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A New Hyperspectral Anomaly Detection Method Based on Higher Order Statistics and Adaptive Cosine Estimator Zhuang Li , Student Member, IEEE, Ye Zhang, Member, IEEE Abstract— Hyperspectral anomaly detection is a hot topic in remote sensing applications. Most of the conventional detectors are based on the Reed–Xiaoli (RX) method and assumedly targets and backgrounds follow a Gaussian distribution in which two problems exist: the outliers in the Gaussian distribution statistics limit the detection accuracy of RX method, and the larger proportions between the backgrounds and anomaly targets account for the higher false alarm rate. In this letter, a new hyperspectral anomaly detection method is proposed, which can solve the two problems mentioned above. The new method includes two improved ideas. First, third- and fourth-order moments are used as statistical features to improve the outlier peak values and highlight the targets. Second, the adaptive cosine estimation as the structural assumption for the RX method is used to suppress the backgrounds for anomalous targets. Experiments on real hyperspectral data sets suggest that our proposed method could not only effectively decrease the impact of background statistics but also improve the detection ability of such outlier values. Furthermore, comparative experimental results revealed that the proposed method achieves higher detection rates with lower false alarm rates. Index Terms— Adaptive cosine estimation, anomaly detection, hyperspectral data set, outlier peak values, statistical features.
I. I NTRODUCTION YPERSPECTRAL images are of significant interest for anomaly detection as they provide a rich source of continuous spectral information about the materials in a scene [1]–[4]. Detection of anomalies in digital images is generally achieved by finding irregular differences between the pixel under test and its surrounding pixels [5]. In a hyperspectral image, anomalous targets usually appear as spots of pixels that have typical spectral and spatial features compared to their surrounding pixels. Anomaly detection plays a significant role in hyperspectral imaging applications because it does not require any prior information regarding the ground materials to detect potential targets [6]. In recent years, many hyperspectral anomaly detection algorithms have been developed. One of the most widely used detectors is the Reed–Xiaoli (RX) detector [7]. For anomaly detection, the RX algorithm assumes that the background
H
Manuscript received May 7, 2019; revised June 18, 2019; accepted July 14, 2019. This work was supported in part by the National Science Foundation of China under Grant 61871150. (Corresponding author: Ye Zhang.) The authors are with the Department of Information Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2019.2929314
conforms to the same multivariate normal distribution as well as computes the mean and the covariance matrix of the background. The Mahalanobis distance is used to estimate the difference between the pixel under test and its reference background and make a decision through the generalized-likelihood ratio test (GLRT). However, the performance of the RX detection algorithm is adversely affected by low target peak values, which lead to missed detections and, thereby, limiting the accuracy [8]. Moreover, for the traditional RX detector, the background and the target are assumed to have the same distribution. The target (especially small ones) can be contaminated by the background, which leads to an increase in the false alarm rate. Several improvements to the RX method have been proposed. For example, Kwon and Nasrabadi [9] proposed a nonlinear version of RXD, named kernel RX (KRX) detector, that the data samples are mapped to a feature space by a kernel function, and the GLRT is performed in that feature space [10]. Another commonly used RX detector variant is the subspace RX (SSRX) detector which seeks to eliminate high-variance principal components representing the background and detect low-variance principal components which represent the target [11]. Weighted-RXD (W-RXD) can improve the background to provide better estimations of the background information in the RXD-based anomaly detection [12]. In addition, other robust detectors, such as the collaborative representation-based detector (CRD), use the collaborative representation to estimate the background and then predict the targets by subtracting the approximated background from the original hyperspectral image [13]. The tensor decomposition-based detector finds out that the pixels are with not only spectral anomaly but also a spatial anomaly in anomaly detection [14]. The anomaly detection algorithm of abundance- and dictionary-based low-rank (ADLR) decomposition is constructed by spectral unmixing, and the original image is decomposed into two parts—background and abnormal target—by using the sparse characteristics of the anomaly targets [15], [16]. The convolutional neural networkbased detection (CNND) trains the network by learning the difference between the anomaly target and the background of different hyperspectral images from the same sensor, then the trained CNN framework applies to the detection. Reportedly, anomaly detection is based on the estimation of the score of the detected pixel and the similarity of the background [17]. In hyperspectral anomaly detection, the main limitations of the RX detector are the susceptibility to noise and the low target peaks. A notable approach for hyperspectral anomaly detection is to exploit higher order moments [18], such as third-order and
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fourth-order moments, to suppress noise, increase useful signal information, and improve the target peak values in the RX algorithm. Moreover, to suppress the background and alleviate the RX detector contamination for anomalous targets that its size is small, an adaptive cosine estimator (ACE) hypothesis can be incorporated into the detection algorithm. Through the exploitation of the higher order moments and the ACE hypothesis, the targets can be effectively highlighted, while the backgrounds can be suppressed, and better anomaly detection performance can be attained. Thus, the contribution of this letter is to propose an anomaly detection method for small anomaly targets with several pixels. The peak of the signal is increased by higher moments, and the ACE hypothesis is adopted to attain the impact of suppressing the background and highlighting the target. Perhaps, this method could provide robust information support for military target reconnaissance and precision strike, which has notable application value. The rest of this letter is organized as follows. Section II introduces the traditional RX algorithm. Higher order moments, adaptive cosine estimation, and their application to the new algorithm for hyperspectral anomaly detection are described in Section III. To show the effectiveness of the proposed algorithm, experimental results are reported and discussed in Section IV. Finally, conclusions and future directions are presented in Section V.
