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An Approach for Subpixel Anomaly Detectionin Hyperspectral Images

Safa Khazai, Abdolreza Safari, Barat Mojaradi, Member, IEEE, and Saeid Homayouni, Member, IEEE

Abstract—Fast detecting difficult targets such as subpixel objectsis a fundamental challenge for anomaly detection (AD) in hyper-spectral images. In an attempt to solve this problem, this paperpresents a novel but simple approach based on selecting a singlefeature forwhich the anomaly value is themaximum. The proposedapproach applied in the original feature space has been evaluatedand comparedwith relevant state-of-the-art ADmethods on TargetDetection Blind Test data sets. Preliminary results suggest that theproposed method can achieve better detection performance thanits counterparts. The results also show that the proposed methodis computationally expedient.

Index Terms—Hyperspectral images, anomaly detection, singlefeature, single band.

I. INTRODUCTION

H YPERSPECTRAL images have a great potential to auto-matic target detection, since they deliver valuable infor-

mation about the spectral characteristics of the earth surface ob-jects and phenomena. Anomaly detection (AD) is a well-knownunsupervised approach for automatic target detection. It has at-tracted significant interest in hyperspectral remote sensing ap-plications such as detecting crop stress locations in precisionagriculture, rare minerals in geology, oil pollution in environ-mental researches, landmines in the public safety and defensedomain, and man-made objects in reconnaissance and surveil-lance applications [1].The general aim of AD is to find the targets which have dis-

tinct spectral behavior from their background without a prioriknowledge of the target signature and any form of atmosphericcompensation. The anomalies are normally defined with refer-ence to a model of background. The local and global spectralanomalies are respectively defined as the observations that differin some way from the neighboring background and from theoverall, or a substantial part of, the scene. Generally, the localmethods for AD yield more reliable results. However, they willsometimes cause a higher false alarm rate (FAR) for isolatedspectral anomalies [2], [3]. For example, an isolated tree in a lo-cally homogeneous patch of grass is detected as a local anomalyeven if the whole image contains a forest [1].

Manuscript received January 04, 2012; revised March 16, 2012, May 02,2012; accepted July 16, 2012. This work was supported by the University ofTehran under Grant 8103618/1/07.S. Khazai, A. Safari, and S. Homayouni are with the Department of Surveying

and Geomatics Engineering, College of Engineering, University of Tehran,Tehran 14395-515, Iran (corresponding author e-mail: khazai@ut.ac.ir).B. Mojaradi is with School of Civil Engineering, Iran University of Science

and Technology, Tehran, Iran.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.2012.2210277

The AD algorithms are typically compared with Reed-Xi-aoli (RX) algorithm [4], which is based on the well-knownMahalanobis distance between the spectral vectors of an inputtest pixel and its surrounding neighbors. The most well-knownproblem with the RX is the small-sample size, which concernsthe estimation of a local background covariance matrix from asmall number of samples in high-dimensional space [5]. Underthis circumstance, the result is a badly-conditioned and unstableestimation. In the case of an -dimensional data, at leastsamples are needed to estimate a covariance matrix. However,many more samples (possibly about ) must actuallybe used to obtain a reliable estimation [6]. To overcome thesmall-sample size problem without considering dimensionreduction solution, two approaches have been explored in thehyperspectral AD literature.The first approach consists of improved versions of the RX

algorithm. It basically aimed at improving estimations of thecovariance matrix. The conventional algorithm, denoted asglobal RX (GRX), estimates a global covariance matrix fromthe entire image. However, Regularized-RX (RRX) [7] is anew algorithm that regularizes the local covariance matrix.Also, Segmented-RX (SRX) [5] is a recent algorithm whichmodels the background using a clustering of all image pixels.Moreover, the topology-based RX (TRX) [8] can be consideredas a state-of-the-art improved-RX algorithm that uses the Topo-logical Anomaly Detector (TAD) for background modeling ofimages [9].The second approach involves the kernel-based (i.e., non-

linear) AD methods, which are based on the kernel theory forextending the original space to a higher dimensional featurespace, the so-called Hilbert space. Kernel functions are used toimplicitly compute the dot products in Hilbert space, withoutactually mapping the input vectors into that space [10]. Hence,the kernel-based methods can deal very well with high dimen-sional data. The kernel-based AD methods can be groupedinto two categories: parametric and non-parametric algorithms.The well-known parametric kernel-based algorithm is theKernel-RX (KRX) [11], which is the nonlinear version of theRX. Moreover, a state-of-the-art non-parametric kernel-basedalgorithm is the support vector data description (SVDD) [12].The study [13] introduced a general framework to use theSVDD for AD in hyperspectral images.Detecting subpixel (i.e., mixed) objects, such as mineral and

crop species, which occupy a portion of the pixel area andwhosesignatures slightly deviate from the background, is one of thefundamental challenges for hyperspectral imaging applications[2]. This study presents a novel but simple AD approach aimed

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at quickly detecting the subpixel targets in hyperspectral im-ages. The proposed approach uses only a single feature that re-veals the anomaly better than others. The performance of theproposed approach is compared to that of the improved RX al-gorithms and the kernel-based AD methods.The rest of this paper is organized as follows. Section II pro-

vides more details about the algorithms used for AD. The pro-posed approach is presented in Section III. Experimental re-sults are presented in Section IV. Lastly, concluding remarksare given in Section V.

II. THE ANOMALY DETECTORS

A. The Improved RX Algorithms

The RX test statistic is the Mahalanobis distance between thepixel being tested with the spectrum and the local backgroundwith mean spectrum as follows:

(1)

where is the estimated covariance matrix from samples sur-rounding the test pixel. The RRX regularizes by adding ascaled identity matrix before inversion. The scale (i.e., regular-ization) factor is given by computing the median eigenvalue ofthe covariance matrix estimated on the whole image. Moreover,the SRX requires that a clustering of the image to be performed.After obtaining a thematic map, the covariance matrix is es-timated over each cluster. Then, the well-known Mahalanobisdistance is computed between the target pixel and its spectrallynearest cluster. The covariance matrix required can be either es-timated over the cluster that is most common in the pixel neigh-borhood, or by computing a mixture of the covariance matricesof the clusters represented in the pixel neighborhood [5].On the other hand, the TRX requires that the global covari-

ance matrix is computed using only those pixels that are as-signed into the background by the TAD [9]. In addition to pro-viding a novel AD algorithm, the TAD incorporates a topo-logical approach to modeling hyperspectral data. However, inTRX, the TAD is used to remove the anomalous pixels from in-fluencing the background covariance matrix. The TAD buildsa graph whose edges connect close pairs of pixels. The back-ground pixels are the pixels in the largest components of thisgraph [8].

B. The Kernel-Based AD Algorithms

The most effective kernel function for kernel-based methodsis Gaussian given by , whereand are two objects in the original space, and denotes

the Gaussian kernel width (i.e., the sigma).The KRX: The KRX is the nonlinear version of the RX that

adopts the Gaussian model in the Hilbert space. The KRX iscompactly given by [11]:

(2)

where

(3)

(4)

where is the centered kernel matrix of training pixels (i.e.,local background pixels), and is thetotal number of training pixels.The SVDD: The basic idea of SVDD to find a hypersphere

withminimum volume that encloses all the normal training sam-ples of the target class [12]. Given training samplesbelonging to a class of interest and a kernel function , theSVDD problem becomes the following:

(5)

with two constraints on Lagrange multipliers (and . The user-specified parameteris called the rejection rate indicating an upper bound on thefraction of outliers. The objects with nonzero are calledthe support vectors (SVs). To test if a new object is outsidethe obtained hypersphere, the squared Euclidean distance to thecenter of the hypersphere, called the SVDD test statistic, has tobe calculated. A test object is rejected when this statistic isbigger than the squared radius [12]:

(6)

where can be determined by calculating the distance from thecenter of the hypersphere to any of the SVs on the boundary,i.e., objects with . The study [13] showedthat, to improve the detection performance of the SVDD for ADin the hyperspectral images, a normalized version of the SVDDtest statistic can be derived through dividing the original teststatistic by the squared radius.

III. PROPOSED APPROACH

Let us consider a case study application in which the subpixelobjects are the interesting targets for AD. From a physical stand-point, a subpixel object is usually strongly mixed with its neigh-boring objects in most spectral regions of its pixel spectrum.Nevertheless, there is almost a high probability that a subpixelobject, especially a man-made one, has a significant impact onsome spectral regions of its pixel spectrum. Therefore, such atarget may be easily discriminated from the local backgroundin those spectral regions or bands. Evidently, the target-back-ground discrimination may be improved using some featuresextracted from the spectral bands. The features can be obtainedby linear transformations such as the principal component anal-ysis (PCA) and kernel-based methods such as the kernel-PCA.However, it may be concluded that the success of a subpixel ADalgorithm depends on its ability to find and use the best featurefor discriminating between the target pixel and the local back-ground pixels. We have appealed to this important point in ourproposal for the AD, which we named the single-feature basedanomaly detector (SFAD).

