A Robust Nonlinear Hyperspectral Anomaly Detection Approach

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IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 4, APRIL 2014

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A Robust Nonlinear Hyperspectral Anomaly Detection Approach Rui Zhao, Bo Du, Member, IEEE, and Liangpei Zhang, Senior Member, IEEE

Abstract—This paper proposes a nonlinear version of an anomaly detector with a robust regression detection strategy for hyperspectral imagery. In the traditional Mahalanobis distance-based hyperspectral anomaly detectors, the background statistics are easily contaminated by anomaly targets, resulting in a poor detection performance. The traditional detectors also often fail to detect anomaly targets when the samples in the image do not conform to a Gaussian normal distribution. In order to solve these problems, this paper proposes a robust nonlinear anomaly detection (RNAD) method by utilizing robust regression analysis in the kernel feature space. Using the robust regression detection strategy, this method can suppress the contamination of the detection statistics by anomaly targets. Moreover, in this anomaly detection method, the input data are implicitly mapped into an appropriate high-dimensional kernel feature space by nonlinear mapping, which is associated with the selected kernel function. Experiments were conducted on synthetic data and an airborne AVIRIS hyperspectral image, and the experimental results indicate that the proposed hyperspectral anomaly detection approach in this paper outperforms three state-of-art commonly used anomaly detection algorithms. Index Terms—Anomaly detection, hyperspectral, kernel-based learning, Mahalanobis distance, nonlinear version, robust regression analysis.

I. INTRODUCTION YPERSPECTRAL remote sensing technology is now a useful and popular type of observation technique for recognizing ground surface materials. Differing from the traditional panchromatic and multispectral remote sensing images, hyperspectral images can provide almost continuous spectral curves of the materials on the ground surface [1], [2]. This is because the spectral resolution of most hyperspectral spectrometers is less than 10 nm, and they can form a dataset “cube,” combining the spatial and spectral dimensions [3]–[5]. With the spectral dimension, we can acquire continuous and reliable spectra of surface materials, which is the basis for hyperspectral ground object identification techniques such as target detection and classification [6], [7].

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Manuscript received June 30, 2013; revised February 03, 2014; accepted March 08, 2014. Date of publication April 06, 2014; date of current version April 18, 2014. This work was supported in part by the National Basic Research Program of China (973 Program) under Grant 2011CB707105 and Grant 2012CB719905 and in part by the National Natural Science Foundation of China under Grant 61102128. R. Zhao and L. Zhang are with the Remote Sensing Group, State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing, Wuhan University, Wuhan 430079, China (e-mail: zhaoruiwinton@ gmail.com; [email protected]). B. Du is with the School of Computer Science, Wuhan University, Wuhan 430072, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTARS.2014.2311995

Anomaly detection is essentially a binary classification problem which divides an image into anomaly targets relative to the background. In anomaly detection processing, we want to detect the ground surface targets with spectra that are very different to the other ground surface materials. Anomaly detection is an unsupervised target detection technique in which we do not need any prior spectral information about the background or target. In most cases, it is difficult to obtain the spectra of the ground surface objects covered by a hyperspectral image, so anomaly detection methods are more practical in many applications. Moreover, spectral variability needs to be addressed in supervised target detection [8]. The multiplicity of possible spectra associated with the objects of interest and the complications of atmospheric compensation have led to the development and application of anomaly detectors. In recent years, hyperspectral anomaly detection techniques have been successfully applied in many application domains, such as mineral reconnaissance, border monitoring, and search and rescue [9]. Anomaly detection from hyperspectral imagery has become a popular research topic in the remote sensing image processing field [3]–[7], [9]. Many different hyperspectral anomaly detection methods have been proposed in the last 20 years. The RX-algorithm proposed by Reed and Yu [10] is acknowledged to be a benchmark Mahalanobis distance-based algorithm and has been widely applied to multispectral and hyperspectral imagery. However, in the RX anomaly detector, the statistical variables of the Mahalanobis distance-based model, such as the mean value and covariance matrix, are susceptible to contamination by anomaly targets. As a result, the RX anomaly detector can have an unstable performance if there are enough anomaly targets to affect the stability of the Mahalanobis distance-based model. In order to enhance the robustness and the stability of the detection processing, robust analysis technology has been applied to anomaly detection. Billor et al. [11] proposed the blocked adaptive computationally efficient outlier nominator (BACON) anomaly detector, which is a robust anomaly detector. The BACON detector can effectively suppress the contamination of the statistical variables by anomaly targets in the background model. Most traditional Mahalanobis distance-based anomaly detectors, such as RX, the cluster-based anomaly detector (CBAD) [12], and BACON, assume that the hyperspectral image data conform to a univariate Gaussian distribution or multivariate Gaussian distribution. However, due to the low spatial resolution of hyperspectral imagery, the categories and components of the surface objects in each pixel are usually complex, and mixed pixels are common in hyperspectral images. In addition, in the process of receiving signals from surface features, the signal may reflect multiple times before reaching the remote hyperspectral

