Spectral–Spatial Complementary Decision Fusion for Hyperspectral Anomaly Detection

Abstract

Hyperspectral anomaly detection has become an important branch of remote–sensing image processing due to its important theoretical value and wide practical application prospects. However, some anomaly detection methods mainly exploit the spectral feature and do not make full use of spatial features, thus limiting the performance improvement of anomaly detection methods. Here, a novel hyperspectral anomaly detection method, called spectral–spatial complementary decision fusion, is proposed, which combines the spectral and spatial features of a hyperspectral image (HSI). In the spectral dimension, the three–dimensional Hessian matrix was first utilized to obtain three–directional feature images, in which the background pixels of the HSI were suppressed. Then, to more accurately separate the sparse matrix containing the anomaly targets in the three–directional feature images, low–rank and sparse matrix decomposition (LRSMD) with truncated nuclear norm (TNN) was adopted to obtain the sparse matrix. After that, the rough detection map was obtained from the sparse matrix through finding the Mahalanobis distance. In the spatial dimension, two–dimensional attribute filtering was employed to extract the spatial feature of HSI with a smooth background. The spatial weight image was subsequently obtained by fusing the spatial feature image. Finally, to combine the complementary advantages of each dimension, the final detection result was obtained by fusing all rough detection maps and the spatial weighting map. In the experiments, one synthetic dataset and three real–world datasets were used. The visual detection results, the three–dimensional receiver operating characteristic (3D ROC) curve, the corresponding two–dimensional ROC (2D ROC) curves, and the area under the 2D ROC curve (AUC) were utilized as evaluation indicators. Compared with nine state–of–the–art alternative methods, the experimental results demonstrate that the proposed method can achieve effective and excellent anomaly detection results.

1. Introduction

In the hyperspectral imaging process, the spectral information and spatial information of the ground targets can be obtained at the same time. The obtained tens or hundreds of spectral bands can be regarded as approximately continuous [1,2,3,4,5]. Hence, the obtained hyperspectral image (HSI) can more comprehensively reflect the subtle differences among different ground targets that cannot be detected by broadband remote sensing [6]. With this feature, fine detection of ground targets can be achieved.

Hyperspectral target detection can detect targets of interest in an HSI. According to whether the spectral information of the background and the targets under test are exploited, hyperspectral target detection can be divided into target detection and anomaly detection [7,8]. Since the prior spectral information of the targets under test is difficult to obtain in practical applications, anomaly detection that does not require the prior spectral information of the targets under test are more suitable for practical applications. For instance, hyperspectral anomaly detection has been applied to mineral exploration [9], precision agriculture [10], environmental monitoring [11] and identification of man–made targets [12].

In the past thirty years, various anomaly detection methods have been proposed. According to the types of the applied features, hyperspectral anomaly detection methods are roughly divided into two categories: spectral feature–based methods and spectral–spatial feature–based methods [13,14].

For the spectral feature–based hyperspectral anomaly detection methods, the most famous anomaly detection method is RX [15], which was proposed by Reed and Xiaoli. The RX method extracts anomaly targets by calculating the Mahalanobis distance between the spectrum of the pixel under test and the mean spectrum of the background window. However, due to the contamination caused by noise and anomaly targets, there are some false alarms in the detection results. In order to improve the performance of the RX method, a series of improved methods are proposed. The local RX (LRX) method [16] entails replacing the mean spectrum of the global background with the mean spectrum of the local background, which improves the detection effect. The weighted RX (WRX) method [17] can improve the accuracy of the detection results by applying different weights to the spectra of background pixels, noise pixels and anomaly target pixels. Carlotto proposed cluster–based anomaly detection (CBAD) [18], which clusters the HSI and then performs the RX detection method in each category. The fractional Fourier entropy (FrFE)–based anomaly detection method proposed by Tao et al. [19] improves the distinction between anomalies and backgrounds through the intermediate domain. Liu et al. [20] proposed two adaptive detectors, namely the one–step generalized likelihood ratio test (1S–GLRT) and the two–step generalized likelihood ratio test (2S–GLRT), which treat multiple pixels as an unknown pattern in anomaly detection. Chen et al. proposed the component decomposition analysis sparsity cardinality (CDASC) method [21] to improve anomaly detection performance. This method represents the original HSI as a linear orthogonal decomposition of principal components, independent components, and a noise component. Tu et al. combined Shannon entropy, joint entropy, and relative entropy to propose the ensemble entropy metric–based detector (EEMD) [22], which adopts a new information theory perspective. Furthermore, considering the nonlinear information of the HSI, Kwon et al. proposed the kernel RX (KRX) method [23]. The essence of this method is to conduct the RX method in the high–dimensional feature space where the original HSI data are mapped. The cluster kernel RX (CKRX) [24] is a fast version of KRX that reduces the time cost of the KRX method. In order to solve the high dimension and redundant information in this HSI, Xie et al. proposed a structure tensor and a guided filtering–based (STGF) [25] detector. The above methods usually need to model complex backgrounds, but these background models do not fit the actual background distribution very well.

In addition to the traditional RX–based hyperspectral anomaly detection method, the representation–based method [26,27] has received widespread attention in recent years. Chen et al. first proposed the hyperspectral target detection based on sparse representation (SR) [28]. The collaborative representation–based anomaly detection (CRD) [29] method proposed by Li et al. uses all neighboring pixels of the pixel to be measured to represent the pixel to be measured. Zhang et al. proposed the low–rank and sparse matrix decomposition–based Mahalanobis distance method, which is referred to as LSMAD [30]. According to Mahalanobis distance, this method calculates background statistics through the low–rank matrix obtained by low–rank and sparse matrix decomposition (LRSMD). Xu et al. performed low–rank and sparse representation (LRASR) [31] on the original HSI based on the background dictionary constructed by clustering, and achieved remarkable results. Li et al. [32] proposed the LSDM–MoG method, which uses the low–rank and sparse decomposition model under the assumption of mixture of Gaussian distribution to improve the characterization of complex backgrounds. By combining the fractional Fourier transform and LRSMD, Ma et al. [33] proposed the hyperspectral anomaly detection method based on feature extraction and background purification (FEBP). The FEBP method can extract intrinsic features in an HSI. Qu et al. [34] utilized spectral unmixing and low–rank decomposition to deal with the mixed–pixels problem in HSIs. Li et al. combined low–rank, sparse, and piecewise smooth priors in the prior-based tensor approximation (PTA) method [35], and the truncated nuclear norm regularization (TNNR) is used instead of traditional nuclear norm regularization to improve the accuracy of low–rank decomposition. In addition, by using the low–rank manifold in an HSI, Cheng et al. proposed the graph and total variation regularized low–rank representation (GTVLRR) [36] method. More relevant methods are statistically analyzed in [37,38]. The above anomaly detection methods have achieved satisfactory results. The representation–based method mainly focuses on the comparison of spectral information between pixels, and these approaches do not utilize the spatial information efficiently.

Compared with the above anomaly detection methods based on spectral features, the anomaly detection methods based on spectral–spatial features have been proven to improve detection performance [39,40]. At present, relevant researchers have proposed anomaly detection methods based on spectral–spatial features. Du et al. [41] selected multiple local windows in the neighborhood of the pixel under test to make full use of spatial feature information. Zhang et al. [42] utilized tensor decomposition to analyze the spectral and spatial characteristics of the HSI. Lei et al. [43] adopted deep brief network and filtering methods to extract spectral and spatial features from the HSI, respectively. Yao and Zhao [44] proposed the bilateral filter–based anomaly detection (BFAD) by combining spatial weights and spectral weights. Zhao et al. [45] proposed the spectral–spatial anomaly detection method based on fractional Fourier transform and saliency weighted collaborative representation (SSFSCRD) to reduce background contamination from noise and anomaly targets. Recently, deep–learning methods [46] using spatial information and spectral information have also received widespread attention. Although the above anomaly detection methods all exploit spectral–spatial information of the HSI, they do not make full use of the three–dimensional spatial structure, spatial attribute features, and spectral characteristics of anomaly targets in HSIs. Therefore, determining how to combine these unique spatial features and spectral features in HSIs requires in–depth research.

In this paper, to better utilize the spectral and spatial features in HSIs, a hyperspectral anomaly detection method on the basis of spectral–spatial complementary decision fusion is proposed. The main contributions of this paper are as follows.

(1)

A spectral–spatial complementary decision fusion (SCDFSCDF) framework was constructed for hyperspectral anomaly detection. In the entire framework, the spectral features and spatial features were fused to ensure satisfactory detection results.

