ink 
hip 
and 
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inal 
ank 
ank 
17) 
id f; 
18) 
hich 
e, is 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXX V, Part B7. Istanbul 2004 
where x, is a referent element for Xj. All referent elements 
consist to a referent variable for the variable Xj. 
A in Eq. (13) corresponding to Kendall's rank correlation 
coefficient can be written as. 
' > 
  
m : m (g, g,) m (x y-C,y 
A= s 1e (x) = $5 SAPE = Sy ALL 
2 AK / 2 b (s. e s. Xg,'g, 2 y A,B 
(20) 
where LS (x, y Lo, nn+1)’ 
MOIS dx, Sint Ba a AE TM is 
n e n 
pug (x; Dn * 1) 
el 2 
To find a numerical solution to the ordinal principal component, 
a mathematical optimization model should be constructed. We 
formulate the optimal problem as an integer programming (IP) 
model: 
Given 
X, Xu Xi ^ Xs 
XY = X, = [x WA SE Xj X3 A X, (21) 
M M MA M 
M Xie Sun. A abus, 
Max 
eeu 22) 
^ nd A,B 
Subject to 
y7[y YA, ] y, 602A ,njand y, zy, Vis j,ij l2A,n 
(23) 
The combination optimization problem stated above can be 
solved by using a branch and bound approach (Winston, 1991) 
that is a basic technique for solving integer and discrete 
programming problems. The method is based on the 
observation that the enumeration of integer solutions has a tree 
structure. The main idea behind the branch and bound method is 
to avoid growing the whole tree as much as possible, because 
the entire tree is just too big in most real problems. Instead 
branch and bound grows the tree in stages, and grows only the 
most promising nodes at any stage. It determines which node is 
the most promising one by estimating a bound on the best value 
of the objective function that can be obtained by growing that 
node to later stages. 
In this paper, we model our integer programming problem 
above as a variable-integer assignment problem. We have n 
integers, 1 thought n, to assign to n variables, y; thought y, 
Each variable can take exactly one integer, and all integers must 
have an assigned variable. The object is to maximum the total 
profit described by Eq. (22). In general, when there are n 
variables and » integers there are / possible assignments. Figure 
2 shows an example of structure tree for a three-variable-integer 
assignment problem. 
1171 
  
e 12.» 
2 
3 
3 
e 1.23 
f 21 
eo {12,3} 
2 
> 
3 
e 125 
i 
e (1.23 
2 
3 
  
Fig. 2. The structure tree for the 3 variable- 
integer assignment problem. 
@ : Root Node = All Solutions. 
o : Bud Node = Partial Assignments. 
@ : Leaf Node = Complete Assignments. 
me A mm ce EE : V3. 
  
  
  
The formal branch and bound formation follows. 
e Root node in the branch and bound tree: all solutions. 
® Bud node: a partial assignment of integer to variable. For 
example, a partial assignment {1, ?, ?} represents the 
assignment of integer | to variable y,. 
e Leaf node: a complete assignment of integers to variables, 
¢.g., à complete assignment [1, 2, 3} represents the 
assignment of integer 1 to y;, 2 to V», and 3 to ys. 
* Objective function: for each leaf node, its objection 
function can be computed by Eq. (22). 
® Bounding function: for each bud node, we first create a 
pseudo-leaf by combining the partial assignment for this 
bud node and the maximum unassigned integer. For 
example, for the bud node (1, ?, ?}, its pseudo-leaf node is 
(1,3, 3j. Then we use the pseudo-leaf and Eq.'(22) to 
compute the bounding function for the bud node. 
* Bud node selection policy: global best value of the 
bounding function. 
e Variable selection policy: choose the next variable in the 
nature order y; to yy. 
Terminating rule: when the best solution objective function 
value is better than or equal to the bounding function value 
associated with all of the bud nodes. 
4. COLOUR MORPHOLOGY 
4.1 Definitions of Infimum and Supremum 
To extend the vector ordering approach to colour imagery, it is 
necessary to define the colours as vectors. In this paper, a 
colour C in RGB colour space is represented as a vector X. = 
[Res Ge Be], where R,, G,, B, denote the red, green, and blue 
components of the colour C, respectively. Therefore, a colour 
image can be viewed as a vector field. Given a colour image C 
and a pixel p in C in which the colour is X,74R,:.G,. B, Let 
W bethe N — nx n window consisted of the neighbourhood 
of the pixel p. All N colours in JW can be written as a (N x 3) 
data matrix X including N observations and three variables, that 
is,