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Coincidence of the Rao Test, Wald Test and GLRT for anomaly detection in hyperspectral imagery
Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery
Journal Pre-proof
Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery Yutong Feng, Jun Liu, Weijian Liu PII: DOI: Reference:
S0165-1684(19)30468-2 https://doi.org/10.1016/j.sigpro.2019.107416 SIGPRO 107416
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
6 July 2019 28 October 2019 2 December 2019
Please cite this article as: Yutong Feng, Jun Liu, Weijian Liu, Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery, Signal Processing (2019), doi: https://doi.org/10.1016/j.sigpro.2019.107416
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery Yutong Fenga
Jun Liua,
∗
Weijian Liub
December 5, 2019
Abstract Anomaly detection methods are designed to detect targets (small anomalies) without a priori information on the target spectral signature. In this letter, we deal with the problem of anomaly detection for hyperspectral images based on the Gaussian model assuming that the background obeys a realvalued Gaussian multivariate distribution with unknown covariance matrix. This model is widely used in hyperspectral images. We derive the corresponding Rao and Wald tests, and show that both the two tests are equivalent to the generalized likelihood ratio test. Keywords – Anomaly detection, Gaussian model, hyperspectral images, Rao test, Wald test, generalized likelihood ratio test (GLRT).
∗
Corresponding author (e-mail: [email protected]). Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, China. b Wuhan Electronic Information Institute, Wuhan 430019, China. a
1
1
Introduction
Hyperspectral images are collected by hyperspectral imaging sensors that accumulate spectra information from each point of a scene [1, 2]. Due to the high resolutions of the spectra of hyperspectral images, the spectral bands can be considered contiguous for every pixel. Thus, hyperspectral imaging data exploitation makes it possible to detect desired targets in the scene based on their spectral signatures. In recent years, spectral signature-based target detection has been studied by many researchers [1, 3–5]. In most of such work, it is assumed that the target spectral signatures are known. The goal is to detect a signal with known signature but unknown amplitude from the background clutter. However, in practical applications, the atmospheric compensation and other complicating factors may lead to significant deviations between the original and measured spectral signatures. This may lead to severe mismatch losses in spectral signature-based detection methods with requiring known spectral signatures [6]. In order to avoid this problem, the anomaly detection methods are proposed and applied to hyperspectral images [7–9]. Anomaly detection of multiband signals have been widely studied, which seeks to distinguish unusual targets (small anomalies) from the background without a priori knowledge about spectral signatures. In hyperpectral images, the classical anomaly detection can be considered as a binary hypothesis testing problem, and the typical solution is the Neyman-Pearson (NP) approach. Based on the NP criterion, the well-known Reed-Xiaoli detector (RXD) [10] was derived from the generalized likelihood ratio test (GLRT) principle, by assuming that the disturbance obeys a real-valued Gaussian multivariate distribution with unknown covariance matrix [8]. Because of its mathematical tractability and its constant false alarm ratio (CFAR) property, the RXD is considered as the benchmark anomaly detector for multi/hyperspectral data. Moreover, it is suitable for real-time applications, since the background parameters in the RXD are estimated by local data. It is worth mentioning that, besides the RXD, other methods have been proposed to deal with the anomaly detection problem, such as the so-called kernel RX [11], the RX-uniform target detector [12], and the component projection and separation optimized filter [13]. As alternative strategies with respect to the GLRT, the Rao and Wald tests might require a smaller computational complexity and be more robust in some realistic operating situations [14]. It is noteworthy that the aforementioned three tests are asymptotically equivalent [15]. Due to these interesting properties, the Rao and Wald tests are widely studied by researchers [14, 16–19]. For instance, the authors systematically analyzed the relationship between the aforementioned three tests, and derived the invariance, coincidence, and statistical equivalence properties of the GLRT, the Rao and Wald tests for radar applications [14, 17]. 2
In this letter, we derive the Rao and Wald tests for the anomaly detection in hyperspectral images under the Gaussian model. Interestingly, it is found that the proposed Rao and Wald tests are statistically equivalent to the RXD for anomaly detection problems in hyperspectral images. This letter is organized as follows. Section 2 provides the problem formulation. In Section 3, we design the Rao and Wald tests and prove the two tests coincide with the RXD. Conclusion and final remarks are given in Section 4.