Double-window structure for the RX algorithm.
algorithm as follows: R X (x) = (x − μb )T −1 (x − μb )
≥ η <
(3)
where μb is the mean of the background samples, is the background covariance matrix, and η is the threshold of the test. The mean μb and covariance of the background are defined as μb = =
N 1 xi N
(4)
1 N
(5)
i=1 N
(x i − μb )(x i − μb )T
i=1
III. P ROPOSED M ETHODS
II. RX AND D OUBLE W INDOW The RX method is based on a local anomaly detection algorithm with a double-window structure: a large background window encloses an anomaly target window, where the window pixel dimensions are odd integers. The pixel of interest is located at the common center of the background and target windows. The images are assumed to be whitened in space and follow a Gaussian distribution. The RX detector essentially computes the mean, covariance, and other parameters for the background window, calculates the corresponding RX operator, and determines through a threshold scheme whether the detected point represents an anomalous target. Given a hyperspectral image with N pixels, where each pixel has P bands, let X P×N = [x 1 , x 2 , . . . , x N ] be the corresponding hyperspectral cube. For each pixel x i = [x 1 , x 2 , . . . , x P ], a local double-window structure is set up and centered at this pixel. The surrounding pixels are considered to be the ones outside the inner window and inside the outer one. If the sizes of the outer and inner windows are wout ×wout and win × win , respectively, then the number of surrounding pixels is (wout × wout − win × win ). Fig. 1 shows the double-window structure for the RX method. The hyperspectral anomaly detection problem at a point x can be posed as the following null and alternative statistical hypotheses: H0 : x = v(Anomalous target absent) H1 : x = Dα + v(Anomalous target present)
Fig. 1.
(1) (2)
where x is the point of interest, v is the background noise, and α is the anomalous target spectrum. On the one hand, when H0 is established (D = 0), the anomalous target is absent. On the other hand, when H1 is established (D > 0), an anomalous target is detected. In a binary classification hypothesis-testing model, the GLRT method can be used to implement the RX
A. Higher Order Moments In the traditional RX algorithm, only low-order background statistics are used, namely, the mean (μb ) and the covariance (). To improve the hyperspectral anomaly detection performance of the RX algorithm, we propose to use higher order moments of the background distribution. In particular, we use the background skewness (third-order moment) and kurtosis (fourth-order moment). Therefore, the GLRT-based decision function of (3) can be reformulated as follows for higher order moments R X (x) = (x 3 − μ3b )T −1 (x 3 − μ3b )
≥ η <
(6)
where the background skewness and kurtosis are given by μ3b = E[x − m]3 =
N 1 (x i − μb )3 N
(7)
i=1
= E[x − m]4 N 1 = [(x i −μb )(x i −μb )T (x i −μb )(x i −μb )T ]. N
(8)
i=1
where m represents the mean of the outer background window. We use the third-order and fourth-order moments to improve the target peak values of the RX algorithm and, hence, enhance the detection performance. B. Adaptive Cosine Estimation With Improved RX Algorithm One key assumption of the traditional RX method is that the background covariance matrix is the same for the null and alternative hypotheses. However, for anomaly and subpixel targets, the background areas for the two hypotheses are
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different. Therefore, the hypotheses of (1) and (2) should be revised as follows: H0 : x = v 1 (Anomalous target absent) H1 : x = Dαd + v 2 (Anomalous target present)
(9) (10)
where x is the hyperspectral pixel vector, v 1 and v 2 are the zero-mean P-dimensional normal random vectors that have different variances, D is the potential abnormal target signal matrix, αd is the potential abnormal target signal abundance matrix, and d is the anomaly target spectrum. Equations (9) and (10) constitute the statistical hypothesis model of an unstructured background based on adaptive cosine estimation, where the background signal is separated from the noise signal during the detection process. Here, we use an ACE-based statistical hypothesis model of the background and target distributions to derive an improved RX algorithm. In particular, the background is assumed to have the same covariance structure under the null and alternative hypotheses. Also, the variance is directly related to the target filling factor, i.e., the ratio of the target to the pixel area. This statistical hypothesis model can be formulated as H0 : x ∼ N(0, σ02 ) H1 : x ∼ N(Dαd , σ12 )
(11) (12)
where is the background covariance matrix, σ02 and σ12 are the variances of the background and target, respectively. Let {y1 , y2 , · · · , y N } be N independent and identically distributed background vectors with zero mean and covariance of , yi ∼ N(0, ). If yi and x are independent, then L(x, Y |H0)
1 T −1 x T −1 x − yi yi = A exp − 2 2 2σ0 i=1 N
L(x, Y |H1)
(13)
(x − Dαd )T −1 (x − Dαd ) 1 T −1 − yi yi = A exp − 2 2σ12 i=1 N
Using the improved RX algorithm [see (6)] with the doublewindow structure and the higher order background statistics [see (7) and (8)], the ACE-based decision function [see (19)] can be changed to D(x) =
((x 3 − μ3b )T −1 d)2 ≥ η (20) 3 T −1 3 T −1 < ((x − μ3b ) (x − μ3b ))(d d)
The decision function D(x) is the ratio of the Mahalanobis distance between the anomalous target and the background calculated by the RX algorithm and the higher order background statistics. Algorithm 1 Proposed Method Input: Hyperspectral image Steps: For each pixel x, 1. Setup the double-window structure centered at x. 2. Calculate the 3rd-order moment (μ3b ) by (7) and the 4th-order moment ( ) by (8). 3. Calculate the ACE-based decision function value D(x) and determine whether an anomalous target is present or not using (20). 4. Calculate the detector output. Output:Detection result
IV. E XPERIMENTAL R ESULTS AND A NALYSIS To assess the detection performance and robustness of the proposed algorithm, experiments were conducted on two real hyperspectral data sets. We verify that the peak of an anomaly point can be strengthened by using higher moments. In anomaly detection with small targets, the backgrounds can be suppressed and the targets can be highlighted by the combination of higher order statistics and adaptive cosine estimation hypothesis. The receiver operating characteristics (ROC) curve and the area under the ROC curve (AUC) are used for quantitative assessment of the hyperspectral anomaly detection performance.