(2)

For the first time, because a three–dimensional Hessian matrix can exploit the spectral information and the three–dimensional spatial structure information of the HSI, it is introduced to hyperspectral anomaly detection to obtain the directional feature images, which can highlight the anomaly targets and suppress the background pixels in HSIs. The three–dimensional Hessian matrix not only contains the spectral information of the HSI, but also does not break the overall structural information of the HSI.

(3)

The truncated nuclear norm approximates the rank function more accurately by minimizing the sum of a few of the smallest singular values. Therefore, to more accurately separate the sparse matrix containing anomaly targets from the directional feature images, we exploited LRSMD with the truncated nuclear norm (TNN) and sparse regular terms for the first time in hyperspectral anomaly detection. In the LRSMD, TNN replaced the nuclear norm to better approximate the rank function. In addition, a sparse regular term of the l_2,1–norm was added.

The rest of this paper is organized as follows. The proposed approach is described in detail in Section 2. The experimental datasets and the evaluation indicators are introduced in Section 3. The parameter analysis, the experimental results and the performance analysis of the proposed method are discussed in Section 4. In Section 5, the work is summarized.

2. Proposed Approach

This article combines the spectral–spatial features and proposes a hyperspectral anomaly detection method based on spectral–spatial complementary decision fusion. The processing of spectral–spatial features is regarded as completed in their respective dimensions. These dimensions are named the spectral dimension and spatial dimension, respectively. In this section, we describe this method in detail.

2.1. SCDF Framework

The proposed SCDF–based hyperspectral anomaly detection scheme is shown in Figure 1, which includes three parts: the spectral dimension, the spatial dimension and the result complementary fusion. In the spectral dimensions, three–directional feature images are extracted through the three–dimensional Hessian matrix. After that, the sparse matrices containing anomaly targets are obtained by executing LRSMD with the TNN and sparse regular terms on the three–directional feature images. Finally, the rough detection maps are extracted from the obtained sparse matrices. In the spatial dimension, the spatial feature of HSI is extracted through attribute filtering. Then, the spatial weight map is obtained from the spatial feature. In the complementary fusion of results, the final detection result is obtained by fusing the spatial weight map and the rough detection maps.

2.2. Spectral Dimension

The spectral dimension we named mainly includes the three–dimensional Hessian matrix of HSI, LRSMD–TNN, and the extraction of rough detection results.

2.2.1. Three–Dimensional Hessian Matrix of the HSI

Anomalies usually only occupy a few pixels in the HSI and can be regarded as a small target area relative to the local homogeneous background. The determinant of the Hessian matrix can express the local structural features of the image [47,48,49]. It can be employed to highlight the target structure in the image and filter out other information. Therefore, the determinant of the Hessian matrix is used in anomaly detection to find the anomaly target structure in the HSI. Considering that the two–dimensional Hessian matrix does not exploit the spectral information and the three–dimensional spatial structure information of the HSI, the three-dimensional Hessian matrix is finally exploited.

The three–dimensional Hessian matrix of the HSI is actually used to obtain the second–order partial derivative of the HSI. Furthermore, according to the linear scale space theory [50], the derivative of a function is equal to the convolution of the function and the derivative of the Gaussian function. As such, only the second–order partial derivatives of the Gaussian function in each direction are required, then the second–order partial derivatives of the HSI in each direction can be obtained.

In the three–dimensional Hessian matrix, the Gaussian function can be represented as:

$$f(x,y,z)={(\frac{1}{\sqrt{2\pi }\sigma })}^3{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hspace{1em}\forall x,y,z\in W$$

(1)

where σ represents the standard deviation and W represents the size of the window as w × w × w. The x and y represent the two–dimensional spatial position coordinates of the HSI, respectively. The z represents the spectral dimension coordinate of the HSI. The second–order partial derivative $\frac{{\partial }^{2}f(x,y,z)}{\partial x^2},\frac{{\partial }^{2}f(x,y,z)}{\partial y^2},\frac{{\partial }^{2}f(x,y,z)}{\partial z^2},\frac{{\partial }^{2}f(x,y,z)}{\partial x\partial y},\frac{{\partial }^{2}f(x,y,z)}{\partial x\partial z}$ and $\frac{{\partial }^{2}f(x,y,z)}{\partial y\partial z}$ of the Gaussian function with respect to x, y, and z is shown as:

$$\left\{\begin{array}{c}\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x^2}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{x^2-{\sigma }^2}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{x^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ \frac{{\partial }^{2}f\left(x,y,z\right)}{\partial y^2}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{y^2-{\sigma }^2}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{y^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ \frac{{\partial }^{2}f\left(x,y,z\right)}{\partial z^2}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{z^2-{\sigma }^2}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{z^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ \frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x\partial y}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{xy}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{xy}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ \frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x\partial z}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{xz}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{xz}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ \frac{{\partial }^{2}f\left(x,y,z\right)}{\partial y\partial z}=\frac{1}{{(\sqrt{2\pi }\sigma )}^3}\frac{yz}{{\sigma }^4}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}=\frac{yz}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \end{array}\right.$$

(2)

In this paper, HSI is represented by the bold letter H = (h₁, h₂, …, h_m×n)$\in$R_m×n×b, where m × n represents the number of all pixels in the HSI, and b represents the number of bands of the HSI. The second–order partial derivatives I_xx, I_yy, I_zz, I_xy, I_xz, and I_yz of the HSI H with respect to x, y, and z can be obtained by:

$$\left\{\begin{array}{c}{\mathbf{I}}_{xx}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x^2}=\mathbf{H}\left(x,y,z\right)*\frac{x^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ {\mathbf{I}}_{yy}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial y^2}=\mathbf{H}\left(x,y,z\right)*\frac{y^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ {\mathbf{I}}_{zz}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial z^2}=\mathbf{H}\left(x,y,z\right)*\frac{z^2-{\sigma }^2}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ {\mathbf{I}}_{xy}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x\partial y}=\mathbf{H}\left(x,y,z\right)*\frac{xy}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ {\mathbf{I}}_{xz}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial x\partial z}=\mathbf{H}\left(x,y,z\right)*\frac{xz}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \\ {\mathbf{I}}_{yz}=\mathbf{H}\left(x,y,z\right)*\frac{{\partial }^{2}f\left(x,y,z\right)}{\partial y\partial z}=\mathbf{H}\left(x,y,z\right)*\frac{yz}{{(\sqrt{2\pi })}^3{\sigma }^7}{e}^{-\frac{x^2+y^2+z^2}{2{\sigma }^2}}\hfill \end{array}\right.$$

(3)

where * is defined as a convolution operation. It is worth noting that the second–order partial derivatives of the HSI at (x, y, z) in the y–x, z–x, and z–y directions are equal to the second–order partial derivatives in the x–y, x–z, and y–z directions, respectively. Then, the determinant of the three–dimensional Hessian matrix in the x–y, x–z, and y–z directions can be calculated by:

$$\left\{\begin{array}{l}{\mathbf{D}}_{xy}=\left({\mathbf{I}}_{xx}\odot {\mathbf{I}}_{yy}-{\mathbf{I}}_{xy}\odot {\mathbf{I}}_{xy}\right)\odot \left({\mathbf{I}}_{xx}\odot {\mathbf{I}}_{yy}-{\mathbf{I}}_{xy}\odot {\mathbf{I}}_{xy}\right)\\ {\mathbf{D}}_{xz}=\left({\mathbf{I}}_{xx}\odot {\mathbf{I}}_{zz}-{\mathbf{I}}_{xz}\odot {\mathbf{I}}_{xz}\right)\odot \left({\mathbf{I}}_{xx}\odot {\mathbf{I}}_{zz}-{\mathbf{I}}_{xz}\odot {\mathbf{I}}_{xz}\right)\\ {\mathbf{D}}_{yz}=\left({\mathbf{I}}_{yy}\odot {\mathbf{I}}_{zz}-{\mathbf{I}}_{yz}\odot {\mathbf{I}}_{yz}\right)\odot \left({\mathbf{I}}_{yy}\odot {\mathbf{I}}_{zz}-{\mathbf{I}}_{yz}\odot {\mathbf{I}}_{yz}\right)\end{array}\right.$$

(4)

where D_xy, D_xz and D_yz have the same size as the original HSI H, $\odot$ represents the matrix dot product. We employ the determinant in each direction to express the directional feature of the HSI.

