검색
검색 팝업 닫기

Ex) Article Title, Author, Keywords

Article

Split Viewer

Article

Current Optics and Photonics 2020; 4(3): 210-220

Published online June 25, 2020 https://doi.org/10.3807/COPP.2020.4.3.210

Copyright © Optical Society of Korea.

Weighted Collaborative Representation and Sparse Difference-Based Hyperspectral Anomaly Detection

Qianghui Wang1, Wenshen Hua1,*, Fuyu Huang1, Yan Zhang1, and Yang Yan2

1Electronic and Optical Engineering Department, Army Engineering University, Shijiazhuang 050000, China, 2Unit 31681 of the People’s Liberation Army, Tianshui 741000, China

Corresponding author: huawensh@126.com

Received: December 2, 2019; Revised: January 20, 2020; Accepted: February 18, 2020

Aiming at the problem that the Local Sparse Difference Index algorithm has low accuracy and low efficiency when detecting target anomalies in a hyperspectral image, this paper proposes a Weighted Collaborative Representation and Sparse Difference-Based Hyperspectral Anomaly Detection algorithm, to improve detection accuracy for a hyperspectral image. First, the band subspace is divided according to the band correlation coefficient, which avoids the situation in which there are multiple solutions of the sparse coefficient vector caused by too many bands. Then, the appropriate double-window model is selected, and the background dictionary constructed and weighted according to Euclidean distance, which reduces the influence of mixing anomalous components of the background on the solution of the sparse coefficient vector. Finally, the sparse coefficient vector is solved by the collaborative representation method, and the sparse difference index is calculated to complete the anomaly detection. To prove the effectiveness, the proposed algorithm is compared with the RX, LRX, and LSD algorithms in simulating and analyzing two AVIRIS hyperspectral images. The results show that the proposed algorithm has higher accuracy and a lower false-alarm rate, and yields better results.

Keywords: Spectroscopy, Hyperspectral images, Anomaly detection, Spectral information, Collaborative representation method

A hyperspectral image (HSI) is a three-dimension image containing spectral and spatial information. Its spectral resolution is extremely high. It includes spectral information from hundreds of continuous bands, from visible light to the midinfrared and even long-wave infrared. The spectral curve is approximately continuous and has been widely used in spectral demixing [1], image classification [2], and target detection [3]. In recent years, with the development of remote-sensing technology and the increasing demand for accurate location of anomalous targets in an image, target detection has achieved rapid development. According to whether the target spectrum is grasped in advance as a priori information, target detection is divided into spectral-matching detection, which requires a priori information, and anomaly detection, which does not [4]. Due to the difficulty of obtaining prior information in practical applications, anomaly detection that does not require such has become a research hotspot [5].

Anomaly detection refers to the detection of anomalous pixels using the spectral information in the image, when prior information about the target is unknown. The essence is a classification problem, that is, dividing the pixels in the image into background pixels and anomalous pixels. Background pixels are one or more types of pixels that occupy most of the image. The concept of “anomalous pixels” has not yet been clearly defined, and generally refers to pixels that deviate from the background distribution to some extent. The purpose of anomaly detection is to “dig” such pixels from the image. Anomaly-detection algorithms have seen great progress. The Reed-Xiaoli Yu (RX) algorithm [6, 7] is a classic anomaly-detection algorithm, proposed by Reed and Yu. It assumes that the background obeys a multivariate Gaussian distribution, and that the anomalous pixels deviate from this distribution. Anomaly detection is performed by calculating the Mahalanobis distance between the pixel spectrum to be tested and the mean background-pixel spectrum. Because the background for the algorithm is obtained from the entire image, it is inevitable that anomalous pixels are mixed into that background, resulting in low detection accuracy and high false-alarm rate. Reference [8] analyzed the influence of background uncertainty on anomaly detection, while reference [9] pointed out that only 0.5% of data pollution will cause background covariance distortion and seriously affect the detection accuracy of the RX algorithm. To obtain the background data more accurately and improve detection, reference [10] proposed the Local RX (LRX) algorithm, which makes the background extraction more accurate and improves detection accuracy by establishing a sliding double-window model [11].

In recent years, the sparse representation method has achieved good results in the field of target detection. In 2011, reference [12] first applied sparse representation to the field of target detection. This method considers that the spectral vector of each pixel in the hyperspectral image is represented by the complete-dictionary linearly. The complete dictionary is made up of atom vectors and atom vectors represent the spectrum of pure matter contained in the image. The complete dictionary contains the target dictionary and the background dictionary, and the linear combination can be represented by the sparse coefficient vector. The entire hyperspectral image can be represented by constructing a complete dictionary and sparse coefficient vectors. In 2014, reference [13] applied sparse representation to anomaly detection, and proposed the Local Sparsity Divergence Index (LSD) algorithm. By constructing a sliding double-window model, background pixels were selected between the inner and outer windows to construct the background dictionary. The pixel to be tested is linearly represented by the background dictionary, and the classification of the pixel to be tested is determined according to the characteristics of the sparse coefficient vector. The algorithm constructs the background dictionary by randomly selecting some pixels, and inevitably mixes anomalous pixels into it, which affects the detection accuracy. In addition, the selection of the size of the double window greatly affects the detection results; inappropriate size affects the solution of the sparse coefficient vector. Aiming to solve that problem, this paper proposes a Weighted Collaborative Representation and Sparsity Divergence-Based (WCRSD) Anomaly Detection algorithm. First, the band subspace is divided to ensure that the sparse coefficient vector has a solution, and the appropriate double-window size is selected. Then, the background dictionary is constructed and weighted according to Euclidean distance, which reduces the influence of anomalous components in the background dictionary on the solution of the sparse coefficient vector. Finally, the method of collaborative representation is used to solve the sparse coefficient vector, and the sparse difference index is calculated to complete the anomaly detection.

