IEEE Geoscience and Remote Sensing Magazine - June 2019 - 104

L (x) =

f (x | H 1)
,
f (x | H 0)

(2)

where f (x | H 0) and f (x | H 1) are conditional probability distributions of x under the associated hypotheses. The LR will
be large when it is more likely that x belongs to H 1 and small
otherwise. The LR determines which hypothesis is more likely
to be correct using the statistical test described in Figure 2.
For the LR test (LRT), y = L (x) is compared with threshold h to determine the predicted hypothesis. Hypothesis
H 1 is chosen if L (x) $ h, and H 0 is chosen if L (x) # h. The
detection statistic y can be modeled using a probability density function, as described in Figure 3. The performance of
the detector is entirely dependent on the distribution between f (y | H 0) and f (y | H 1).
The choice of threshold h is crucial to the detection
performance. A good detector should have a high number of correct detections (denoted as D) and a small number of incorrect detections [i.e., false alarms (denoted as
FAs)]. The relevant performance metrics are quantified as
the probability of detection p D and the probability of a
false alarm p FA and are defined as follows:
p D(h) _

#h 3 f (y | H 1)dy =1 - #-h3 f (y | H 1) dy =1 - Fy(h| H 1),

p FA(h) _

#h

(3a)

3

f (y | H 0) dy =1 - #

h

-3

f (y | H 0) dy =1 - Fy(h| H 0),
(3b)

where f (y | H 0) and f (y |H 1) are the cumulative distribution
functions of the output statistic y. Figure 3 shows that, for
a small h, both p D and p FA will be very high. Similarly, for
a high h, both quantities will be very low. The Neyman-
Pearson criterion for threshold selection is typically used
in hyperspectral image processing. Using this criterion, a

maximum p FA is defined, and a corresponding threshold
is selected.
A great variety of detection algorithms have been proposed for HSI remote sensing applications, and extensive
reviews may be found in [16]-[18]. For military and defense
applications, the current automatic target recognition capabilities are extended to situations in which the targets are
smaller, more diverse, more fleeting, and less distinctive,
while the background clutter is more complex [19]. Under
such circumstances, traditional spatial-based image processing techniques are generally not applicable and may not perform well if they are applied [1].
DETECTION WITHOUT PRIOR KNOWLEDGE
For strategic deployments during conflicts, the type and
the number of an opponent's targets being deployed in the
field are unknown. These targets are generally deployed
in a background that is distinct from other man-made objects, such as military vehicles or unexploded ordnance, and
can be readily detected as anomalies using HSI techniques,
even though they may have been camouflaged [10]. For the
anomaly detection (AD) problem, we seek pixels in the image that are unusual compared to the background pixels
[17]. These pixels generally correspond to small, rare objects
that occupy a very small fraction of the background area in
which they are embedded [16]. To solve this problem, we
consider a pixel as an anomaly if d(x, b) 2 h, where b is a
background model, d($) is a distance measure, and h is a
threshold. The detector is defined by
D(x | b) =[d(x, b) 2 h] .

A common, simplified model consists of the mean spectral
vector n and the Euclidean distance measure, i.e.,
D(x | b) =[< x - n < 2 h].

Detection
Statistics
y = L(x)

x

y

Threshold
η

y> η
y< η

Target
No Target

FIGURE 2. The LRT, which is performed when f ( x | H 0 ) and

f ( x | H 1 ) are known.

Rejection Region

Detection Region
f (y |H1)

f (y |H0)

Threshold η

Detection Statistic y

Probability of Detection (pD)
Probability of False Alarm (pFA)
FIGURE 3. Distributions under the hypotheses H 1 and H 0 .,

104

(4)

(5)

The background data may include all of the image pixels (i.e.,
global background) or only the pixels in the spatial neighborhood of the pixel under test (i.e., local background), thus
leading to global and local detection algorithms, respectively. In local detection, the window of pixels surrounding
the pixel under test is selected to characterize the local background. The window size is a compromise between statistical precision in the estimates of the background properties
(e.g., mean vector or correlation/covariance matrix), which
favors a large window, and spatial nonstationarity of the
background, which favors a small window [17].
The AD technique has received considerable attention
within the remote sensing community [10], [16]-[18].
Many of these methods are based on the Reed-Xiaoli detector (RXD) [10],
1
RXD =(x - n)T C -background
(x - n),

(6)

where C background is the band-to-band spectral covariance
of the background, modeled as a multivariate Gaussian
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2019