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sensor [13], which means that the data in hyperspectral images may not conform to a Gaussian distribution. In this case, the Gaussian distribution, which is assumed in the Mahalanobis distance-based anomaly detectors, cannot accurately describe the hyperspectral data, and the corresponding detection performance will be poor. Establishing nonlinear anomaly detectors by describing the non-Gaussian distribution in hyperspectral images can result in better detection performance. A linear non-Gaussian model in the low-dimensional original data space can easily be extended to a nonlinear Gaussian domain by mapping the input data into a potentially infinite feature space, which can be efficiently implemented by the kernel learning methods [14], [15]. This basic idea behind the kernel learning methods has been widely used to exploit the nonlinear characteristics of data in machine learning. In the kernel methods, the learning is performed in a high-dimensional feature space where the discrimination between the classes in the data can be enhanced, subsequently generating simpler decision rules and improving the generalization performance [16]. Recently, kernel versions of data description techniques have been introduced in detection and other applications, such as the kernel-RX algorithm [17], kernel orthogonal subspace projection (KOSP) [18], the kernel matched signal detector [19], the kernel matched subspace detector (KMSD) [16], kernel eigenspace separation transform (KEST) [20], kernel principal component analysis (KPCA) [21], [30], kernel support vector machine [22], the sparse kernel-based anomaly detector [23], and the kernel target-constrained interference-minimized filter (KTCIMF) [24]. Nonlinear Gaussian data description-based anomaly and target detection algorithms may achieve better performance than the linear nonGaussian model-based detectors. This can be found in [16], [17], [19], [23], and [24] that nonlinear Gaussian data description in high-dimensional kernel space can make some contributions to anomaly or target detection. In this paper, we combine the idea of kernel-based learning [25] and robust regression analysis [26] to develop a new nonlinear robust detection strategy-based anomaly detection method for hyperspectral imagery, the robust nonlinear anomaly detection (RNAD) method. First, we project the hyperspectral image data in the original low-dimensional space to an appropriate high-dimensional feature space, which is called the kernel feature space. In this feature space, we can obtain Gaussiandistributed data transformed from the non-Gaussian modelbased data in the original data space. Robust regression analysis is then undertaken on the data in the feature space to derive the iterative corresponding generalized likelihood ratio test (GLRT) expressions. The GLRT expressions in the feature space have to be kernelized before they can be implemented, because of the high dimensionality of the feature space. The kernelization procedures are presented in this paper. The aim of the RNAD method is to suppress the contamination of the Mahalanobis distance-based features by anomaly targets in the kernel feature space. The rest of the paper is organized as follows. In Section II, a brief introduction to the kernel learning methods is presented. Section III outlines the robust regression analysis strategy and the formation of the RNAD algorithm. Section IV describes the experiments with the proposed algorithm and compares it with