2.1. Sparse Representation Model

Suppose x is the spectral vector of the pixel to be tested in the image, and D represents an M-dimensional complete dictionary composed of N vectors. The complete dictionary D can be divided into the target dictionary T and background dictionary B. The target dictionary contains all anomalous target atom vectors, and the background dictionary contains all background atom vectors. x can be approximated by the atom vector in the complete dictionary D

where D is a matrix of dimension M × N , composed of NM-dimensional atomic spectra and N = Nt +Nb; T =[t1, t2,···, tNt] ; B = [b1, b2,···, bNb]; αt represents an Nt-dimensional sparse coefficient vector of the target dictionary; and αb represents an Nb-dimensional sparse coefficient vector of the background dictionary. α represents an N-dimensional sparse coefficient vector and reflects the proportion of each atom in the dictionary corresponding to x. The complete dictionary can be obtained from the partial pixels of the hyperspectral image.

2.2. Local Sparse Difference Index Algorithm

The local sparse difference index includes the local spectral sparse difference index and the local spatial sparse difference index. The construction of the background dictionary is completed using the sliding double-window model, as shown in Fig. 1.

As can be seen from Fig. 1, the sliding double-window model includes two windows, inner and outer. The pixels between them are assumed to be background pixels, while the central pixel is the one to be tested. Both the inner and outer windows are represented by squares with odd edge length and the edge length made up of number of pixels. The edge length of the outer window is usually the distance between the two anomalous targets. If the size is too large, a large number of anomalous pixels may be mixed into the background pixels, which will affect the detection accuracy and cause a serious false alarm. Too small a size causes the number of background pixels to be smaller than the image spectral dimension, which will affect the solution of the sparse coefficient. The edge length of the inner window is usually the size of the anomalous target. Too large or too small a window edge length will reduce detection accuracy or detection efficiency.

Figure 1.The sliding double-window model.

The sliding double-window model is used to randomly select some pixels between the inner and outer windows to construct the background dictionary. If the pixel to be tested is an anomalous target pixel, the distribution of the sparsity coefficient is scattered, it is difficult for the central pixel to be sparsely represented by the background dictionary, and the reconstruction error and the sparse difference index are large. If the pixel to be tested is a background pixel, it is accurately represented by a sparse background dictionary, the distribution of the sparse coefficient is concentrated, and the reconstruction error and sparse difference index are small. Suppose N pixels are selected between the inner and outer windows as the background dictionary B, and x is represented by B. We judge the attributes of the pixel to be tested according to the characteristics of the sparse coefficient vector α

The local spectral sparse difference index is defined as

From the above formula, we find that when the pixel to be tested is a background pixel, only a few coefficients in the sparse coefficient vector α have nonzero values, so the local spectral sparse difference index is close to zero. When the pixel to be tested is an anomalous pixel, it cannot be accurately represented by the background dictionary, and the obtained sparse coefficient distribution is scattered, resulting in a large local sparse difference index.

In addition to spectral correlation, hyperspectral images also have strong spatial correlation, which can be reflected by adjacent pixels. Figure 2 depicts the four-neighborhood pixel of the hyperspectral image, and its spectral curves.

Figure 2.The four-neighborhood pixel of the hyperspectral image, and its spectral curves.

From the figure above, we find that the spectral curves of the four-neighborhood pixel are very similar, so there is a great possibility that they all belong to background or anomaly. Applying image spatial correlation to anomaly detection can greatly improve detection efficiency and accuracy.

Similar to the local spectral sparse difference index, the local spatial sparse difference index also uses a doublewindow model to construct the background dictionary, which acts on a single band. For the j-band hyperspectral image, we have

where represents the j-band background dictionary, and represents the sparse coefficient vector corresponding to the j-band background dictionary. N represents the number of atoms in the background dictionary, represents the ith dictionary atom in the jth band, and represents the sparse coefficient corresponding to the ith dictionary atom in the jth band.

The j-band local spatial sparse difference index is defined as

The local sparse difference index can be expressed as

where M represents the total number of bands. I contains both spectral information and spatial information. By setting a threshold, the anomalous pixel is extracted to achieve anomaly detection.

The collaborative representation method also uses a sliding double-window model to obtain the background dictionary, so that the pixel to be tested can be linearly represented by the background dictionary to obtain an approximate value. The objective function of the collaborative representation method is

where y represents the pixel to be tested and λ is the Lagrange multiplier. Xs represents the background dictionary, consisting of the background pixel between the inner and outer windows: Xs = [x1, x2 ···, xs]M×S. M represents the spectral dimension, S the number of pixels in the background dictionary, and α the sparse coefficient vector. The purpose of this method is to find the sparse coefficient vector α and minimize the objective function. Taking the derivate of α and setting it equal to zero, we obtain

To ensure that the pixels in the background dictionary similar to the pixel to be tested have larger weights, and that those not similar have smaller weights, a diagonal matrix Γy is used to adjust the weights of the pixels in the background dictionary. The expression is

Here x1, x2 ···, xs represents the background pixel’s spectral vector in the background dictionary Xs, and the diagonal represents the Euclidean distance between the pixel to be tested and the background pixel. The improved objective function can be expressed as