Figure 2 shows the spectral curves before and after the transformation of the three–dimensional Hessian matrix of three random pixels in the MUUFL Gulfport HSI. As shown in Figure 2a, the covers corresponding to pixels 1 and 3 are two different mixed ground surfaces, respectively, and the cover corresponding to pixel 2 is a tree. It can be seen from Figure 2b–d that after the three–dimensional Hessian matrix transformation, the background is suppressed to a certain extent. The MUUFL Gulfport HSI is introduced in detail in Section 3.

2.2.2. LRSMD–TNN

After obtaining the directional feature images of the HSI, we performed the LRSMD on the directional feature images to obtain the sparse matrix containing anomaly targets. Each directional feature image needed to be transformed into a two–dimensional matrix D$\in$R_b×mn in advance, and then the LRSMD was performed. The typical LRSMD model [51] can be written as:

$$\begin{array}{cc}\underset{\mathbf{L},\mathbf{S}}{\min }\hfill & \mathrm{rank}(\mathbf{L})+\lambda \Vert \mathbf{S}\Vert {}_1\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & \mathbf{D}=\mathbf{L}+\mathbf{S}\hfill \end{array}$$

(5)

where D represents the matrix to be decomposed. L$\in$R_b×mn represents the decomposed low–rank matrix, which mainly contains background information. S$\in$R_b×mn represents the decomposed sparse matrix, which contains anomaly targets information. rank(·) represents the rank function, ‖S‖₁ = ∑_i,j|S_i,j| represents the l₁–norm and λ is a parameter that balances low–rank and sparse matrices. To solve problem (5), most existing hyperspectral anomaly detection based on LRSMD applies the nuclear norm to approximate the rank function. As such, the optimized LRSMD model can be written as:

$$\begin{array}{cc}\underset{\mathbf{L},\mathbf{S}}{\min }\hfill & \Vert \mathbf{L}\Vert {}_{*}+\lambda \Vert \mathbf{S}|{|}_1\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & \mathbf{D}=\mathbf{L}+\mathbf{S}\hfill \end{array}$$

(6)

where ‖·‖_∗ indicates the nuclear norm, which is equal to the sum of the singular values of the matrix.

Although the LRSMD method based on the nuclear norm has achieved satisfactory results in hyperspectral anomaly detection, the nuclear norm cannot accurately approximate the rank function [52]. Specifically, in the rank function, all non–zero singular values have the same contribution to the rank of the matrix. However, in the nuclear norm, when the sum of all singular values is minimized, each non–zero singular value has a different contribution to the rank of the matrix [35,52,53]. In addition, these methods do not consider the intrinsic sparsity property in the low–rank matrix [54,55,56]. These factors will cause the decomposition result of the LRSMD method based on the nuclear norm to be inaccurate to some extent. In order to obtain the more accurate decomposition results, the TNN can replace the nuclear norm to approximate the rank function. The truncated nuclear norm only minimizes the sum of a few of the smallest singular values, because the rank of the matrix only corresponds to the first few non–zero singular values, and large non–zero singular values contribute little to the rank of the matrix [57,58]. Simultaneously, an extra sparse regular term on the low–rank matrix formed by l_2,1–norm is introduced to consider the sparsity of the low–rank matrix in an intermediate transform domain. The low–rank matrix is considered sparse in the intermediate transform domain. We assume that g(·) represents the transform operator that transforms into the intermediate transform domain. Moreover, we used the l_2,1–norm instead of the l₁–norm to make the columns of S tend to be sparse. Thus, in our method, the LRSMD model is:

$$\begin{array}{cc}\underset{\mathbf{L},\mathbf{S}}{\min }\hfill & \Vert L\Vert {}_r+\lambda \Vert \mathrm{S}\Vert {}_{2,1}+\beta \Vert g(\mathrm{L})\Vert {}_{2,1}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & \mathbf{D}=\mathbf{L}+\mathbf{S}\hfill \end{array}$$

(7)

where β is a parameter that balances the sparsity of the low–rank matrix in the intermediate transform domain. ‖L‖_r is the TNN of L. It is defined as the sum of the minimum of m and n minus the r minimum singular values of L. ‖·‖_2,1 represents the l_2,1–norm, which is defined as the sum of the l₂–norm of each column in the matrix. g(·) represents the transform operator in the intermediate transform domain. Their specific form is as follows:

$$\Vert L\Vert {}_{\mathrm{r}}={\displaystyle \sum _{o=r+1}^{\min (m,n)}{\mathsf{\delta }}_o}$$

(8)

$$\Vert \mathbf{S}\Vert {}_{2,1}={\displaystyle \sum _{i=1}^{mn}\sqrt{{\displaystyle \sum _{j=1}^b{({\mathbf{S}}_{j,i})}^2}}}$$

(9)

where δ_o represents the o–th smallest singular value of L. S_j,i represents the element in the j–th row and i–th column of the matrix S. To solve Equation (7), a new two–stage iterative method [58] was adopted.

In the first stage, since solving Equation (7) is NP–hard, it cannot be solved directly. According to Theorem 1 in [58], suppose the singular value decomposition of D′ is:

$${\mathbf{D}}^{\prime }=\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^T$$

(10)

where U = (u₁, u₂, …, u_m)$\in$R_b×b, Σ$\in$R_b×mn and V = (v₁, v₂, …, v_n)$\in$R_mn×mn. After taking out the first r singular values, we assume:

$$\begin{array}{l}\mathbf{A}={\left(u_1,u_2, \dots ,u_r\right)}^T\in {\mathbf{R}}_{r\times b}\\ \mathbf{B}={\left(v_1,v_2, \dots ,v_r\right)}^T\in {\mathbf{R}}_{r\times mn}\end{array}$$

(11)

Then, we can obtain:

$$\begin{array}{cc}\hfill \mathrm{Tr}(\mathbf{A}{\mathbf{D}}^{\prime }{\mathbf{B}}^T)& =\mathrm{Tr}({(u_1,u_2,\dots ,u_r)}^T\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^T(v_1,v_2,\dots ,v_r))\hfill \\ & =\mathrm{Tr}(\mathrm{diag}({\delta }_1,{\delta }_2,\dots ,{\delta }_r))\hfill \\ & ={\displaystyle \sum _{o=1}^r{\delta }_o}\hfill \end{array}$$

(12)

where Tr(·) represents the trace of the matrix, and diag(·) stands for the diagonal matrix. Combined with Equation (12), the TNN is redefined as:

$$\Vert \mathbf{L}\Vert {}_r={\displaystyle \sum _{o=r+1}^{\min (m,n)}{\delta }_o}=\Vert \mathbf{L}\Vert {}_{*}-{\displaystyle \sum _{o^{\prime }=1}^r{\delta }_{o^{\prime }}}=\Vert \mathbf{L}\Vert {}_{*}-\mathrm{Tr}(\mathbf{A}\mathbf{L}{\mathbf{B}}^T)$$

(13)

Thus, the final optimized Equation (7) is as follows:

$$\begin{array}{cc}\underset{\mathbf{L},\mathbf{S}}{\min }\hfill & \Vert \mathbf{L}\Vert {}_{*}-\mathrm{Tr}(\mathbf{A}\mathbf{L}{\mathbf{B}}^T)+\lambda \Vert \mathbf{S}\Vert {}_{2,1}+\beta \Vert g(\mathbf{L})|{|}_{2,1}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & \mathbf{D}=\mathbf{L}+\mathbf{S}\hfill \end{array}$$

(14)

Through iterative calculation of Equation (14), the final low–rank matrix L and sparse S can be obtained. Among them, since the low–rank matrix L is unknown, matrix D replaces the low–rank matrix L to calculate matrices A and B in each iteration. The iterative convergence condition is:

$${‖{\mathbf{L}}_{t+1}+{\mathbf{S}}_{t+1}-{\mathbf{D}}_t‖}_F\le {\tau }_1$$

(15)

where ‖·‖_F represents the Frobenius norm of the matrix, τ₁ is the reconstruction error, and t is the number of iterations.

In the second stage, to solve problem (14), it could first be divided into several sub–problems, and then we applied the idea of the alternating direction method of multipliers (ADMM) [59] to update each variable.