(14) where A = (2π)−(1/2)L(N+1)||−(1/2)(N+1) (σ02 )−(1/2)L and the maximum likelihood estimates of αd , σ02 and σ12 are ∧
αd = (D T −1 D)−1 D T −1 x (15) ∧ T −1 x x σ02 = (16) L ∧ (x − Dαd )T −1 (x − Dαd ) . (17) σ12 = L Using the likelihood functions [see (13) and (14)], and the maximum likelihood estimates [see (15)–(17)], the GLRT becomes DACE (x) =
x T −1 D(D T −1 D)−1 D T −1 x ≥ η . < ACE x T −1 x
(18)
When D = d, it suggests that the detected pixel is the anomaly target point we are looking for, so this test is simplified to: DACE (x) =
(x T −1 d)2 ≥ ηACE . (x T −1 x)(d T −1 d) <
(19)
A. Hyperspectral Data Sets For our anomaly detection experiments, we applied two real hyperspectral data sets. The first data set [Fig. 2(a)] was captured by an airborne visible/infrared imaging spectrometer sensor over an area of the San Diego International Airport, CA, USA. The airport scene consists of 400 × 400 pixels with a spatial resolution of 3.5 m. The data set has 126 spectral bands, where the water absorption bands have been removed. We used two subimages of the original image, which describe 3 and 38 anomalous targets, respectively. The size of each subimage is 100 × 100 pixels. The two subimages are shown in Fig. 2(b) and (d) and their ground truth maps are shown in Fig. 2(c) and (e), respectively. The second data set is the Avon data set, which was collected using a push-broom hyperspectral ProSpec TIR-VS sensor during the “SpecTIR Hyperspectral Airborne Experiment 2012” (SHARE 2012) data collection campaign [19]. From the Avon morning reflectance data, a region of 330 × 330 pixels was selected for experiments, as shown in Fig. 3(a). This region covers a driving park in Avon, south of Rochester,
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Fig. 5. Detection results for the San Diego data set (1). (a) RX. (b) Improved RX.
Fig. 2. San Diego data set. (a) Whole image. (b) Subimage with three anomalous targets. (c) Ground truth data of (b). (d) Subimage with 38 anomalous targets. (e) Ground truth data of (d).
Fig. 3.
Avon data set. (a) False color image. (b) Ground truth.
Fig. 4. Anomalous target peak surface plot of the (a) traditional RX detector and (b) improved RX detector.
NY, USA, with a spatial resolution of 1 m. This data set has 360 bands covering the spectral range of 400–2450 nm, with a spectral resolution of 5 nm. There are 25 tarps and 3 red or blue felts with 68 target pixels to be detected in the scene, as shown in Fig. 3(b). B. Performance Comparison Between the Traditional and Improved RX Algorithm Here, we compare the effectiveness of the RX algorithm and the improved one based on higher order moments for anomaly detection. We used the first subimage of the San Diego data set (with three anomalous targets) [Fig. 2(b)] and a double window whose inner and outer widths are 3 and 5 pixels, respectively. Fig. 4 shows a 3-D surface map of the anomalous target peaks for the two algorithms. It can be realized that the target peaks of our improved RX algorithm are significantly higher than those of the traditional RX algorithm. Hence, our approach leads to stronger target highlighting in anomaly detection. Fig. 5 shows that the improved RX algorithm can detect the targets more apparently and accurately in a smaller window, while the output of the traditional RX algorithm is not obvious or strong.
Fig. 6. Detection results for San Diego data set (2) using different methods. (a) RX. (b) Improved RX. (c) Proposed method. (d) KRX. (e) SSRX. (f) W-RX. (g) ADLR. (h) CNND.