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Fig. 1. The true color composite of the HyMap radiance image. The left box specifies the ROI-1 and contains 6 self-test targets The right box shows the ROI-2and contains 20 synthetic targets.

The SFAD determines the anomaly value of target pixel inthe feature , as follows:

(7)

where is the value of feature for target pixel , and isthe median value of local background pixels of in the feature, and is obtained as follows:

(8)

where is the value of the feature for the neighboring pixel. Note that is the median absolute deviation (MAD) of thefeature . The MAD, which is a robust measure of dispersion[14], is defined as the median of the absolute deviations fromthe data’s median. However, SFAD test statistic for the targetpixel is obtained as follows:

(9)

where is the number of appropriate features. Note that theSFAD evaluates all the features, and for each target, tries to se-lect a specific feature for which the anomaly value is the max-imum. We have named this feature the discriminative feature. Itis worthwhile to note that the SFAD applies a kind of weightingto the anomaly values, where the discriminative feature that pro-vides the maximum value takes , and other featurestake . In this work, we employ the physical (i.e.,primary) variant of the SFAD, called the single-band based AD(SBAD), which evaluates all the spectral bands in order to se-lect a discriminative band.As with any AD algorithm, removal of low signal-to-noise

ratio (SNR) channels, i.e., bad bands, increases the detectionperformance. However, the SBAD is more sensitive to removalof these bands than the conventional AD methods, which useall spectral bands. Since the SBAD chooses one band to per-form AD, the discriminative bands found may be discriminativebecause of the presence of bad bands, not because of the pres-ence of the anomalies. Note that the bad bands caused mainlyby sensor failure or strong water-vapor absorption provide ei-ther little useful information or no information to be extractedabout the anomalies. Therefore, removing such bands in the datais required before performing the SBAD.

IV. EXPERIMENTAL RESULTS

A. Data Sets

The Target Detection Blind Test data sets [15], [16] includetwo HyMap radiance and reflectance images of Cooke City inMontana, USA (see Fig. 1). The images were collected by an air-borne HyMap sensor, which has 126 spectral bands. The groundresolution of imagery data is approximately 3 m. In the imagescenes, some real targets were located in an open grass regionduring the image acquisition.Evidently, to deal with large hyperspectral images, the AD

methods suffer from high computational costs. To overcomethis problem, one can partition the whole image into some ap-proximately equal subsets, and then apply AD methods on eachsubset independent of the others by parallel processing. Here,it is worthwhile to refer the study [17] that developed a generalstrategy to automatically map parallel hybrid AD algorithms inhyperspectral images. However, based on a partitioning proce-dure for which the size of subsets is 90 90, two regions ofinterest (ROI), ROI-1 and ROI-2, are selected in the HyMapimages.Real Targets: The ROI-1 consists of the twelve real targets.

Two of the real targets are at the full pixel (i.e., resolved) size;the other ten are at the subpixel sizes. The targets included sixfabric panels for the self-test and six for the blind-test. Fig. 1shows the location of six self-test targets, while Table I brieflydescribes each target in more detail.Synthetic Targets: In this study, we used the target implant

method used in [18] to evaluate the performance of AD algo-rithms on a wide variety of subpixel targets. Based on a linearmixing model, a synthetic subpixel anomaly with spectrum isgenerated by fractionally implanting a desired pixel target witha reflectance spectrum, in a given pixel in the background (i.e.,host pixel) with a reflectance spectrum as follows:

(10)

where denotes the implant fraction, and are thereflectance and implanted spectrum of the -th neighbor of thehost pixel, respectively, and is the spatial Euclidean distancebetween the -th adjacent pixel and the host pixel.

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To generate the synthetic targets, we used the reflectance ver-sion of the HyMap image and took the spectrum of the purepixel targets F1 and F2 as two different target types. Afterward,to create synthetic targets in the ROI-2, each of the spectra wasimplanted separately at 20 host pixels with consideration of theiradjacent effects in a constant window size of 5 5. Moreover,each implant is accomplished at three fractions: 1/9, 2/9, and4/9. Note that at each implant fraction two synthetic data setsare generated using the targets F1 and F2.

B. Preprocessing

Before the performing of the AD algorithms, some prepro-cessing steps should be done as follows:Removing Saturated Pixels: Saturated pixels caused tem-

porarily by, e.g., sun glint, bright cloud, bright land or snow/ice,contain signals that are saturated in some spectral bands [19].Such pixels appear as targets to hyperspectral detection al-gorithms; however, they are not genuine targets [20]. Thus,AD algorithms would produce false alarms on the saturatedpixels. Hence, to improve the performance of AD, removingthe saturated pixels should be taken into consideration beforeperforming all the algorithms. Note that since the saturatedbands will surely result as discriminative bands, the SBAD ismore sensitive than the other AD algorithms to the problemof saturated pixels. The saturated pixels can be easily detectedby searching for the radiance values of the data that are biggerthan a given value (e.g., 90%) of saturation levels defined [21].It is worthwhile to note that the saturation levels are sensordependent and based on industry specifications or instrumentcharacterization runs in calibration facilities [21]. In fact, theywould depend upon the particular settings of the instrument atthe time of collection. In this study, since the saturation levelsfor the data sets used are not known, we manually investigatedthe existence of the saturated pixels by searching for exces-sively large spectral values in the radiance image. The resultsshowed that there are no saturated pixels in the images.Removing the Bad Bands in the Radiance Image: The bad

bands may be specified a priori. However, in this study to dis-tinguish the bad bands, a heuristic method is employed. Theheuristic method used assigns a loss level for each band equal tothe inverse of the variance of the data in that band. Obviously,the bad bands have a value of the variance that is very close tozero. Thus, they will have excessively large loss levels. Con-sidering that most of the bands are normal, in this study the badbands are defined as the bands that have the loss levels more thana 2.5 MAD away from the median of the obtained loss levels ofall bands. By using this solution, three outlier bands of 63, 64,and 126 were found in the HyMap radiance image. After re-moving these bands, the number of remaining bands was 123.Normalization of the Radiance Image: All pixel vectors, in

the whole of the radiance image, are normalized by a maximumspectral value in the image, so that the entries of the normalizedpixel vectors fit into the interval of spectral values between zeroand one. Note that the reflectance image does not need data nor-malization, as the values in this image range from zero to one.Detection Window Setting: An important decision for the AD

methods is the specifying the local background pixels for eachtarget pixel. This decision is usually made through a dual con-

centric windowwhich separates the local area around each pixelinto two regions, a small inner window region (IWR) centeredwithin a larger outer window region (OWR) [22]. The IWR isused to enclose the target of interest to be detected, while theOWR is employed to model the local background around thetarget region. In the case of subpixel anomalies, the size of IWRis always 1 1 based on the assumption that each target of in-terest completely enclosed in a pixel. Hence, only the size ofOWR, called the detection window, should be set. Since there isno a specific method to choose the size of the detection window[13], a detection windows of 5 5 is used for all the experi-ments.The Sigma Setting: When using the Gaussian kernel, the

main problem of the kernel-based AD algorithms, the SVDDand the KRX, is the optimal setting of the sigma parameter. Inthis study, we used the sigma estimation method given by [18]:

(11)

where is the maximum Euclidean spectral distance be-tween training instances (i.e., surrounding pixels of the targetpixel). The parameter is set experimentally by users. We setto 0.1, which means 10% pixels in the detection window are

allowed to be outliers.The SRX Setting: In order to segment the datacube, we

employed the well-known K-means algorithm. However, themain problem with using clustering algorithms such as theK-means one is the choice of the optimum number of clusters.In this study, we experimentally set the number of clusters inthe HyMap images to five.The TRX Setting: Inspired by [23], for the purpose of mod-

eling the background using the TAD, the distance between everypair of pixels in a random set of 2% of the image is first com-puted. Then, a graph is constructed by adding an edge betweenthe closest 10% of pairs of pixels. Afterwards, the largest com-ponents, those components containing greater than 2% of thesample pixels, are designated as background.

