International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 
  
r f(p)- ex(p)- f(p)» eos f(G»)- f(»v) © 
This operation can extract darker objects smaller than the 
structure element from the original image. 
Structure element sequence: N dilated by itself successively to 
form a structure element sequence as follows: 
{N,N®N.,.....,nN} (7) 
In order to record conveniently, the result dilated n times nN is 
replaced by n in the following calculation. 
Granulometry and  unti-granulometry: — Granulometry 
proposed by Matheron is used for analysis of objects and 
image 
structures in images. An 
b». f(pYn123.—. } obtained 
with a series of element structures {nN, w= Las , defined 
sequence 
by opening operations 
by equation (7), is taken as granulometry. If opening is replaced 
by closing, the image sequence lo, f(p)n 113 } is 
called anti- granulometry. 
2.2 Extraction of morphological features 
The key step in the method proposed in this paper is to extract 
the morphological feature because the rule of segmention is 
based on the assumption that one connected component in the 
image will hold the same morphological feature. If a greyscale 
image is interpreted as topographical relief, pixels in the image 
can be approximately classified into three patterns: peak(lighter 
region),valley(darker region) and plain , which are respectively 
labeled as P ^ V and P' .There are many objects with 
different sizes and shapes in an image, and some of them may 
have a high response to an element structure in a given size and 
a lower response for other size. In order to find out the most 
sensitive size for every pixel, the granulometry and 
unti-granulometry are used to get two image sequences 
trip) v4 elo. ern: n] and vo, (p) VÀ «|o. Ss ai n] 
with the same structure element sequence. The derivative 
sequence of opening images can be defined as follows: 
Ay(p)- tha (p): ^ra) Ira (0) - vaio vA el on) 
(7) 
48 
So there is a derivative vector Ay(x) at every pixel x: 
Ay(x)={a7;(e) vA elt] (8) 
Ay; (x) is grey scale of Ay; (p) at pixel x. There is the 
greatest grey scale change when this image is processed by 
opening with the structure element corresponding to the 
supremum V Ay(x) at pixel x, i.e. the pixel x is most sensitive 
for this size structure element for the opening operation. So the 
multiscale opening feature at x is defined as: 
e,()- lar G)- vare) ©) 
Similarly, the multiscale closing feature at x is defined as: 
®, (x)= 2:40; (1)=vapls) (10 
Based on the features above, the image segmentation algorithm 
can be denoted as: 
k,, 2D, (x): vAy(x)» vAg(x) 
D(x)= ke, o, (x): vAy(x)<vAg(x) (11) 
k; =0: vAy(x) = vAg(x) 
where (x )is the feature function. For the same structure 
element sequence, the greatest change of grey scale induced by 
opening operation is bigger than that by closing operation. 
When (x) e [1,2 ++" "-- n], the pixel is considered as a lighter 
point, i.e peak; while o(x) € [+ 12,27 n] , the pixel is taken 
as a darker point, i.e valley. For d(x) = 0, the pixel belongs 
to a plain. - The eigenvalue is concluded in a 
set = R,..— 51012 4] . The pixels with the same 
feature are considered to be in the same region. 
In case of uncertainty or ambiguity in distinguishing between 
scene foreground and background , it is also possible to soften 
the conditions of the morphological features by rewriting (11) 
as: 
k; j=l, (x): vAy(x)- vAo(x) 20 
o(x)- ET o, (x): vAq(x)- vAy(x)» o (12) 
ky =0:{v Ay(x)-vAg{]<o