C. Anomaly Detection for Small Targets We investigate here the effectiveness of the proposed algorithm for anomaly detection given small targets. We performed the experiments using the second San Diego subimage [Fig. 2(d)] that contains 38 small aircraft targets. The detection results of the traditional RX algorithm, the improved RX algorithm, the proposed method, KRX, SSRX, W-RX, ADLR, and CNND are shown in Fig. 6. The results in Fig. 6 show that the improved RX algorithm is superior to the traditional RX algorithm and can effectively highlight the targets, but the high false alarm rate persists. The ACE algorithm can roughly detect anomaly targets, but the targets are not clear. The proposed algorithm can clearly detect all 38 anomalous targets in the result. This shows the importance of the higher order moments for improving the signal peaks, and the importance of the ACE hypothesis for background suppression. In comparison to our results, the KRX and SSRX outcomes are more noisy and nonuniform. The false alarm rate of the W-RX is higher, and the target is not clear. ADLR and CNND have more interference, and the pixels in the background that are not the target are detected as targets, which affects the target’s positioning and tracking. D. ROC Analysis of the Detection Performance We carried on ROC analysis and computed AUC measures on the San Diego (the second subimage) and Avon data sets to quantitatively compare the performance of seven detectors, namely the RX, SSRX, KRX, proposed detector, W-RX, ADLR, and CNND. Fig. 7(a) shows the ROC curves of the four algorithms for the San Diego data set (the second subimage). The SSRX and KRX detectors outperform the traditional RX detector. W-RX, ADLR, and CNND are better than KRX and SSRX. Our proposed achieves the best overall performance among
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Fig. 7. ROC curves of different anomaly detection methods on two data sets. (a) San Diego data set (the second subimage). (b) Avon data set.
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The use of these higher order moments as the improvement of the RX method increased the target peak values and thus helped highlight anomalous targets more effectively. The ACE detection hypothesis was applied to the improved RX detection algorithm, and a new hyperspectral anomaly detection algorithm was designed. This proposed detector can suppress the backgrounds while highlighting the targets even more effectively. The validity of the proposed method was verified by experiments on the two real data sets. The experimental results show that the method has superior performance and robustness in hyperspectral anomaly detection. For future work, the proposed method can be exploited for anomalous targets tracking in heavily cluttered environments.
TABLE I AUC M ETRICS ON THE S AN D IEGO D ATA S ET ( THE S ECOND S UBIMAGE ) ON D IFFERENT M ETHODS AND W INDOW S IZES
TABLE II AUC M ETRICS ON THE AVON D ATA S ET ON D IFFERENT M ETHODS AND W INDOW S IZES
the compared methods. Fig. 7(b) shows similar results on the Avon data set. Tables I and II illustrate the AUC scores of the compared methods for the San Diego (the second subimage) and the Avon data set, respectively. A higher AUC value indicates a better detection performance. Meanwhile, the detection performance of the seven detectors is compared for four different double-window widths. As shown in Table I and Table II, it can be seen that the proposed method obtains the highest AUC for both data sets. For example, given the San Diego data set (the second subimage) and the double window with widths (3, 5), the AUC score obtained by the proposed method is 0.9632, which is higher than those of the RX, KRX, SSRX, W-RX, ADLR, and CNND detectors. This demonstrates the capacity of our method to highlight the targets and suppress the backgrounds. V. C ONCLUSION In this letter, we proposed a new hyperspectral anomaly detection method based on higher order statistics and ACE. Higher order moments were used to replace the mean and variance as the statistical features of the hyperspectral image.
R EFERENCES [1] A. Plaza et al., “Recent advances in techniques for hyperspectral image processing,” Remote Sens. Environ., vol. 113, pp. S110–S122, Sep. 2009. [2] M. Imani, “RX Anomaly detector with rectified background,” IEEE Geosci. Remote Sens. Lett., vol. 14, no. 8, pp. 1313–1317, Aug. 2017. [3] S. Matteoli, M. Diani, and G. Corsini, “A tutorial overview of anomaly detection in hyperspectral images,” IEEE Aerosp. Electron. Syst. Mag., vol. 25, no. 7, pp. 5–28, Jul. 2010. [4] S. M. Schweizer and J. M. F. Moura, “Hyperspectral imagery: Clutter adaptation in anomaly detection,” IEEE Trans. Inf. Theory, vol. 46, no. 5, pp. 1855–1871, Aug. 2000. [5] C. Zhao, X.