1.1
Notation
In the following, scalars, vectors and matrices are represented by lightfaced lowercase, boldfaced lowercase and boldfaced uppercase letters, respectively. ln(·) denotes the natural logarithm. E[·] is the statistical expectation. The notation , means “defined as”. A−1 and |A| stand for the inverse and determinant of matrix A, respectively. ∂f /∂a denotes the partial derivative of the scalar function f with respect to the vector a. vec(C) denotes the vectorization of the matrix C. The symbol (·)T stands for transpose operation. 0M,N denotes the M × N -dimensional matrices of
zeros. Finally, R denotes the set of real numbers, and RM ×N denotes the set of M × N -dimensional real matrices.
2
Problem Formulation
The problem of anomaly detection in hyperspectral images can be formulated as the following binary hypothesis test [10]:
H0 : y(i) = n(i),
i = 1, . . . , N,
H : y(i) = n(i) + sa(i), 1
(1)
i = 1, . . . , N,
where y(i) ∈ RM ×1 , i = 1, 2, . . . , N denote observed spectra containing the potential anomaly target with known shape (spatial pattern) and unknown position; M is the number of spectral bands, N represents the number of pixels under test with N > M ; s ∈ RM ×1 denotes the unknown spectral signature (can be deemed as a steering vector) of the possible target, and a(i) is the known target amplitude. It should be noted that for radar target detection, the steering vector s is known and the target amplitude a(i) is unknown [14, 17]. As to the background clutter vector n(i) ∈ RM ×1 , we assume that n(i) and n(j), i 6= j, are independent, zero-mean, Gaussian random vectors with the positive-definite covariance matrix M ∈
3
RM ×M , that is
E[n(i)n(j)T ] = M, E[n(i)n(j)T ] = 0,
i=j
(2)
i 6= j.
The covariance matrix M is assumed to be unknown and needs to be estimated. In compact forms, we can combine the vectors into matrices as Y = [y(1), y(2), . . . , y(N )] ∈ RM ×N ,
(3)
N = [n(1), n(2), . . . , n(N )] ∈ RM ×N .
(4)
and
Then, the binary hypotheses test can be recast as H0 : Y = N, where
H : Y = N + saT , 1
a = [a(1), a(2), . . . , a(N )]T ∈ RN ×1
(5)
(6)
is an amplitude vector standing for the spatial pattern of target. Note that the spectral signature s is unknown, whereas the signal pattern a is assumed to be known. Under the above model, the GLRT has been derived in [10], which is the well-known RXD. It takes the form TRXD =
(Ya)T (YYT )−1 (Ya) H1 ≷ λRXD , aT a H0
(7)
where λRXD is the detection threshold. It should be mentioned that if no a priori information about the target spatial pattern is available, the spatial pattern can be neglected here (i.e., just one anomaly pixel is intended to be detected). In this situation, the amplitude vector can be written as a = [0, . . . , 0, 1, 0, . . . , 0]T , where 1 is at n-th position. Then the original RXD reduces to the Mahalanobis distance ˆ −1 y(n), TRXD = y(n)T M
(8)
ˆ is the estimated background clutter covariance matrix. It can be written as where the term M ˆ = 1 YYT . M N
(9)
This pixel-wised form of the RXD is also widely considered in open literatures [6, 8, 9, 12]. In next section, we derive the Rao and Wald tests for the detection problem in (5), and show that the two tests coincide with the RXD. 4