C. Detection Performance

Evaluation Criteria: The primary way used to analyze theability of the AD methods is a two-dimensional display of thedetection results. The detection results can also be thresholdedto distinguish between likely targets and backgrounds. For agiven threshold, the detection performance of the algorithmscan be measured by the ratio of the false alarms to the detectedtargets. Based on assumption the distribution for the anomalyvalues of the background is Gaussian, a cutoff threshold can beused for comparing the detection performance of the algorithms.Such threshold is desired for remote sensing applications due tothe various environmental conditions a scene can be exposedto, and to the diverse clutter background in different geographiclocations [24]. Inspired by [24], an adaptive cutoff threshold canbe obtained as follows:

(12)

where is the cutoff threshold value at a given significant levelof and are the mean and standard deviation of the

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TABLE ITHE CHARACTERISTICS OF REAL TARGETS

anomaly values obtained for all pixels, respectively, andis the statistic at the significant level of , which controls thenumber of pixels declared as anomaly.In general, the detection performance of the AD algorithms

is evaluated using the experimental Receiver Operating Charac-teristic (ROC) curves. Nevertheless, an ROC curve using a fewdistinct targets is not a valid ROC curve in a statistical sense.Therefore, ROC curves cannot be used for evaluating the per-formance of the AD methods on the ROI-1, which contains the12 different real targets. In order to assess the relative detec-tion performance of the algorithms on the ROI-1, the FAR canbe measured separately for each target. However, in this study,the average FAR (AFAR) [25] is used as the main criterion forevaluating the detection performance of the algorithms over thesubpixel targets in both the ROIs.Results for the ROI-1: Fig. 2 shows the normalized detection

results using the AD algorithms where the positions of the self-test targets are superimposed. From Fig. 2, it can be observedthat the functionality of the SBAD is different from that of otheralgorithms.Fig. 3 shows the FAR obtained for each combination of al-

gorithm and target. As can be seen in Fig. 3, the SBAD is thebest algorithm for detecting the subpixel targets F3a, F4b, F6b,and F7a. Moreover, the SVDD provides the best performancefor detecting the subpixel targets F3b, F5b, and F7b. Also, theSRX is the best algorithm for discovering the subpixel targetsF4a, F5a, and F6a. Note that, the SBAD and the SVDD performthe worst out of all the AD algorithms at full pixel targets F1and F2, respectively. For detecting these targets, the SRX andthe TRX provide the best results, respectively.The AFAR and the average rank across all the subpixel targets

for each algorithm are reported in Table II. The results presentedin Table II demonstrate that the SBAD algorithm performs thebest out of the seven algorithms. Moreover, Fig. 4 shows thedetection map of the algorithms using the cutoff threshold at a(reasonable) significant level of 0.001 . It also showsthe detected targets and the false alarms obtained for each algo-rithm. From Fig. 4, the ratio of the number of the false alarmsto the number of detected targets is 36.2, 23.0, 17.7, 12.5, 11.7,11.3, and 8.1 for the RRX, KRX, GRX, SRX, SVDD, TRX,and SBAD, respectively. Comparative results indicate that theSBAD performs the best out of all the AD algorithms consid-ered at the significant level of 0.001.

Fig. 2. 2D detection results for ROI-1 using the AD algorithms. To obtain afair visual comparison, each detection map is normalized by its maximum value.The ground truth of the self-test targets have been superimposed on the detectionmaps.

Results for the ROI-2: Fig. 5 shows the AFAR values ob-tained by the AD methods at each implant fraction. As can beseen in Fig. 5, while all the algorithms, except the KRX andRRX, have the same performance at fraction 4/9, the SBADyields better results at fractions 1/9 and 2/9. Compared to theSVDD, which is the second best algorithm, the SBAD improves(i.e., decreases) the AFAR values by about 14% and 7% at frac-tions 1/9 and 2/9, respectively. Therefore, based on the exam-ined data, the detection performance of the SBAD is the best outof all the seven algorithms at small fractions.

D. Discussions About the Functionality of SBAD

In this section, we discuss below some of the functional char-acteristics of the SBAD algorithm.Discriminative Bands Found: As can be observed in Fig. 3,

the detection sequence of the real targets obtained by the SBADis F4a, F7a, F1, F4b, F3a, F3b, F6b, F6a, F2, F5a, F7b, andF5b. For this detection sequence, the discriminative bands are

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Fig. 3. The FAR obtained for each combination of algorithm and target on the ROI-1.

18, 97, 19, 18, 123, 76, 6, 3, 6, 19, 96, and 121, respectively.This demonstrates that the discriminative bands for each targetmay be different and each band may have a different anomalyvalue for a specific target.Sensitivity to the Bad Bands: To illustrate the sensitivity of

the SBAD to the bad bands, we conducted an experiment onROI-1 using all 126 bands. The experimental results show thatthe AFAR value obtained on the subpixel targets is 0.05 for theSBAD algorithm. Thus, in comparison with using all the bands,the bad band removal improves the AFAR value by 3%. This re-sult indicates that removing the bad bands results in a moderateimprovement in the detection performance of the SBAD.In Support of Using Only One Band: To support the hypoth-

esis that only the discriminative band should be used for sub-pixel AD, the SBAD is compared to a method that uses all spec-tral bands. Such a method is introduced as a uniform averagingof the anomaly values, called mean-band based AD (MBAD),which is very similar to the method proposed in [26]. The re-sults over the ROI-1 show that, the MBAD obtains an AFAR of0.12, which is 10% lower than the SBAD.To visually assess the detection functionality of the SBAD

compared to that of the MBAD, Fig. 6 shows the obtainedanomaly values of the bands for two subpixel targets F6a andF7b, each of which has a different material and size. It alsoshows the mean anomaly values of the bands for the localbackground pixels of the targets (which tend to belong to thesame class). The SBAD finds the discriminative bands of 3 and4 on target F6a and its local background, respectively. It alsofinds the discriminative bands of 96 and 1 on target F7b andits local background, respectively. However, as can be seenin Fig. 6, the target-background separation achieved by theSBAD is two and three times bigger than that of obtained bythe MBAD over the targets F6a and F7b, respectively.The above results demonstrate the superiority of the SBAD

over the MBAD. In other words, using only one band (i.e.,the discriminative band) for subpixel AD is more efficient thanusing all bands.Sensitivity to the Noise: The influence of the SNR on the de-

tection performance of the AD algorithms is evaluated over theROI-1. Inspired by the method proposed in [27], the noisy datacan be simulated by adding a noise to the reflectance image at

predefined SNR values. In our experiments, the noise is spec-trally correlated where its covariance matrix is Toeplitz matrix,whose elements for two arbitrary bands and are computed asfollows:

(13)

where is -dimensional vectors standing for signal data, andis the factor that controls the spectral correlation. In this study,we set to 0.8 and simulated noisy data at three SNR of 30, 50,and 70 dB. Fig. 7 shows the AFAR values achieved by the ADalgorithms at the three SNR values.The results show that the SBAD improves the AFAR value by

about 5% and 3% compared to the TRX and SVDD algorithms,both of which are the next best algorithms at SNR values of70 and 50 dB, respectively. Besides, as can be seen in Fig. 7, todetect the subpixel targets at low SNR of 30 dB, the best (i.e., thesmallest) AFAR values are obtained for the SBAD and SVDDalgorithms. From Fig. 7, it is worthwhile to note that none ofthe AD algorithms are robust against significant additive noisewhere SNR values are smaller than 50 dB.

E. Computational Efficiency

A main goal in developing an anomaly detector for hyper-spectral imagery is to have a computationally efficient algo-rithm. Hence, in the following, the computational complexityof the considered AD algorithms is discussed.The computational complexity of the RX of every pixel stems

from the inversion of the local covariance matrix, which in gen-eral is third in power in the number of bands [28]. In additionto this complexity, the RRX is required to perform eigenvaluedecomposition of the image in order to compute the regulariza-tion factor. Moreover, in the KRX, the time complexity is thirdin power in the number of pixels in the local background region.This is because of calculating the (pseudo) inverse of the cen-tered kernel matrices. On the other hand, the complexity of theGRX, SRX, and the TRX is linear in the number of the spectralbands. Note that the background modeling is a time-consumingprocess for the SRX because of its clustering procedure. Notealso that for the SRX, the number of clusters has to be selected a

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Fig. 4. Detection maps obtained by the AD methods at the statistically significant level of 0.001. The detected targets are marked by red points, while the blackpoints show false alarms.

TABLE IITHE AFAR AND THE AVERAGE RANK ACROSS ALL THE 10 SUBPIXEL TARGETSFOR EACH ALGORITHM. THE CORRESPONDING STANDARD DEVIATIONS (STD)

ARE ALSO REPORTED FOR EACH ALGORITHM

priori. Moreover, the backgroundmodeling in the TRX is highlydependent upon the size of the random set of the image.The complexity of the SVDD of every pixel is also third in

power in the number of pixels in the local background [29].Moreover, the free-parameter should be selected a priori. Incontrast to the AD algorithms considered, except the GRX, thecomputational efficiency of the SBAD is remarkable. This isbecause the SBAD requires neither preprocessing nor a free-parameter setting. Moreover, the computational complexity ofthe SBAD is linear in the number of the spectral bands. Thecomputational complexity of the AD algorithms considered and

Fig. 5. AFAR values of the algorithms obtained on ROI-2 at three implant frac-tions.

associated characteristics affecting algorithm performance aresummarized in Table III.As regards the SRX and TRX which require preprocessing

steps, in order to obtain a fair comparison between the ADalgorithms used, the processing time of the algorithms in the

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Fig. 6. The target-background separation obtained by the SBAD and MBAD on two targets F6a and F7b and their local backgrounds.