Since the transform operator g(L) in the intermediate transform domain of L is not conducive to function decoupling [60], we introduced the variable C, and let C = g(L). In addition, we assumed that A_t and B_t were obtained after the t–th iteration. Then, problem (14) can be redefined as:

$$\begin{array}{cc}\underset{\mathbf{L},\mathbf{S}}{\min }\hfill & \Vert \mathbf{L}\Vert {}_{*}-Tr({\mathbf{A}}_t\mathbf{L}{\mathbf{B}}_t^T)+\lambda \Vert \mathbf{S}|{|}_{2,1}+\beta \Vert \mathbf{C}|{|}_{2,1}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & \mathbf{D}=\mathbf{L}+\mathbf{S}\hfill \\ & \mathbf{C}=g(\mathbf{L})\hfill \end{array}$$

(16)

The augmented Lagrangian function of problem (16) is expressed as:

$$\begin{array}{l}\hspace{1em}\pounds (\mathbf{L},\mathbf{S},\mathbf{C},{\mathbf{X}}_1,{\mathbf{X}}_2,\mu )\\ =\Vert \mathbf{L}\Vert {}_{*}-Tr({\mathbf{A}}_t\mathbf{L}{\mathbf{B}}_t^T)+\lambda \Vert \mathbf{S}|{|}_{2,1}+\beta \Vert \mathbf{C}\Vert {}_{2,1}\\ \hspace{1em}+〈{\mathbf{X}}_1,\mathbf{D}-\mathbf{L}-\mathbf{S}〉+〈{\mathbf{X}}_2,\mathbf{C}-g(\mathbf{L})〉\\ \hspace{1em}+\frac{\mu }{2}(\Vert \mathbf{D}-\mathbf{L}-\mathbf{S}\Vert {}_F^2+\Vert \mathbf{C}-g(\mathbf{L})\Vert {}_F^2)\end{array}$$

(17)

where X₁ and X₂ represent Lagrange multipliers. The penalty coefficient is represented by μ > 0. The inner product of the matrix is represented by $〈\cdot ,\hspace{1em}\cdot 〉$.

For the multivariate in problem (17), the alternate update method can be utilized. When we update a variable, we keep other variables unchanged. Therefore, problem (17) can be broken down into the following subproblems in the k–th iteration under the t–th iteration of the first stage.

When we keep S and C unchanged and update L, it is:

$${\mathbf{L}}_{k+1}=\arg \hspace{0.17em}\underset{\mathbf{L}}{\min }\Vert {\mathbf{L}}_k\Vert {}_{*}+\frac{{\mu }_k}{2}{‖\mathbf{D}-{\mathbf{L}}_k-{\mathbf{S}}_k+\frac{{\mathbf{A}}_t^T{\mathbf{B}}_t}{{\mu }_k}+\frac{{({\mathbf{X}}_1)}_k}{{\mu }_K}‖}_F^2+\frac{{\mu }_k}{2}{‖{\mathbf{C}}_k-g({\mathbf{L}}_k)+\frac{{({\mathbf{X}}_2)}_k}{{\mu }_K}‖}_F^2$$

(18)

In order to solve the problem of the transform operator g(·), combined with the theorem of Parseval [60] and Theorem 2 in [58], we know that the sum (or integral) of the squares of a function is equal to the sum (or integral) of the squares of its Fourier transform. For example, ${‖\xi ‖}_F^2={‖\zeta ‖}_F^2$, where $\xi =\chi (\zeta )$, and $\chi (\cdot )$ is a unitary transform. Hence, we suppose that g(·) is a unitary transform and G(·) is the inverse transform of g(·). By applying the inverse transform G(·) to problem (18), we can get:

$${\mathbf{L}}_{k+1}=\arg \hspace{0.17em}\underset{\mathbf{L}}{\min }\Vert {\mathbf{L}}_k\Vert {}_{*}+{\mu }_k{‖{\mathbf{L}}_k-\frac{1}{2}\left[\mathbf{D}-{\mathbf{S}}_k+\frac{{\mathbf{A}}_t^T{\mathbf{B}}_t+{({\mathbf{X}}_1)}_k}{{\mu }_k}+G\left({\mathbf{C}}_k+\frac{{({\mathbf{X}}_2)}_k}{{\mu }_K}\right)\right]‖}_F^2$$

(19)

Problem (19) can be solved using the classical singular value shrinkage operator. Then, the objective function can finally be written as:

$${\mathbf{L}}_{k+1}=\mathrm{SVT}{}_{\frac{1}{2{\mu }_k}}\left\{\frac{1}{2}\left[\mathbf{D}-{\mathbf{S}}_k+\frac{1}{{\mu }_k}\left[{\mathbf{A}}_t^T{\mathbf{B}}_t+{({\mathbf{X}}_1)}_k\right]+G\left[{\mathbf{C}}_k+\frac{{({\mathbf{X}}_2)}_k}{{\mu }_k}\right]\right]\right\}$$

(20)

where the definition of SVT_δ′(·) is as follows: when given a matrix Q and δ′ > 0, SVT_δ′(·) can be expressed as $\mathrm{SVT}{}_{ \delta \prime }(\mathbf{Q})={\mathbf{U}}^{\prime }\mathrm{diag}[\max (\omega - \delta \prime )]{{\mathbf{V}}^{\mathbf{\prime }}}^{\mathrm{T}}$, ${\mathbf{U}}^{\prime }\in {\mathbf{R}}^{m\times r},\hspace{1em}{\mathbf{V}}^{\prime }\in {\mathbf{R}}^{r\times n}$ and $\omega ={({\delta }_1,{\delta }_2,\dots ,{\delta }_r)}^T$ is the first r singular values of the matrix Q.

When we keep L and C unchanged and update S, the objective function is:

$${\mathbf{S}}_{k+1}=\arg \hspace{0.17em}\underset{\mathbf{S}}{\min }\lambda \Vert {\mathbf{S}}_k\Vert {}_{2,1}+\frac{{\mu }_k}{2}{‖\mathbf{D}-{\mathbf{S}}_k-{\mathbf{L}}_{k+1}+\frac{{({\mathbf{X}}_1)}_k}{{\mu }_k}‖}_F^2$$

(21)

When we keep L and S unchanged and update C, the objective function is:

$${\mathbf{C}}_{k+1}=\arg \hspace{0.17em}\underset{\mathbf{C}}{\min }\beta \Vert \mathbf{C}\Vert {}_{2,1}+\frac{{\mu }_k}{2}{‖\mathbf{C}-g({\mathbf{L}}_{k+1})+\frac{{({\mathbf{X}}_2)}_k}{{\mu }_k}‖}_F^2$$

(22)

The updated method of Lagrange multipliers X₁ and X₂ is:

$$\left\{\begin{array}{l}{({\mathbf{X}}_1)}_{k+1}={({\mathbf{X}}_1)}_k+{\mu }_k(\mathbf{D}-{\mathbf{L}}_{k+1}-{\mathbf{S}}_{k+1})\\ {({\mathbf{X}}_2)}_{k+1}={({\mathbf{X}}_2)}_k+{\mu }_k\left[{\mathbf{C}}_{k+1}-g({\mathbf{L}}_{k+1})\right]\end{array}\right.$$

(23)

The updated method of penalty coefficient μ is:

$${\mu }_{k+1}=\min (\epsilon {\mu }_k,{\mu }_{\max })$$

(24)

where ε > 1 is a given fixed value that is utilized to update the penalty coefficient μ. μ_max is the maximum value of the given penalty coefficient μ.

The convergence condition is:

$${‖{\mathbf{L}}_{k+1}-{\mathbf{L}}_k‖}_F\le {\tau }_2 \mathrm{or} {‖{\mathbf{S}}_{k+1}-{\mathbf{S}}_k‖}_F\le {\tau }_2$$

(25)

where τ₂ is the reconstruction error.

The detailed solution method of the LRSMD–TNN is shown in Algorithm 1. After performing the LRSMD–TNN on the three–directional feature images D_xy, D_xz and D_yz, we can obtain the three corresponding sparse matrices S_l (l = 1, 2, 3).

2.2.3. The Extraction of Rough Detection Results

After obtaining the sparse matrix S_l, we extract the rough detection result through finding the Mahalanobis distance. For the measured pixel ${{s}}_i^l$ in the sparse matrix S_l, the Mahalanobis distance is:

$${\mathbf{E}}_l={({\mathit{s}}_i^l-{\mathit{u}}_g^l)}^T{\mathbf{C}}_g^l({\mathit{s}}_i^l-{\mathit{u}}_g^l),\hspace{1em}i=1,2,\dots ,m\times n$$

(26)

where E_l is the rough detection map, ${\mathit{u}}_g^l$ is the mean vector of the sparse matrix S_l and ${\mathbf{C}}_g^l$ is the covariance matrix of the sparse matrix S_l.

Algorithm 1 Low–rank and sparse matrix decomposition with truncated nuclear norm

Input: Directional feature image D.

