D 
o o mo 
HH 05 
  
  
  
  
Minimizing with respect to a and dy. respectively gives 
SO ami Te moras acini velid.enidQuy c0 (6) 
: ; 1*1 : : 
X j eds ej 
and 
Y (8.7 =08,004 7D*ecs f - & 7 ) = 0 (7) 
> 6 14 pi^ A, 
Si -T- : ; 
ince 85%. zc0 , when i:* j 
dri l8 (8) 
i i >i 
where), = 378 
k k"k' 
Substitution gives 
1 Mie z 
gor CC a =-À-.a (9) 
Equation 9 is an usual eigenvalue equation, which may be 
solved by standard routines. After computing the vectors a,» 
the base vectors dy. can be obtained using equation 8. 
It is an advantage if the eigenvalues M and the 
corresponding eigenvectors a, are sorted in decreasing 
order. If the p largest eigenvalues are used in the 
expansion, it can be shown that the Euclidian norm of the 
residual matrix R, is equal to the sum of the eigenvalues 
ort tte where m is the rank of the C matrix. In this 
way, an optimum set of orthogonal base functions a; can be 
selected, minimizing the loss of information. 
3. KARHUNEN-LOEVE EXPANSION FOR TERRAIN CLASSIFICATION 
In this study, the correlation functions of "terrain 
elevations are analyzed using the K-L expansion. For this 
purpose the six ISPRS DEM test areas, as described by 
- 685 =