Fig. 7. AFAR values of the algorithms obtained on the subpixel targets inROI-1 at three SNR values.

whole of HyMap image, is computed. The algorithms are imple-mented using MATLAB and a personal computer (CPU speed:2.5-GHz; RAM memory: 4-GB). For comparison purposes, wehave reported only the core processing times. The results showthat the SBAD, GRX, and TRX algorithms have a processingtime of approximately 5.5 minutes. In the TRX, the backgroundmodeling takes about 12 seconds. Moreover, the results showthat the SRX, RRX, and KRX have processing times of 12, 22,

TABLE IIICOMPUTATIONAL COMPLEXITY AND ALGORITHM CHARACTERISTICS

AFFECTING ALGORITHM PERFORMANCE

and 24 minutes, respectively. As regards the SRX, the K-meansalgorithm takes 6.5 minutes to cluster the HyMap image. Ad-ditionally, the SVDD has a run-time of 54 minutes. Based onthese results, it can be concluded that the RRX, SRX, KRX,and SVDD are computationally infeasible for real-time AD ap-plications. Compared to the RRX, SRX, KRX, and SVDD, theSBAD saves the processing time by a factor of about 54%, 75%,77%, and 90%, respectively. This demonstrates that the SBADis computationally expedient. It is worthwhile to note that usingthe standard IDL/ENVI programming environment and parallelprocessing methods over the image subsets will significantly

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speed up the processing times. Consequently, the SBAD maybe applied in real-time AD.

V. CONCLUSION

In order to detect the anomalies at the subpixel level, thispaper introduces a novel approach, called SFAD. In this study,we evaluated the original variant of the SFAD, the SBAD, com-pared to the improved RX algorithms and the kernel-based ADmethods. The experimental results obtained on the real and syn-thetic data sets showed that, compared to the GRX and the fivestate-of-the-art algorithms considered, the SVDD, KRX, SRX,TRX, and RRX, the SBAD provides superior detection perfor-mance for detecting of subpixel anomalies. Moreover, the re-sults showed that the SBAD is computationally expedient incomparison with the SVDD, KRX, SRX, and RRX. However,based on the examined synthetic noisy data, efforts to achieve arobust detection performance for the real-time application of theSBAD (and all the algorithms used) are prone to false alarms insituations in which the data contain significant noise. Moreover,the SBAD would seem to be very sensitive to unknown sensorartifacts and scene conditions, which would lead to questionableperformance in real-world applications. To remedy these prob-lems, in future work, we will evaluate the use of the SFAD inreduced dimensional spaces obtained by approaches such as thePCA and the minimum noise fraction (MNF).

ACKNOWLEDGMENT

The authors would like to thank the Digital Imaging and Re-mote Sensing group Center for Imaging Science, Rochester In-stitute of Technology, Rochester, NY, for providing the TargetDetection Test data sets, Dr. J. P. Kerekes for his valuable helpin providing truth locations of the blind-test targets, and Dr. D.M. J. Tax (Delft University of Technology) for kindly offeringthe MATLAB dd_tools.

REFERENCES

[1] S. Matteoli, M. Diani, and G. Corsini, “A tutorial overview of anomalydetection in hyperspectral images,” IEEE A&E Sys. Mag., vol. 25, no.7, pp. 5–27, Jul. 2010.

[2] D. Manolakis and G. Shaw, “Detection algorithms for hyperspectralimaging application,” IEEE Signal Process. Mag., vol. 19, pp. 29–43,Jan. 2002.

[3] D. W. J. Stein, S. G. Beaven, L. E. Ho, E. M. Winter, A. P. Schaum,and A. D. Stocker, “Anomaly detection from hyperspectral imagery,”IEEE Signal Process. Mag., vol. 19, no. 1, pp. 58–69, Jan. 2002.

[4] I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection ofan optical pattern with unknown spectral distribution,” IEEE Trans.Acoust. Speech Signal Process., vol. 38, no. 10, pp. 1760–1770, Oct.1990.

[5] S. Matteoli, M. Diani, and G. Corsini, “Improved estimation of localbackground covariance matrix for anomaly detection in hyperspectralimages,” Opt. Eng., vol. 49, no. 4, 2010.

[6] J. A. Richards and X. Jia, Remote Sensing Digital Image Processing.New York: Springer-Verlag, 1993.

[7] N. M. Nasrabadi, “Regularization for spectral matched filter and RXanomaly detector,” in Proc. SPIE, 2008, vol. 6966, pp. 1–12.

[8] B. D. Bartlett, A. Schlamm, C. Salvaggio, and D. W. Messinger,“Anomaly detection of man-made objects using spectro-polarimetricimagery,” in Algorithms and Technologies for Multispectral, Hyper-spectral, and Ultraspectral Imagery XVII, Proc. SPIE, 2011, vol.8048, p. 80480B.

[9] W. F. Basener and D. W. Messinger, “Anomaly detection usingtopology,” in Algorithms and Technologies for Multispectral, Hyper-spectral, and Ultraspectral Imagery XIII, SPIE 6565, 2007.

[10] J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Anal-ysis. Cambridge, U.K.: Cambridge Univ. Press, 2004.

[11] H. Kwon and N. M. Nasrabadi, “Kernel RX-algorithm: A nonlinearanomaly detector for hyperspectral imagery,” IEEE Trans. Geosci. Re-mote Sens., vol. 43, no. 2, pp. 388–397, Feb. 2005.

[12] D. M. J. Tax and R. P. W. Duin, “Support vector domain description,”Pattern Recog. Lett., vol. 20, no. 11–13, pp. 1191–1199, Nov. 1999.

[13] A. Banerjee, P. Burlina, and C. Diehl, “A support vector method foranomaly detection in hyperspectral imagery,” IEEE Trans. Geosci. Re-mote Sens., vol. 44, no. 8, pp. 2282–2291, Aug. 2006.

[14] R. A.Maronna, R. D.Martin, and V. J. Yohai, Robust Statistics: Theoryand Methods. New York: Wiley, 2006.

[15] J. P. Kerekes and D. Snyder, “Target Detection Blind Test,” [Online].Available: http://dirsapps.cis.rit.edu/blindtest/

[16] D. Snyder, J. P. Kerekes, I. Fairweather, R. Crabtree, J. Shive, andS. Hager, “Development of a web-based application to evaluate targetfinding algorithms,” in Proc. IGARSS, 2008, vol. 2, pp. 915–918.

[17] J. M. Molero, A. Paz, E. M. Garzon, J. A. Martinez, A. Plaza, andI. Garcia, “Fast anomaly detection in hyperspectral images with RXmethod on heterogeneous clusters,” J. Supercomput., vol. 58, no. 3,Dec. 2011.

[18] S. Khazai, S. Homayouni, A. Safari, and B. Mojaradi, “Anomaly de-tection in hyperspectral images based on an adaptive support vectormethod,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 2, pp. 646–650,Jul. 2011.

[19] “MERIS Product Handbook,” European Space Agency, 2011, no. 3.[20] E. J. Ientilucci, “Hyperspectral Sub-Pixel Target Detection Using

Hybrid Algorithms and Physics Based Modeling,” Ph.D. dissertation,Rochester Inst. Technol., Rochester, NY, 2005.

[21] M. Bachmann, J. Biesemans, J. Hanus, M. Kneubuehler, I. Perez Gon-zalez, and T. Ruhtz, “Quality Layers for VITO, DLR, INTA and PML,”EUFAR FP7—report DJ2.2.2, 2011, DLR-DFD.

[22] H. Kwon, S. Z. Der, and N. M. Nasrabadi, “Dual-window-basedanomaly detection for hyperspectral imagery,” in Proc. SPIE, 2003,vol. 5094, p. 148.

[23] W. F. Basener and D. W. Messinger, “Enhanced detection and visual-ization of anomalies in spectral imagery,” Algorithms and Technologiesfor Multispectral, Hyperspectral, and Ultraspectral Imagery XV, SPIE7334(1), p. 73341Q, 2009.

[24] D. S. Rosario, “Algorithm Development for Hyperspectral AnomalyDetection,” Ph.D. dissertation, Univ. Maryland, Baltimore, 2008.

[25] P. Bajorski, E. J. Ientilucci, and J. R. Schott, “Comparison of ba-sisvector selection methods for target and background subspaces asapplied to subpixel target detection,” in Proc. SPIE, 2004, vol. 5425,pp. 97–108.