3
Design of Rao and Wald Tests
3.1
Rao Test
In order to derive the Rao test, we first introduce an (M 2 + M )-dimensional parameter vector θ, given by T T T θ = θ Tr , θ Ts = s , vecT (M) ,
(10)
where the spectral signature θ r = s is set to be the relevant parameter, and θ s = vec(M) is set to be the nuisance parameter. The real-valued Rao test with nuisance parameters is given by [15] TRao
h i ∂lnf (Y; s, M) T −1 ˜ I ( θ ) = 0 ˜ ∂θ r θr θr θ=θ 0
H1 ∂lnf (Y; s, M) × ≷ λR , ∂θ r ˜ 0 H0 θ=θ
(11)
˜ 0 = θ T , θ T T is the maximum likelihood estimate (MLE) of where λR is the detection threshold, θ r0 s0
θ under hypothesis H0 , f (Y; s, M) is the probability density function of Y under hypothesis H1 , i.e., MN 2
N
|M|− 2 i 1 h −1 T T T × exp − tr M Y−sa Y−sa . 2
f (Y; s, M) =(2π)−
The Fisher information matrix (FIM) I(θ) is defined as [20] ∂lnf (Y; s, M) ∂lnf (Y; s, M) I(θ) = E . ∂θ ∂θ T Further, we can partition I(θ) into the following form Iθr θr (θ) Iθr θs (θ) , I(θ) = Iθs θr (θ) Iθs θs (θ)
(12)
(13)
(14)
where Iθr θr (θ) is the block matrix corresponding to the parameter θ r in I(θ). Thus, I−1 (θ) θr θr
can be expressed as
I−1 (θ) θr θr =
−1 Iθr θr (θ) − Iθr θs (θ)I−1 . θ s θ s (θ)Iθ s θ r (θ)
(15)
Taking the logarithm of (12) and calculating its derivative with respect to θ r and θ Tr , we get ∂lnf (Y; s, M) = M−1 Y − saT a, ∂θ r 5
(16)
and ∂lnf (Y; s, M) T T T = a Y − sa M−1 , ∂θ Tr
(17)
respectively. After some algebra, one can readily verify that in (14), Iθr θs (θ) is a null matrix, and Iθr θr (θ) = aT aM−1 .
(18)
As a consequence, we have
M I−1 (θ) θr θr = I−1 θ r θ r (θ) = aT a .
(19)
Note that, the MLE of θ under H0 is T 1 T T T ˜ YY . θ 0 = 0M,1 , vec N
(20)
Plugging (16), (17), (19), (20) into (11), with the constant scalar dropped, yields the Rao test as TRao =
(Ya)T (YYT )−1 (Ya) H1 ≷ λRao , aT a H0
(21)
where λRao denotes a detection threshold. Comparing (7) and (21), we can immediately find that the Rao test has the same form as the RXD.
3.2
Wald Test
The real-valued Wald test is given by [15] T
˜ TWald = θ r1
h
˜ 1 )θ θ I−1 (θ r r
i−1
H1
˜ r1 ≷ λW , θ
(22)
H0
˜ 1 = θ T , θ T T is the MLE of θ under hypothesis H1 . where λW is the detection threshold, θ r1 s1 Following the similar derivations of the FIM for the Rao test, one can readily verify that h
˜ 1 )θ θ I−1 (θ r r
i−1
˜ 1 ). = Iθ r θ r (θ
(23)
Moreover, the MLE of θ under H1 hypothesis is given by [10] " T #T T 1 Ya T T T ˜1 = θ , vec Y−sa Y−sa . aT a N Combining (18), (22), (23), (24), we can equivalently write the Wald test as T −1 YaaT YaaT T (Ya) Y− aT a Y− aT a (Ya) TWald = aT a H1
≷ λWald ,
H0
6
(24)
(25)
where λWald is a detection threshold. Observe that T YaaT YaaT Y− T Y− T a a a a T aaT aaT IN − T YT = Y IN − T a a a a
(26)
T = YP⊥ aY ,
where IN denotes the N × N identity matrix, and T −1 T P⊥ a , IN − a(a a) a ,
(27)
is a projection operator. Thus, the Wald test can be written as TWald
T (Ya)T YP⊥ aY = aT a
−1
(Ya) H1 ≷ λWald .
(28)
H0
In the following, we show that the Wald test in (28) is equivalent to the RXD and the Rao test. First, performing a whitening procedure on Y, we define 1
Z = M− 2 Y.
(29)
Plugging (29) into (28), the Wald test takes the form as TWald
T (Za)T ZP⊥ aZ = aT a
−1
(Za)
.