-F. Yao, and Y. Yan, “Modified kernel RX algorithm based on background purification and inverse-of-matrix-free calculation,” IEEE Geosci. Remote Sens. Lett., vol. 14, no. 4, pp. 544–548, Apr. 2017. [6] J. Theiler and G. Grosklos, “Problematic projection to the in-sample subspace for a kernelized anomaly detector,” IEEE Geosci. Remote Sens. Lett., vol. 13, no. 4, pp. 485–489, Apr. 2016. [7] I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech Signal Process., vol. 38, no. 10, pp. 1760–1770, Oct. 1990. [8] R. Zhao, B. Du, and L. Zhang, “Hyperspectral anomaly detection via a sparsity score estimation framework,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 6, pp. 3208–3222, Jun. 2017. [9] H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: A nonlinear anomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 2, pp. 388–397, Feb. 2005. [10] C. Zhao and Y. Xi-Feng, “Fast real-time kernel RX algorithm based on Cholesky decomposition,” IEEE Geosci. Remote Sens. Lett., vol. 15, no. 11, pp. 1760–1764, Nov. 2018. [11] A. P. Schaum, “Hyperspectral anomaly detection beyond RX,” Proc. SPIE, vol. 14, May 2007, Art. no. 656502. [12] Q. Guo, B. Zhang, Q. Ran, L. Gao, J. Li, and A. Plaza, “WeightedRXD and linear filter-based RXD: Improving background statistics estimation for anomaly detection in hyperspectral imagery,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 7, no. 6, pp. 2351–2366, Jun. 2014. [13] L. Wei and D. Qian, “Collaborative representation for hyperspectral anomaly detection,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 3, pp. 1463–1474, Mar. 2015. [14] X. Zhang, G. Wen, and W. Dai, “A tensor decomposition-based anomaly detection algorithm for hyperspectral image,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 10, pp. 5801–5820, Oct. 2016. [15] Y. Qu et al., “Hyperspectral anomaly detection through spectral unmixing and dictionary-based low-rank decomposition,” IEEE Trans. Geosci. Remote Sens., vol. 56, no. 8, pp. 4391–4405, Aug. 2018. [16] Y. Yuan, D. Ma, and Q. Wang, “Hyperspectral anomaly detection via sparse dictionary learning method of capped norm,” IEEE Access, vol. 7, pp. 16132–16144, 2019. [17] W. Li, G. Wu, and Q. Du, “Transferred deep learning for anomaly detection in hyperspectral imagery,” IEEE Geosci. Remote Sens. Lett., vol. 14, no. 5, pp. 597–601, May 2017. [18] E. Hasanbelliu, L. S. Giraldo, and J. C. Príncipe, “Information theoretic shape matching,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 36, no. 12, pp. 2436–2451, Dec. 2014. [19] Q. Ling, Y. Guo, Z. Lin, and W. An, “A constrained sparse representation model for hyperspectral anomaly detection,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 4, pp. 2358–2371, Apr. 2019.
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A New Hyperspectral Anomaly Detection Method Based on Higher Order Statistics and Adaptive Cosine Estimator Zhuang Li , Student Member, IEEE, Ye Zhang, Member, IEEE Abstract— Hyperspectral anomaly detection is a hot topic in remote sensing applications. Most of the conventional detectors are based on the Reed–Xiaoli (RX) method and assumedly targets and backgrounds follow a Gaussian distribution in which two problems exist: the outliers in the Gaussian distribution statistics limit the detection accuracy of RX method, and the larger proportions between the backgrounds and anomaly targets account for the higher false alarm rate. In this letter, a new hyperspectral anomaly detection method is proposed, which can solve the two problems mentioned above. The new method includes two improved ideas. First, third- and fourth-order moments are used as statistical features to improve the outlier peak values and highlight the targets. Second, the adaptive cosine estimation as the structural assumption for the RX method is used to suppress the backgrounds for anomalous targets. Experiments on real hyperspectral data sets suggest that our proposed method could not only effectively decrease the impact of background statistics but also improve the detection ability of such outlier values. Furthermore, comparative experimental results revealed that the proposed method achieves higher detection rates with lower false alarm rates. Index Terms— Adaptive cosine estimation, anomaly detection, hyperspectral data set, outlier peak values, statistical features.
I. I NTRODUCTION YPERSPECTRAL images are of significant interest for anomaly detection as they provide a rich source of continuous spectral information about the materials in a scene [1]–[4]. Detection of anomalies in digital images is generally achieved by finding irregular differences between the pixel under test and its surrounding pixels [5]. In a hyperspectral image, anomalous targets usually appear as spots of pixels that have typical spectral and spatial features compared to their surrounding pixels. Anomaly detection plays a significant role in hyperspectral imaging applications because it does not require any prior information regarding the ground materials to detect potential targets [6]. In recent years, many hyperspectral anomaly detection algorithms have been developed. One of the most widely used detectors is the Reed–Xiaoli (RX) detector [7]. For anomaly detection, the RX algorithm assumes that the background
H
Manuscript received May 7, 2019; revised June 18, 2019; accepted July 14, 2019. This work was supported in part by the National Science Foundation of China under Grant 61871150. (Corresponding author: Ye Zhang.) The authors are with the Department of Information Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2019.2929314