three traditional anomaly detection algorithms. Finally, Section V draws our conclusion. II. KERNEL LEARNING THEORY A. Problems With the Mahalanobis Distance-Based Anomaly Detection Algorithms In general, most of the Mahalanobis distance-based anomaly detectors, such as RX and CBAD, assume that the data in hyperspectral images are homogeneously distributed, which is that all the classes of the ground surface materials in the image conform to the same model [3], [6], [9]. The hyperspectral anomaly detection techniques also usually assume that the hyperspectral data conform to a univariate or multivariate normal Gaussian-distributed model [10], [12]. However, in most cases, hyperspectral image data obtained by remote sensors do not conform to this assumption. The complexity of the hyperspectral data distribution means that the Gaussian-distributed models fail to describe the actual distribution of the hyperspectral image data. Non-Gaussian-distributed data in hyperspectral images in anomaly detection applications are mainly caused by the following reasons: 1) hyperspectral image data are always composed of multiple types of ground surface materials, which cannot be regarded as homogeneous [9] and 2) the spectra of pixels are not simply mixed by the various types of surface features’ spectra with a linear mixture model, but have some higher-order relationship between the spectral bands at different wavelengths [17]. The signals of hyperspectral remote or airborne sensors may reflect or refract more than once on the ground surface materials. Moreover, the signals may reflect or refract on one or more than one type of ground surface material. The complexity of the ground surface in a hyperspectral image may then result in a nonGaussian distribution of the background and anomaly targets. By utilizing kernel learning theory, the RNAD detector proposed in this paper can handle the non-Gaussian distribution problem. B. Introduction to Kernel Learning Theory Kernel learning theory is usually used in conditions that require a more appropriate data distribution in a specific application, just as we want hyperspectral image data to be Gaussian-distributed with the Mahalanobis distance-based anomaly detection algorithms. In Fig. 1, the red points represent anomaly targets, and the blue points represent the background pixels in the hyperspectral image. The simplified two-dimensional (2-D) original space on the left represents the original hyperspectral data space, and the simplified three-dimensional (3-D) feature space on the right represents an appropriate highdimensional kernel feature space. As shown in Fig. 1, it is common that in practical hyperspectral anomaly detection applications, hyperspectral image data have a non-Gaussian distribution. As a result, it is not possible to find an elliptical hyperplane for the Mahalanobis distance-based anomaly detectors to describe the hyperspectral background. By utilizing kernel learning theory, we can project the hyperspectral image data from the original hyperspectral data space to a high-dimensional feature space termed the kernel feature space, in which the

ZHAO et al.: ROBUST NONLINEAR HYPERSPECTRAL ANOMALY DETECTION APPROACH

Fig. 1. Kernel feature projection by kernel learning methods for hyperspectral anomaly detection.

non-Gaussian-distributed hyperspectral data can conform to a global or local Gaussian distribution, as illustrated in Fig. 1. In an appropriate kernel feature space, the Gaussian-distributed hyperspectral background can benefit the Mahalanobis distance-based anomaly detection. In the following, kernel learning theory is introduced. Kernel learning methods map the original data space to an appropriate high-dimensional feature space by a selected kernel function . In the high-dimensional kernel feature space, Mahalanobis distance-based algorithms such as RX and CBAD operated on the original data can be expanded to their nonlinear versions and obtain a Gaussian data distribution. Using such kernel feature mappings, and certain kernel functions, data in the original data space can be implicitly transformed into a high-dimensional kernel feature space [27]. For the original data space ( RJ ), the input vectors ( ) can be mapped into a much higher dimensional feature space by a nonlinear mapping function , and the data in the feature space will then be . In particular

Mapping the data using into is useful in many ways. The most significant benefit is that it is possible to define a similarity measure using the dot product in in terms of a function of the corresponding data in the input space. Thus, it is possible to write

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Fig. 2. Anomaly interference with Mahalanobis distance-based detection. (a) Ideal decision boundary. (b) Decision boundary contaminated by anomaly targets.

feature space now consist of two Gaussian distributions, thus modeling the two hypotheses as

The corresponding RX-algorithm in the feature space is now represented as

With the eigenvector decomposition and kernel methods, we can get the kernelization of the RX-algorithm in the kernel feature space [17] as

In (6),

,

, and

Here, spectral image and

are implemented as

are the samples from the hyperis the number of samples.