To improve the stability of the solution, the constraint that the sum is 1 is applied to the sparse coefficient vector α. y = [y;1], Xs = [Xs ;E], where E is the row vector of 1×s of 1. Thus we can obtain an expression for α in the new objective function

The collaborative representation method exhibits higher accuracy in solving the sparse coefficient vector α, but there are some problems. When the pixel to be tested is anomalous and there are also anomalous pixels in the background dictionary, the sparse coefficient is relatively concentrated, which reduces the accuracy of the solution. To avoid this problem and reduce the influence of anomalous pixels in the background dictionary on the detection results, the weighted collaborative representation method is proposed. The pixels in the background dictionary are weighted according to the Euclidean distance, so the weight of anomalous pixels in the background dictionary is reduced. The weighted objective function is expressed as

where

The expression for α' after weighting can be obtained

Applying the weighted collaborative representation method to the local sparse difference index to solve the sparse coefficient vector, the Weighted Collaborative Representation and Sparse Difference-based Hyperspectral Anomaly Detection algorithm is obtained. The specific steps and processes are as follows:

  • (1) Input the hyperspectral image, divide the band subspace according to the correlation coefficient between the bands, and detect each subspace separately.

  • (2) Select the appropriate double-window size and Lagrange multiplier, and use the weighted collaborative representation method to calculate the sparse coefficient vector.

  • (3) Calculate the local sparse difference index according to the sparse coefficient vector, and obtain the detection results for each subspace.

  • (4) Superimpose the detection results of each subspace to obtain the final detection result.

To verify the effectiveness and reliability of the algorithm, two groups of AVIRIS hyperspectral data are adopted for the simulation experiment in this paper. The simulation environment consists of: CPU Intel Core i7-7700HQ, dominant frequency 2.80 GHz, 8 GB RAM, software Matlab R2017a.

4.1. Experimental Data

Data-1 selected part of the data from the US San Diego Naval Airport, taken by the Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) sensor. The wavelength range is 0.4-1.8 μm. The bands with low signal-to-noise ratio and severe water-vapor absorption are removed; 126 bands are reserved. The spatial resolution is 3.5 m. The size of the intercepted area is 100 × 100 pixels, and there are 38 anomalous targets. Data-2 selected part of the data from the US Los Angeles Airport taken by the AVIRIS sensor. The wavelength range is 0.37-2.51 μm. The bands with low signal-to-noise ratio and severe water-vapor absorption are removed; 205 bands are reserved. The spatial resolution is 7.1 m. The size of the intercepted area is 100 × 100 pixels, and there are 17 anomalous targets. Figures 3 and 4 show the 50th band image and target distribution of the two groups of data:

Figure 3.Data-1 50th band image and target distribution.
Figure 4.Data-2 50th band image and target distribution.

4.2. Test Results and Analysis

First, we calculate the correlation coefficient between the bands according to Eq. (16), where xi and xj represent the spectral values of the ith and jth band images. mi and mj are the mean values of xi and xj respectively. Rij is the correlation coefficient of the ith and jth bands, with a value between 0 and 1; the closer to 1, the stronger the correlation between the two bands is. Correlation coefficients of adjacent bands for two groups of data are shown in Fig. 5. The two groups of data are divided into different band subspaces based on the correlation coefficients of adjacent bands

Figure 5.Correlation coefficients of adjacent bands for two groups of data.

  • Data-1:1-9, 10-71, 72-93, 94-114, 115-126

  • Data-2:1-104, 105-146, 147-154, 155-205

Second, the appropriate double-window size and Lagrange multiplier are selected for each band subspace of the two groups of data, and the algorithm proposed in this paper is used for detection. The detected pixel number of Data-1 is set to 400, and that of Data-2 is set to 150. The detection results are shown in Figs. 6 and 7.

Figure 6.Data-1 subspace detection results.
Figure 7.Data-2 subspace detection results.

The detection results for the subspaces are superimposed to obtain the detection results for the algorithm proposed in this paper. We also run the RX algorithm, LRX algorithm, and LSD algorithm to detect the image. The detection results for the four algorithms are shown in Figs. 8 and 9:

Figure 8.Data-1 detection results for four algorithms.
Figure 9.Data-2 detection results for four algorithms.

From the results for the two groups of data above, it can be seen that for Data-1 the RX algorithm shows the worst detection. Most of the detected anomalous targets are background, and the false-alarm rate is the highest. The LRX algorithm greatly reduces the false-alarm rate and its detection results are greatly improved, compared to the RX algorithm, because the LRX algorithm uses a double-window model in the detection process to select background pixels more accurately. The detection accuracy of both the LSD algorithm and the algorithm proposed in this paper is better still. The recognition of the target contour is more accurate, and the detection effect is the best. For Data-2, the RX and LRX algorithms yield poor detection results, and it is almost difficult to identify anomalous targets. This is because these two anomalydetection algorithms are based on Gaussian models, and it is difficult to detect images with poor Gaussian character and uneven distribution of anomalous targets. Although the LSD algorithm has detected most anomalous targets, it also has a large area of false alarms, and the false-alarm rate is high. Compared to the three prior algorithms, the proposed algorithm not only detects almost all anomalous targets, but also has the lowest false-alarm rate, because the proposed algorithm increases the screening process of background pixels, compared to the LSD algorithm. For images with uneven distributions of anomalous targets, it can greatly reduce the interference of anomalous components in the background pixels on the detection results, and improve detection accuracy.