conforms to the same multivariate normal distribution as well as computes the mean and the covariance matrix of the background. The Mahalanobis distance is used to estimate the difference between the pixel under test and its reference background and make a decision through the generalized-likelihood ratio test (GLRT). However, the performance of the RX detection algorithm is adversely affected by low target peak values, which lead to missed detections and, thereby, limiting the accuracy [8]. Moreover, for the traditional RX detector, the background and the target are assumed to have the same distribution. The target (especially small ones) can be contaminated by the background, which leads to an increase in the false alarm rate. Several improvements to the RX method have been proposed. For example, Kwon and Nasrabadi [9] proposed a nonlinear version of RXD, named kernel RX (KRX) detector, that the data samples are mapped to a feature space by a kernel function, and the GLRT is performed in that feature space [10]. Another commonly used RX detector variant is the subspace RX (SSRX) detector which seeks to eliminate high-variance principal components representing the background and detect low-variance principal components which represent the target [11]. Weighted-RXD (W-RXD) can improve the background to provide better estimations of the background information in the RXD-based anomaly detection [12]. In addition, other robust detectors, such as the collaborative representation-based detector (CRD), use the collaborative representation to estimate the background and then predict the targets by subtracting the approximated background from the original hyperspectral image [13]. The tensor decomposition-based detector finds out that the pixels are with not only spectral anomaly but also a spatial anomaly in anomaly detection [14]. The anomaly detection algorithm of abundance- and dictionary-based low-rank (ADLR) decomposition is constructed by spectral unmixing, and the original image is decomposed into two parts—background and abnormal target—by using the sparse characteristics of the anomaly targets [15], [16]. The convolutional neural networkbased detection (CNND) trains the network by learning the difference between the anomaly target and the background of different hyperspectral images from the same sensor, then the trained CNN framework applies to the detection. Reportedly, anomaly detection is based on the estimation of the score of the detected pixel and the similarity of the background [17]. In hyperspectral anomaly detection, the main limitations of the RX detector are the susceptibility to noise and the low target peaks. A notable approach for hyperspectral anomaly detection is to exploit higher order moments [18], such as third-order and
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fourth-order moments, to suppress noise, increase useful signal information, and improve the target peak values in the RX algorithm. Moreover, to suppress the background and alleviate the RX detector contamination for anomalous targets that its size is small, an adaptive cosine estimator (ACE) hypothesis can be incorporated into the detection algorithm. Through the exploitation of the higher order moments and the ACE hypothesis, the targets can be effectively highlighted, while the backgrounds can be suppressed, and better anomaly detection performance can be attained. Thus, the contribution of this letter is to propose an anomaly detection method for small anomaly targets with several pixels. The peak of the signal is increased by higher moments, and the ACE hypothesis is adopted to attain the impact of suppressing the background and highlighting the target. Perhaps, this method could provide robust information support for military target reconnaissance and precision strike, which has notable application value. The rest of this letter is organized as follows. Section II introduces the traditional RX algorithm. Higher order moments, adaptive cosine estimation, and their application to the new algorithm for hyperspectral anomaly detection are described in Section III. To show the effectiveness of the proposed algorithm, experimental results are reported and discussed in Section IV. Finally, conclusions and future directions are presented in Section V.
Double-window structure for the RX algorithm.
algorithm as follows: R X (x) = (x − μb )T −1 (x − μb )
≥ η <
(3)
where μb is the mean of the background samples, is the background covariance matrix, and η is the threshold of the test. The mean μb and covariance of the background are defined as μb = =
N 1 xi N
(4)
1 N
(5)
i=1 N
(x i − μb )(x i − μb )T
i=1
III. P ROPOSED M ETHODS
II. RX AND D OUBLE W INDOW The RX method is based on a local anomaly detection algorithm with a double-window structure: a large background window encloses an anomaly target window, where the window pixel dimensions are odd integers. The pixel of interest is located at the common center of the background and target windows. The images are assumed to be whitened in space and follow a Gaussian distribution. The RX detector essentially computes the mean, covariance, and other parameters for the background window, calculates the corresponding RX operator, and determines through a threshold scheme whether the detected point represents an anomalous target. Given a hyperspectral image with N pixels, where each pixel has P bands, let X P×N = [x 1 , x 2 , . . . , x N ] be the corresponding hyperspectral cube. For each pixel x i = [x 1 , x 2 , . . . , x P ], a local double-window structure is set up and centered at this pixel. The surrounding pixels are considered to be the ones outside the inner window and inside the outer one. If the sizes of the outer and inner windows are wout ×wout and win × win , respectively, then the number of surrounding pixels is (wout × wout − win × win ). Fig. 1 shows the double-window structure for the RX method. The hyperspectral anomaly detection problem at a point x can be posed as the following null and alternative statistical hypotheses: H0 : x = v(Anomalous target absent) H1 : x = Dα + v(Anomalous target present)
Fig. 1.
