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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
is used to colour edge detection and building roof extraction 
from remotely sensed images. 
The paper is organized as follows. Section 2 is devoted to a 
series of background notions in vector ordering, multivariate 
data analysis, and colour morphology. In Section 3, a new 
reduced ordering based on ordinal first principal component 
analysis is introduced. The basic morphological operations such 
as dilation, erosion, closing, and opening based on the new 
vector ordering are proposed in Section 4. Section 5 is devoted 
to the applications of the proposed morphological operators to 
colour edge detection and building roof extraction. In Section 6, 
preliminary results of building extraction from pansharpened 
Ikonos and QuickBird and colour aerial imagery are given 
followed by discussion and outlook in Section 7. 
2. BACKGROUND 
2.1 Ordering Vector 
A set of multivariate data consisting of n m-dimension random 
vectors can be modeled as an n x m data matrix X. The rows of 
the matrix X will be written with X,, X,,A X, corresponding 
2? 7 
to n observations. The columns of the matrix X will be written 
with X,,X, ‚A X, corresponding to p variables. The 
element locating at the row / and the column / in the matrix X is 
x; representing jth variable on the ith observation, i.e., 
X, Xn Xp A X. 
X, Xa XA x, (1) 
SEM [xxl 7-5 
M : M MA M 
X, Xo X2 A X om 
where, X. s [v x0 A x]; G2 ,2,A n) (2) 
LG = 1,2,A m) (3) 
The aim of ordering multivariate data X is to arrange them to 
the form X, TT X, TA 1 X, according to each variable Xo 
and x, — [nA X 
=v 
= 1,2, …, m, where the symbol 7t means less preferred to and 
the subscripts i, j, ..., k range over mutually exclusive and 
exhaustive subsets of integers 1,2, ..., n. 
Unfortunately, ^ ordering ^ multivariate data are not 
straightforward, because there is not the notion of the natural 
ordering in a vector field as in the one-dimensional case. 
Although there is no unambiguous form of multivariate data 
ordering scheme, much work has still been done to order the 
data. Barnett (1976) proposed the so-called sub-ordering 
principles to rule the ordering. The sub-ordering principles are 
classified in four groups: (1) marginal ordering (M-ordering), in 
which multivariate data is ordered along each one of its m- 
dimensions independently; (2) condition ordering (C-ordering), 
in which the multivariate vectors are ordered conditionally on 
one of components. Thus, one of the components is tanked and 
other components of each vector are listed according to the 
position of their ranked component; (3) partial ordering (P- 
ordering), in which multivariate data is to partition the vectors 
into groups, such that the groups can be distinguished with 
respect to order, rank, or extremeness (Titterington, 1978); and 
(4) reduced ordering (R-ordering), it reduces vectors to a scalar 
value according to a measure criterion. Mardia (1976) further 
developed the sub-classification of reduced ordering: distance 
ordering and projection ordering. The distance ordering refers 
to the use of any specific measures of distance, and the 
projection ordering considers ordering the sample by using the 
first principal component (PC1) or higher. 
2.2 First Principal Component Analysis 
An obvious extension of the univariate notion of mean and 
variance leads to the following definitions (Mardia et al., 
T. ; 2 y 
1979). The mean of jth variable, X; = [3 532 ^ y] ; 
is defined as 
1 n 
x; = ca, (4) 
Be 
; : : : i 4 Kal 
The variance of the jth variable, x, = [<> x, À Ay] ; 
is defined as 
] D oe] ^) 2 x ; 
$ -—» (x,-X,) =s, pei123 m (5) 
nt 
The covariance between the ith variable, 
7 : : 
x, = [xj 1. 9A ras] ; and jth variables, 
To. s 
X. [X jr X A For] , 1s defined as 
SS 
n 
EN E, M ox) LE LD H1. (0) 
n rs] 
The vector of means, or mean of the matrix Xs 
1 n 
E Wu = kia E ni 
[T A UTE 
1 
r=] n 
X'I (7) 
: ^ . T 
where 7 is a column vector of n one, i.e., | = [LLA A] . Also 
the variance-covariance matrix S of the matrix X is 
S=[s,]= “3X, — X(X,- X)= lxux ® 
€ n 
where pm] s bp denotes the centering matrix and I 
N 
denotes identity. 
It is obvious thát S is symmetric and position semi-definite. By 
the spectral decomposition theorem, the variance-covariance 
matrix $ may be written in the form 
S-GLG (9) 
where L is a diagonal matrix of the eigenvalues of S, that is 
A 10 A 0 
f. © + A 0 (10) 
M MA M 
0. 10.4. A, 
where À, > A; 2A 2 A, 2 0, G is an orthogonal matrix, its 
column vectors g; (i = 1, 2, ..., m) is the standardized 
eigenvectors corresponding to the eigenvalues 4, (i = 1, 2, ..., 
m) of S, i.e., 
G=[g.8,.A sad (11) 
According to the eigenvectors, PC1 is defined as 
y, (X - IX)g, (12) 
The eigenvalue 4, of the sample variance-covariance matrix S 
can be written as 
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