## 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:

Possible presence of targets in the background estimation data lead to the corruption of background covariance matrix by target spectra. This may lead to significant performance degradation; therefore, it is extremely important that, the estimation of $\mu $ and $\sum $ should be done using a set of “target-free” pixels that accurately characterize the background. Some approaches to attain this objective include: 1. run a detection algorithm, remove a set of pixels that score high, recompute the covariance with the remaining pixels, and “re-run” the detection algorithm and 2. before computing the covariance, remove the pixels with high projections onto the target subspace.^{3}

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.

## 2.1.

### Classification Algorithms

The goal of an HSI statistical classification algorithm is to determine whether a test pixel is likely made of the same material as a target pixel. Define the conditional probability density of the test pixel, $x$, as realized under the alternative hypothesis, ${H}_{a}$ (same as the target pixel), as ${f}_{a}(x|{H}_{a})$; and the conditional probability density of test pixel, $x$, as realized under the null hypothesis, ${H}_{0}$ (not the same as the target pixel), as ${f}_{0}(x|{H}_{0})$. The corresponding likelihood ratio test is

If $F(x)$ is greater than the user-defined threshold, $t$, then the null hypothesis is rejected, meaning that the test pixel is considered a target pixel; otherwise, the null hypothesis cannot be rejected, implying that the test pixel is not considered a target pixel.^{6}That is, if $F(x)>t$, the test pixel is considered a target pixel, or if $F(x)\le t$, the test pixel is not considered a target pixel.

The classification algorithm utilized in this article is the full-pixel AMF as defined by Manolakis and Shaw.^{7} The algorithm assumes the target spectra and background spectra have a common covariance matrix, $\mathrm{\Sigma}$:

^{3}

## 2.2.

### Variants of the AMF

The standard AMF and three variants of the AMF are implemented in this research. The first method is the standard AMF as described above, where the mean vector and covariance matrix are taken from the entire image. The first variant, called robust AMF, is suggested by the quote from Manolakis et al.^{3} in the Introduction. In this method, an anomaly detector analyzes the image and then the mean vector and covariance matrix are estimated from the image without the detected anomalies. The second variant is referred to as clustered AMF. In this method, anomalies are removed as in robust AMF, and then the image is clustered without the detected anomalies. Each of the clusters yields a mean vector and covariance matrix estimate. Since a target pixel may be surrounded by pixels of various types (e.g., grass, trees, etc.), the cluster most represented in the collection will be referred to as the modal class The corresponding background statistics for the pixels to be classified are determined through the modal class of its neighbors. A similar idea has been proposed in anomaly detection.^{8} Due to the time-consuming nature of determining which cluster a pixel is located in, a third variant, called largest cluster AMF, is developed. This method removes the anomalies and clusters the resulting data as is done in clustered AMF. However, the mean vector and covariance matrix for the pixels to be classified are estimated from the single largest cluster of data in the image. Table 1 is provided to show the source of mean vectors and covariance matrices for the implemented AMFs.

## Table 1

Source of mean vectors and covariance matrices for the implemented AMFs.

Source of mean vector/covariance matrix | |
---|---|

Standard AMF | Entire image |

Robust AMF | “Target-free” data |

Clustered AMF | Corresponding cluster of “target-free” data |

Largest cluster AMF | Largest cluster of “target-free” data |

## 2.3.

### Atmospheric Compensation

Spectral signature matching within HSI typically incorporates a spectral library consisting of ground-measured reflectance data from objects of interest. The difficulty with spectral signature matching is that hyperspectral images are collected using a sensor that collects pupil-plane radiance, which includes reflected and radiated energy as well as atmospheric distortions. Before spectral signatures from an image can be compared to target signatures, atmospheric compensation must be performed to bring the spectral library from the reflectance space to the pupil-plane radiance space. Since radiance data is a function of atmospheric conditions, which vary greatly by collection time, the spectral library must be processed with each image separately.^{9}