(30)
Define ˜= a
a 1
(aT a) 2
,
(31)
then one can readily verify that
Thus we have
⊥ ˜a ˜T . P⊥ ˜ = IN − a a = Pa
(32)
−1 T TWald = (Z˜ a)T ZP⊥ (Z˜ a). ˜Z a
(33)
Define an N × N orthonormal matrix
T ˜ , UT , G= a
(34)
where U is an (N − 1) × N matrix with orthonormal row vectors such that ˜T UT = 01,N −1 . a 7
(35)
˜, Z and P⊥ Applying the transformation G to a ˜ , yields a ˜T GT = [1, 0, ..., 0], a
(36)
V , ZGT = [v1 , v2 , ..., vN ],
(37)
and
T Q , GP⊥ ˜G = a
0
01,N −1
0N −1,1
IN −1
,
(38)
respectively, where IN −1 is an (N − 1)-dimensional identity matrix. Then the Wald test function in (33) reduces to TWald = v1T VQVT
−1
v1 .
(39)
˜ where V ˜ = [v2 , . . . , vN ], one can readily verify By partitioning matrix V into two parts V = [v1 , V], that the Wald test has the following form −1 ˜V ˜T TWald = v1T V v1 .
It has been shown in [10, eq. (33)] that the RXD can be formulated as −1 ˜V ˜T v1T V v1 TWald . = TRXD = −1 1 + TWald ˜V ˜T v1 1 + v1T V
(40)
(41)
Hence, the Wald test is equivalent to the RXD, because any monotonous transformation on the test statistic does not change the detection performance. In summary, the Rao test, Wald test and RXD coincide for the detection problem in (5).
4
Conclusion
Anomaly detection for hyperspectral images under the Gaussian model has been studied in this letter. The corresponding Rao and Wald tests were derived. We found that the proposed Rao and Wald tests coincide with the RXD. This interesting property is not common for radar target detection in homogeneous environments [21].
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: 8
Acknowledgements This work was partially supported by the National Natural Science Foundation of China under Grants 61871469 and 61771442, the Youth Innovation Promotion Association CAS (CX2100060053), the Fundamental Research Funds for the Central Universities (WK2100000006), the Key Research Program of the Frontier Sciences, CAS, under Grant QYZDY-SSW-JSC035, and the National Key Research and Development Program of China (No. 2018YFB1801105).
References [1] H. Li and J. H. Michels, “Parametric adaptive signal detection for hyperspectral imaging,” IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2704–2715, July 2006. [2] N. M. Nasrabadi, “Hyperspectral target detection : An overview of current and future challenges,” IEEE Signal Processing Magazine, vol. 31, no. 1, pp. 34–44, January 2014. [3] D. G. Manolakis, “Taxonomy of detection algorithms for hyperspectral imaging applications,” Optical Engineering, vol. 44, no. 6, pp. 1 – 11 – 11, 2005. [4] D. Manolakis, D. Marden, D. Shaw, and T. A., “Hyperspectral image processing for automatic target detection applications,” IEEE Signal Processing Magazine, vol. 14, no. 1, pp. 79–116, January 2003. [5] B. Du, Y. Zhang, L. Zhang, and D. Tao, “Beyond the sparsity-based target detector: A hybrid sparsity and statistics-based detector for hyperspectral images,” IEEE Transactions on Image Processing, vol. 25, no. 11, pp. 5345–5357, November 2016. [6] D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, and A. D. Stocker, “Anomaly detection from hyperspectral imagery,” IEEE Signal Processing Magazine, vol. 19, no. 1, pp. 58–69, January 2002. [7] D. Manolakis and G. Shaw, “Detection algorithms for hyperspectral imaging applications,” IEEE Signal Processing Magazine, vol. 19, no. 1, pp. 29–43, January 2002. [8] S. Matteoli, M. Diani, and G. Corsini, “A tutorial overview of anomaly detection in hyperspectral images,” IEEE Aerospace and Electronic Systems Magazine, vol. 25, no. 7, pp. 5–28, July 2010.
