9085	0.6180	0.6893	0.9281
	${\mathrm{AUC}}_{\left(\mathrm{D},\tau \right)}\uparrow$	0.4440	0.4611	0.4061	0.2357	0.0781	0.4445	0.3519	0.4470	0.6803	0.2803
	${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}\downarrow$	0.3539	0.4127	0.3971	0.1358	0.0955	0.4177	0.1543	0.3956	0.6219	0.0714
	${\mathrm{AUC}}_{\mathrm{TD}}\uparrow$	1.1312	1.0530	0.9185	0.9151	0.5299	0.9937	1.2604	1.0650	1.3696	1.2084
	${\mathrm{AUC}}_{\mathrm{BS}}\uparrow$	0.3334	0.1791	0.1153	0.5436	0.3563	0.1315	0.7541	0.2225	0.0674	0.8567
	${\mathrm{AUC}}_{\mathrm{TDBS}}\uparrow$	0.0901	0.0484	0.0090	0.0998	0.0175	0.0268	0.1976	0.0514	0.0583	0.2089
	${\mathrm{AUC}}_{\mathrm{ODP}}\uparrow$	0.7774	0.6403	0.5214	0.7792	0.4344	0.5760	1.1061	0.6695	0.7477	1.1370

↑ indicates a higher result is better, ↓ indicates a lower result is better, bold indicates the best result, and underline indicates the second-best result.

. The detection maps of WMS-LRTR are closest to the ground truth, with PTA achieving the second-best performance. In addition, the other methods exhibit poor performance, being severely affected by noise, and detect almost no anomalies. As shown in Table 6, WMS-LRTR achieves a relatively high ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ and ${\mathrm{AUC}}_{\mathrm{ODP}}$ value even in the presence of severe noise interference. In addition, in noisy environments, WMS-LRTR demonstrates the best background suppression ability, with both ${\mathrm{AUC}}_{\left(\mathrm{F},\tau \right)}$ and ${\mathrm{AUC}}_{\mathrm{BS}}$ achieving optimal values, which can be attributed to the PnP denoising prior. On the two noisy datasets, the ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ of WMS-LRTR is higher by 0.2% and 2.2% compared to the second-best PTA method. In summary, the proposed WMS-LRTR method has excellent noise resistance ability and still maintains superior AD performance even in severe noisy cases.

4.4.3. Ablation Analysis

Ablation experiments are carried out on four datasets in this section.

Table 7. The comparison of ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ values for ablation experiments.

Dataset	Without PCA		Without WTNN		Without WMS		Without PnP Prior		WMS-LRTR
	${\mathbf{AUC}}_{{\displaystyle (}\mathbf{D},\mathbf{F}{\displaystyle )}}$	Time (s)	${\mathbf{AUC}}_{{\displaystyle (}\mathbf{D},\mathbf{F}{\displaystyle )}}$	Time (s)	${\mathbf{AUC}}_{{\displaystyle (}\mathbf{D},\mathbf{F}{\displaystyle )}}$	Time (s)	${\mathbf{AUC}}_{{\displaystyle (}\mathbf{D},\mathbf{F}{\displaystyle )}}$	Time (s)	${\mathbf{AUC}}_{{\displaystyle (}\mathbf{D},\mathbf{F}{\displaystyle )}}$	Time (s)
Airport-1	0.9076	14,801.741	0.8294	226.314	0.9350	46.210	0.9240	195.416	0.9435	222.551
Airport-2	0.9322	12,431.595	0.9585	96.175	0.9627	26.713	0.9441	62.733	0.9704	92.963
Airport-3	0.9274	15,124.256	0.9529	271.938	0.9546	132.316	0.9533	179.844	0.9579	297.676
Airport-4	0.9779	13,547.221	0.9914	138.498	0.9952	41.520	0.9906	96.701	0.9961	130.514
Average	0.9363	13,976.203	0.9331	183.231	0.9619	61.690	0.9530	133.674	0.9670	185.926

Bold is used to indicate the best result.

summarizes the ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ values without PCA, WTNN, WMS, and PnP denoising prior, respectively. In the case without WMS, we apply the $L_{F,1}$-norm which can explore tensor sparsity as an alternative. As shown in Table 7, ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ of WMS-LRTR is higher by 3.3%, 3.6%, 0.5%, and 1.5%, respectively, in the four cases, which convinces the effectiveness of WMS-LRTR. In addition, the corresponding runtime is also provided in Table 7, from which it is evident that PCA significantly reduces the computational load required for detection. This is the reason why we employ PCA as a data preprocessing step.