Linear and model-based approaches are available to transform data from the reflectance space to the radiance space. Model-based approaches, such as MODTRAN,^{10} require prior knowledge about the scene collection. Precise computation of radiance from reflectance values is elusive. A linear method called the empirical line method (ELM) was chosen for its simplicity and its ability to work well in controlled scenes such as the ones used here.^{9} Although more computationally intensive atmospheric compensation methods are available, ELM, the normalized difference vegetation index (NDVI), and the bare soil index (BI) offer a way to compute a linear, and intuitively understandable, transformation between reflectance to radiance without complicated formulas, models, or nonlinear methods. Additionally, ELM is a reputable and well-known method; it is noted as offering simplicity and quality in controlled environments.^{4}^{,}^{9}^{,}^{11}12.13.^{–}^{14} This research is not focused on developing or testing new atmospheric compensation methods; rather, it is focused on developing a new classification method.

Linear methods assume that atmospheric content is a linear addition where the pupil-plane radiance is a function of reflectance with a scaling multiplier and offset:

where ${\rho}_{i}$ is a reflectance signature to be transformed into the ${L}_{i}$ radiance space with gain, $a$, and offset, $b$, as calculated by where ${\rho}_{1}$ and ${\rho}_{2}$ are known reflectance signatures from the spectral library, and ${L}_{1}$ and ${L}_{2}$ are the corresponding radiance measurements from the scene. Linear methods are comprised of two general types: methods such as ELM, which require known objects of interest to be within the spectral library and located within the image; and vegetation normalization methods, which use expected radiance and reflectance of vegetation in place of specific known objects. Both permit atmospheric compensation to be conducted for the remaining objects in the spectral library.^{9}

For situations lacking prior knowledge of scene content, methods such as vegetation normalization are appropriate, where the linear method in Eq. (3) is applied with radiance measurements for materials expected in the scene. The NDVI and the BI are two methods that allow atmospheric compensation to be performed depending on the scene landcover.^{9}

## 2.3.1.

#### Normalized difference vegetation index

NDVI^{15} is a method that is used to determine whether a pixel within a hyperspectral image is green vegetation. It does this by comparing the radiance of the near infrared (NIR) spectrum to the red spectrum

^{9}In this research, we used bands corresponding to 660 nm for red and 860 nm for NIR.

NDVI is calculated for each pixel within an image, and pixels with the highest scores can be used as vegetation within the radiance space. Hence, the vegetation in the spectral library can be used as ${\rho}_{2}$ in Eqs. (4) and (5), and the average spectral signature of the pixels with the highest NDVI score can be used as ${L}_{2}$. ${L}_{1}$ can be determined from the shadows within an image which can be estimated by the spectral signature, which is calculated by taking the minimum value from each band in the image across all pixels. Finally, ${\rho}_{1}$ is set as a vector of zeros, and interpreted as the ideal minimum radiance in the image.^{9}

## 2.3.2.

#### Bare soil index

BI^{16} is a method similar to NDVI; however, it is designed for bare soil within a hyperspectral image. It can be employed in the same fashion as NDVI, assuming that there is a soil measurement within the spectral library:

## (7)

$$\mathrm{BI}=\frac{(\mathrm{SWIR}+\text{red})-(\mathrm{NIR}+\text{blue})}{(\mathrm{SWIR}+\text{red})+(\mathrm{NIR}+\text{blue})},$$^{9}In this research, we used the band corresponding to 470 nm for blue, 660 nm for red, 860 nm for NIR, and 2280 nm for SWIR.

## 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 (PCA^{17} 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.

## 3.1.

### RX Detector

The RX algorithm, developed by Reed and Yu,^{18} detects anomalies through the use of a moving window. The pixel in the center of the window is scored against the other pixels in the window. The window is then shifted by one pixel and the process is repeated until each pixel, $x$, has received an $\mathrm{RX}(x)$ score based on

## (8)

$$\mathrm{RX}(x)={(x-\mu )}^{T}{\left[\right(\frac{N}{N+1})\sum +(\frac{1}{N+1})(x-\mu ){(x-\mu )}^{T}]}^{-1}(x-\mu ),$$## 3.2.

