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)