(1) (2)
where x is the point of interest, v is the background noise, and α is the anomalous target spectrum. On the one hand, when H0 is established (D = 0), the anomalous target is absent. On the other hand, when H1 is established (D > 0), an anomalous target is detected. In a binary classification hypothesis-testing model, the GLRT method can be used to implement the RX
A. Higher Order Moments In the traditional RX algorithm, only low-order background statistics are used, namely, the mean (μb ) and the covariance (). To improve the hyperspectral anomaly detection performance of the RX algorithm, we propose to use higher order moments of the background distribution. In particular, we use the background skewness (third-order moment) and kurtosis (fourth-order moment). Therefore, the GLRT-based decision function of (3) can be reformulated as follows for higher order moments R X (x) = (x 3 − μ3b )T −1 (x 3 − μ3b )
≥ η <
(6)
where the background skewness and kurtosis are given by μ3b = E[x − m]3 =
N 1 (x i − μb )3 N
(7)
i=1
= E[x − m]4 N 1 = [(x i −μb )(x i −μb )T (x i −μb )(x i −μb )T ]. N
(8)
i=1
where m represents the mean of the outer background window. We use the third-order and fourth-order moments to improve the target peak values of the RX algorithm and, hence, enhance the detection performance. B. Adaptive Cosine Estimation With Improved RX Algorithm One key assumption of the traditional RX method is that the background covariance matrix is the same for the null and alternative hypotheses. However, for anomaly and subpixel targets, the background areas for the two hypotheses are
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different. Therefore, the hypotheses of (1) and (2) should be revised as follows: H0 : x = v 1 (Anomalous target absent) H1 : x = Dαd + v 2 (Anomalous target present)
(9) (10)
where x is the hyperspectral pixel vector, v 1 and v 2 are the zero-mean P-dimensional normal random vectors that have different variances, D is the potential abnormal target signal matrix, αd is the potential abnormal target signal abundance matrix, and d is the anomaly target spectrum. Equations (9) and (10) constitute the statistical hypothesis model of an unstructured background based on adaptive cosine estimation, where the background signal is separated from the noise signal during the detection process. Here, we use an ACE-based statistical hypothesis model of the background and target distributions to derive an improved RX algorithm. In particular, the background is assumed to have the same covariance structure under the null and alternative hypotheses. Also, the variance is directly related to the target filling factor, i.e., the ratio of the target to the pixel area. This statistical hypothesis model can be formulated as H0 : x ∼ N(0, σ02 ) H1 : x ∼ N(Dαd , σ12 )
(11) (12)
where is the background covariance matrix, σ02 and σ12 are the variances of the background and target, respectively. Let {y1 , y2 , · · · , y N } be N independent and identically distributed background vectors with zero mean and covariance of , yi ∼ N(0, ). If yi and x are independent, then L(x, Y |H0)
1 T −1 x T −1 x − yi yi = A exp − 2 2 2σ0 i=1 N
L(x, Y |H1)
(13)
(x − Dαd )T −1 (x − Dαd ) 1 T −1 − yi yi = A exp − 2 2σ12 i=1 N
Using the improved RX algorithm [see (6)] with the doublewindow structure and the higher order background statistics [see (7) and (8)], the ACE-based decision function [see (19)] can be changed to D(x) =
((x 3 − μ3b )T −1 d)2 ≥ η (20) 3 T −1 3 T −1 < ((x − μ3b ) (x − μ3b ))(d d)
The decision function D(x) is the ratio of the Mahalanobis distance between the anomalous target and the background calculated by the RX algorithm and the higher order background statistics. Algorithm 1 Proposed Method Input: Hyperspectral image Steps: For each pixel x, 1. Setup the double-window structure centered at x. 2. Calculate the 3rd-order moment (μ3b ) by (7) and the 4th-order moment ( ) by (8). 3. Calculate the ACE-based decision function value D(x) and determine whether an anomalous target is present or not using (20). 4. Calculate the detector output. Output:Detection result
IV. E XPERIMENTAL R ESULTS AND A NALYSIS To assess the detection performance and robustness of the proposed algorithm, experiments were conducted on two real hyperspectral data sets. We verify that the peak of an anomaly point can be strengthened by using higher moments. In anomaly detection with small targets, the backgrounds can be suppressed and the targets can be highlighted by the combination of higher order statistics and adaptive cosine estimation hypothesis. The receiver operating characteristics (ROC) curve and the area under the ROC curve (AUC) are used for quantitative assessment of the hyperspectral anomaly detection performance.
(14) where A = (2π)−(1/2)L(N+1)||−(1/2)(N+1) (σ02 )−(1/2)L and the maximum likelihood estimates of αd , σ02 and σ12 are ∧
αd = (D T −1 D)−1 D T −1 x (15) ∧ T −1 x x σ02 = (16) L ∧ (x − Dαd )T −1 (x − Dαd ) . (17) σ12 = L Using the likelihood functions [see (13) and (14)], and the maximum likelihood estimates [see (15)–(17)], the GLRT becomes DACE (x) =
x T −1 D(D T −1 D)−1 D T −1 x ≥ η . < ACE x T −1 x
(18)
When D = d, it suggests that the detected pixel is the anomaly target point we are looking for, so this test is simplified to: DACE (x) =
(x T −1 d)2 ≥ ηACE . (x T −1 x)(d T −1 d) <
(19)
A. Hyperspectral Data Sets For our anomaly detection experiments, we applied two real hyperspectral data sets. The first data set [Fig. 2(a)] was captured by an airborne visible/infrared imaging spectrometer sensor over an area of the San Diego International Airport, CA, USA. The airport scene consists of 400 × 400 pixels with a spatial resolution of 3.5 m. The data set has 126 spectral bands, where the water absorption bands have been removed. We used two subimages of the original image, which describe 3 and 38 anomalous targets, respectively. The size of each subimage is 100 × 100 pixels. The two subimages are shown in Fig. 2(b) and (d) and their ground truth maps are shown in Fig. 2(c) and (e), respectively. The second data set is the Avon data set, which was collected using a push-broom hyperspectral ProSpec TIR-VS sensor during the “SpecTIR Hyperspectral Airborne Experiment 2012” (SHARE 2012) data collection campaign [19]. From the Avon morning reflectance data, a region of 330 × 330 pixels was selected for experiments, as shown in Fig. 3(a). This region covers a driving park in Avon, south of Rochester,
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS
Fig. 5. Detection results for the San Diego data set (1). (a) RX. (b) Improved RX.
