The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
304 
= K LxÁ r ~v) (2) 
Where r is the pixel spectral vector, p is the mean spectral 
vector for the area of interest (the mean of each spectral band), 
L is the number of spectral bands, and K is the spectral 
covariance matrix. 
It is interesting to note that mathematically, RX algorithm can 
be considered to be an inverse procedure of the PCA algorithm 
which searches for targets in minor components. In this case, a 
small eigenvalue will create a large value of d RX (r). This is 
comparable to searching for minor components by finding 
smaller eigenvalue of K. It offers explanation of why RX 
algorithm works for anomaly detection. Other types of RX-base 
algorithms derived and developed from this basic algorithm to 
improve detection performance which will be discussed. 
N J 
Twi(n -Pw)( r i -M w ) 
is _ _l 
w ~ Ñ 
I 
1 
N 
T.m 
_ i 
Mw 
N 
I <7/ 
1 
(5) 
(6) 
Where w, and q, are weight scalars for each pixel in the image 
defined by: 
2.2 Normalized RX (NRX) & Modified RX (MRX) 
It is interesting to see that the RX equation by (2) performs 
some kind of a matched filter that its performance is entirely 
depend on two parameters: the matched signal 
( (r-ju) T K L x L ' ) and the scale constant (tc=l) which seems 
before the matched filter. According to this advantage, two 
alternatives of the RX referred as normalized RX and modified 
RX, those are denoted by (3) and (4) can be developed by 
setting k = [(r-p) T (r-p)] 1 and k = [(r- p) T (r-p)J' 1/2 in this way 
(Chang, 2002): 
1 
(7) 
1 
Wi = l + h-p w \\ 
Finally RX filter with weighted covariance matrix can be 
applied to the image. 
ô WRXi r ) = (r-p w f K~ X (r-p w ) (8) 
5 NRX {r) 
0 MRX ^ “ 
(r-pYK L l L (r-p) 
(r-p? {r-p) 
(r-pf K ll L {r-p) 
^ 2.4 Uniform Target Detector algorithm (UTD) 
Another type of anomaly detector, referred to the low 
probability target detector (LPTD), it was developed by 
Harsanyi (1993) and given by: 
(4) 
¿LPTD( r ) = hxL T r I\ L r (9) 
2.3 Weighted RX algorithm (WRX) 
In view of the fact that anomaly targets are generally small and 
the background is homogeneous, the sample covariance matrix 
of the entire image can be viewed as background sample 
covariance matrix Also the anomalies or small man-made 
targets can be separated as outlier in image data cube. 
Consequently, the RX algorithm uses Mahanalobis distance to 
find them by using sample covariance matrix to whiten the 
background pixel, then those anomaly pixels become outliers. It 
would not be a problem when the number of anomaly pixels is 
few. But since this algorithm assumes Gaussian noise and uses 
sample covariance matrix for data whitening, when percentage 
of the anomaly pixels is relatively large, the sample covariance 
matrix cannot represent the background distribution. In this case 
the RX algorithm will not perform well. So to solve this 
problem, it is proposed to use weighted covariance matrix 
(Hsuan, 2005). 
In weighted RX algorithm, proper weight is assigned to each 
pixel in the sample covariance matrix using its distance to the 
data center for better representation of background distribution. 
So weighted matrix can be written as: 
This detector was designed base on the sample correlation 
matrix R. If R is replaced with the sample covariance matrix K, 
an alternative LPTD could be develop using sample covariance 
matrix K, referred to as uniform target detector (UTD) which is 
given by: 
^UTD = Olxl ~ A) ^LxL ~ Y) (10) 
Where \ LxI is the L dimensional unity vector with ones in all the 
elements. Thus an anomalous target is assumed to have uniform 
distribution of radiance over all the spectral bands. Therefore it 
is predictable to extract background signatures which are 
uniformly distributed in the image scene. 
In this case it is remarkable to note that the background 
subtraction could enhance the RX detection algorithm 
performance as shown by Ashton and Schaum (1998). By 
incorporating the UTD into basic RX, the background can be 
removed as well as noise to improve the performance of basic 
RX detector. This advantage enables us to develop a new type 
of anomaly detector by subtracting UTD from RX as follows 
(Chang, 2003): 
S RX-UTD^ zz(<r ~^ TK LxL^ r ~^ t 11 )