The detection results are analyzed qualitatively above, and quantitatively below. The number of anomalous targets, the number of target pixels, and the number of false alarms detected by the algorithm are three important indicators to measure the performance of the algorithm. The test results are shown in Table 1

TABLE 1. The test results for two groups of data


From the table above, it is not difficult to find that, compared to the other three algorithms, the proposed algorithm has the characteristics of high detection rate, low false-alarm rate, and better detection. Receiver Operating Characteristic (ROC) curves and Area Under the Curve (AUC) values are used to measure the detection results. The ROC curve reflects the relationship between the detection rate and the false-alarm rate; the abscissa indicates the false-alarm rate Pf, and the ordinate indicates the detection rate Pd.

where Nd represents the number of detected true anomalous target pixels, Nt the total number of anomalous target pixels in the image, Nmiss the number of false-alarm pixels detected, and Ntot the total number of pixels in the image. The more the ROC curve is bent to the upper left, the larger the AUC value, the better the detection result of the algorithm, and the higher the accuracy; the straighter the ROC curve, the smaller the AUC value, the worse the detection results, and the lower the accuracy. Figure 10 shows the ROC curves for the four algorithms, for the two groups of data. The AUC values and running times of the algorithms are shown in Table 2.

Figure 10.The ROC curves for the four algorithms, for the two groups of data.

TABLE 2. AUC values and running times for the four algorithms


According to the ROC curve and AUC value, we can find that the detection performance of the proposed algorithm is significantly better than that of the other three algorithms, and has better applicability. It can be seen from the ROC curve that the proposed algorithm still has good detection performance and strong background-suppression ability even when the false-alarm rate is low; furthermore, as the running time is not too long, it is completely acceptable.

Aiming at the problem that the detection accuracy of the Local Sparse Difference Index algorithm is not high, this paper proposes a Weighted Collaborative Representation and Sparse Difference-based Hyperspectral Anomaly Detection algorithm. The weighted collaborative representation method is used to solve the sparse coefficient vector, which reduces the influence of the anomalous components in the background dictionary on detection. Analysis of experimental data shows that the algorithm can effectively improve the detection accuracy and reduce the false-alarm rate. Compared to three other algorithms, it exhibits better detection performance.

  1. J. Yuan, Y. J. Zhang, and F. P. Gao, "A overview on linear hyperspectral unmixing," J. Infrared. Millimeter. Waves 37, 553-571 (2018).
  2. Y. F. Qi, and Z. Y. Ma, "Hyperspectral image classification method based on cooperative representation of neighborhood spectral probabilities," Laser. Technol. 43, 12-16 (2019).
  3. Shan L. X.. Hyperspectral image sub-pixel small target detection, M. S. Thesis (Xidian University, Shaanxi 2017).
  4. P. Bajorski, "Target detection under misspecified models in hyperspectral images," IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 5, 470-477 (2012).
    CrossRef
  5. L. Sun, J. H. Bao, and Y. C. Liu, "Analysis of target detection algorithm for hyperspectral images," Sci. Surv. Mapp. 37, 131-132+108 (2012).
  6. 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 38, 1760-1770 (1990).
    CrossRef
  7. X. L. Yu, L. E. Hoff, I. S. Reed, A. M. Chen, and L. B. Stotts, "Automatic target detection and recognition in multiband imagery: A unified ML detection and estimation approach," IEEE Trans. Image Process 6, 143-156 (1997).
    CrossRef
  8. S. S. Du, S. Y. Li, and Z. Y. Zeng, "Influence of background uncertainty on hyperspectral anomaly target detection," J. PLA Univ. Sci. Technol. (Natural Science Edition) 17, 598-604 (2016).
  9. T. E. Smetek, and K. W. Bauer, "Finding hyperspectral anomalies using multivariate outlier detection," in Proc. IEEE Aerospace Conference (Big Sky, MT, USA, March 2007).
  10. Y. P. Taitano, B. A. Geier, and K. W. Bauer, "A locally adaptable iterative RX detector," EURASIP. J. Adv. Signal. Process. 2010, 341908 (2010).
    CrossRef
  11. W. M. Liu, and C. I. Chang, "Multiple-window anomaly detection for hyperspectral imagery," IEEE. J. Sel. Top. Appl. Earth. Obs. Remote. Sens. 6, 644-658 (2013).
    CrossRef
  12. Y. Chen, N. M. Naarabadi, and T. D. Tran, "Hyperspectral image classification using dictionary-based sparse representation," IEEE Trans. Geosci. Remote Sens. 49, 3973-3985 (2011).
    CrossRef
  13. Z. Yuan, H. Sun, K. Ji, Z. Li, and H. Zou, "Local sparsity divergence for hyperspectral anomaly detection," IEEE Geosci. Remote Sens. Lett. 11, 1697-1701 (2014).
    CrossRef

Article

Article

Current Optics and Photonics 2020; 4(3): 210-220

Published online June 25, 2020 https://doi.org/10.3807/COPP.2020.4.3.210

Copyright © Optical Society of Korea.