[26] C. E. Caefer, S. R. Rotman, J. Silverman, and P. W. Yip, “Algorithmsfor point target detection in hyperspectral imagery,” in Proc. SPIE,2002, vol. 4816.

[27] J. M. Bioucas-Diasa and J. M. P. Nascimento, “Hyperspectral subspaceidentification,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 8, pp.2435–2445, Aug. 2008.

[28] S. M. Schweizer and J. M. F. Moura, “Efficient detection in hyperspec-tral imagery,” IEEE Trans. Image Process., vol. 10, no. 4, pp. 584–597,Apr. 2001.

[29] Y.-H. Liu, Y.-C. Liu, and Y.-J. Chen, “Fast support vector data descrip-tions for novelty detection,” IEEE Trans. Neural Netw., vol. 21, no. 8,pp. 1296–1313, Aug. 2010.

Safa Khazai received the B.S. degree in surveyingand geomatics engineering from Imam Hosein Uni-versity, Tehran, Iran, in 1999, and the M.S. and Ph.D.degrees in photogrammetry and remote sensing fromthe University of Tehran, Tehran, Iran, in 2002 and2012, respectively.His main research interests include anomaly and

target detection in hyperspectral images, hyperspec-tral image simulation, and kernel methods.

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Abdolreza Safari received the B.S. degree insurveying and geomatics engineering, and the M.S.and Ph.D. degrees in geodesy from the Departmentof Surveying and Geomatics Engineering, Collegeof Engineering, University of Tehran, Iran, in 1993,1998, and 2004, respectively.He was an Assistant Professor in the Department

of Surveying and Geomatics Engineering of TehranUniversity from 2004 to 2011 and currently is anAssociate Professor in the same department. Hisresearch interests are the mathematical modeling of

remote sensing and geodetic data.Dr. Safari is a member of the Center of Excellence for Surveying Engineering

in Natural Disaster Management, Department of surveying and Geomatics En-gineering, College of Engineering, University of Tehran.

Barat Mojaradi (M’12) received the B.S. degreein surveying and geomatics engineering from TabrizUniversity, Iran, in 1998, and the M.S. degree, andPh.D. degree in remote sensing, from K.N. ToosiUniversity of Technology, Tehran, Iran, in 2000 and2009, respectively.He is currently an Assistant Professor at Iran Uni-

versity of Science and Technology, Tehran, Iran. Hismain research interests are in remote sensing, patternrecognition, image processing and soft computing.

Saeid Homayouni (M’05) received the B.S. degreein surveying and geomatics engineering fromUniver-sity of Isfahan, Isfahan, Iran, in 1996, theM.S. degreein remote sensing and geographic information sys-tems from Tarbiat Modaress University, Tehran, Iran,in 1999, and the Ph.D. degree in signal and imagefrom Telecom of Paris, Paris, France, in 2006.He is currently with Department of Geomatics and

Surveying, College of Engineering, University ofTehran, Iran, as an Assistant Professor. His researchactivities are focused on remote sensing image

analysis for urban and agro-environmental applications.

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An Approach for Subpixel Anomaly Detectionin Hyperspectral Images

Safa Khazai, Abdolreza Safari, Barat Mojaradi, Member, IEEE, and Saeid Homayouni, Member, IEEE

Abstract—Fast detecting difficult targets such as subpixel objectsis a fundamental challenge for anomaly detection (AD) in hyper-spectral images. In an attempt to solve this problem, this paperpresents a novel but simple approach based on selecting a singlefeature forwhich the anomaly value is themaximum.The proposedapproach applied in the original feature space has been evaluatedand comparedwith relevant state-of-the-art ADmethods onTargetDetection Blind Test data sets. Preliminary results suggest that theproposed method can achieve better detection performance thanits counterparts. The results also show that the proposed methodis computationally expedient.

Index Terms—Hyperspectral images, anomaly detection, singlefeature, single band.

I. INTRODUCTION

H YPERSPECTRAL images have a great potential to auto-matic target detection, since they deliver valuable infor-

mation about the spectral characteristics of the earth surface ob-jects and phenomena. Anomaly detection (AD) is a well-knownunsupervised approach for automatic target detection. It has at-tracted significant interest in hyperspectral remote sensing ap-plications such as detecting crop stress locations in precisionagriculture, rare minerals in geology, oil pollution in environ-mental researches, landmines in the public safety and defensedomain, and man-made objects in reconnaissance and surveil-lance applications [1].The general aim of AD is to find the targets which have dis-

tinct spectral behavior from their background without a prioriknowledge of the target signature and any form of atmosphericcompensation. The anomalies are normally defined with refer-ence to a model of background. The local and global spectralanomalies are respectively defined as the observations that differin some way from the neighboring background and from theoverall, or a substantial part of, the scene. Generally, the localmethods for AD yield more reliable results. However, they willsometimes cause a higher false alarm rate (FAR) for isolatedspectral anomalies [2], [3]. For example, an isolated tree in a lo-cally homogeneous patch of grass is detected as a local anomalyeven if the whole image contains a forest [1].

Manuscript received January 04, 2012; revised March 16, 2012, May 02,2012; accepted July 16, 2012. This work was supported by the University ofTehran under Grant 8103618/1/07.S. Khazai, A. Safari, and S. Homayouni are with the Department of Surveying

and Geomatics Engineering, College of Engineering, University of Tehran,Tehran 14395-515, Iran (corresponding author e-mail: khazai@ut.ac.ir).B. Mojaradi is with School of Civil Engineering, Iran University of Science

and Technology, Tehran, Iran.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.2012.2210277

The AD algorithms are typically compared with Reed-Xi-aoli (RX) algorithm [4], which is based on the well-knownMahalanobis distance between the spectral vectors of an inputtest pixel and its surrounding neighbors. The most well-knownproblem with the RX is the small-sample size, which concernsthe estimation of a local background covariance matrix from asmall number of samples in high-dimensional space [5]. Underthis circumstance, the result is a badly-conditioned and unstableestimation. In the case of an -dimensional data, at leastsamples are needed to estimate a covariance matrix. However,many more samples (possibly about ) must actuallybe used to obtain a reliable estimation [6]. To overcome thesmall-sample size problem without considering dimensionreduction solution, two approaches have been explored in thehyperspectral AD literature.The first approach consists of improved versions of the RX

algorithm. It basically aimed at improving estimations of thecovariance matrix. The conventional algorithm, denoted asglobal RX (GRX), estimates a global covariance matrix fromthe entire image. However, Regularized-RX (RRX) [7] is anew algorithm that regularizes the local covariance matrix.Also, Segmented-RX (SRX) [5] is a recent algorithm whichmodels the background using a clustering of all image pixels.Moreover, the topology-based RX (TRX) [8] can be consideredas a state-of-the-art improved-RX algorithm that uses the Topo-logical Anomaly Detector (TAD) for background modeling ofimages [9].The second approach involves the kernel-based (i.e., non-

linear) AD methods, which are based on the kernel theory forextending the original space to a higher dimensional featurespace, the so-called Hilbert space. Kernel functions are used toimplicitly compute the dot products in Hilbert space, withoutactually mapping the input vectors into that space [10]. Hence,the kernel-based methods can deal very well with high dimen-sional data. The kernel-based AD methods can be groupedinto two categories: parametric and non-parametric algorithms.The well-known parametric kernel-based algorithm is theKernel-RX (KRX) [11], which is the nonlinear version of theRX. Moreover, a state-of-the-art non-parametric kernel-basedalgorithm is the support vector data description (SVDD) [12].The study [13] introduced a general framework to use theSVDD for AD in hyperspectral images.Detecting subpixel (i.e., mixed) objects, such as mineral and

crop species, which occupy a portion of the pixel area andwhosesignatures slightly deviate from the background, is one of thefundamental challenges for hyperspectral imaging applications[2]. This study presents a novel but simple AD approach aimed

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at quickly detecting the subpixel targets in hyperspectral im-ages. The proposed approach uses only a single feature that re-veals the anomaly better than others. The performance of theproposed approach is compared to that of the improved RX al-gorithms and the kernel-based AD methods.The rest of this paper is organized as follows. Section II pro-

vides more details about the algorithms used for AD. The pro-posed approach is presented in Section III. Experimental re-sults are presented in Section IV. Lastly, concluding remarksare given in Section V.