III. ROBUST DETECTION STRATEGY IN THE KERNEL SPACE Equation (2) is commonly referred to in machine learning literature as the kernel trick [25] and was first used in [28], where represents the kernel function. Equation (3) is the Gaussian RBF kernel function which the RNAD algorithm proposed in this paper utilizes. The nonlinear mapping could potentially generate a feature space in which it is not computationally feasible to directly implement any algorithm. Fortunately, the kernel trick helps circumvent this problem. C. Kernel-RX Algorithm In the kernel feature space, there are the same assumptions as those used in the RX-algorithm, i.e., the mapped input data in the

Mahalanobis distance-based detectors such as the RX detector are the most commonly used detection strategy in hyperspectral anomaly detection. The statistics of the Mahalanobis distance-based anomaly detectors will be easily affected if there is a large quantity of anomaly target pixels in the image, because the Mahalanobis distance-based features such as the mean value and covariance will be seriously contaminated by the large number of anomalies. Fig. 2 illustrates anomaly interference with Mahalanobis distance-based detection. In Fig. 2(a) and (b), the hyperspectral data space is simply represented by a 2-D space, where the blue dots represent the background pixels, the red dots represent anomaly targets, and the dashed line represents the decision

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boundary. From Fig. 2(b), we can see that the decision boundary for the anomaly detection deviates severely from the ideal decision boundary, because of the contamination by anomaly targets. This explains why the Mahalanobis distance-based anomaly detectors such as RX usually have a poor stability [11], [29]. Due to the instability of the Mahalanobis distance-based anomaly detectors, we need a robust strategy for anomaly detection processing. The RNAD algorithm proposed in this paper adopts the following strategy of computing the background statistic features to look for pixels in the hyperspectral image that do not contain anomaly target pixels. First, set a few background pixels as the fundamental dataset before starting the iteration calculation. Then, in each iteration, add the pixels whose detected values are tolerated by the threshold resulting from the Mahalanobis distance statistic, and constrain via a chisquared distribution. The iteration calculation terminates when there are no new pixels added to the background dataset. In this way, we will obtain a pixel dataset in the hyperspectral image that is not contaminated by anomaly target pixels. The whole procedure of this strategy is operated in an appropriate kernel feature space by the Gaussian RBF kernel function. The steps for the RNAD algorithm are as follows. 1) The hyperspectral data should first be normalized to [0, 1]. 2) Apply the global kernel-RX [17] detector to the hyperspectral image data and select the pixels which have the minimum detection values to build the initial background dataset. The volume of the initial dataset is usually set as three to four times the number of bands, by experience. 3) Apply the clustering operation to the background dataset and get the cluster centers , where is the number of cluster classes. K-means is employed as the clustering method. We then apply the cluster centers to compute the Gram matrix, as shown in (10), and we calculate the Mahalanobis distance in the kernel feature space, as shown in (12), for pixels outside of the background dataset

where

and where respectively

is implemented with (11)

are implemented with (13) and (14),

degrees of freedom [26], where is the number of hyperspectral image bands. The threshold value must be multiplied by a correction factor, as shown in (15), in advance [26]

where

size of the current background dataset; number of samples in the hyperspectral image;

5) If there are new pixels added to the background dataset, then go to step (3) for robust regression analysis, else if there are no new pixels added to the background dataset, then this is the final background dataset, and go to step (6). 6) Cluster the final background dataset to compute the Gram matrix, as in step (3), for the kernel-RX detector, which is applied to the whole image dataset to export the output detected values. 7) Set the detection threshold to segment the detected result to a binary image. IV. EXPERIMENTAL RESULTS In the experiments, we choose three state-of-the-art anomaly detection algorithms for comparison: RX, KRX, and BACON. The reasons why we choose these three anomaly detection methods are: 1) The RX anomaly detector is an acknowledged benchmark anomaly detection algorithm which exploits the Gaussian background statistics but does not have any operations to prevent the contamination by anomalies. 2) The KRX detector is the most famous kernel anomaly detection method. It assumes that the Gaussian distribution may not be true but that it can be satisfied by a proper mapping to the high-dimensional kernel feature space. 3) BACON is a detector that uses robust background statistics. It exploits the iterative analysis of the background statistics to prevent the contamination by anomalies. By a comparison with the above three methods, the proposed RNAD method can be evaluated, and whether combining the robust analysis and the kernel method is efficient or not can be proved. A. Nonlinear Synthetic Data