9
[9] J. Frontera-Pons, M. A. Veganzones, F. Pascal, and J. Ovarlez, “Hyperspectral anomaly detectors using robust estimators,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 9, no. 2, pp. 720–731, February 2016. [10] I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no. 19, pp. 1760–1770, October 1990. [11] Heesung Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: A nonlinear anomaly detector for hyperspectral imagery,” IEEE Transactions on Geoscience and Remote Sensing, vol. 43, no. 2, pp. 388–397, February 2005. [12] C. I. Chang and S. S. Chiang, “Anomaly detection and classification for hyperspectral imagery,” IEEE Transactions on Geoscience and Remote Sensing, vol. 40, no. 6, pp. 1314–1325, June 2002. [13] S. Chang, B. Du, and L. Zhang, “Baso: A background-anomaly component projection and separation optimized filter for anomaly detection in hyperspectral images,” IEEE Transactions on Geoscience and Remote Sensing, vol. 56, no. 7, pp. 3747–3761, July 2018. [14] A. De Maio and S. Iommelli, “Coincidence of the Rao test, Wald test, and GLRT in partially homogeneous environment,” IEEE Signal Processing Letters, vol. 15, no. 4, pp. 385–388, April 2008. [15] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory.
Upper Saddle
River, NJ: Prentice Hall, 1998. [16] A. De Maio, “Rao test for adaptive detection in Gaussian interference with unknown covariance matrix,” IEEE Transactions on Signal Processing, vol. 55, no. 7, pp. 3577–3584, July 2007. [17] A. De Maio, S. M. Kay, and A. Farina, “On the invariance, coincidence, and statistical equivalence of the GLRT, Rao test, and Wald test,” IEEE Transactions on Signal Processing, vol. 58, no. 4, pp. 1967–1979, April 2010. [18] S. K. Kay and Z. Zhu, “The complex parameter Rao test,” IEEE Transactions on Signal Processing, vol. 64, no. 24, pp. 6580–6588, December 2016. [19] W. Liu, Y. Wang, and W. Xie, “Fisher information matrix, Rao test, and Wald test for complexvalued signals and their applications,” Signal Processing, vol. 94, no. 1, pp. 1–5, January 2014.
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[20] W. Liu, W. Xie, J. Liu, D. Zou, H. Wang, and Y. Wang, “Detection of a distributed target with direction uncertainty,” IET Radar, Sonar Navigation, vol. 8, no. 9, pp. 1177–1183, December 2014. [21] W. Liu, W. Xie, J. Liu, and Y. Wang, “Adaptive double subspace signal detection in Gaussian background–Part I: Homogeneous environments,” IEEE Transactions on Signal Processing, vol. 62, no. 9, pp. 2345–2357, May 2014.
11
Journal Pre-proof
Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery Yutong Feng, Jun Liu, Weijian Liu PII: DOI: Reference:
S0165-1684(19)30468-2 https://doi.org/10.1016/j.sigpro.2019.107416 SIGPRO 107416
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
6 July 2019 28 October 2019 2 December 2019
Please cite this article as: Yutong Feng, Jun Liu, Weijian Liu, Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery, Signal Processing (2019), doi: https://doi.org/10.1016/j.sigpro.2019.107416
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Coincidence of the Rao Test, Wald Test and GLRT for Anomaly Detection in Hyperspectral Imagery Yutong Fenga
Jun Liua,
∗
Weijian Liub
December 5, 2019
Abstract Anomaly detection methods are designed to detect targets (small anomalies) without a priori information on the target spectral signature. In this letter, we deal with the problem of anomaly detection for hyperspectral images based on the Gaussian model assuming that the background obeys a realvalued Gaussian multivariate distribution with unknown covariance matrix. This model is widely used in hyperspectral images. We derive the corresponding Rao and Wald tests, and show that both the two tests are equivalent to the generalized likelihood ratio test. Keywords – Anomaly detection, Gaussian model, hyperspectral images, Rao test, Wald test, generalized likelihood ratio test (GLRT).
