The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
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2.5 Causal RX algorithm (CRX)
Since RX detector involves mean and covariance matrix
computation, it can not be implemented in real-time. Hence a
real-time processing version of the RX is introduced where the
sample correlation matrix (R) is used instead of the sample
covariance matrix (AT). It is called "Causal" which means that
the information used for data processing is up to the pixel being
processed and updated only based on the pixels that were
already processed.
m diff - m out m in
Cdiff = c in ~ Cout
(15)
Consequently anomalies can be extracted by projecting the
differential mean between two windows on to the eigenvector
associated with the largest positive eigenvalue of differential
covariance matrix (Kwon, 2003) by:
m) = jZf- l nn T
(12)
0 DWEST
(16)
Since the computation of the inverse of a sample correlation
matrix can be carried out in parallel via QR matrix
decomposition method, this algorithm can be implemented in a
real-time manner (Chang, 2003).
SCRX (He) = {rk) r
(13)
2.6 Adaptive Causal Anomaly Detector algorithm (ACAD)
ACAD algorithm is a developed version of causal RX model. In
this algorithm, strong signatures of detected anomalies are
removed during detection process due to their undesirable
effects on detection of subsequent anomalies. Because one
major problem encountered in CRX algorithms is that if an
earlier detected anomaly has an intense signature it may have
considerable impact on the detection of later anomalies. This
occurrence is mainly caused by an inappropriate use of sample
correlation matrix. According to Chang (2003), a proper sample
correlation matrix should be one that removes all the earlier
detected anomaly pixels being included in the sample
correlation matrix. For this reason, the (R) in Causal RX
equation should be replaced with a sample correlation matrix
that removes all detected anomalies defined by (Hsueh, 2004):
Also in order to implement the RX for the dual windows, the
RX in equation is modified as:
,RX-DW
00 = | mdiff (r) 1
Cout( r )
w (r)
(17)
2.8 Nested Spatial Window-base Target Detector (NSWTD)
NSWTD model implements a nested three local windows,
entitled inner, middle and outer windows where the first two
windows are used to extract smallest and largest anomalies
respectively, while the outer window is used to model the local
background. Moreover the other main distinction of this model
from the DWEST and RX-base algorithms is using the
Orthogonal Projection Divergence (OPD) as measurement
criterion instead of eigenvector projection or sample covariance
matrix (Liu, 2004) by:
OPD (si,sj) = J(sj P Sj \ + sj Ps^sj )
P ^ k =1 LxL- s k( s l s k) l s k
(18)
hrk) = R(rk)~ZtjeA(k) t fj
(14)
S ACAD = { R-
Hc
n<
He
Where A(k) is the set of earlier detected anomalous target pixels
tj prior to the currently being processed image pixel (r k ).
Also the mentioned Rx-base algorithms are known as Global or
Local anomaly detector if the mean spectrum is derived from
the full image data or from a local window around each pixel
during detection process.
2.7 Dual Window-base Eigen Separation Transform
anomaly detector (DWEST)
DWEST model implements two local windows, entitled inner
and outer windows which are used to maximize the separation
between anomalies and background. The idea of using the inner
window is to detect an anomaly present in it, whereas the
purpose of the outer window is to model the background of the
anomaly assumed in the inner window[5]. By moving these two
local windows entire the image, local mean (m in , m out ) and
covariance matrix (C in , C out ) of each window and their
differences are calculated as below:
Since three nested windows used in this algorithm, the inner
window implanted in the middle window which is in turn nested
in outer window, the OPD must be implemented twice. First
between inner and middle windows is specified by
s 2W-NSW (r)= QpD ^ m in {r\m diff (19)
Where m di ffj is the mean of the outer window with subtraction
of the inner window. The second OPD is between the middle
and outer windows is specified by
§ 2W-NSW (r)= 0pD {^ m mid {r\m diff 2 (r)j (20)
Where m di ff 2 is the mean of the outer window with subtraction
of the middle window. Finally, a 3-window NSWTD, denoted
by 5 3W ~ NSW ( r ):
s 3W-NSW (r)= max/=u { s*r-h*W ir)) (21)