Moreover,

Table 8. The ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ values for a single direction and multiple directions.

Dataset	Index	Horizontal	Vertical	Spectral	Multidirectional
Airport-1	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9428	0.9415	0.9412	0.9435
Airport-2	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9679	0.9686	0.9629	0.9704
Airport-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9547	0.9526	0.9577	0.9579
Airport-4	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9958	0.9956	0.9950	0.9961

Bold is used to indicate the best result.

further demonstrates the superiority of WMS. We compare the ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ results of WMS with those obtained by considering only a single direction, respectively. Since WMS simultaneously explores the sparsity of abnormal local structural features from three dimensions, horizontal, vertical, and spectral, and considers the spatial characteristics of HSI, it achieves better results than only considering one direction.

In addition,

Table 9. The runtime (s) of the compared methods.

Dataset	RX [14]	LRASR [26]	GTVLRR [28]	AUTO-AD [43]	RGAE [44]	DeCNN-AD [53]	PTA [36]	PCA-TLRSR [37]	LARTVAD [38]	WMS-LRTR
Airport-1	0.102	36.594	214.276	53.080	151.695	56.391	41.515	5.185	46.579	222.551
Airport-2	0.387	52.394	223.684	24.520	144.780	61.706	30.623	5.322	51.808	92.963
Airport-3	0.089	47.754	171.489	20.675	152.961	73.266	36.622	21.625	40.472	297.676
Airport-4	0.092	40.181	180.609	26.694	156.080	77.445	33.261	22.451	55.220	130.514
Average	0.168	44.231	197.515	31.242	151.379	67.202	35.505	13.646	48.520	185.926

presents the runtime in seconds of compared methods. It can be seen that the runtime of WMS-LRTR is slightly slower, primarily due to the computational overhead caused by WMS and PnP denoising. Nonetheless, WMS-LRTR demonstrates superior detection ability compared to other methods. Therefore, although there is a trade-off in execution time, the performance improvement justifies this compromise within reasonable limits.

4.4.4. Block Size Analysis

This section discusses the configuration of the block size involved in WMS. As shown in

Table 10. The selection of block size in weighted multidirectional sparsity.

Dataset	Index	2 × 2	3 × 3	4 × 4	5 × 5
Airport-1	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9435	0.9424	0.9379	0.9335
	Time (s)	222.551	240.222	313.199	506.953
Airport-2	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9704	0.9663	0.9659	0.9612
	Time (s)	92.963	111.301	132.522	204.909
Airport-3	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9579	0.9597	0.9587	0.9595
	Time (s)	297.676	358.482	419.583	494.698
Airport-4	${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$	0.9961	0.9960	0.9953	0.9955
	Time (s)	130.514	150.457	179.795	278.522

Bold is used to indicate the best result.

, increasing the block size does not lead to a significant improvement in detection capability. On the contrary, as the block size increases, the number of pixels involved in each calculation grows, resulting in a heavier computational burden. Therefore, considering both detection performance and computational time, we set the block size to $2\times 2$.

4.4.5. Convergence Analysis

This section presents the relative errors with the number of iterations on all selected datasets. It follows from Figure 17 that fluctuations in the relative errors are observed within the initial 40 iterations, and the convergence curves gradually become smoother in the subsequent iterations. Furthermore, the relative errors approach zero within 100 iterations for all datasets. This observation further validates the convergence capability of WMS-LRTR.

5. Conclusions

This paper introduces a WMS-LRTR method for AD. Specifically, WTNN estimates the background effectively, WMS can exploit the local structure through tensor unfolding, while PnP denoising prior can effectively preserve low-rank property of the background. By integrating them in a tensor approximation framework, we ensure the low-rank property of the background, local structured sparsity of the anomaly, and the robustness of AD. Comprehensive experiments with nine methods on eight datasets demonstrate that WMS-LRTR outperforms the others, with the average ${\mathrm{AUC}}_{\left(\mathrm{D},\mathrm{F}\right)}$ increased by approximately 1.6% compared to the second-best PCA-TLRSR method. Moreover, the statistical separability, noise resistance, ablation, and convergence have been discussed in detail.

Although WMS-LRTR achieves better detection results, it often takes more time. In the future, we are interested in developing faster optimization algorithms to reduce the computational cost. In addition, we will combine it with advanced deep learning methods to further improve the performance.