Initialization: parameter λ > 0 and β > 0, number of singular values r < min(m,n), penalty coefficient μ0 and μmax, parameter ε > 1, reconstruction error τ1 and τ2, number of iterations t = k = 1, L1 = S1 = X1 = X1 = C1 = 0, D1 = D.

While Equation (15) does not converge do

(1) Update At and Bt according to Equations (10) and (11).

While Equation (25) does not converge do

(2) Update Lk+1 according to Equation (20).

(3) Update Sk+1 according to Equation (21).

(4) Update Ck+1 according to Equation (22).

(5) Update (X1)k+1 and (X2)k+1 according to Equation (23).

(6) Update μk+1 according to Equation (24).

End

Return Lt+1, St+1 and Dt+1.

End

Return L and S.

2.3. Spatial Dimension

The above method mainly considers the spectral characteristics of anomaly targets in the HSI, but do not consider their spatial characteristics. In the spatial dimension, anomaly targets usually appear in the form of low probability and small area [61]. When the spectra of the anomaly targets are similar to those of the background, the spatial feature can be exploited to distinguish the anomaly targets from the background. At present, the anomaly detection based on attribute and edge–preserving filters (AED) [62] has achieved excellent results in extracting the spatial features of the HSI. The attribute filtering can remove the connected bright and dark part of the HSI. Therefore, we further combine the spatial characteristics of the HSI through the attribute filtering.

In this section, we perform attribute filtering and differential operation on each band of the original HSI to extract spatial feature image M$\in$R_m×n×b, which can be obtained by:

$${\mathbf{M}}_j=\left|{\mathbf{H}}_j-A_{\eta }({\mathbf{H}}_j)\right|+\left|{\mathbf{H}}_j-A_{\theta }({\mathbf{H}}_j)\right|$$

(27)

where M_j (j = 1, 2, …, b) is the j–th band of the obtained spatial feature image M. H_j is the j–th band of the HSI, and A_η and A_θ represent attribute thinning and thickening performed on the regions with values greater than a given threshold and regions with values less than a given threshold in the HSI, respectively, which are described in detail in the background section of [55]. Through these two operations, the spatial features in the HSI can be preserved. Then, the spatial anomaly detection result T is obtained through the l₂–norm, as shown below:

$$\mathbf{T}(i)=\sqrt{{\displaystyle \sum _{j=1}^b{({\widehat{\mathit{m}}}_{j,i})}^2}},\hspace{1em}i=1,2,\dots ,m\times n$$

(28)

where T(i) is the i–th pixel of the spatial anomaly detection result T, and ${\widehat{\mathit{m}}}_{j,i}$ is the i–th pixel in the j–th band in the spatial feature image M. Since the spatial anomaly detection result T is used as the complementary fusion weight of the aforementioned rough detection map E_l, we named the spatial anomaly detection result as the spatial weight map.

2.4. Complementary Fusion

In the final result fusion stage, the final detection result is obtained by fusing each dimension detection result. The spatial features and spectral features can be further complemented through fusion. The specific operation is to merge the detection result of each spectral dimension with the detection result of the spatial dimension. This can ensure that the detection results of other dimensions are supplemented to a certain extent when the detection effect of a certain dimension is not satisfactory. In this way, the spectral characteristics and the spatial characteristics are complementary to achieve satisfactory results. Hence, the final detection map F is obtained by:

$$\mathbf{F}=\mathbf{T}\odot \left({\displaystyle \sum _{l=1}^3{\mathbf{E}}_l}\right)$$

(29)

where $\odot$ represents the matrix dot product.

Through the above methods, the spectral characteristics and spatial characteristics can be combined to obtain optimized detection results. The fusion method can further suppress background information, which is also verified by the experimental results. The detailed steps of the proposed SCDF method are shown in Algorithm 2.

Algorithm 2 Spectral–spatial complementary decision fusion

Input: HSI H.

Initialization: standard deviation σ, window size w, number of singular values r, predefined logical predicate Tρ.

(1) Calculate the directional feature images Dxy, Dxz and Dyz by Equations (2)–(4).

(2) Calculate the sparse matrix Sl by Algorithm 1.

(3) Get the rough detection map El by Equation (26).

(4) Extract the spatial feature image M by Equation (27).

(5) Calculate the spatial weight map T by Equation (28).

(6) Obtain the final detection map F by Equation (29).

Return the final detection map F.

3. Experimental Dataset and Evaluation Indicator

In this section, we first discuss the synthetic and three real datasets in brief. Secondly, the evaluation indicators used in the experiments are introduced.

3.1. Experimental Dataset

Synthetic dataset: The synthetic data [63] are provided by Tan et al. The background of the synthetic dataset is cropped from the Salinas dataset, collected from Salinas Valley, CA, USA, by Airborne Visible/Infrared Imaging Spectrometer (AVIRIS). The Salinas dataset has a size of 512 × 217 and contains 224 bands. In the experiment, a 120 × 120 area was selected as the background of the synthetic data. In addition, after removing the water absorption and bad bands (108–112, 154–167 and 224), 204 bands were retained. The anomaly target is embedded in the background through the target implantation method.

$${\mathit{z}}_{\mathrm{sp}}=f_{\mathrm{saf}}{\mathit{t}}_{\mathrm{ts}}+(1-f_{\mathrm{saf}}){\mathit{b}}_{\mathrm{bs}}$$

(30)

where z_sp is the spectrum of the synthesized target, f_saf is the abundance fraction and b_bs is the spectrum of the background. The range of f_saf is from 0.04 to 1. The synthetic data and the corresponding ground truth map are shown in Figure 3.

Urban: The second dataset was collected from an urban area by a hyperspectral digital imagery collection experiment (HYDICE) sensor [64]. The original dataset is a 400 × 400 HSI with a spatial resolution of 2 m per pixel. There are 210 bands in the original HSI. In our experiment, we cropped an 80 × 80 area, as shown in Figure 4a. After removing the water absorption bands and the low signal–to–noise ratio (SNR) bands (1–4, 76, 87, 101–111, 136–153, 198–210), 162 available bands were reserved. The corresponding ground truth map is shown in Figure 4b.

MUUFL Gulfport: The third dataset was collected from the University of Southern Mississippi Gulf Park Campus in Long Beach, Mississippi, USA, by the hyperspectral compact airborne spectrographic imager (CASI–1500) [65]. The original dataset is a 325 × 220 HSI. There are 72 bands in the original HSI. After the first four and last four noise bands are removed, the remaining 62 bands are reserved in the experiment. In our experiment, we crop a 200 × 200 area, as shown in Figure 5a. In this HSI, six cloth panels of different sizes are considered anomaly targets. The corresponding ground truth map is shown in Figure 5b.

Chikusei: The fourth dataset was acquired by Headwall Hyperspec–VNIR–C imaging sensor [66] in Chikusei, Ibaraki, Japan. The dataset contains 128 bands ranging from 363 nm to 1018 nm. The size of the original HSI with a spatial resolution of 2.5 m per pixel is 2517 × 2335. We cropped an area of 100 × 100 as the experimental image. All bands are included in the experiment. The corresponding ground truth map is made with the help of the Environment for Visualizing Images (ENVI) software. The false–color map of the Chikusei scene and the corresponding ground truth map are shown in Figure 6.

3.2. Evaluation Indicator

The alternative methods that we used in the experiment are the RX [15], the LRX [16], the LSMAD [30], the LSDM–MoG [32], the CDASC [21], the 2S–GLRT [20], the BFAD [44], the SSFSCRD [45], and the AED [62] methods. In the parameter analysis and discussion of experimental results, the three–dimensional receiver operating characteristic (3D ROC) [67] curve and the corresponding two–dimensional ROC (2D ROC) curves, and the area under the 2D ROC curve (AUC) are utilized as evaluation indicators to quantitatively evaluate the performance of the proposed method. With the same threshold τ, the 3D ROC curve (P_D, P_F, τ) can more accurately evaluate the performance of each method. P_D is the probability of detection and P_F is the false alarm probability. The corresponding 2D ROC curves are (P_D, P_F), (P_D, τ), and (P_F, τ). The AUC values of the three 2D ROC curves are AUC_(D,F), AUC_(D,τ), and AUC_(F,τ). The AUC_(D,F), AUC_(D,τ), and AUC_(F,τ) indicate the effectiveness, target detectability (TD), and background suppressibility (BS) of the detection method, respectively. In addition to these three AUC values, another five AUC values [63] are also proposed to evaluate the performance of the detection method. They are represented as AUC_TD, AUC_BS, AUC_SNPR, AUC_TDBS, and AUC_ODP, respectively. Among them, AUC_TD calculates the effectiveness and TD of a detection method.