Fig. 2. San Diego data set. (a) Whole image. (b) Subimage with three anomalous targets. (c) Ground truth data of (b). (d) Subimage with 38 anomalous targets. (e) Ground truth data of (d).
Fig. 3.
Avon data set. (a) False color image. (b) Ground truth.
Fig. 4. Anomalous target peak surface plot of the (a) traditional RX detector and (b) improved RX detector.
NY, USA, with a spatial resolution of 1 m. This data set has 360 bands covering the spectral range of 400–2450 nm, with a spectral resolution of 5 nm. There are 25 tarps and 3 red or blue felts with 68 target pixels to be detected in the scene, as shown in Fig. 3(b). B. Performance Comparison Between the Traditional and Improved RX Algorithm Here, we compare the effectiveness of the RX algorithm and the improved one based on higher order moments for anomaly detection. We used the first subimage of the San Diego data set (with three anomalous targets) [Fig. 2(b)] and a double window whose inner and outer widths are 3 and 5 pixels, respectively. Fig. 4 shows a 3-D surface map of the anomalous target peaks for the two algorithms. It can be realized that the target peaks of our improved RX algorithm are significantly higher than those of the traditional RX algorithm. Hence, our approach leads to stronger target highlighting in anomaly detection. Fig. 5 shows that the improved RX algorithm can detect the targets more apparently and accurately in a smaller window, while the output of the traditional RX algorithm is not obvious or strong.
Fig. 6. Detection results for San Diego data set (2) using different methods. (a) RX. (b) Improved RX. (c) Proposed method. (d) KRX. (e) SSRX. (f) W-RX. (g) ADLR. (h) CNND.
C. Anomaly Detection for Small Targets We investigate here the effectiveness of the proposed algorithm for anomaly detection given small targets. We performed the experiments using the second San Diego subimage [Fig. 2(d)] that contains 38 small aircraft targets. The detection results of the traditional RX algorithm, the improved RX algorithm, the proposed method, KRX, SSRX, W-RX, ADLR, and CNND are shown in Fig. 6. The results in Fig. 6 show that the improved RX algorithm is superior to the traditional RX algorithm and can effectively highlight the targets, but the high false alarm rate persists. The ACE algorithm can roughly detect anomaly targets, but the targets are not clear. The proposed algorithm can clearly detect all 38 anomalous targets in the result. This shows the importance of the higher order moments for improving the signal peaks, and the importance of the ACE hypothesis for background suppression. In comparison to our results, the KRX and SSRX outcomes are more noisy and nonuniform. The false alarm rate of the W-RX is higher, and the target is not clear. ADLR and CNND have more interference, and the pixels in the background that are not the target are detected as targets, which affects the target’s positioning and tracking. D. ROC Analysis of the Detection Performance We carried on ROC analysis and computed AUC measures on the San Diego (the second subimage) and Avon data sets to quantitatively compare the performance of seven detectors, namely the RX, SSRX, KRX, proposed detector, W-RX, ADLR, and CNND. Fig. 7(a) shows the ROC curves of the four algorithms for the San Diego data set (the second subimage). The SSRX and KRX detectors outperform the traditional RX detector. W-RX, ADLR, and CNND are better than KRX and SSRX. Our proposed achieves the best overall performance among
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Fig. 7. ROC curves of different anomaly detection methods on two data sets. (a) San Diego data set (the second subimage). (b) Avon data set.
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The use of these higher order moments as the improvement of the RX method increased the target peak values and thus helped highlight anomalous targets more effectively. The ACE detection hypothesis was applied to the improved RX detection algorithm, and a new hyperspectral anomaly detection algorithm was designed. This proposed detector can suppress the backgrounds while highlighting the targets even more effectively. The validity of the proposed method was verified by experiments on the two real data sets. The experimental results show that the method has superior performance and robustness in hyperspectral anomaly detection. For future work, the proposed method can be exploited for anomalous targets tracking in heavily cluttered environments.
TABLE I AUC M ETRICS ON THE S AN D IEGO D ATA S ET ( THE S ECOND S UBIMAGE ) ON D IFFERENT M ETHODS AND W INDOW S IZES
TABLE II AUC M ETRICS ON THE AVON D ATA S ET ON D IFFERENT M ETHODS AND W INDOW S IZES
the compared methods. Fig. 7(b) shows similar results on the Avon data set. Tables I and II illustrate the AUC scores of the compared methods for the San Diego (the second subimage) and the Avon data set, respectively. A higher AUC value indicates a better detection performance. Meanwhile, the detection performance of the seven detectors is compared for four different double-window widths. As shown in Table I and Table II, it can be seen that the proposed method obtains the highest AUC for both data sets. For example, given the San Diego data set (the second subimage) and the double window with widths (3, 5), the AUC score obtained by the proposed method is 0.9632, which is higher than those of the RX, KRX, SSRX, W-RX, ADLR, and CNND detectors. This demonstrates the capacity of our method to highlight the targets and suppress the backgrounds. V. C ONCLUSION In this letter, we proposed a new hyperspectral anomaly detection method based on higher order statistics and ACE. Higher order moments were used to replace the mean and variance as the statistical features of the hyperspectral image.
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