II. THE ANOMALY DETECTORS

A. The Improved RX Algorithms

The RX test statistic is the Mahalanobis distance between thepixel being tested with the spectrum and the local backgroundwith mean spectrum as follows:

(1)

where is the estimated covariance matrix from samples sur-rounding the test pixel. The RRX regularizes by adding ascaled identity matrix before inversion. The scale (i.e., regular-ization) factor is given by computing the median eigenvalue ofthe covariance matrix estimated on the whole image. Moreover,the SRX requires that a clustering of the image to be performed.After obtaining a thematic map, the covariance matrix is es-timated over each cluster. Then, the well-known Mahalanobisdistance is computed between the target pixel and its spectrallynearest cluster. The covariance matrix required can be either es-timated over the cluster that is most common in the pixel neigh-borhood, or by computing a mixture of the covariance matricesof the clusters represented in the pixel neighborhood [5].On the other hand, the TRX requires that the global covari-

ance matrix is computed using only those pixels that are as-signed into the background by the TAD [9]. In addition to pro-viding a novel AD algorithm, the TAD incorporates a topo-logical approach to modeling hyperspectral data. However, inTRX, the TAD is used to remove the anomalous pixels from in-fluencing the background covariance matrix. The TAD buildsa graph whose edges connect close pairs of pixels. The back-ground pixels are the pixels in the largest components of thisgraph [8].

B. The Kernel-Based AD Algorithms

The most effective kernel function for kernel-based methodsis Gaussian given by , whereand are two objects in the original space, and denotes

the Gaussian kernel width (i.e., the sigma).The KRX: The KRX is the nonlinear version of the RX that

adopts the Gaussian model in the Hilbert space. The KRX iscompactly given by [11]:

(2)

where

(3)

(4)

where is the centered kernel matrix of training pixels (i.e.,local background pixels), and is thetotal number of training pixels.The SVDD: The basic idea of SVDD to find a hypersphere

withminimum volume that encloses all the normal training sam-ples of the target class [12]. Given training samplesbelonging to a class of interest and a kernel function , theSVDD problem becomes the following:

(5)

with two constraints on Lagrange multipliers (and . The user-specified parameteris called the rejection rate indicating an upper bound on thefraction of outliers. The objects with nonzero are calledthe support vectors (SVs). To test if a new object is outsidethe obtained hypersphere, the squared Euclidean distance to thecenter of the hypersphere, called the SVDD test statistic, has tobe calculated. A test object is rejected when this statistic isbigger than the squared radius [12]:

(6)

where can be determined by calculating the distance from thecenter of the hypersphere to any of the SVs on the boundary,i.e., objects with . The study [13] showedthat, to improve the detection performance of the SVDD for ADin the hyperspectral images, a normalized version of the SVDDtest statistic can be derived through dividing the original teststatistic by the squared radius.

III. PROPOSED APPROACH

Let us consider a case study application in which the subpixelobjects are the interesting targets for AD. From a physical stand-point, a subpixel object is usually strongly mixed with its neigh-boring objects in most spectral regions of its pixel spectrum.Nevertheless, there is almost a high probability that a subpixelobject, especially a man-made one, has a significant impact onsome spectral regions of its pixel spectrum. Therefore, such atarget may be easily discriminated from the local backgroundin those spectral regions or bands. Evidently, the target-back-ground discrimination may be improved using some featuresextracted from the spectral bands. The features can be obtainedby linear transformations such as the principal component anal-ysis (PCA) and kernel-based methods such as the kernel-PCA.However, it may be concluded that the success of a subpixel ADalgorithm depends on its ability to find and use the best featurefor discriminating between the target pixel and the local back-ground pixels. We have appealed to this important point in ourproposal for the AD, which we named the single-feature basedanomaly detector (SFAD).

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Fig. 1. The true color composite of the HyMap radiance image. The left box specifies the ROI-1 and contains 6 self-test targets The right box shows the ROI-2and contains 20 synthetic targets.

The SFAD determines the anomaly value of target pixel inthe feature , as follows:

(7)

where is the value of feature for target pixel , and isthe median value of local background pixels of in the feature, and is obtained as follows:

(8)

where is the value of the feature for the neighboring pixel. Note that is the median absolute deviation (MAD) of thefeature . The MAD, which is a robust measure of dispersion[14], is defined as the median of the absolute deviations fromthe data’s median. However, SFAD test statistic for the targetpixel is obtained as follows:

(9)

where is the number of appropriate features. Note that theSFAD evaluates all the features, and for each target, tries to se-lect a specific feature for which the anomaly value is the max-imum. We have named this feature the discriminative feature. Itis worthwhile to note that the SFAD applies a kind of weightingto the anomaly values, where the discriminative feature that pro-vides the maximum value takes , and other featurestake . In this work, we employ the physical (i.e.,primary) variant of the SFAD, called the single-band based AD(SBAD), which evaluates all the spectral bands in order to se-lect a discriminative band.As with any AD algorithm, removal of low signal-to-noise

ratio (SNR) channels, i.e., bad bands, increases the detectionperformance. However, the SBAD is more sensitive to removalof these bands than the conventional AD methods, which useall spectral bands. Since the SBAD chooses one band to per-form AD, the discriminative bands found may be discriminativebecause of the presence of bad bands, not because of the pres-ence of the anomalies. Note that the bad bands caused mainlyby sensor failure or strong water-vapor absorption provide ei-ther little useful information or no information to be extractedabout the anomalies. Therefore, removing such bands in the datais required before performing the SBAD.

IV. EXPERIMENTAL RESULTS

A. Data Sets

The Target Detection Blind Test data sets [15], [16] includetwo HyMap radiance and reflectance images of Cooke City inMontana, USA (see Fig. 1). The imageswere collected by an air-borne HyMap sensor, which has 126 spectral bands. The groundresolution of imagery data is approximately 3 m. In the imagescenes, some real targets were located in an open grass regionduring the image acquisition.Evidently, to deal with large hyperspectral images, the AD

methods suffer from high computational costs. To overcomethis problem, one can partition the whole image into some ap-proximately equal subsets, and then apply AD methods on eachsubset independent of the others by parallel processing. Here,it is worthwhile to refer the study [17] that developed a generalstrategy to automatically map parallel hybrid AD algorithms inhyperspectral images. However, based on a partitioning proce-dure for which the size of subsets is 90 90, two regions ofinterest (ROI), ROI-1 and ROI-2, are selected in the HyMapimages.Real Targets: The ROI-1 consists of the twelve real targets.

Two of the real targets are at the full pixel (i.e., resolved) size;the other ten are at the subpixel sizes. The targets included sixfabric panels for the self-test and six for the blind-test. Fig. 1shows the location of six self-test targets, while Table I brieflydescribes each target in more detail.Synthetic Targets: In this study, we used the target implant

method used in [18] to evaluate the performance of AD algo-rithms on a wide variety of subpixel targets. Based on a linearmixing model, a synthetic subpixel anomaly with spectrum isgenerated by fractionally implanting a desired pixel target witha reflectance spectrum, in a given pixel in the background (i.e.,host pixel) with a reflectance spectrum as follows:

(10)

where denotes the implant fraction, and are thereflectance and implanted spectrum of the -th neighbor of thehost pixel, respectively, and is the spatial Euclidean distancebetween the -th adjacent pixel and the host pixel.

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To generate the synthetic targets, we used the reflectance ver-sion of the HyMap image and took the spectrum of the purepixel targets F1 and F2 as two different target types. Afterward,to create synthetic targets in the ROI-2, each of the spectra wasimplanted separately at 20 host pixels with consideration of theiradjacent effects in a constant window size of 5 5. Moreover,each implant is accomplished at three fractions: 1/9, 2/9, and4/9. Note that at each implant fraction two synthetic data setsare generated using the targets F1 and F2.