Weighted Collaborative Representation and Sparse Difference-Based Hyperspectral Anomaly Detection

Qianghui Wang1, Wenshen Hua1,*, Fuyu Huang1, Yan Zhang1, and Yang Yan2

1Electronic and Optical Engineering Department, Army Engineering University, Shijiazhuang 050000, China, 2Unit 31681 of the People’s Liberation Army, Tianshui 741000, China

Correspondence to:huawensh@126.com

Received: December 2, 2019; Revised: January 20, 2020; Accepted: February 18, 2020

Abstract

Aiming at the problem that the Local Sparse Difference Index algorithm has low accuracy and low efficiency when detecting target anomalies in a hyperspectral image, this paper proposes a Weighted Collaborative Representation and Sparse Difference-Based Hyperspectral Anomaly Detection algorithm, to improve detection accuracy for a hyperspectral image. First, the band subspace is divided according to the band correlation coefficient, which avoids the situation in which there are multiple solutions of the sparse coefficient vector caused by too many bands. Then, the appropriate double-window model is selected, and the background dictionary constructed and weighted according to Euclidean distance, which reduces the influence of mixing anomalous components of the background on the solution of the sparse coefficient vector. Finally, the sparse coefficient vector is solved by the collaborative representation method, and the sparse difference index is calculated to complete the anomaly detection. To prove the effectiveness, the proposed algorithm is compared with the RX, LRX, and LSD algorithms in simulating and analyzing two AVIRIS hyperspectral images. The results show that the proposed algorithm has higher accuracy and a lower false-alarm rate, and yields better results.

Keywords: Spectroscopy, Hyperspectral images, Anomaly detection, Spectral information, Collaborative representation method

I. INTRODUCTION

A hyperspectral image (HSI) is a three-dimension image containing spectral and spatial information. Its spectral resolution is extremely high. It includes spectral information from hundreds of continuous bands, from visible light to the midinfrared and even long-wave infrared. The spectral curve is approximately continuous and has been widely used in spectral demixing [1], image classification [2], and target detection [3]. In recent years, with the development of remote-sensing technology and the increasing demand for accurate location of anomalous targets in an image, target detection has achieved rapid development. According to whether the target spectrum is grasped in advance as a priori information, target detection is divided into spectral-matching detection, which requires a priori information, and anomaly detection, which does not [4]. Due to the difficulty of obtaining prior information in practical applications, anomaly detection that does not require such has become a research hotspot [5].

Anomaly detection refers to the detection of anomalous pixels using the spectral information in the image, when prior information about the target is unknown. The essence is a classification problem, that is, dividing the pixels in the image into background pixels and anomalous pixels. Background pixels are one or more types of pixels that occupy most of the image. The concept of “anomalous pixels” has not yet been clearly defined, and generally refers to pixels that deviate from the background distribution to some extent. The purpose of anomaly detection is to “dig” such pixels from the image. Anomaly-detection algorithms have seen great progress. The Reed-Xiaoli Yu (RX) algorithm [6, 7] is a classic anomaly-detection algorithm, proposed by Reed and Yu. It assumes that the background obeys a multivariate Gaussian distribution, and that the anomalous pixels deviate from this distribution. Anomaly detection is performed by calculating the Mahalanobis distance between the pixel spectrum to be tested and the mean background-pixel spectrum. Because the background for the algorithm is obtained from the entire image, it is inevitable that anomalous pixels are mixed into that background, resulting in low detection accuracy and high false-alarm rate. Reference [8] analyzed the influence of background uncertainty on anomaly detection, while reference [9] pointed out that only 0.5% of data pollution will cause background covariance distortion and seriously affect the detection accuracy of the RX algorithm. To obtain the background data more accurately and improve detection, reference [10] proposed the Local RX (LRX) algorithm, which makes the background extraction more accurate and improves detection accuracy by establishing a sliding double-window model [11].

In recent years, the sparse representation method has achieved good results in the field of target detection. In 2011, reference [12] first applied sparse representation to the field of target detection. This method considers that the spectral vector of each pixel in the hyperspectral image is represented by the complete-dictionary linearly. The complete dictionary is made up of atom vectors and atom vectors represent the spectrum of pure matter contained in the image. The complete dictionary contains the target dictionary and the background dictionary, and the linear combination can be represented by the sparse coefficient vector. The entire hyperspectral image can be represented by constructing a complete dictionary and sparse coefficient vectors. In 2014, reference [13] applied sparse representation to anomaly detection, and proposed the Local Sparsity Divergence Index (LSD) algorithm. By constructing a sliding double-window model, background pixels were selected between the inner and outer windows to construct the background dictionary. The pixel to be tested is linearly represented by the background dictionary, and the classification of the pixel to be tested is determined according to the characteristics of the sparse coefficient vector. The algorithm constructs the background dictionary by randomly selecting some pixels, and inevitably mixes anomalous pixels into it, which affects the detection accuracy. In addition, the selection of the size of the double window greatly affects the detection results; inappropriate size affects the solution of the sparse coefficient vector. Aiming to solve that problem, this paper proposes a Weighted Collaborative Representation and Sparsity Divergence-Based (WCRSD) Anomaly Detection algorithm. First, the band subspace is divided to ensure that the sparse coefficient vector has a solution, and the appropriate double-window size is selected. Then, the background dictionary is constructed and weighted according to Euclidean distance, which reduces the influence of anomalous components in the background dictionary on the solution of the sparse coefficient vector. Finally, the method of collaborative representation is used to solve the sparse coefficient vector, and the sparse difference index is calculated to complete the anomaly detection.