$$0\le \mathrm{AUC}{}_{\mathrm{TD}}=\mathrm{AUC}{}_{(\mathrm{D},\mathrm{F})}+\mathrm{AUC}{}_{(\mathrm{D},\tau )}\le 2$$

(31)

By combining the BS of a detection method, AUC_BS is defined as:

$$-1\le \mathrm{AUC}{}_{\mathrm{BS}}=\mathrm{AUC}{}_{(\mathrm{D},\mathrm{F})}-\mathrm{AUC}{}_{(\mathrm{F},\tau )}\le 1$$

(32)

In order to comprehensively consider the TD and BS of a detection method, AUC_TDBS is defined as:

$$-1\le \mathrm{AUC}{}_{\mathrm{TDBS}}=\mathrm{AUC}{}_{(\mathrm{D},\tau )}-\mathrm{AUC}{}_{(\mathrm{F},\tau )}\le 1$$

(33)

The signal–to–noise probability ratio AUC_SNPR is defined as:

$$0\le \mathrm{AUC}{}_{\mathrm{SNPR}}=\frac{\mathrm{AUC}{}_{(\mathrm{D},\tau )}}{\mathrm{AUC}{}_{(\mathrm{F},\tau )}}$$

(34)

The overall detection probability AUC_ODP is defined as:

$$0\le \mathrm{AUC}{}_{\mathrm{TDBS}}=\mathrm{AUC}{}_{(\mathrm{D},\mathrm{F})}+\mathrm{AUC}{}_{(\mathrm{D},\tau )}-\mathrm{AUC}{}_{(\mathrm{F},\tau )}\le 2$$

(35)

It is worth noting that the smaller the AUC_(F,τ) value, the better the performance of the detection method. The larger the other AUC values, the better the performance of the detection method.

4. Discussion

In this section, the parameters of the proposed SCDF method are first analyzed in detail. Then, the effectiveness and superiority of the proposed SCDF method is demonstrated by qualitative and quantitative comparison with alternative methods. Finally, the dimension performance and the performance of TNN of the proposed SCDF method are analyzed and discussed.

4.1. Parameter Analysis

In the experiment, the uncertain parameters in the alternative methods were selected according to the AUC_(D,F) value or the parameter values provided by the original authors. On the four experimental datasets, the values of the uncertain parameters of all alternative methods are shown in

Table 1. Uncertain parameter settings for different alternative methods on four datasets.

Method	Parameter Setting
LRX	The size of window: (3, 5), (11, 15), (13, 31), (13, 15)
LSMAD	The rank: 5, 6, 6, 30
	The cardinality: 10, 1, 1, 5
LSDM–MoG	The rank: 60, 60, 30, 60
	The number of mixture Gaussian noise: 4, 4, 4, 4
CDASC	The value of virtual dimensionality: 13, 10, 15, 10
	The number of principal components: 2, 4, 6, 7
	The number of independent components: 11, 6, 9, 3
2S–GLRT	The size of window: (3, 5), (5, 7), (13, 25), (5, 7)
BFAD	The size of window: (3, 5), (3, 5), (13, 31), (5, 7)
	The outputting sensitivity of spatial distance weight: 0.8, 5, 10, 0.5
	The outputting sensitivity of spectral distance weight: 0.3, 1, 1, 0.3
SSFSCRD	The size of window: (3, 5), (5, 7), (13, 21), (3, 5)
	The fractional order: 0.8, 0.5, 0.8, 0.9
	The Lagrange multiplier: 0.5, 1, 0.1, 0.1
	The weighting coefficient: 0.5, 0.1, 0.1, 0.1
AED	The band number: 3, 3, 3, 3
	The thresholding number: 5, 25, 125, 25
	The domain transform recursive filtering: (1, 0.1). (5, 1), (5, 1), (5, 1)

.

The parameters of the proposed SCDF method were window size w, the standard deviation σ, parameters λ and β, number of singular values r, penalty coefficient μ₀ and μ_max, parameter ε, reconstruction errors τ₁ and τ₂, and the predefined logical predicate T_ρ. Although there were many parameters in the proposed SCDF method, some parameters had little effect on the performance of the proposed method, such as penalty coefficient μ₀ and μ_max, parameter ε, and reconstruction errors τ₁ and τ₂. Therefore, we set μ₀ = 0.005, μ_max = 10,000, ε = 1.01, and τ₁ = τ₂ = 10⁻⁶. In addition, for the parameters λ and β, according to the recommendations of the previous studies and the analysis of experimental results [52,53,58], we finally set the two parameters to λ = 0.0005 and β = 0.00009. Consequently, the remaining uncertain parameters include window size w, the standard deviation σ, the number of singular values r, and the predefined logical predicate T_ρ.

To determine the optimal values of these uncertain parameters, when analyzing one of the uncertain parameters of the proposed SCDF method in the experiment, the other parameters are kept constant. In the experiment, the AUC_(D,F) value is used as the evaluation index. At the beginning of the experiment, the default parameters are set to w = 3, σ = 0.9, r = 20, and T_ρ = 105. The analysis results of the impact of the four parameters on the performance of the proposed SCDF method are shown in Figure 7. These uncertain parameters are mainly determined according to the AUC_(D,F) values in Figure 7. Finally, the window size w is set to w = 5 in the synthetic dataset and the urban dataset. In the MUUFL Gulfport dataset and the Chikusei dataset, w is set to w = 3. For the standard deviation σ, σ is set to σ = 0.9 in the urban dataset and the Chikusei dataset. In the synthetic dataset, σ is set to σ = 0.5. In the MUUFL Gulfport dataset, σ is set to σ = 1. In all datasets, the number of singular values r is uniformly set to r = 20. The predefined logical predicate T_ρ, is set to T_ρ = 5 in the synthetic dataset and the urban dataset, and T_ρ = 145 in the Chikusei dataset and the MUUFL Gulfport dataset.

4.2. Experimental Results and Discussion

In the experiment, we evaluated the effectiveness and advancement of the proposed SCDF method from both visual detection effects and quantitative comparison. On different datasets, the detection results of the alternative methods and the proposed SCDF method are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

It can be seen from Figure 8 that in addition to the BFAD method, other methods have better suppressed the background. In the detection results of the RX, the LRX, and the LSMAD methods, not all targets are well–separated from the background. The proposed SCDF method, as well as the LSDM–MoG, the CDASC, the 2S–GLRT, the BFAD, the SSFSCRD, and the AED methods can well–separate anomaly targets from the background. In addition, the anomaly targets in the detection results of the proposed SCDF method are more prominent.

As shown in Figure 9, on the urban dataset, the backgrounds of the detection results of the RX, the LSDM–MoG, and the BFAD methods are not well–suppressed. The background suppression effect is not very satisfactory in the detection result of the AED method. The detection results of other methods are relatively similar.

On the MUUFL Gulfport dataset, it can be seen from Figure 10 that all methods have detected anomaly targets, but they all contain a certain number of false targets. In the detection results of the LRX and the 2S–GLRT methods, the anomaly targets are not well–highlighted from the background. The detection results of the LSMAD, the LSDM–MoG, and the CDASC methods are similar to those of the proposed SCDF method. Among them, the targets in the detection results of the LSDM–MoG are relatively more obvious than the detection results of the proposed SCDF method, and the false alarms are slightly less. However, the proposed SCDF method still obtains satisfactory detection results.

Figure 11 shows the results of different methods on the Chikusei dataset. The detection results of the LRX, the CDASC, and the SSFSCRD methods are not particularly prominent. The background suppression is not particularly satisfactory in the detection results of the LSDM–MoG and the BFAD methods. The detection results of other methods are similar to those of the proposed SCDF method. It can be seen from the visual detection results of the four datasets that the proposed SCDF method obtains satisfactory detection results.

The 3D ROC curves and the corresponding 2D ROC curves of the detection results of different methods are shown in Figure 12, Figure 13, Figure 14 and Figure 15. The different AUC values calculated according to the 3D ROC curves on different datasets are shown in

Table 2. Comparison of AUC performance of different methods on the synthetic dataset.