∗
Corresponding author (e-mail: [email protected]). Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, China. b Wuhan Electronic Information Institute, Wuhan 430019, China. a
1
1
Introduction
Hyperspectral images are collected by hyperspectral imaging sensors that accumulate spectra information from each point of a scene [1, 2]. Due to the high resolutions of the spectra of hyperspectral images, the spectral bands can be considered contiguous for every pixel. Thus, hyperspectral imaging data exploitation makes it possible to detect desired targets in the scene based on their spectral signatures. In recent years, spectral signature-based target detection has been studied by many researchers [1, 3–5]. In most of such work, it is assumed that the target spectral signatures are known. The goal is to detect a signal with known signature but unknown amplitude from the background clutter. However, in practical applications, the atmospheric compensation and other complicating factors may lead to significant deviations between the original and measured spectral signatures. This may lead to severe mismatch losses in spectral signature-based detection methods with requiring known spectral signatures [6]. In order to avoid this problem, the anomaly detection methods are proposed and applied to hyperspectral images [7–9]. Anomaly detection of multiband signals have been widely studied, which seeks to distinguish unusual targets (small anomalies) from the background without a priori knowledge about spectral signatures. In hyperpectral images, the classical anomaly detection can be considered as a binary hypothesis testing problem, and the typical solution is the Neyman-Pearson (NP) approach. Based on the NP criterion, the well-known Reed-Xiaoli detector (RXD) [10] was derived from the generalized likelihood ratio test (GLRT) principle, by assuming that the disturbance obeys a real-valued Gaussian multivariate distribution with unknown covariance matrix [8]. Because of its mathematical tractability and its constant false alarm ratio (CFAR) property, the RXD is considered as the benchmark anomaly detector for multi/hyperspectral data. Moreover, it is suitable for real-time applications, since the background parameters in the RXD are estimated by local data. It is worth mentioning that, besides the RXD, other methods have been proposed to deal with the anomaly detection problem, such as the so-called kernel RX [11], the RX-uniform target detector [12], and the component projection and separation optimized filter [13]. As alternative strategies with respect to the GLRT, the Rao and Wald tests might require a smaller computational complexity and be more robust in some realistic operating situations [14]. It is noteworthy that the aforementioned three tests are asymptotically equivalent [15]. Due to these interesting properties, the Rao and Wald tests are widely studied by researchers [14, 16–19]. For instance, the authors systematically analyzed the relationship between the aforementioned three tests, and derived the invariance, coincidence, and statistical equivalence properties of the GLRT, the Rao and Wald tests for radar applications [14, 17]. 2
In this letter, we derive the Rao and Wald tests for the anomaly detection in hyperspectral images under the Gaussian model. Interestingly, it is found that the proposed Rao and Wald tests are statistically equivalent to the RXD for anomaly detection problems in hyperspectral images. This letter is organized as follows. Section 2 provides the problem formulation. In Section 3, we design the Rao and Wald tests and prove the two tests coincide with the RXD. Conclusion and final remarks are given in Section 4.
1.1
Notation
In the following, scalars, vectors and matrices are represented by lightfaced lowercase, boldfaced lowercase and boldfaced uppercase letters, respectively. ln(·) denotes the natural logarithm. E[·] is the statistical expectation. The notation , means “defined as”. A−1 and |A| stand for the inverse and determinant of matrix A, respectively. ∂f /∂a denotes the partial derivative of the scalar function f with respect to the vector a. vec(C) denotes the vectorization of the matrix C. The symbol (·)T stands for transpose operation. 0M,N denotes the M × N -dimensional matrices of
zeros. Finally, R denotes the set of real numbers, and RM ×N denotes the set of M × N -dimensional real matrices.
2
Problem Formulation
The problem of anomaly detection in hyperspectral images can be formulated as the following binary hypothesis test [10]:
H0 : y(i) = n(i),
i = 1, . . . , N,
H : y(i) = n(i) + sa(i), 1
(1)
i = 1, . . . , N,
where y(i) ∈ RM ×1 , i = 1, 2, . . . , N denote observed spectra containing the potential anomaly target with known shape (spatial pattern) and unknown position; M is the number of spectral bands, N represents the number of pixels under test with N > M ; s ∈ RM ×1 denotes the unknown spectral signature (can be deemed as a steering vector) of the possible target, and a(i) is the known target amplitude. It should be noted that for radar target detection, the steering vector s is known and the target amplitude a(i) is unknown [14, 17]. As to the background clutter vector n(i) ∈ RM ×1 , we assume that n(i) and n(j), i 6= j, are independent, zero-mean, Gaussian random vectors with the positive-definite covariance matrix M ∈
3
RM ×M , that is
E[n(i)n(j)T ] = M, E[n(i)n(j)T ] = 0,
i=j
(2)
i 6= j.
The covariance matrix M is assumed to be unknown and needs to be estimated. In compact forms, we can combine the vectors into matrices as Y = [y(1), y(2), . . . , y(N )] ∈ RM ×N ,
(3)
N = [n(1), n(2), . . . , n(N )] ∈ RM ×N .
