International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
    
   
   
    
     
   
   
   
   
   
   
   
    
   
   
   
    
   
   
    
   
  
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
1 M 
My tu I) (1) 
i=] 
0,,01,--0,, 
Cors (2) 
01052055 
where -——Y 60) - m), - m) E 7L... M, M is the number of pixels in the 
k 
dataset, 7, and m, are the means of the ih band and the jth band, respectively. x, is the 
value in the ith band of the kth pixel. 
Step 4: Define the Manhalobis distance metric of each pixel in D by the mean and covariance. 
The numerical expression of the anomaly detector in each iteration is given as: 
T 
d(9(x,)) — ($(,) -m)Cs (9) -m,) (3) 
With the algebra computation as well as the kernelized trick, the distance can be done by dot 
product of pixels in original low dimension feature space. By spectrum decomposition, the 
background covariance matrix can be transferred into: 
CST EAE (4) 
where V, and A are the eigenvectors and eigenvalues matrix, respectively. It is proved that 
each eigenvectors in the feature space can be expressed as the linear composition of the 
centralized input samples in the feature space [5]. 
. M . . 
Va s $,8/9(x) = Xf 
= (5) 
where X, is the composed of the kernelized input dataset samples and P /is the eigenvectors 
of the centered kernel matrix (Gram matrix). Due to the physical structure in high dimension feature 
space and some formulas computation, all the eigenvectors with nonzero eigenvalues: 
V. - XB © 
By substitute and into, and some similar algebra computation, the final expression of the 
detector is: 
d((x,)) -(K7 - K;, ) K, (K7; - Kj.) (7) 
where each item is a centralized Gram matrix [5], which can be figured out by the kernel function 
on original samples.