B. Preprocessing

Before the performing of the AD algorithms, some prepro-cessing steps should be done as follows:Removing Saturated Pixels: Saturated pixels caused tem-

porarily by, e.g., sun glint, bright cloud, bright land or snow/ice,contain signals that are saturated in some spectral bands [19].Such pixels appear as targets to hyperspectral detection al-gorithms; however, they are not genuine targets [20]. Thus,AD algorithms would produce false alarms on the saturatedpixels. Hence, to improve the performance of AD, removingthe saturated pixels should be taken into consideration beforeperforming all the algorithms. Note that since the saturatedbands will surely result as discriminative bands, the SBAD ismore sensitive than the other AD algorithms to the problemof saturated pixels. The saturated pixels can be easily detectedby searching for the radiance values of the data that are biggerthan a given value (e.g., 90%) of saturation levels defined [21].It is worthwhile to note that the saturation levels are sensordependent and based on industry specifications or instrumentcharacterization runs in calibration facilities [21]. In fact, theywould depend upon the particular settings of the instrument atthe time of collection. In this study, since the saturation levelsfor the data sets used are not known, we manually investigatedthe existence of the saturated pixels by searching for exces-sively large spectral values in the radiance image. The resultsshowed that there are no saturated pixels in the images.Removing the Bad Bands in the Radiance Image: The bad

bands may be specified a priori. However, in this study to dis-tinguish the bad bands, a heuristic method is employed. Theheuristic method used assigns a loss level for each band equal tothe inverse of the variance of the data in that band. Obviously,the bad bands have a value of the variance that is very close tozero. Thus, they will have excessively large loss levels. Con-sidering that most of the bands are normal, in this study the badbands are defined as the bands that have the loss levelsmore thana 2.5 MAD away from the median of the obtained loss levels ofall bands. By using this solution, three outlier bands of 63, 64,and 126 were found in the HyMap radiance image. After re-moving these bands, the number of remaining bands was 123.Normalization of the Radiance Image: All pixel vectors, in

the whole of the radiance image, are normalized by a maximumspectral value in the image, so that the entries of the normalizedpixel vectors fit into the interval of spectral values between zeroand one. Note that the reflectance image does not need data nor-malization, as the values in this image range from zero to one.DetectionWindow Setting: An important decision for the AD

methods is the specifying the local background pixels for eachtarget pixel. This decision is usually made through a dual con-

centric windowwhich separates the local area around each pixelinto two regions, a small inner window region (IWR) centeredwithin a larger outer window region (OWR) [22]. The IWR isused to enclose the target of interest to be detected, while theOWR is employed to model the local background around thetarget region. In the case of subpixel anomalies, the size of IWRis always 1 1 based on the assumption that each target of in-terest completely enclosed in a pixel. Hence, only the size ofOWR, called the detection window, should be set. Since there isno a specific method to choose the size of the detection window[13], a detection windows of 5 5 is used for all the experi-ments.The Sigma Setting: When using the Gaussian kernel, the

main problem of the kernel-based AD algorithms, the SVDDand the KRX, is the optimal setting of the sigma parameter. Inthis study, we used the sigma estimation method given by [18]:

(11)

where is the maximum Euclidean spectral distance be-tween training instances (i.e., surrounding pixels of the targetpixel). The parameter is set experimentally by users. We setto 0.1, which means 10% pixels in the detection window are

allowed to be outliers.The SRX Setting: In order to segment the datacube, we

employed the well-known K-means algorithm. However, themain problem with using clustering algorithms such as theK-means one is the choice of the optimum number of clusters.In this study, we experimentally set the number of clusters inthe HyMap images to five.The TRX Setting: Inspired by [23], for the purpose of mod-

eling the background using the TAD, the distance between everypair of pixels in a random set of 2% of the image is first com-puted. Then, a graph is constructed by adding an edge betweenthe closest 10% of pairs of pixels. Afterwards, the largest com-ponents, those components containing greater than 2% of thesample pixels, are designated as background.

C. Detection Performance

Evaluation Criteria: The primary way used to analyze theability of the AD methods is a two-dimensional display of thedetection results. The detection results can also be thresholdedto distinguish between likely targets and backgrounds. For agiven threshold, the detection performance of the algorithmscan be measured by the ratio of the false alarms to the detectedtargets. Based on assumption the distribution for the anomalyvalues of the background is Gaussian, a cutoff threshold can beused for comparing the detection performance of the algorithms.Such threshold is desired for remote sensing applications due tothe various environmental conditions a scene can be exposedto, and to the diverse clutter background in different geographiclocations [24]. Inspired by [24], an adaptive cutoff threshold canbe obtained as follows:

(12)

where is the cutoff threshold value at a given significant levelof and are the mean and standard deviation of the

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TABLE ITHE CHARACTERISTICS OF REAL TARGETS

anomaly values obtained for all pixels, respectively, andis the statistic at the significant level of , which controls thenumber of pixels declared as anomaly.In general, the detection performance of the AD algorithms

is evaluated using the experimental Receiver Operating Charac-teristic (ROC) curves. Nevertheless, an ROC curve using a fewdistinct targets is not a valid ROC curve in a statistical sense.Therefore, ROC curves cannot be used for evaluating the per-formance of the AD methods on the ROI-1, which contains the12 different real targets. In order to assess the relative detec-tion performance of the algorithms on the ROI-1, the FAR canbe measured separately for each target. However, in this study,the average FAR (AFAR) [25] is used as the main criterion forevaluating the detection performance of the algorithms over thesubpixel targets in both the ROIs.Results for the ROI-1: Fig. 2 shows the normalized detection

results using the AD algorithms where the positions of the self-test targets are superimposed. From Fig. 2, it can be observedthat the functionality of the SBAD is different from that of otheralgorithms.Fig. 3 shows the FAR obtained for each combination of al-

gorithm and target. As can be seen in Fig. 3, the SBAD is thebest algorithm for detecting the subpixel targets F3a, F4b, F6b,and F7a. Moreover, the SVDD provides the best performancefor detecting the subpixel targets F3b, F5b, and F7b. Also, theSRX is the best algorithm for discovering the subpixel targetsF4a, F5a, and F6a. Note that, the SBAD and the SVDD performthe worst out of all the AD algorithms at full pixel targets F1and F2, respectively. For detecting these targets, the SRX andthe TRX provide the best results, respectively.The AFAR and the average rank across all the subpixel targets

for each algorithm are reported in Table II. The results presentedin Table II demonstrate that the SBAD algorithm performs thebest out of the seven algorithms. Moreover, Fig. 4 shows thedetection map of the algorithms using the cutoff threshold at a(reasonable) significant level of 0.001 . It also showsthe detected targets and the false alarms obtained for each algo-rithm. From Fig. 4, the ratio of the number of the false alarmsto the number of detected targets is 36.2, 23.0, 17.7, 12.5, 11.7,11.3, and 8.1 for the RRX, KRX, GRX, SRX, SVDD, TRX,and SBAD, respectively. Comparative results indicate that theSBAD performs the best out of all the AD algorithms consid-ered at the significant level of 0.001.

Fig. 2. 2D detection results for ROI-1 using the AD algorithms. To obtain afair visual comparison, each detection map is normalized by its maximum value.The ground truth of the self-test targets have been superimposed on the detectionmaps.

Results for the ROI-2: Fig. 5 shows the AFAR values ob-tained by the AD methods at each implant fraction. As can beseen in Fig. 5, while all the algorithms, except the KRX andRRX, have the same performance at fraction 4/9, the SBADyields better results at fractions 1/9 and 2/9. Compared to theSVDD, which is the second best algorithm, the SBAD improves(i.e., decreases) the AFAR values by about 14% and 7% at frac-tions 1/9 and 2/9, respectively. Therefore, based on the exam-ined data, the detection performance of the SBAD is the best outof all the seven algorithms at small fractions.

D. Discussions About the Functionality of SBAD

In this section, we discuss below some of the functional char-acteristics of the SBAD algorithm.Discriminative Bands Found: As can be observed in Fig. 3,

the detection sequence of the real targets obtained by the SBADis F4a, F7a, F1, F4b, F3a, F3b, F6b, F6a, F2, F5a, F7b, andF5b. For this detection sequence, the discriminative bands are

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Fig. 3. The FAR obtained for each combination of algorithm and target on the ROI-1.

18, 97, 19, 18, 123, 76, 6, 3, 6, 19, 96, and 121, respectively.This demonstrates that the discriminative bands for each targetmay be different and each band may have a different anomalyvalue for a specific target.Sensitivity to the Bad Bands: To illustrate the sensitivity of

the SBAD to the bad bands, we conducted an experiment onROI-1 using all 126 bands. The experimental results show thatthe AFAR value obtained on the subpixel targets is 0.05 for theSBAD algorithm. Thus, in comparison with using all the bands,the bad band removal improves the AFAR value by 3%. This re-sult indicates that removing the bad bands results in a moderateimprovement in the detection performance of the SBAD.In Support of Using Only One Band: To support the hypoth-

esis that only the discriminative band should be used for sub-pixel AD, the SBAD is compared to a method that uses all spec-tral bands. Such a method is introduced as a uniform averagingof the anomaly values, called mean-band based AD (MBAD),which is very similar to the method proposed in [26]. The re-sults over the ROI-1 show that, the MBAD obtains an AFAR of0.12, which is 10% lower than the SBAD.To visually assess the detection functionality of the SBAD

compared to that of the MBAD, Fig. 6 shows the obtainedanomaly values of the bands for two subpixel targets F6a andF7b, each of which has a different material and size. It alsoshows the mean anomaly values of the bands for the localbackground pixels of the targets (which tend to belong to thesame class). The SBAD finds the discriminative bands of 3 and4 on target F6a and its local background, respectively. It alsofinds the discriminative bands of 96 and 1 on target F7b andits local background, respectively. However, as can be seenin Fig. 6, the target-background separation achieved by theSBAD is two and three times bigger than that of obtained bythe MBAD over the targets F6a and F7b, respectively.The above results demonstrate the superiority of the SBAD

over the MBAD. In other words, using only one band (i.e.,the discriminative band) for subpixel AD is more efficient thanusing all bands.Sensitivity to the Noise: The influence of the SNR on the de-

tection performance of the AD algorithms is evaluated over theROI-1. Inspired by the method proposed in [27], the noisy datacan be simulated by adding a noise to the reflectance image at

predefined SNR values. In our experiments, the noise is spec-trally correlated where its covariance matrix is Toeplitz matrix,whose elements for two arbitrary bands and are computed asfollows:

