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MITIGATION OF CORRELATION AND HETEROGENEITY EFFECTS
IN HYPERSPECTRAL DATA
JASON P. WILLIAMS TREVOR J. BIHL
Dept of Operations Research Dept of Operations Research
Air Force Institute of Technology Air Force Institute of Technology
Wright-Patterson AFB, Ohio Wright-Patterson AFB, Ohio
KENNETH W. BAUER
Dept of Operations Research
Air Force Institute of Technology
Wright-Patterson AFB, Ohio
Abstract
The RX anomaly detector is well known for its unsupervised ability
to detect anomalies in hyperspectral images (HSI). However, the RX
method assumes the data is uncorrelated and homogeneous, both of
which are not inherent in HSI data. To defeat the correlation and
homogeneity, a new method dubbed Iterative Linear RX is proposed.
Rather than the test pixel being inside a window used by RX,
Iterative Linear RX employs a line of pixels above and below the test
pixel. Through the use of Receiver Operating Characteristic (ROC)
curves, this paper presents Iterative Linear RX alongside the standard
RX algorithm, the newly introduced Iterative RX, a successful
advancement to the benchmark RX HSI detector, and the global
Support Vector Data Description (SVDD) algorithm, a promising
new HSI detector, to show the results of the newly proposed method.
INTRODUCTION
The RX anomaly detector is well known for its unsupervised ability to detect
anomalies in hyperspectral images (HSI). However, the RX method assumes the data is
uncorrelated and homogeneous, both of which are not inherent in HSI data. The purpose
of this paper is to present a modification to the newly successful Iterative RX introduced
by Taitano (2007). The new method, dubbed Iterative Linear RX, has the ability to
defeat some of the correlation and homogeneity problems hindering Iterative RX. Rather
than the test pixel being inside a window used by RX, Linear RX employs a line of pixels
above and below the test pixel. Through the use of Receiver Operating Characteristic
(ROC) curves, this paper presents Iterative Linear RX alongside the standard RX
algorithm, the Iterative RX algorithm, a successful advancement to the benchmark RX
HSI detector, and the global Support Vector Data Description (SVDD) algorithm, a
promising new HSI detector, to show the results of the newly proposed method.
SUPPORT VECTOR DATA DESCRIPTION
Banerjee, Burlina, and Dieh (2006) took the original SVDD algorithm by Tax
and Duin (1999) and applied it as a HSI anomaly detector. Linear SVDD attempts to
determine the minimum hypersphere that explains the region of the corresponding data.
When applied to HSI data, a set of M background pixels are randomly selected from the
image as the training data. The goal is to find the minimum inclosing hypersphere of the
M pixels, described by equation 1.
2
min( ) subject to , 1,...,
i
R x S i M
∈ =
(1)
The radius R and center a of the hypersphere are determined by optimizing the
Lagrangian in equation 2.
(
)
{
}
2 2
( , , ) , 2 , ,
i i i i i
i
L R a R R x x a x a a
α α
= − − − +
∑
(2)
After optimizing L with respect to α
i
, the kernel trick can be applied which
leads to the SVDD statistic in equation 3.
( )
2
22
( ) ( , ) 2 ( , )
where ( , ) exp /
i i
i
SVDD y R K y y K y x
K x y x y
α
σ
= − +
= − −
∑
(3)
The variable σ
2
is a radial basis function parameter used as a scaling factor to
direct the size of the boundaries, hence adjusting how well the SVDD algorithm
generalizes the incoming data. The algorithm is processed in the following steps:
1. Randomly select M pixels from the image.
2. Estimate an optimal value for σ
2
using a cross-validation or minimax method.
3. Estimate the parameter (R, a, α
i
) needed to model the hypersphere.
4. Determine whether or not every pixel in the images lies within the hypersphere.
If the pixel is not in the hypersphere it is declared an anomaly.
RX
The RX detector was introduced by Reed and Yu (1990). It detects anomalies
using a moving window approach where the pixel in the center is scored by comparing it
to the additional pixels in the window. The window is then shifted one row or column of
pixels, and the new center pixel is scored. This process is continued until all possible
pixels have been analyzed. Each test pixel is scored using a generalized likelihood ratio
test which is simplified to equation 4 if the pixels are assumed to be normal with mean µ
and covariance Σ. It should also be noted that as N goes to infinity the RX score becomes
the squared Mahalanobis distance between the test pixel and the mean of the background
pixels.
1
1
( ) ( ) ( )( ) ( )
1 1
T T
N
RX x x x x x
N N
µ µ µ µ
−
= − ∑+ − − −
+ +
(4)
The pixels are determined to be anomalies if their corresponding RX score is
greater than
χ
α, ρ
where
α
and
ρ
are the corresponding quantile and degrees of freedom of
the Chi-squared distribution.
ITERATIVE RX
This standard RX detector has been improved and dubbed Iterative RX by
Taitano (2007). The Iterative RX detector works by running the RX detector multiple
times, with each iteration calculating better estimates of the true mean and covariance of
the background pixels. The Iterative RX algorithm is processed in the following steps:
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1.