4) Compare the Mahalanobis distance values in the kernel feature space to a threshold, which is the square root of an appropriate quantile from the chi-squared distribution with

In this section, we evaluate the detection performance of the RX, BACON, kernel-RX (KRX), and RNAD algorithms. A nonlinear synthetic dataset with a size of pixels is used in this experiment, as shown in Fig. 3(a). We plant nine anomaly targets of pixels in this synthetic data, with the centers of these targets located at (25, 25), (25, 50), (25, 75), (50, 25), (50, 50), (50, 75), (75, 25), (75, 50), and (75, 75), respectively, as shown in Fig. 3(b). The spectra of the background and anomaly target pixels are selected from the spectral library in

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Fig. 3. Nonlinear synthetic data. (a) First band of the nonlinear synthetic data. (b) Reference for the synthetic data.

ENVI 4.7. The background pixels consist of three endmembers of natural vegetation in “veg_lib,” which are “Aspen_Leaf,” “Blue_Spruce,” and “Pinon_Pine.” The spectrum of the anomaly endmember mineral is “Actinolite,” selected from “usgs_min.” The spectra of the endmembers are shown in Fig. 4. Here, in the nonlinear synthetic data, we apply nonlinear spectral mixing to obtain non-Gaussian-distributed data, and we make the background and anomaly targets very close. The Hapke approximate equation [31] is applied to generate nonlinearly mixed pixels in this experiment. In general, we assume that the spectral data of remote sensing have the characteristic of bi-directional reflectance, so we should use (16) to transform this bi-directional reflectance to single scattering albedo (SSA)

where represents the angle of incidence, represents the angle of view, , , and represent the function of multi-directional scattering among the surface features

The steps for generating nonlinear synthetic data are as follows. 1) Transform the spectral value of the endmembers to the corresponding reflectance. 2) Transform the reflectance to SSA by (16) and (17). 3) For the background pixels: generate three random number between [0, 1], with the constraint of

, as the

abundance of the background endmembers. For the anomaly target pixels, in a similar way to the background pixels, control the abundance of the anomaly target endmember to be less than 25%, then set power of 1 or 2 as the spectra of the background endmembers, and set power of 2 as the spectrum of the anomaly target endmember. The mixed albedo is then as follows. Background pixels

Fig. 4. Spectra of the endmembers in the nonlinear synthetic data.

Anomaly target pixels

4) Transform the mixed albedo to mixed reflectance by (18) and (19). Add random normal noise to to generate the synthetic data by (20)

where

represents the signal-to-noise ratio and are random normal vectors that have zero mean and variance of unity. We set the anomaly endmember abundance of the nine anomaly target groups as 10%–90%, respectively, to investigate the ability of the sub-pixel anomaly target detection for RNAD. Fig. 5 demonstrates that both the RX and BACON detectors have a poor ability to suppress the background information, which is because the data in the original data space do not conform to a Gaussian distribution, so that background with features similar to the anomaly targets is not suppressed. In the detection results, KRX and RNAD enlarge the separation of the background and targets, so KRX and RNAD can suppress the background information more effectively. The ROC curves in Fig. 6 indicate that the RNAD algorithm has a much better detection performance than the RX, BACON, and KRX algorithms. We can demonstrate from this experiment that hyperspectral image data which is projected to the kernel feature space can be Gaussian-distributed, and robust regression iterative analysis can suppress the contamination of the detection statistics by anomaly targets. B. Airborne Hyperspectral Image Data In this experiment, we use an airborne hyperspectral image collected by the airborne visible infrared imaging spectrometer (AVIRIS) imaging sensor from San Diego airport. The original image dataset has a size of pixels, with 224 bands, as

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Fig. 5. Detection results with the synthetic data: (a) RX; (b) BACON; (c) KRX; and (d) RNAD.

Fig. 6. Detection accuracy evaluation for the nonlinear synthetic data. (a) ROC curves. (b) AUC values.

Fig. 8. (a) True color image of a subset of the airport hyperspectral image. (b) Reference for the anomaly targets.