II. SPARSE REPRESENTATION MODEL AND LOCAL SPARSE DIFFERENCE INDEX ALGORITHM

2.1. Sparse Representation Model

Suppose x is the spectral vector of the pixel to be tested in the image, and D represents an M-dimensional complete dictionary composed of N vectors. The complete dictionary D can be divided into the target dictionary T and background dictionary B. The target dictionary contains all anomalous target atom vectors, and the background dictionary contains all background atom vectors. x can be approximated by the atom vector in the complete dictionary D

where D is a matrix of dimension M × N , composed of NM-dimensional atomic spectra and N = Nt +Nb; T =[t1, t2,···, tNt] ; B = [b1, b2,···, bNb]; αt represents an Nt-dimensional sparse coefficient vector of the target dictionary; and αb represents an Nb-dimensional sparse coefficient vector of the background dictionary. α represents an N-dimensional sparse coefficient vector and reflects the proportion of each atom in the dictionary corresponding to x. The complete dictionary can be obtained from the partial pixels of the hyperspectral image.

2.2. Local Sparse Difference Index Algorithm

The local sparse difference index includes the local spectral sparse difference index and the local spatial sparse difference index. The construction of the background dictionary is completed using the sliding double-window model, as shown in Fig. 1.

As can be seen from Fig. 1, the sliding double-window model includes two windows, inner and outer. The pixels between them are assumed to be background pixels, while the central pixel is the one to be tested. Both the inner and outer windows are represented by squares with odd edge length and the edge length made up of number of pixels. The edge length of the outer window is usually the distance between the two anomalous targets. If the size is too large, a large number of anomalous pixels may be mixed into the background pixels, which will affect the detection accuracy and cause a serious false alarm. Too small a size causes the number of background pixels to be smaller than the image spectral dimension, which will affect the solution of the sparse coefficient. The edge length of the inner window is usually the size of the anomalous target. Too large or too small a window edge length will reduce detection accuracy or detection efficiency.

Figure 1. The sliding double-window model.

The sliding double-window model is used to randomly select some pixels between the inner and outer windows to construct the background dictionary. If the pixel to be tested is an anomalous target pixel, the distribution of the sparsity coefficient is scattered, it is difficult for the central pixel to be sparsely represented by the background dictionary, and the reconstruction error and the sparse difference index are large. If the pixel to be tested is a background pixel, it is accurately represented by a sparse background dictionary, the distribution of the sparse coefficient is concentrated, and the reconstruction error and sparse difference index are small. Suppose N pixels are selected between the inner and outer windows as the background dictionary B, and x is represented by B. We judge the attributes of the pixel to be tested according to the characteristics of the sparse coefficient vector α

The local spectral sparse difference index is defined as

From the above formula, we find that when the pixel to be tested is a background pixel, only a few coefficients in the sparse coefficient vector α have nonzero values, so the local spectral sparse difference index is close to zero. When the pixel to be tested is an anomalous pixel, it cannot be accurately represented by the background dictionary, and the obtained sparse coefficient distribution is scattered, resulting in a large local sparse difference index.

In addition to spectral correlation, hyperspectral images also have strong spatial correlation, which can be reflected by adjacent pixels. Figure 2 depicts the four-neighborhood pixel of the hyperspectral image, and its spectral curves.

Figure 2. The four-neighborhood pixel of the hyperspectral image, and its spectral curves.

From the figure above, we find that the spectral curves of the four-neighborhood pixel are very similar, so there is a great possibility that they all belong to background or anomaly. Applying image spatial correlation to anomaly detection can greatly improve detection efficiency and accuracy.

Similar to the local spectral sparse difference index, the local spatial sparse difference index also uses a doublewindow model to construct the background dictionary, which acts on a single band. For the j-band hyperspectral image, we have

where represents the j-band background dictionary, and represents the sparse coefficient vector corresponding to the j-band background dictionary. N represents the number of atoms in the background dictionary, represents the ith dictionary atom in the jth band, and represents the sparse coefficient corresponding to the ith dictionary atom in the jth band.

The j-band local spatial sparse difference index is defined as

The local sparse difference index can be expressed as

where M represents the total number of bands. I contains both spectral information and spatial information. By setting a threshold, the anomalous pixel is extracted to achieve anomaly detection.

III. WEIGHTED COLLABORATIVE REPRESENTATION AND SPARSITY DIVERGENCE-BASED ANOMALY DETECTION ALGORITHM

The collaborative representation method also uses a sliding double-window model to obtain the background dictionary, so that the pixel to be tested can be linearly represented by the background dictionary to obtain an approximate value. The objective function of the collaborative representation method is

where y represents the pixel to be tested and λ is the Lagrange multiplier. Xs represents the background dictionary, consisting of the background pixel between the inner and outer windows: Xs = [x1, x2 ···, xs]M×S. M represents the spectral dimension, S the number of pixels in the background dictionary, and α the sparse coefficient vector. The purpose of this method is to find the sparse coefficient vector α and minimize the objective function. Taking the derivate of α and setting it equal to zero, we obtain

To ensure that the pixels in the background dictionary similar to the pixel to be tested have larger weights, and that those not similar have smaller weights, a diagonal matrix Γy is used to adjust the weights of the pixels in the background dictionary. The expression is

Here x1, x2 ···, xs represents the background pixel’s spectral vector in the background dictionary Xs, and the diagonal represents the Euclidean distance between the pixel to be tested and the background pixel. The improved objective function can be expressed as

To improve the stability of the solution, the constraint that the sum is 1 is applied to the sparse coefficient vector α. y = [y;1], Xs = [Xs ;E], where E is the row vector of 1×s of 1. Thus we can obtain an expression for α in the new objective function