Reduce the dimensionality of the data by running principal component analysis
on the whole data set and retaining the p largest principal components. This is
necessary due to the vast amount of data being processed within each iteration.
2.
Run the standard RX algorithm on the data set, but remove any pixels that are
in the set of anomalies from the data being used to estimate the mean and
covariance of the background. For the first iteration the set of anomalies is
empty.
3.
Determine the current set of anomalous pixels from the newly calculated RX
scores that are greater than
χ
α, ρ.
This allows for pixels to enter and exit the set
of anomalous pixels each iteration.
4.
If the set of pixels identified as anomalies is identical to the previous iteration
or the maximum number of iterations has been reached, stop, otherwise return
to step 2.
ITERATIVE LINEAR RX
Iterative Linear RX is similar to Iterative RX except instead of a window being
moved through the image, it employs a vertical line of pixels above and below the test
pixel. If the number of pixels above or below the test pixel exceeds the height of the
image, the required pixels are taken from the bottom of the previous column or from the
top of the following column. There are two major advantages of the using a line over a
window. First, the average distance between the test pixel and the pixels used to estimate
the mean and covariance is drastically increased, as depicted in figure 1, thus decreasing
the effect of correlation and heterogeneity inherent in the data. Second, the use of the
line gives the ability to test every pixel in an image, which the RX method cannot do to
the required window.
Figure 1: Average Distance from Test Pixel
METHODOLOGY
All four of the anomaly detectors were compared using images from the ARES
desert and forest radiance collections. The images consist of 210 wavelength bands,
primarily from the visible spectrum, however only 145 are used due to atmospheric
distortion. This paper will depict two of the six images tested in the experiments, ARES
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1F and ARES 2D, displayed with their corresponding truth masks in figure 2. The pixels
surrounding the anomalies, called border pixels, contain background and anomaly data;
therefore they are not considered in either class when scoring the detectors. ARES 1F
consists of total 30,560 pixels, of which 1,007 are considered anomalies and ARES 2D
consists of 22,360 pixels with 523 anomalous pixels.
ARES 1F ARES 2D
Figure 2: ARES 1F and ARES 2D with corresponding Truth Masks
The SVDD algorithm was run with
σ
2
= 905 based off of the minimax method
described by Banerjee, Burlina, and Diehl (2006) and M = 500. Standard and Iterative
RX was run using a square window of 21 pixels based on the recommendations of
Taitano (2007). Iterative Linear RX used a line size equal to the number of pixels in the
vertical direction of the image, based on preliminary results, and both were run using a
maximum of 30 iterations. Prior to any processing with the RX detectors, principal
component analysis was performed and the seven largest principal components were
retained based on previous experiments.
The goal of the experiments was to compare the promising SVDD detector, the
Standard and Iterative RX detector, and our Iterative Linear RX detector. Each of the
detectors were run using various values of alpha for the Chi-squared distribution, and
their results were collected and used to generate ROC curves to visually score them.
RESULTS
The majority of the images the Iterative Linear RX tested provided better
results than Iterative RX and SVDD. In the few cases where Iterative Linear RX was not
the clear winner, due to known issues, methods are being implemented to overcome the
problems.
When viewing the ROC curves, which are displayed in figure 3, it is easily seen
that Iterative Linear RX is performing with the highest abilities. In both of the images
SVDD fairly successful in detecting outliers, however, Iterative Linear RX is showing
even better results. In the ARES 1F image, Standard and Iterative RX does a poor job
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with the large anomalies, because when the window is centered on an anomalous pixel on
the target the majority of the window also contains the anomaly. This completely defeats
the purpose of the window, which is to give a good estimate of the true background of the
image; hence, the pixel being analyzed appears like the other pixels in the window and is
not classified as an anomaly. Iterative Linear RX defeats this problem by only containing
a small portion of even a large anomaly due to the vertical line used to estimate the
background. In the ARES 2D image, the Standard and Iterative RX methods do a poor
job due to its inability to classify targets on the border of the image. In the tests, the
window size was set to 21 by 21 pixels which generates a border of 10 pixels all the way
around the image that it cannot test and in this particular image creates problems. On the
other hand Iterative Linear RX can test every pixel in the image, and thereby does a much
better job of classifying the anomalous pixels.
Figure 3: ARES 1F and ARES 2D ROC Curves
DISCUSSION
This paper presented an update to the newly introduced Iterative RX algorithm
by altering the method of determining the mean and covariance estimates of background
pixels within HSI data. The new method also accounts for a portion of the correlation
and heterogeneity that is inherent in this data and assumed to be nonexistent by the
Standard and Iterative RX algorithm. It has shown to be successful in classifying
anomalies at a higher rate than the Iterative RX method, which is a reasonable competitor
to the SVDD algorithm. It is also a fully unsupervised method, whereas SVDD is a
supervised method that requires the random selection of background pixels. In low
anomaly to background environments that are tested in this image SVDD is likely to get a
good estimate for the background, however, due to the random selection process, it is
possible to get a bad draw.
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