### Iterative RX Detector

The iterative RX (IRX) detector was introduced by Taitano et al.^{19} as an extension to the RX detector in an attempt to mitigate the effects that anomalies have on mean vector and covariance matrix calculations. The algorithm runs RX in an iterative fashion, each time removing pixels flagged in the previous iteration as anomalous from the mean vector and covariance statistics used to calculate the RX scores. This process continues until the set of anomalies in the previous iteration matches the set from the current iteration or the maximum number of iterations has been completed.^{19}

## 3.3.

### Linear RX and Iterative Linear RX Detectors

The linear RX (LRX) and iterative linear RX (ILRX) detectors^{20} function in the same manner as the RX and IRX except that a vertical line of data is used as opposed to a window. If the number of pixels selected for the line size is larger than the image, then pixels are taken from the bottom of the previous column and the top of the subsequent column. These methods are better than the previously described methods because they increase the average distance between the test pixel and the pixels used to estimate the background statistics, thereby decreasing the effects of correlation due to spatial proximity.^{20}

## 3.4.

### Autonomous Global Anomaly Detector

The AutoGAD^{21} is an independent component analysis (ICA)^{22}^{,}^{23} based detector that is processed in four phases. Phase I reduces the dimensionality of the data through PCA^{17}, using the geometry of the eigenvalue curve to determine the number of PCs to retain. Phase II conducts ICA on the retained PCs from Phase I via the FastICA algorithm.^{24} Phase III determines the independent components (ICs) that potentially contain anomalies using two filters: the potential anomaly signal-to-noise ratio and the maximum pixel score. Phase IV smooths the background noise in the ICs selected in Phase III using an adaptive Wiener filter^{25} in an iterative fashion, and then locates the potential anomalies using the zero bin method described by Chiang et al.^{26}

## 3.5.

### Support Vector Data Description

Support vector data description (SVDD) was originally applied to HSI data by Banerjee et al.^{27}^{,}^{28} SVDD is a semi-supervised algorithm that requires a training set of background data. Since HSI images are usually assumed to contain few anomalies, the training set is generated by randomly selecting pixels from the image. The minimum volume hypersphere about the training set, $S=\{x:{\Vert x-a\Vert}^{2}<{R}^{2}\}$, is then calculated with center $a$ and radius $R$. The hypersphere is determined through constrained Lagrangian optimization that simplifies to

^{27}

^{,}

^{28}

## 3.6.

### Blocked Adaptive Computationally Efficient Outlier Nominators

Blocked adaptive computationally efficient outlier nominators (BACON) is a statistical outlier detector created by Billor et al.^{29} It attempts to locate outliers in a data set through the use of iterative estimates of the model with a robust starting point. The algorithm is computationally efficient, regularly requiring fewer than five iterations to converge, so it is applicable to HSI data. The basic idea is to start with a small subset of outlier-free data and iteratively add blocks of data to the data set until all data points not considered outliers are in the data set. The final data set is then assumed to be outlier free and thus can be used to generate robust mean vector and covariance matrix estimates.^{29}

The BACON algorithm begins by selecting an initial basic subset of data with $m=cp$ data points with the smallest Mahalanobis distance, where in the case of HSI $p$ is equal to the number of bands within image and $c=4$ or 5, as suggested by Billor et al.,^{29} so long as $m\ge n$ where $n$ is the number of pixels in the image. Next, the Mahalanobis distances are calculated for each of the pixels, remembering that $\mu $ and $\mathrm{\Sigma}$ are now the mean vector and covariance matrix of the basic subset. A new basic subset of all of the pixels with distances less than ${c}_{\mathrm{npr}}{\chi}_{p,\alpha /2}^{2}$ is selected, where ${\chi}_{p,\alpha /n}^{2}$ is the $1-(\alpha /n)$ significance level of a chi-squared distribution with $p$ degrees of freedom and ${c}_{\mathrm{npr}}={c}_{\mathrm{np}}+{c}_{\mathrm{hr}}$ is the correction factor, where

Here, $r$ is the size of the current basic subset, $n$ is the number of observations, or pixels, and $p$ is the dimensionality of the data, or bands. If the size of the new basic subset is the same size of the basic subset from the previous iteration, the algorithm is terminated. Otherwise, a new basic subset is calculated.^{29}

## 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 equipment^{32} 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.