(13)

where is -dimensional vectors standing for signal data, andis the factor that controls the spectral correlation. In this study,we set to 0.8 and simulated noisy data at three SNR of 30, 50,and 70 dB. Fig. 7 shows the AFAR values achieved by the ADalgorithms at the three SNR values.The results show that the SBAD improves the AFAR value by

about 5% and 3% compared to the TRX and SVDD algorithms,both of which are the next best algorithms at SNR values of70 and 50 dB, respectively. Besides, as can be seen in Fig. 7, todetect the subpixel targets at low SNR of 30 dB, the best (i.e., thesmallest) AFAR values are obtained for the SBAD and SVDDalgorithms. From Fig. 7, it is worthwhile to note that none ofthe AD algorithms are robust against significant additive noisewhere SNR values are smaller than 50 dB.

E. Computational Efficiency

A main goal in developing an anomaly detector for hyper-spectral imagery is to have a computationally efficient algo-rithm. Hence, in the following, the computational complexityof the considered AD algorithms is discussed.The computational complexity of the RX of every pixel stems

from the inversion of the local covariance matrix, which in gen-eral is third in power in the number of bands [28]. In additionto this complexity, the RRX is required to perform eigenvaluedecomposition of the image in order to compute the regulariza-tion factor. Moreover, in the KRX, the time complexity is thirdin power in the number of pixels in the local background region.This is because of calculating the (pseudo) inverse of the cen-tered kernel matrices. On the other hand, the complexity of theGRX, SRX, and the TRX is linear in the number of the spectralbands. Note that the background modeling is a time-consumingprocess for the SRX because of its clustering procedure. Notealso that for the SRX, the number of clusters has to be selected a

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Fig. 4. Detection maps obtained by the AD methods at the statistically significant level of 0.001. The detected targets are marked by red points, while the blackpoints show false alarms.

TABLE IITHE AFAR AND THE AVERAGE RANK ACROSS ALL THE 10 SUBPIXEL TARGETSFOR EACH ALGORITHM. THE CORRESPONDING STANDARD DEVIATIONS (STD)

ARE ALSO REPORTED FOR EACH ALGORITHM

priori. Moreover, the backgroundmodeling in the TRX is highlydependent upon the size of the random set of the image.The complexity of the SVDD of every pixel is also third in

power in the number of pixels in the local background [29].Moreover, the free-parameter should be selected a priori. Incontrast to the AD algorithms considered, except the GRX, thecomputational efficiency of the SBAD is remarkable. This isbecause the SBAD requires neither preprocessing nor a free-parameter setting. Moreover, the computational complexity ofthe SBAD is linear in the number of the spectral bands. Thecomputational complexity of the AD algorithms considered and

Fig. 5. AFAR values of the algorithms obtained on ROI-2 at three implant frac-tions.

associated characteristics affecting algorithm performance aresummarized in Table III.As regards the SRX and TRX which require preprocessing

steps, in order to obtain a fair comparison between the ADalgorithms used, the processing time of the algorithms in the

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Fig. 6. The target-background separation obtained by the SBAD and MBAD on two targets F6a and F7b and their local backgrounds.

Fig. 7. AFAR values of the algorithms obtained on the subpixel targets inROI-1 at three SNR values.

whole of HyMap image, is computed. The algorithms are imple-mented using MATLAB and a personal computer (CPU speed:2.5-GHz; RAM memory: 4-GB). For comparison purposes, wehave reported only the core processing times. The results showthat the SBAD, GRX, and TRX algorithms have a processingtime of approximately 5.5 minutes. In the TRX, the backgroundmodeling takes about 12 seconds. Moreover, the results showthat the SRX, RRX, and KRX have processing times of 12, 22,

TABLE IIICOMPUTATIONAL COMPLEXITY AND ALGORITHM CHARACTERISTICS

AFFECTING ALGORITHM PERFORMANCE

and 24 minutes, respectively. As regards the SRX, the K-meansalgorithm takes 6.5 minutes to cluster the HyMap image. Ad-ditionally, the SVDD has a run-time of 54 minutes. Based onthese results, it can be concluded that the RRX, SRX, KRX,and SVDD are computationally infeasible for real-time AD ap-plications. Compared to the RRX, SRX, KRX, and SVDD, theSBAD saves the processing time by a factor of about 54%, 75%,77%, and 90%, respectively. This demonstrates that the SBADis computationally expedient. It is worthwhile to note that usingthe standard IDL/ENVI programming environment and parallelprocessing methods over the image subsets will significantly

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speed up the processing times. Consequently, the SBAD maybe applied in real-time AD.

V. CONCLUSION

In order to detect the anomalies at the subpixel level, thispaper introduces a novel approach, called SFAD. In this study,we evaluated the original variant of the SFAD, the SBAD, com-pared to the improved RX algorithms and the kernel-based ADmethods. The experimental results obtained on the real and syn-thetic data sets showed that, compared to the GRX and the fivestate-of-the-art algorithms considered, the SVDD, KRX, SRX,TRX, and RRX, the SBAD provides superior detection perfor-mance for detecting of subpixel anomalies. Moreover, the re-sults showed that the SBAD is computationally expedient incomparison with the SVDD, KRX, SRX, and RRX. However,based on the examined synthetic noisy data, efforts to achieve arobust detection performance for the real-time application of theSBAD (and all the algorithms used) are prone to false alarms insituations in which the data contain significant noise. Moreover,the SBAD would seem to be very sensitive to unknown sensorartifacts and scene conditions, which would lead to questionableperformance in real-world applications. To remedy these prob-lems, in future work, we will evaluate the use of the SFAD inreduced dimensional spaces obtained by approaches such as thePCA and the minimum noise fraction (MNF).

ACKNOWLEDGMENT

The authors would like to thank the Digital Imaging and Re-mote Sensing group Center for Imaging Science, Rochester In-stitute of Technology, Rochester, NY, for providing the TargetDetection Test data sets, Dr. J. P. Kerekes for his valuable helpin providing truth locations of the blind-test targets, and Dr. D.M. J. Tax (Delft University of Technology) for kindly offeringthe MATLAB dd_tools.

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Safa Khazai received the B.S. degree in surveyingand geomatics engineering from Imam Hosein Uni-versity, Tehran, Iran, in 1999, and the M.S. and Ph.D.degrees in photogrammetry and remote sensing fromthe University of Tehran, Tehran, Iran, in 2002 and2012, respectively.His main research interests include anomaly and

target detection in hyperspectral images, hyperspec-tral image simulation, and kernel methods.

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Abdolreza Safari received the B.S. degree insurveying and geomatics engineering, and the M.S.and Ph.D. degrees in geodesy from the Departmentof Surveying and Geomatics Engineering, Collegeof Engineering, University of Tehran, Iran, in 1993,1998, and 2004, respectively.He was an Assistant Professor in the Department

of Surveying and Geomatics Engineering of TehranUniversity from 2004 to 2011 and currently is anAssociate Professor in the same department. Hisresearch interests are the mathematical modeling of

remote sensing and geodetic data.Dr. Safari is a member of the Center of Excellence for Surveying Engineering

in Natural Disaster Management, Department of surveying and Geomatics En-gineering, College of Engineering, University of Tehran.

Barat Mojaradi (M’12) received the B.S. degreein surveying and geomatics engineering from TabrizUniversity, Iran, in 1998, and the M.S. degree, andPh.D. degree in remote sensing, from K.N. ToosiUniversity of Technology, Tehran, Iran, in 2000 and2009, respectively.He is currently an Assistant Professor at Iran Uni-

versity of Science and Technology, Tehran, Iran. Hismain research interests are in remote sensing, patternrecognition, image processing and soft computing.

Saeid Homayouni (M’05) received the B.S. degreein surveying and geomatics engineering fromUniver-sity of Isfahan, Isfahan, Iran, in 1996, the M.S. degreein remote sensing and geographic information sys-tems from Tarbiat Modaress University, Tehran, Iran,in 1999, and the Ph.D. degree in signal and imagefrom Telecom of Paris, Paris, France, in 2006.He is currently with Department of Geomatics and

Surveying, College of Engineering, University ofTehran, Iran, as an Assistant Professor. His researchactivities are focused on remote sensing image

analysis for urban and agro-environmental applications.