Method	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
RX	0.8073	0.2143	0.0314	1.0216	0.7759	6.8318	0.1829	0.9903
LRX	0.9895	0.2414	0.00008	1.2309	0.9894	3003.0358	0.2413	1.2309
LSMAD	0.8911	0.3640	0.0191	1.2551	0.8720	19.0536	0.3449	1.2360
LSDM–MoG	0.9713	0.5205	0.0342	1.4918	0.9370	15.2135	0.4863	1.4576
CDASC	0.9919	0.3725	0.0013	1.3644	0.9906	280.9193	0.3712	1.3631
2S–GLRT	0.9491	0.1744	0.0049	1.1235	0.9441	35.3608	0.1695	1.1185
BFAD	0.9603	0.9340	0.2910	1.8943	0.6693	3.2093	0.6430	1.6033
SSFSCRD	0.9916	0.4713	0.0083	1.4629	0.9833	56.8626	0.4630	1.4546
AED	0.9895	0.6785	0.0030	1.6680	0.9865	226.3940	0.6755	1.6650
SCDF	0.9982	0.4851	0.0005	1.4833	0.9976	844.0422	0.4845	1.4828

,

Table 3. Comparison of AUC performance of different methods on the urban dataset.

Method	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
RX	0.9628	0.3859	0.0509	1.3487	0.9120	7.5857	0.3350	1.2978
LRX	0.9812	0.2582	0.0075	1.2395	0.9738	34.5067	0.2507	1.2320
LSMAD	0.2788	0.0172	0.0340	0.2960	0.2448	0.5068	−0.0168	0.2620
LSDM–MoG	0.9956	0.7360	0.0566	1.7317	0.9391	13.0127	0.6795	1.6751
CDASC	0.9893	0.5917	0.0641	1.5811	0.9252	9.2315	0.5276	1.5170
2S–GLRT	0.9922	0.3169	0.0097	1.3091	0.9824	32.5407	0.3072	1.2994
BFAD	0.9960	0.8330	0.0856	1.8290	0.9104	9.7369	0.7474	1.7434
SSFSCRD	0.9994	0.4556	0.0086	1.4550	0.9908	53.0610	0.4471	1.4465
AED	0.9959	0.5454	0.0629	1.5413	0.9331	8.6753	0.4825	1.4784
SCDF	0.9988	0.5791	0.0021	1.5779	0.9967	269.8049	0.5769	1.5758

,

Table 4. Comparison of AUC performance of different methods on the MUUFL Gulfport dataset.

Method	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
RX	0.9978	0.2247	0.0162	1.2225	0.9816	13.8612	0.2085	1.2063
LRX	0.9840	0.0442	0.0010	1.0282	0.9830	45.1625	0.0433	1.0272
LSMAD	0.9923	0.2735	0.0170	1.2658	0.9753	16.0725	0.2565	1.2488
LSDM–MoG	0.9992	0.6008	0.0581	1.6000	0.9411	10.3378	0.5427	1.5419
CDASC	0.9985	0.3661	0.0123	1.3646	0.9862	29.8252	0.3539	1.3524
2S–GLRT	0.9702	0.0405	0.0003	1.0107	0.9699	135.2632	0.0402	1.0104
BFAD	0.9459	0.1092	0.0184	1.0551	0.9275	5.9332	0.0908	1.0367
SSFSCRD	0.9865	0.1605	0.0181	1.1469	0.9684	8.8850	0.1424	1.1289
AED	0.9919	0.2947	0.0486	1.2866	0.9433	6.0614	0.2461	1.2380
SCDF	0.9949	0.0420	0.0005	1.0368	0.9944	84.7453	0.0415	1.0363

and

Table 5. Comparison of AUC performance of different methods on the Chikusei dataset.

Method	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
RX	0.9991	0.3306	0.0337	1.3297	0.9654	9.8166	0.2969	1.2960
LRX	0.9886	0.1396	0.0079	1.1282	0.9807	17.5729	0.1317	1.1203
LSMAD	0.9985	0.2645	0.0281	1.2629	0.9704	9.4273	0.2364	1.2349
LSDM–MoG	0.9929	0.4163	0.0411	1.4092	0.9518	10.1363	0.3752	1.3681
CDASC	0.9996	0.2190	0.0052	1.2186	0.9944	42.0535	0.2137	1.2134
2S–GLRT	0.9869	0.2192	0.0151	1.2061	0.9718	14.5301	0.2041	1.1910
BFAD	0.9939	0.8599	0.2064	1.8538	0.7875	4.1651	0.6534	1.6474
SSFSCRD	0.9977	0.3300	0.0409	1.3278	0.9568	8.0682	0.2891	1.2869
AED	0.9972	0.4820	0.0353	1.4792	0.9619	13.6584	0.4467	1.4439
SCDF	0.9999	0.4060	0.0019	1.4059	0.9980	211.9936	0.4041	1.4041

. First, on the synthetic dataset, Figure 12 shows that the 3D ROC curve of the proposed SCDF method is relatively close to those of most alternative methods, and the corresponding 2D ROC curves (P_D, τ) and (P_F, τ) can also draw similar conclusions. The corresponding 2D ROC curve (P_D, P_F) is slightly higher than those of other alternative methods. The corresponding AUC values are shown in Table 2. It can be seen from Table 2 that the AUC_(D,F) and AUC_(B,S) values of the proposed method are larger than those of the alternative methods, which are 0.9982 and 0.9976, respectively. Among the alternative and proposed methods, the AUC_(F,τ) = 0.00008 and AUC_SNPR = 3003.0358 of the LRX method are the largest, the AUC_(D,τ) = 0.9340 and AUC_TD = 1.8943 of the BFAD method are the largest, the AUC_TDBS = 0.6755, and AUC_ODP = 1.6650 of the AED method are the largest. Although other AUC values do not reach the maximum values, they are still relatively close to the maximum AUC values. For example, AUC_(F,τ) = 0.0005, AUC_TD = 1.4833, AUC_TDBS = 0.4845, and AUC_ODP = 1.4828 of the proposed SCDF method are close to the corresponding optimal AUC values of 0.00008, 1.8943, 0.6755, and 1.6650.

On the urban dataset, it can be seen from Figure 13 that the 3D ROC curve of the proposed SCDF method is higher than those of most alternative methods. The corresponding 2D ROC curves (P_D, P_F) are also very close to the 2D ROC curves of the BFAD and SSFSCRD methods. The 2D ROC curve (P_D, τ) of the proposed SCDF method is slightly lower than those of the LSDM–MoG, the BFAD, and the CDASC methods. The 2D ROC curve (P_F, τ) of the proposed SCDF method is lower than those of all alternative methods. Combining Table 3, we can see that the AUC_(F,τ) = 0.0021, AUC_BS = 0.9967 and AUC_SNPR = 269.8049 of the proposed SCDF method are superior to all alternative methods. Among them, the AUC_(D,τ) = 0.8330, AUC_TD = 1.8290, AUC_TDBS = 0.7474, and AUC_ODP = 1.7434 of the BFAD method are the largest among all methods. The AUC_(D,F) = 0.9994 of the SSFSCRD method is the largest among all methods. However, the AUC values of the proposed SCDF method are also very close to the AUC values of the BFAD method. Therefore, the proposed SCDF method obtains a stable and excellent detection result on the urban dataset.

For the MUUFL Gulfport dataset, combining Figure 14 and Table 4, it can be seen that the comprehensive performance of the LSDM–MoG method is the best among the methods used in this work. The AUC_BS = 0.9944 of the proposed SCDF method is better than those of the alternative methods. The AUC_(D,F) = 0.9992, AUC_(D,τ) = 0.6008, AUC_TD = 1.6000, AUC_TDBS = 0.5427, and AUC_ODP = 1.5419 of the LSDM–MoG method are the largest among all alternative and proposed methods. The AUC_(F,τ) = 0.0003 and AUC_SNPR = 135.2632 of the 2S–GLRT method are the largest among all alternative and proposed methods. Although the other AUC values of the proposed SCDF method are not the best, the difference between these AUC values and the optimal AUC values is very small. Therefore, the proposed SCDF method obtains an effective and stable detection result.

For the Chikusei dataset, it can be clearly seen from Figure 15 that the 2D ROC curves (P_D, P_F) and (P_F, τ) are better than those of alternative methods. The 3D ROC curve and 2D ROC curve (P_D, τ) of the BFAD method are superior to those of other methods. In Table 5, the AUC_(D,F) = 0.9999, AUC_(F,τ) = 0.0019, AUC_BS = 0.9980, and AUC_SNPR = 211.9936 of the proposed SCDF method are better than those of the alternative methods. However, the AUC_(D,τ) = 0.8599, AUC_TD = 1.8538, AUC_TDBS = 0.6534, and AUC_ODP = 1.6474 of the BFAD method are superior to those of other alternative methods and the proposed SCDF method. On the Chikusei dataset, although the detection performance of the proposed SCDF method is not optimal in all aspects, it is also close to the detection performance of the BFAD method.

