International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
  
P 
A - S sri) (13) 
ree 
2 ; : Eus 
where 7 (x, > F1) is the sample correlation coefficient 
between x; and y,. Let a = (a,,a,,À a y be a standardized 
m 
. 1 . . 
vector, Le. à à — |. Then Xa givens 7 observations on a new 
variable defined as a weighted sum of the columns of X. The 
sample variance of this new variable is a'Sa. 
Theorem 1 No standardized linear combination (SLC) of X; (i 
=1, 2, ..., n) has a variance larger than ,, the variance of PCI 
(Mardia et al., 1979). 
From Theorem 1, not surprisingly, the standardized linear 
combination with the largest variance is PCI. 
2.3 Colour Morphology 
Similar to the extension from binary morphology to grayscale 
morphology, the extension of concepts of grayscale 
morphology to colour image processing also need some rules to 
organize colour values. Today, complete lattices are considered 
as the right mathematical framework for colour morphology, 
not only because the framework will retain the basic properties 
of its grayscale counterparts, but also it could be used in 
applications similar to those of the corresponding grayscale 
operations. An inherent difficulty in the framework is that there 
is not an obvious and unambiguous method of fully ordering 
colours (vectors) So far, there is not a unified colour 
morphology theory, due to the variety of ordering schemes and 
colour spaces that can be used. Different approaches to colour 
morphologies can be classified by following the classification 
of ordering schemes stated in Section 2.2: marginal 
morphologies (Talbot et al, 1998), partial morphology 
(Vardavoulia et al., 2002), and reduced morphologies (Comer 
and Del, 1998). 
3. ORDERING VECTORS 
Although principal component analysis is an important and 
essential technique for data reduction and has been widely used 
in remote sensing image processing, when the variables are 
ambiguous, it makes no sense to estimate PCI as the linear 
combination of variables with standardized weights having 
maximal, because the linear combination of ambiguous 
variables is not well defined. However, according to Theorem 1 
in Section 2.2, Eq. (13) provides an alternative way to define a 
sample PC1, ordinal PC1 (Korhonen and Siljamaki, 1998). 
Definition 1 Vector y € R" is PC1 of the centered variables 
m, if y is a solution vector of the following: 
(14) 
peque 
maid 30 (y, x; Jy'y=} 
j=l 
From the Definition 1 on PC1, we can extend the definition of 
the PC1 to cases in which some restrictions are imposed on the 
vector y and variance sj; 
Definition 2 The ordinal PCl based on' fuzzy pair wise 
. . n. ^ 
comparison matrix y EL iva vector, y, € (1,2,A qns 1 
2; wn, and VF VV, Viz k . which determines a rank 
, 
order for observations such that the sum of products on squares 
of some correlation coefficients between the variables x, j = 1, 
2, ..., m, and y with the variance of variable xj, j = 1, 2, ..., m is 
maximal. That is 
mS ssl E) (15) 
i=! 
where r (xj, y) is a correlation coefficients between the variables 
X,j 1,2, m, and y, and s; is the variance of variable x, j 
];2, m 
Definition 3 Let variable X, = [x,;,%,;,A el 
Vi ez [ju (x;)] is 
a fuzzy pair wise comparison matrix describing the fuzzy rank 
order of observation according to variable x; if 
(s, Ay) NS LA, 7 (16) 
v1 «x. The A X7 matrix M(xj)) = 
Hau S; VS 
X; 
  
UnilX;) 
  
  
«it 
  
Fig.1 The membership function for 
fuzzy pair wise comparison matrix. 
  
  
  
As defined in Definition 3, the zu, denotes the membership 
grades representing the comparison relationship between xj; and 
xy; according to the rank C. The value of the membership grades 
calculated by Eq. (16) is in [-1, 1], which shows not only the 
degree of the relationship between x, and x,, but also the fact 
that these two elements are positively or negatively related. We 
have illustrated the membership function of Eq. (16) in Figure 
LE 
To find the ordinal PC1 defined in Definition 2, it is necessary 
to define a rank correlation coefficient. Following the original 
idea of Daniels (1946), there are two ways to compute the rank 
correlation coefficients, Spearman's and Kendall’s rank 
correlation coefficients (Kendall, 1962), as follows. 
v OE J) Ee 
N ee tm fre Fr d 
where ry is called Kendall’s rank correlation coefficient, and f; 
is calculated by 
" = [44,,^ Hw A uio fh (18) 
The variance of variable X; — A m Cr , which 
is used to measure the degree of dispersion for the variable, is 
_ ev 
7520 =x)’ (19) 
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