Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
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.
Qianghui Wang^{1}, Wenshen Hua^{1,*}, Fuyu Huang^{1}, Yan Zhang^{1}, and Yang Yan^{2}
Corresponding author: huawensh@126.com
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
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
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.
Suppose
where
The local sparse difference index includes the local
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.
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
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
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.
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
The
The local sparse difference index can be expressed as
where
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
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
Here
To improve the stability of the solution, the constraint that the sum is 1 is applied to the sparse coefficient vector
The collaborative representation method exhibits higher accuracy in solving the sparse coefficient vector
where
The expression for
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.
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:
First, we calculate the correlation coefficient between the bands according to Eq. (16), where
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.
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:
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
where
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.
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.
Qianghui Wang^{1}, Wenshen Hua^{1,*}, Fuyu Huang^{1}, Yan Zhang^{1}, and Yang Yan^{2}
^{1}
Correspondence to:huawensh@126.com
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
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
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.
Suppose
where
The local sparse difference index includes the local
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.
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
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
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.
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
The
The local sparse difference index can be expressed as
where
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
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
Here
To improve the stability of the solution, the constraint that the sum is 1 is applied to the sparse coefficient vector
The collaborative representation method exhibits higher accuracy in solving the sparse coefficient vector
where
The expression for
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.
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:
First, we calculate the correlation coefficient between the bands according to Eq. (16), where
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.
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:
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
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
where
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.