1.

Introduction

Hyperspectral imagery (HSI) is a method used to collect contiguous data across a large swath of the electromagnetic spectrum. This is accomplished by using a specialized camera mounted on an aircraft or satellite to record the magnitude of the bands within the collected wavelengths of each pixel within the area of interest. The number of pixels in a hyperspectral image depends on the resolution of the camera and the size of the area being imaged. The number of bands recorded is 200 or more,¹ and typically spans the range from ultraviolet to the infrared regions of the electromagnetic spectrum. The vast amount of data contained in HSI affords an excellent opportunity to detect anomalies using multivariate statistical techniques, as each material reflects individual wavelengths of the spectrum differently.²

Target detection algorithms can be divided into two groups: anomaly detection algorithms and classification algorithms.¹ Anomaly detection algorithms do not require the spectral signatures of the anomalies that they are attempting to locate. A pixel is declared an anomaly if its spectral signature is statistically different than the model of the local or global background that it is being tested against. This implies these algorithms cannot distinguish between anomalies. They only make a decision on whether a pixel is anomalous; hence, the application can be considered a two-class classification problem.¹ Classification algorithms attempt to identify targets based on their specific spectral signature; however, to accomplish this, they require additional information in the form of a spectral library.³

Eismann et al.⁴ claim that the mean vector and covariance matrix required for anomaly classification can be estimated globally from the entire image on the assumption that there are a small number of anomalies in the image and this has an insignificant effect on the covariance matrix. This statement is contested in Smetek,⁵ where the potential ill effects of a small number of anomalies on the estimation of the covariance matrix are detailed. Similarly, Manolakis et al.³ state that:

This article demonstrates that classification algorithms, such as the adaptive matched filter (AMF), may be improved by addressing spatial correlation and homogeneity problems inherent to HSI that are often ignored in practice. We begin by showing the benefit of using an anomaly detector to remove potential anomalies from the mean vector and covariance matrix statistics, as suggested by Manolakis et al.³ In addition, we show further benefits by addressing the nonhomogeneity of HSI through the use of cluster analysis prior to classification.

The remainder of this article is organized as follows. Section 2 reviews the basics of classification, describes the AMF as well as AMF variants used in this research, and discusses atmospheric compensation. Section 3 briefly outlines the seven anomaly detectors implemented. Section 4 discusses clustering—more specifically, the $X$-means algorithm. Section 5 details the methodology implemented. Section 6 presents the results of the experiments. Finally, Sec. 7 presents our conclusions.

2.

Classification

This section describes the AMF classification algorithms used in this research. Three new variants to the AMF are introduced that have the ability to classify with improved accuracy by addressing correlation and homogeneity problems inherent to HSI. Elementary atmospheric compensation is also discussed, detailing a method to transform a spectral library into the image space to allow for proper classification.

3.

Anomaly Detection Algorithms

To apply an anomaly detection algorithm to hyperspectral data, first the atmospheric absorption bands should be removed and the data cube must be reshaped into a data matrix. The removal of the absorptions bands in the images employed in this research results in the retention of 145 of the 210 original bands. HSI data is typically stored in a three-dimensional matrix, referred to as an image cube or a data cube, with the first two dimensions of the matrix being the location of the pixel in the image and the third dimension being the magnitude at each of the recorded electromagnetic bands. Therefore, it can be viewed as a stack of images, with each image representing the intensity of a given band. An $n\times p$ data matrix is generated by reshaping the data cube into a matrix with the first dimension containing all $n$ pixels in the image and the second dimension containing all $p$ bands. After the data is in the proper form, principal component analysis (PCA¹⁷ is employed as a data reduction tool. In all the algorithms except autonomous global anomaly detector (AutoGAD), the user is left to determine the number of principal components (PCs) to retain in order to reduce the dimensionality of the data.

4.