## Table 2

Hyperspectral image data.

Image | Size | Total pixels | Anomalous pixels | Anomalies | Unique targets |
---|---|---|---|---|---|

1F | 191×160 | 30,560 | 994 | 10 | 5 |

2F | 312×152 | 47,424 | 281 | 30 | 9 |

3F | 226×136 | 30,736 | 96 | 20 | 11 |

4F | 205×80 | 16,400 | 75 | 29 | 12 |

5F | 470×156 | 73,320 | 440 | 15 | 20 |

6F | 355×150 | 53,250 | 976 | 45 | 10 |

1D | 215×104 | 22,360 | 490 | 46 | 22 |

2D | 156×156 | 24,336 | 417 | 4 | 3 |

3D | 460×78 | 35,880 | 405 | 12 | 12 |

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.

In order to generate receiver operating characteristic (ROC)—like curves^{33} 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.

Anomaly detector | Anomaly classifier | ||
---|---|---|---|

Background | Target | Background | |

Target | False negative (detector) | True positive (classifier) | False negative (classifier) |

Background | True negative (detector) | False positive (classifier) | True negative (classifier) |

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

where $I$ is the number of images analyzed, here $I=9$. Below, the true positive fraction (TPF) (classifier) versus false positive fraction (FPF) (classifier) data point for each ${C}_{{T}_{1}^{i}{T}_{2}^{j}}$ was plotted for a given anomaly detector—anomaly classifier combination, and the convex hull of the resulting data is interpreted as a rough ROC curve, as is shown notionally in Fig. 3, where the circles are data points and the line represents the ROC curve.## 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

## (15)

$$\mathrm{TPF}=\frac{{\mathrm{TP}}_{\mathrm{C}}}{{\mathrm{TP}}_{\mathrm{C}}+{\mathrm{FN}}_{\mathrm{C}}+{\mathrm{FN}}_{\mathrm{D}}}$$## (16)

$$\mathrm{FPF}=\frac{{\mathrm{FP}}_{\mathrm{C}}}{{\mathrm{FP}}_{\mathrm{C}}+{\mathrm{TN}}_{\mathrm{C}}+{\mathrm{TN}}_{\mathrm{D}}},$$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

## (17)

$$\mathrm{TPF}=\frac{{\mathrm{TP}}_{\mathrm{C}}}{{\mathrm{TP}}_{\mathrm{C}}+{\mathrm{FN}}_{\mathrm{C}}}$$## (18)

$$\mathrm{FPF}=\frac{{\mathrm{FP}}_{\mathrm{C}}}{{\mathrm{FP}}_{\mathrm{C}}+{\mathrm{TN}}_{\mathrm{C}}}.$$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.

## References

## Biography

**Jason P. Williams** received a BS degree in mathematics from the University of Illinois, Urbana, in 2001 and MS and PhD degrees in operations research from the Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, in 2007 and 2012, respectively. He is currently with the Headquarters U.S. Air Forces in Europe, Ramstein Air Base, Germany. His research interests are in the areas of multivariate statistics and response surfaces.

**Kenneth W. Bauer** received a BS degree from Miami University, Oxford, OH, in 1976, an MEA degree from the University of Utah, Salt Lake City, UT, in 1980, an MS degree from the Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, in 1981, and a PhD degree from Purdue University, West Lafayette, IN, in 1987. He is currently a professor of operations research with the Air Force Institute of Technology, where he teaches classes in applied statistics and pattern recognition. His research interests lie in the areas of automatic target recognition and multivariate statistics.

**Mark A. Friend** received his BS from Texas Christian University in 1996 and MS and PhD degrees in operations research from the Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, in 1998 and 2007, respectively. He currently serves as a lieutenant colonel in the United States Air Force stationed at Wright-Patterson AFB, OH, where he is an assistant professor in the Department of Operational Sciences at the Air Force Institute of Technology. His research interests lie in the areas of automatic target recognition, applied statistics, and simulation.