Fig. 7. San Diego airport hyperspectral image data.

shown in Fig. 7. We only use 189 spectral bands after removing the low-SNR or water vapor absorption bands (1–6, 33–35, 97, 107–113, 153–166, and 221–224) in this experiment. We cut out a sub-image of the data to be the experimental data, and we then regard the small aircraft as the anomaly targets, as shown in Fig. 8(a). A total of 17 small aircraft with 170 pixels are regarded as the anomaly targets, and the airfield is regarded as the

background surface. The reference for the anomaly targets is shown in Fig. 8(b). In this airborne hyperspectral image, there is a high degree of correlation between the spectral bands. Therefore, we first apply kernel-PCA [21] preprocessing on the image so that we can obtain the useful spectral features of the ground surface materials. The detection results and the accuracy evaluation of RX, BACON, KRX, and RNAD are shown in Figs. 9 and 10, respectively. The ROC curves and AUC values in Fig. 10 demonstrate that the RX, BACON, and kernel-RX anomaly detectors do not perform as well as the RNAD algorithm. Overall, the RNAD algorithm is better at highlighting anomaly targets and suppressing background information. This success can be attributed to

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Fig. 9. Detection results for the airborne hyperspectral image: (a) RX; (b) BACON; (c) KRX; and (d) RNAD.

Fig. 10. Detection accuracy evaluation for the airborne hyperspectral image: (a) ROC curves. (b) AUC values.

the kernel learning in the RNAD algorithm, and the fact that the robust regression analysis can result in a larger separation between background and anomaly targets. V. CONCLUSION This paper proposes a hyperspectral anomaly detection algorithm which utilizes kernel learning theory and a robust regression analysis to get a more appropriate data distribution for hyperspectral anomaly detection, and suppresses contamination of the detection statistics by the anomaly targets. RNAD obtains more appropriate Gaussian-distributed hyperspectral data, with the data mapped to an appropriate kernel feature space, which interprets the nonlinear information in the hyperspectral image data. Experiments with airborne hyperspectral images and synthetic nonlinear data showed that the RNAD algorithm can perform better than the traditional anomaly detectors and can build more robust background features, without contamination by anomaly targets.

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Rui Zhao received the B.S. degree in photogrammetry and remote sensing from Wuhan University, Wuhan, China, in 2012, where he is currently pursuing the Ph. D. degree at the State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing (LIESMARS). His research interests include hyperspectral image processing and machine learning.

Bo Du (M’11) received the B.S. degree from Wuhan University, Wuhan, China, in 2005, and the Ph.D. degree in photogrammetry and remote sensing from the State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing, in 2010. He is currently an Associate Professor with the School of Computer, Wuhan University, Wuhan, China. His research interests include pattern recognition, hyperspectral image processing, and signal processing.

Liangpei Zhang (M’06–SM’08) received the B.S. degree in physics from Hunan Normal University, Changsha, China, in 1982, the M.S. degree in optics from Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, China, in 1988, and the Ph.D. degree in photogrammetry and remote sensing from Wuhan University, Wuhan, China, in 1998. He is currently the Head of the Remote Sensing Division, State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing, Wuhan University. He is also a “Chang-Jiang Scholar” Chair Professor appointed by the Ministry of Education of China. He is currently a Principal Scientist for the China State Key Basic Research Project (2011–2016) appointed by the Ministry of National Science and Technology of China to lead the Remote Sensing Program in China. He has authored and co-authored more than 310 research papers and is the holder of five patents. He edits several conference proceedings, issues, and geoinformatics symposiums. His research interests include hyperspectral remote sensing, high-resolution remote sensing, image processing, and artificial intelligence. Dr. Zhang is a Fellow of the IEE, Executive Member (Board of Governor) of the China National Committee of International Geosphere–Biosphere Programme, Executive Member of the China Society of Image and Graphics, etc. He regularly serves as a Co-Chair of the series SPIE Conferences on Multispectral Image Processing and Pattern Recognition, Conference on Asia Remote Sensing, and many other conferences. He also serves as an Associate Editor of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, International Journal of Ambient Computing and Intelligence, International Journal of Image and Graphics, International Journal of Digital Multimedia Broadcasting, Journal of Geo-Spatial Information Science, and Journal of Remote Sensing.