The collaborative representation method exhibits higher accuracy in solving the sparse coefficient vector α, but there are some problems. When the pixel to be tested is anomalous and there are also anomalous pixels in the background dictionary, the sparse coefficient is relatively concentrated, which reduces the accuracy of the solution. To avoid this problem and reduce the influence of anomalous pixels in the background dictionary on the detection results, the weighted collaborative representation method is proposed. The pixels in the background dictionary are weighted according to the Euclidean distance, so the weight of anomalous pixels in the background dictionary is reduced. The weighted objective function is expressed as

where

The expression for α' after weighting can be obtained

Applying the weighted collaborative representation method to the local sparse difference index to solve the sparse coefficient vector, the Weighted Collaborative Representation and Sparse Difference-based Hyperspectral Anomaly Detection algorithm is obtained. The specific steps and processes are as follows:

  • (1) Input the hyperspectral image, divide the band subspace according to the correlation coefficient between the bands, and detect each subspace separately.

  • (2) Select the appropriate double-window size and Lagrange multiplier, and use the weighted collaborative representation method to calculate the sparse coefficient vector.

  • (3) Calculate the local sparse difference index according to the sparse coefficient vector, and obtain the detection results for each subspace.

  • (4) Superimpose the detection results of each subspace to obtain the final detection result.

IV. EXPERIMENTAL RESULTS AND ANALYSIS

To verify the effectiveness and reliability of the algorithm, two groups of AVIRIS hyperspectral data are adopted for the simulation experiment in this paper. The simulation environment consists of: CPU Intel Core i7-7700HQ, dominant frequency 2.80 GHz, 8 GB RAM, software Matlab R2017a.

4.1. Experimental Data

Data-1 selected part of the data from the US San Diego Naval Airport, taken by the Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) sensor. The wavelength range is 0.4-1.8 μm. The bands with low signal-to-noise ratio and severe water-vapor absorption are removed; 126 bands are reserved. The spatial resolution is 3.5 m. The size of the intercepted area is 100 × 100 pixels, and there are 38 anomalous targets. Data-2 selected part of the data from the US Los Angeles Airport taken by the AVIRIS sensor. The wavelength range is 0.37-2.51 μm. The bands with low signal-to-noise ratio and severe water-vapor absorption are removed; 205 bands are reserved. The spatial resolution is 7.1 m. The size of the intercepted area is 100 × 100 pixels, and there are 17 anomalous targets. Figures 3 and 4 show the 50th band image and target distribution of the two groups of data:

Figure 3. Data-1 50th band image and target distribution.
Figure 4. Data-2 50th band image and target distribution.

4.2. Test Results and Analysis

First, we calculate the correlation coefficient between the bands according to Eq. (16), where xi and xj represent the spectral values of the ith and jth band images. mi and mj are the mean values of xi and xj respectively. Rij is the correlation coefficient of the ith and jth bands, with a value between 0 and 1; the closer to 1, the stronger the correlation between the two bands is. Correlation coefficients of adjacent bands for two groups of data are shown in Fig. 5. The two groups of data are divided into different band subspaces based on the correlation coefficients of adjacent bands

Figure 5. Correlation coefficients of adjacent bands for two groups of data.

  • Data-1:1-9, 10-71, 72-93, 94-114, 115-126

  • Data-2:1-104, 105-146, 147-154, 155-205

Second, the appropriate double-window size and Lagrange multiplier are selected for each band subspace of the two groups of data, and the algorithm proposed in this paper is used for detection. The detected pixel number of Data-1 is set to 400, and that of Data-2 is set to 150. The detection results are shown in Figs. 6 and 7.

Figure 6. Data-1 subspace detection results.
Figure 7. Data-2 subspace detection results.

The detection results for the subspaces are superimposed to obtain the detection results for the algorithm proposed in this paper. We also run the RX algorithm, LRX algorithm, and LSD algorithm to detect the image. The detection results for the four algorithms are shown in Figs. 8 and 9:

Figure 8. Data-1 detection results for four algorithms.
Figure 9. Data-2 detection results for four algorithms.

From the results for the two groups of data above, it can be seen that for Data-1 the RX algorithm shows the worst detection. Most of the detected anomalous targets are background, and the false-alarm rate is the highest. The LRX algorithm greatly reduces the false-alarm rate and its detection results are greatly improved, compared to the RX algorithm, because the LRX algorithm uses a double-window model in the detection process to select background pixels more accurately. The detection accuracy of both the LSD algorithm and the algorithm proposed in this paper is better still. The recognition of the target contour is more accurate, and the detection effect is the best. For Data-2, the RX and LRX algorithms yield poor detection results, and it is almost difficult to identify anomalous targets. This is because these two anomalydetection algorithms are based on Gaussian models, and it is difficult to detect images with poor Gaussian character and uneven distribution of anomalous targets. Although the LSD algorithm has detected most anomalous targets, it also has a large area of false alarms, and the false-alarm rate is high. Compared to the three prior algorithms, the proposed algorithm not only detects almost all anomalous targets, but also has the lowest false-alarm rate, because the proposed algorithm increases the screening process of background pixels, compared to the LSD algorithm. For images with uneven distributions of anomalous targets, it can greatly reduce the interference of anomalous components in the background pixels on the detection results, and improve detection accuracy.