Clustering

Cluster analysis is a multivariate analysis technique for grouping, or clustering, a dataset into smaller subsets known as clusters. The goal of cluster analysis is to maximize the between-cluster variation while minimizing intra-cluster variation.³⁰ $K$-means is a clustering technique where the data is split into $k$ user-defined number of clusters. The $K$-means algorithm is initialized by randomly selecting $k$ starting points, or cluster centers. Next, a random data point is selected and added to the nearest cluster. The corresponding cluster center is then updated with the new data, allowing for currently clustered data to move into other clusters. This process is repeated until all the data points are in one of the $k$ clusters and no data points are moved in an iteration of the algorithm.³⁰

The difficulty when using a clustering algorithm such as $K$-means is selecting $k$, the number of clusters. $X$-means is a clustering algorithm based upon $K$-means and has the advantage of being able to select the number of clusters from a user-defined range as opposed to a single value. This is accomplished by running $K$-means multiple times, splitting each of the original clusters in two, and scoring each possible subset of full and partial clusters using the Bayesian information criterion (BIC) to determine the optimal clustering.³¹

Rather than supplying the $X$-means algorithm the specific number of clusters as in $K$-means, the user defines a range of possible clusters, $k$-lower and $k$-upper. In this research, the $X$-means algorithm was implemented with $k$-lower and $k$-upper set to 3 and 10, respectively. The algorithm begins by running $K$-means on the data set with $k$ equal to $k$-lower. Next, each of the original $k$ clusters are split in two using $K$-means, with $k$ equal to 2. Then all $2^k$ possible combinations of whole clusters and split clusters are analyzed for the corresponding BIC scores. Then $k$ is incremented and the process is repeated until $k$-upper has been analyzed. The algorithm then returns the $k$ cluster centers with the highest corresponding BIC score, and $K$-means is run one last time using the returned cluster centers as the initial starting points.³¹

5.

Methodology

The images used in this research come from the hyperspectral digital imagery collection equipment³² sensor for the Forest Radiance I and Desert Radiance II collection events in 1995. The images were collected with 210 bands between 397 and 2500 nm and the ground reflectance data was collected with 430 bands between 350 and 2500 nm. The collection names allude to images consisting of forest and desert scenes. Nine images were employed in this research: six forest images and three desert images (as shown in Fig. 1 and with image details displayed in Table 2). Atmospheric compensation was performed with NDVI for forest images, and BI was applied for the desert images due to the lack of vegetation. The targets (which constitute the anomalies of interest herein) in the images consists of vehicles, various colored fabrics and tarps, and numerous types of metals and other building materials.

Fig. 1

Hyperspectral images.

Table 2

Hyperspectral image data.

The first step was to analyze an image using one of the seven anomaly detectors described in Sec. 3: RX, IRX, LRX, ILRX, AutoGAD, SVDD, or BACON. Each algorithm has user-defined settings that influence the algorithms’ performance. In these experiments, the RX detectors and SVDD used the best settings as reflected in Williams et al.,²⁰ the settings for AutoGAD were taken from Johnson et al.,²¹ and the settings for BACON were taken from Billor et al.²⁹ The anomalies detected by the anomaly detector were used twice: first, the anomalies were removed from the image to calculate a robust mean vector and covariance matrix, and second, the anomalous pixels served as the test pixels to be classified using one of the four AMF variants described in Sec. 2.2: standard AMF, robust AMF, clustered AMF, and largest cluster AMF.

The following steps were performed to classify anomalies. First, the spectral library, consisting of 30 objects for forest images and 34 objects for desert images, was transformed from reflectance space to radiance space using NDVI or BI atmospheric compensation depending on the image scene, as depicted in the bottom of Fig. 2. Next, one of the seven anomaly detectors then analyzed the image for anomalous pixels, as shown in the top of Fig. 2. Pixels below the anomaly detector threshold ($T_1$) are classified as background. Pixels above $T_1$ are classified as anomalies and are used as the pixels to be classified by the AMFs. The mean vector ($\mu$) and covariance matrix ($\sum$) are estimated using the appropriate set of data for the AMF variant. The standard AMF uses $\mu$ and $\sum$ from the entire image, the robust AMF uses $\mu$ and $\sum$ from the image without the detected anomalies, and the clustered AMF and largest cluster AMF use $\mu$ and $\sum$ from the appropriate cluster of data, as determined by the $X$-means algorithm. Finally, the data is processed through the classifier, and pixels below the classifier threshold ($T_2$) are classified as background and pixels above $T_2$ are classified as appropriate target types.