(4)
and
Then, the binary hypotheses test can be recast as H0 : Y = N, where
H : Y = N + saT , 1
a = [a(1), a(2), . . . , a(N )]T ∈ RN ×1
(5)
(6)
is an amplitude vector standing for the spatial pattern of target. Note that the spectral signature s is unknown, whereas the signal pattern a is assumed to be known. Under the above model, the GLRT has been derived in [10], which is the well-known RXD. It takes the form TRXD =
(Ya)T (YYT )−1 (Ya) H1 ≷ λRXD , aT a H0
(7)
where λRXD is the detection threshold. It should be mentioned that if no a priori information about the target spatial pattern is available, the spatial pattern can be neglected here (i.e., just one anomaly pixel is intended to be detected). In this situation, the amplitude vector can be written as a = [0, . . . , 0, 1, 0, . . . , 0]T , where 1 is at n-th position. Then the original RXD reduces to the Mahalanobis distance ˆ −1 y(n), TRXD = y(n)T M
(8)
ˆ is the estimated background clutter covariance matrix. It can be written as where the term M ˆ = 1 YYT . M N
(9)
This pixel-wised form of the RXD is also widely considered in open literatures [6, 8, 9, 12]. In next section, we derive the Rao and Wald tests for the detection problem in (5), and show that the two tests coincide with the RXD. 4
3
Design of Rao and Wald Tests
3.1
Rao Test
In order to derive the Rao test, we first introduce an (M 2 + M )-dimensional parameter vector θ, given by T T T θ = θ Tr , θ Ts = s , vecT (M) ,
(10)
where the spectral signature θ r = s is set to be the relevant parameter, and θ s = vec(M) is set to be the nuisance parameter. The real-valued Rao test with nuisance parameters is given by [15] TRao
h i ∂lnf (Y; s, M) T −1 ˜ I ( θ ) = 0 ˜ ∂θ r θr θr θ=θ 0
H1 ∂lnf (Y; s, M) × ≷ λR , ∂θ r ˜ 0 H0 θ=θ
(11)
˜ 0 = θ T , θ T T is the maximum likelihood estimate (MLE) of where λR is the detection threshold, θ r0 s0
θ under hypothesis H0 , f (Y; s, M) is the probability density function of Y under hypothesis H1 , i.e., MN 2
N
|M|− 2 i 1 h −1 T T T × exp − tr M Y−sa Y−sa . 2
f (Y; s, M) =(2π)−
The Fisher information matrix (FIM) I(θ) is defined as [20] ∂lnf (Y; s, M) ∂lnf (Y; s, M) I(θ) = E . ∂θ ∂θ T Further, we can partition I(θ) into the following form Iθr θr (θ) Iθr θs (θ) , I(θ) = Iθs θr (θ) Iθs θs (θ)
(12)
(13)
(14)
where Iθr θr (θ) is the block matrix corresponding to the parameter θ r in I(θ). Thus, I−1 (θ) θr θr
can be expressed as
I−1 (θ) θr θr =
−1 Iθr θr (θ) − Iθr θs (θ)I−1 . θ s θ s (θ)Iθ s θ r (θ)
(15)
Taking the logarithm of (12) and calculating its derivative with respect to θ r and θ Tr , we get ∂lnf (Y; s, M) = M−1 Y − saT a, ∂θ r 5
(16)
and ∂lnf (Y; s, M) T T T = a Y − sa M−1 , ∂θ Tr
(17)
respectively. After some algebra, one can readily verify that in (14), Iθr θs (θ) is a null matrix, and Iθr θr (θ) = aT aM−1 .
(18)
As a consequence, we have
M I−1 (θ) θr θr = I−1 θ r θ r (θ) = aT a .
(19)
Note that, the MLE of θ under H0 is T 1 T T T ˜ YY . θ 0 = 0M,1 , vec N
(20)
Plugging (16), (17), (19), (20) into (11), with the constant scalar dropped, yields the Rao test as TRao =
(Ya)T (YYT )−1 (Ya) H1 ≷ λRao , aT a H0
(21)
where λRao denotes a detection threshold. Comparing (7) and (21), we can immediately find that the Rao test has the same form as the RXD.