The detection results are analyzed qualitatively above, and quantitatively below. The number of anomalous targets, the number of target pixels, and the number of false alarms detected by the algorithm are three important indicators to measure the performance of the algorithm. The test results are shown in Table 1

The test results for two groups of data

From the table above, it is not difficult to find that, compared to the other three algorithms, the proposed algorithm has the characteristics of high detection rate, low false-alarm rate, and better detection. Receiver Operating Characteristic (ROC) curves and Area Under the Curve (AUC) values are used to measure the detection results. The ROC curve reflects the relationship between the detection rate and the false-alarm rate; the abscissa indicates the false-alarm rate Pf, and the ordinate indicates the detection rate Pd.

where Nd represents the number of detected true anomalous target pixels, Nt the total number of anomalous target pixels in the image, Nmiss the number of false-alarm pixels detected, and Ntot the total number of pixels in the image. The more the ROC curve is bent to the upper left, the larger the AUC value, the better the detection result of the algorithm, and the higher the accuracy; the straighter the ROC curve, the smaller the AUC value, the worse the detection results, and the lower the accuracy. Figure 10 shows the ROC curves for the four algorithms, for the two groups of data. The AUC values and running times of the algorithms are shown in Table 2.

Figure 10. The ROC curves for the four algorithms, for the two groups of data.
AUC values and running times for the four algorithms

According to the ROC curve and AUC value, we can find that the detection performance of the proposed algorithm is significantly better than that of the other three algorithms, and has better applicability. It can be seen from the ROC curve that the proposed algorithm still has good detection performance and strong background-suppression ability even when the false-alarm rate is low; furthermore, as the running time is not too long, it is completely acceptable.

V. CONCLUSION

Aiming at the problem that the detection accuracy of the Local Sparse Difference Index algorithm is not high, this paper proposes a Weighted Collaborative Representation and Sparse Difference-based Hyperspectral Anomaly Detection algorithm. The weighted collaborative representation method is used to solve the sparse coefficient vector, which reduces the influence of the anomalous components in the background dictionary on detection. Analysis of experimental data shows that the algorithm can effectively improve the detection accuracy and reduce the false-alarm rate. Compared to three other algorithms, it exhibits better detection performance.

Fig 1.

Figure 1.The sliding double-window model.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 2.

Figure 2.The four-neighborhood pixel of the hyperspectral image, and its spectral curves.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 3.

Figure 3.Data-1 50th band image and target distribution.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 4.

Figure 4.Data-2 50th band image and target distribution.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 5.

Figure 5.Correlation coefficients of adjacent bands for two groups of data.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 6.

Figure 6.Data-1 subspace detection results.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 7.

Figure 7.Data-2 subspace detection results.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 8.

Figure 8.Data-1 detection results for four algorithms.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 9.

Figure 9.Data-2 detection results for four algorithms.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210

Fig 10.

Figure 10.The ROC curves for the four algorithms, for the two groups of data.
Current Optics and Photonics 2020; 4: 210-220https://doi.org/10.3807/COPP.2020.4.3.210
The test results for two groups of data

AUC values and running times for the four algorithms

References

  1. J. Yuan, Y. J. Zhang, and F. P. Gao, "A overview on linear hyperspectral unmixing," J. Infrared. Millimeter. Waves 37, 553-571 (2018).
  2. Y. F. Qi, and Z. Y. Ma, "Hyperspectral image classification method based on cooperative representation of neighborhood spectral probabilities," Laser. Technol. 43, 12-16 (2019).
  3. Shan L. X.. Hyperspectral image sub-pixel small target detection, M. S. Thesis (Xidian University, Shaanxi 2017).
  4. P. Bajorski, "Target detection under misspecified models in hyperspectral images," IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 5, 470-477 (2012).
    CrossRef
  5. L. Sun, J. H. Bao, and Y. C. Liu, "Analysis of target detection algorithm for hyperspectral images," Sci. Surv. Mapp. 37, 131-132+108 (2012).
  6. 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 38, 1760-1770 (1990).
    CrossRef
  7. X. L. Yu, L. E. Hoff, I. S. Reed, A. M. Chen, and L. B. Stotts, "Automatic target detection and recognition in multiband imagery: A unified ML detection and estimation approach," IEEE Trans. Image Process 6, 143-156 (1997).
    CrossRef
  8. S. S. Du, S. Y. Li, and Z. Y. Zeng, "Influence of background uncertainty on hyperspectral anomaly target detection," J. PLA Univ. Sci. Technol. (Natural Science Edition) 17, 598-604 (2016).
  9. T. E. Smetek, and K. W. Bauer, "Finding hyperspectral anomalies using multivariate outlier detection," in Proc. IEEE Aerospace Conference (Big Sky, MT, USA, March 2007).
  10. Y. P. Taitano, B. A. Geier, and K. W. Bauer, "A locally adaptable iterative RX detector," EURASIP. J. Adv. Signal. Process. 2010, 341908 (2010).
    CrossRef
  11. W. M. Liu, and C. I. Chang, "Multiple-window anomaly detection for hyperspectral imagery," IEEE. J. Sel. Top. Appl. Earth. Obs. Remote. Sens. 6, 644-658 (2013).
    CrossRef
  12. Y. Chen, N. M. Naarabadi, and T. D. Tran, "Hyperspectral image classification using dictionary-based sparse representation," IEEE Trans. Geosci. Remote Sens. 49, 3973-3985 (2011).
    CrossRef
  13. Z. Yuan, H. Sun, K. Ji, Z. Li, and H. Zou, "Local sparsity divergence for hyperspectral anomaly detection," IEEE Geosci. Remote Sens. Lett. 11, 1697-1701 (2014).
    CrossRef