4.3. Complementary Dimension Performance Analysis

In this section, the complementary dimension performance of the proposed SCDF method is analyzed. The detection results of each dimension are compared with the final detection results from both qualitative and quantitative aspects. In the spectral dimension and the spatial dimension, the rough detection map and the spatial weight map are used as the dimension detection results, respectively. Figure 16 shows the visual detection results of each dimension and the final detection result. It can be clearly seen from Figure 16 that, compared to the detection results of each dimension, the background of the final detection result is cleaner and the anomaly targets of the final detection result are more prominent. In the final test results, there were fewer false alarms. As a consequence, from the visual effect analysis, the desired detection result is reached after the fusion. Additionally, it can be seen that when the detection result of a dimension is relatively poor, the remaining dimensions can be supplemented to obtain the desired detection result.

The AUC performance comparison corresponding to Figure 16 is shown in

Table 6. Comparison of AUC performance of different dimensions on various datasets.

Dataset	Dimension	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
Synthetic data	Dimension 1	0.9905	0.7097	0.0300	1.7002	0.9605	23.6893	0.6797	1.6702
	Dimension 2	0.9721	0.4955	0.0307	1.4676	0.9413	16.1221	0.4648	1.4368
	Dimension 3	0.9672	0.4952	0.0251	1.4624	0.9421	19.7153	0.4700	1.4373
	Dimension 4	0.9997	0.6906	0.0232	1.6903	0.9765	29.7763	0.6674	1.6671
	Final result	0.9982	0.4851	0.0006	1.4833	0.9976	844.0422	0.4845	1.4828
Urban	Dimension 1	0.9930	0.7249	0.0395	1.7179	0.9534	18.3303	0.6854	1.6783
	Dimension 2	0.9771	0.5781	0.0527	1.5552	0.9244	10.9608	0.5253	1.5024
	Dimension 3	0.9914	0.6612	0.0666	1.6526	0.9248	9.9258	0.5946	1.5859
	Dimension 4	0.9982	0.7068	0.0271	1.7050	0.9712	26.1143	0.6797	1.6780
	Final result	0.9988	0.5791	0.0021	1.5779	0.9967	269.8049	0.5769	1.5758
MUUFL Gulfport	Dimension 1	0.9903	0.0737	0.0027	1.0640	0.9876	27.6608	0.0710	1.0613
	Dimension 2	0.9896	0.0756	0.0027	1.0652	0.9869	28.1420	0.0729	1.0625
	Dimension 3	0.9910	0.0882	0.0029	1.0792	0.9881	29.9237	0.0852	1.0762
	Dimension 4	0.9767	0.4696	0.0929	1.4463	0.8837	5.0532	0.3767	1.3534
	Final result	0.9949	0.0420	0.0005	1.0368	0.9944	84.7453	0.0415	1.0363
Chikusei	Dimension 1	0.9999	0.6345	0.0156	1.6344	0.9843	40.7601	0.6190	1.6189
	Dimension 2	0.9998	0.5688	0.0129	1.5686	0.9869	43.9322	0.5558	1.5556
	Dimension 3	0.9997	0.5759	0.0130	1.5756	0.9867	44.2998	0.5629	1.5627
	Dimension 4	0.9973	0.6202	0.0799	1.6175	0.9174	7.7616	0.5403	1.5376
	Final result	0.9999	0.4041	0.0016	1.4040	0.9983	249.2678	0.4024	1.4024

. It can be seen that on the synthetic dataset and the urban dataset, AUC_(D,τ), AUC_TD, AUC_TDBS, and AUC_ODP values of the detection result of dimension 1 are the best. On the Chikusei dataset, AUC_(D,F) = 0.9999, AUC_(D,τ) = 0.6345, AUC_TDBS = 0.6190, and the AUC_ODP = 1.6189 of the detection result of dimension 1 are the best. However, on the MUFL Gulfport dataset, the detection performance of dimension 1 is not the best. On the urban, MUFL Gulfport and Chikusei datasets, AUC_(D,F), AUC_(F,τ), AUC_BS, and AUC_SNPR values of the final detection results are the best. On the synthetic dataset, AUC_(F,τ) = 0.0006, AUC_BS = 0.9976 and AUC_SNPR = 844.0422 of the final detection results are the best. Therefore, the final detection results are more satisfactory overall. At last, by combining Figure 16 and Table 6, we can conclude that the proposed SCDF method makes full use of the information of each dimension from the experimental results, thereby ensuring the superiority and effectiveness of the detection results.

4.4. The Performance Analysis of TNN

In order to verify the performance of TNN, we did not add the TNN in the LRSMD and still used the nuclear norm. We denoted the method of using the nuclear norm as SCDF–noTNN. The proposed method is denoted as SCDF–TNN. The AUC values were used as the evaluation index. The detection results of the two methods on the experimental datasets are shown in Figure 17. The corresponding AUC values are shown in

Table 7. Comparison of AUC performance of the SCDF method with TNN and the SCDF method without TNN on each dataset.

Dataset	Method	AUC(D,F)	AUC(D,τ)	AUC(F,τ)	AUCTD	AUCBS	AUCSNPR	AUCTDBS	AUCODP
Synthetic data	SCDF–noTNN	0.9975	0.4849	0.0008	1.4824	0.9967	616.8351	0.4841	1.4816
	SCDF–TNN	0.9982	0.4851	0.0006	1.4833	0.9976	844.0422	0.4845	1.4828
Urban	SCDF–noTNN	0.9972	0.6562	0.0081	1.6534	0.9891	81.0583	0.6481	1.6453
	SCDF–TNN	0.9988	0.5791	0.0021	1.5779	0.9967	269.8049	0.5769	1.5758
MUFL Gulfport	SCDF–noTNN	0.9921	0.0256	0.0004	1.0176	0.9916	57.2450	0.0251	1.0172
	SCDF–TNN	0.9949	0.0420	0.0005	1.0369	0.9944	84.7453	0.0415	1.0363
Chikusei	SCDF–noTNN	0.9986	0.3551	0.0033	1.3537	0.9953	107.5117	0.3518	1.3504
	SCDF–TNN	0.9999	0.4041	0.0016	1.4040	0.9983	249.2678	0.4024	1.4024

. It can be seen from Figure 17 that the detection results of the two methods are very close. In Table 7, on the urban dataset, AUC_(D,τ) = 0.6562, AUC_TD = 1.6534, AUC_TDBS = 0.6481 and AUC_ODP = 1.6453 of the SCDF–noTNN method are slightly larger than those of the SCDF–TNN method. On the MUUFL Gulfport dataset, AUC_(F,τ) = 0.0004 of the SCDF–noTNN method is slightly smaller than that of the SCDF–TNN method. However, all the AUC values of the SCDF–TNN method on other datasets are larger than those of the SCDF–noTNN method. Although the improvement is slight, there is still a certain degree of improvement in the detection results of the SCDF–TNN method. In addition, it can be seen that the spectral–spatial complementary decision fusion framework that we proposed is helpful for anomaly detection. Therefore, it can be concluded that the proposed method in this paper is effective and competitive.

5. Conclusions

In this paper, we proposed a novel spectral–spatial complementary decision fusion method for hyperspectral anomaly detection. This proposed method is mainly divided into three parts: the spectral dimension, the spatial dimension, and the results fusion. First of all, in the spectral dimension, the three–dimensional Hessian matrix transformation was first introduced to obtain directional feature maps in hyperspectral anomaly detection. Next, LRSMD with TNN was first adopted to decompose the directional feature images to obtain the sparse matrix. Finally, the rough detection map is extracted from the sparse matrix by Mahalanobis distance. In the spatial dimension, attribute filtering is employed to extract the spatial features of the HSI as the complementary spatial weight map of the spectral features. In the decision fusion stage, the spatial weight map and the rough detection maps are fused together to obtain the final anomaly detection result. In this way, the proposed method makes full use of the complementary advantages of the spectral features and spatial features of HSI in each dimension. On the synthetic and real–world datasets, compared with other state–of–the–art comparative methods, experimental results show the advancement and superiority of the method proposed in this paper from qualitative and quantitative analysis.

Additionally, the uncertain parameters in the proposed SCDF method are not determined automatically. This limits the detection performance of the proposed method. Moreover, the proposed SCDF method does not consider the case of mixed pixels. In the future, we will mainly work on adaptive anomaly detection methods to improve the robustness of the anomaly detection methods. Furthermore, we will also consider the impact of mixed pixels on detection performance.