3.2
Wald Test
The real-valued Wald test is given by [15] T
˜ TWald = θ r1
h
˜ 1 )θ θ I−1 (θ r r
i−1
H1
˜ r1 ≷ λW , θ
(22)
H0
˜ 1 = θ T , θ T T is the MLE of θ under hypothesis H1 . where λW is the detection threshold, θ r1 s1 Following the similar derivations of the FIM for the Rao test, one can readily verify that h
˜ 1 )θ θ I−1 (θ r r
i−1
˜ 1 ). = Iθ r θ r (θ
(23)
Moreover, the MLE of θ under H1 hypothesis is given by [10] " T #T T 1 Ya T T T ˜1 = θ , vec Y−sa Y−sa . aT a N Combining (18), (22), (23), (24), we can equivalently write the Wald test as T −1 YaaT YaaT T (Ya) Y− aT a Y− aT a (Ya) TWald = aT a H1
≷ λWald ,
H0
6
(24)
(25)
where λWald is a detection threshold. Observe that T YaaT YaaT Y− T Y− T a a a a T aaT aaT IN − T YT = Y IN − T a a a a
(26)
T = YP⊥ aY ,
where IN denotes the N × N identity matrix, and T −1 T P⊥ a , IN − a(a a) a ,
(27)
is a projection operator. Thus, the Wald test can be written as TWald
T (Ya)T YP⊥ aY = aT a
−1
(Ya) H1 ≷ λWald .
(28)
H0
In the following, we show that the Wald test in (28) is equivalent to the RXD and the Rao test. First, performing a whitening procedure on Y, we define 1
Z = M− 2 Y.
(29)
Plugging (29) into (28), the Wald test takes the form as TWald
T (Za)T ZP⊥ aZ = aT a
−1
(Za)
.
(30)
Define ˜= a
a 1
(aT a) 2
,
(31)
then one can readily verify that
Thus we have
⊥ ˜a ˜T . P⊥ ˜ = IN − a a = Pa
(32)
−1 T TWald = (Z˜ a)T ZP⊥ (Z˜ a). ˜Z a
(33)
Define an N × N orthonormal matrix
T ˜ , UT , G= a
(34)
where U is an (N − 1) × N matrix with orthonormal row vectors such that ˜T UT = 01,N −1 . a 7
(35)
˜, Z and P⊥ Applying the transformation G to a ˜ , yields a ˜T GT = [1, 0, ..., 0], a
(36)
V , ZGT = [v1 , v2 , ..., vN ],
(37)
and
T Q , GP⊥ ˜G = a
0
01,N −1
0N −1,1
IN −1
,
(38)
respectively, where IN −1 is an (N − 1)-dimensional identity matrix. Then the Wald test function in (33) reduces to TWald = v1T VQVT
−1
v1 .
(39)
˜ where V ˜ = [v2 , . . . , vN ], one can readily verify By partitioning matrix V into two parts V = [v1 , V], that the Wald test has the following form −1 ˜V ˜T TWald = v1T V v1 .
It has been shown in [10, eq. (33)] that the RXD can be formulated as −1 ˜V ˜T v1T V v1 TWald . = TRXD = −1 1 + TWald ˜V ˜T v1 1 + v1T V
(40)
(41)
Hence, the Wald test is equivalent to the RXD, because any monotonous transformation on the test statistic does not change the detection performance. In summary, the Rao test, Wald test and RXD coincide for the detection problem in (5).
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Conclusion
Anomaly detection for hyperspectral images under the Gaussian model has been studied in this letter. The corresponding Rao and Wald tests were derived. We found that the proposed Rao and Wald tests coincide with the RXD. This interesting property is not common for radar target detection in homogeneous environments [21].
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: 8
Acknowledgements This work was partially supported by the National Natural Science Foundation of China under Grants 61871469 and 61771442, the Youth Innovation Promotion Association CAS (CX2100060053), the Fundamental Research Funds for the Central Universities (WK2100000006), the Key Research Program of the Frontier Sciences, CAS, under Grant QYZDY-SSW-JSC035, and the National Key Research and Development Program of China (No. 2018YFB1801105).
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