Fig. 2

Classification experimental process graph.

In order to generate receiver operating characteristic (ROC)—like curves³³ for each AMF/anomaly detector pair across all nine test images, the following methodology was employed. Each anomaly detector was run across a range of anomaly detector thresholds ($T_1^i$), $\mathrm{i}=1,2,\dots ,19$ where $T_1^1=0.01$, $T_1^{i+1}=T_1^i+{\mathrm{\Delta }}_1$, and ${\mathrm{\Delta }}_1=0.005$. The pixels flagged as anomalous are then processed through the four AMFs. Each target in the spectral library is compared to a pixel of interest, and the target type with the largest resulting AMF score is declared given its score is above the AMF threshold ($T_2^j$), $j=1,2,\dots ,100$ where $T_2^1=0.01$, $T_2^{j+1}=T_2^j+{\mathrm{\Delta }}_2$, and ${\mathrm{\Delta }}_2=0.01$. The thresholding implemented in this research is created by finding the range of all the AMF scores corresponding to the target type with the largest resulting AMF score and normalizing them between 0 and 1 to allow for consistency among the different AMFs within an image. When a threshold is selected, any pixel with an AMF score less than the threshold is classified as background. Analyzing each of the different detector thresholds across all four of the AMFs, while allowing $T_2^j$ to vary across its range of possible values, enumerates all combinations of $T_1^i$ and $T_2^j$.

As the images are processed, a $2\times 3$ confusion matrix, as displayed in Table 3, was generated for each $T_1^i$, $T_2^j$, image combination, denoted $C_{T_1^i{T}_2^j}(k)$, where $i$ and $j$ represent specific threshold values and $k$ is the image of interest. The confusion matrix is comprised of two sections to allow for the scoring of every pixel in the image. The anomaly detector section reflects the declaration of a test pixel as background or the passing of the pixel to be classified. Relative to the detector declaration of background, there are false negatives (FNs) and true negatives (TNs). The anomaly classifier section then reflects the ultimate classification of the pixels classified as anomalous by the anomaly detector.

Table 3

2×3 Confusion matrix.

After each of the nine images are processed, new $2\times 3$ confusion matrices are generated that consist of the sum across all nine images for each $C_{T_1^i{T}_2^j}^k$ combination defined by

Fig. 3

ROC curve generated from convex hull of data.

6.

Results

The four AMFs and the seven anomaly detectors were scored to produce ROC curves as described in Sec. 5, with the results shown in Figs. 4 and 5. Each figure contains separate graphs of classification performance generated from each anomaly detector. Figure 4 shows the ROC curves for the full process (detection plus classification). This means taking into account the full $2\times 3$ confusion matrix, implying that

Fig. 4

ROC curves for full process.

Fig. 5

ROC curves for AMF results.

To get a better visualization of the data, a set of conditional ROC curves were created. Figure 5 displays the ROC curves which account for the performance of the AMF given detections, implying that

Again, we see the variants outperform the standard AMF, and in most cases, the clustering methods enhance the robust AMF.

7.

Conclusions

This article demonstrates improvements to AMF performance by addressing correlation and nonhomogeneity problems inherent to HSI data, which are often ignored when classifying anomalies. The standard AMF and three variants were employed along with seven different anomaly detectors utilized prior to classification. Manolakis et al.³ state the estimation of the mean vector and covariance matrix should be calculated using “target-free” data, generating a robust AMF. Here, it was shown that accounting for spatial correlation effects in the detector yields nearly “target-free” data for use in an AMF that is greatly improved through the use of cluster analysis methods.