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two French 
geostatistical scientists -- G. Matheron and J. Serra -- in the 
1960s (Matheron, 1975; Serra, 1982). It has since then found 
increasing application in digital image processing. The basic 
morphological operators are dilation and erosion. They are 
defined as follows (see Serra, 1982; Haralick et al, 1987, Li 
and Chen, 1991): 
Dilation: À ® B = {a + b: ae À, beB} = Up eBAb (1) 
Erosion: À © B = {a: a+ beA, beB} =NpeBAp (2) 
where A is the original image and B is called the structuring 
element, which can be considered to be an analogy to the 
kernel in a convolution operation. In Eq.(1) it is called 
"dilation of A by B" and in Eq.(2) "erosion of A by B". 
Examples of these two operators are given in Fig.8, where the 
features are represented by pixels of “1”s and the origin of the 
structuring element is marked with a circle. The structuring 
element is a critical one in these operations. More discussion 
regarding this element will be conducted at a later stage. To 
show how these operators work clearly and exactly, “1” and 
“0” are used to represent the binary images used in this 
discussion. In this diagram, “+” means those becoming 1 after 
dilation and “-” means those becoming 0 after erosion. This 
convention will be used throughout this paper. 
If a symmetric structuring element with origin at the centre is 
used for dilation, then the shape of the original image will be 
expanded uniformly along all directions, thus the dilation in 
this particular case is called expansion. Similarly, the erosion 
in this case is called shrink. These two special operations are 
illustrated in Fig.9. 
0.0.0. 0:0.0:0..0. 0 
0,000000 00 
07O 111000 0 
039/173-3 10 0^0 L Lal 
00:21 L'o 00 6 120 4 
0.0:0/1.0.0 0.0/0 111 
0.-.0,0 0:0 0,0.0.0 
000000000 
(a) Original image A (b) The structuring 
element H 
9000. 000000 0.00 -0.0.0 0 0.0 
O0 + ++++000 0000900000 
0 +111+ +00 00---0000 
0. 1 11r 0 0 00-1--000 
0+1211+4+ +40 0 0.0 .- -- 0.0.0.0 
0 ++ 1+ +000 0 0:°07* 0.00 0..0 
00+++0000 0° 0.0.0:0°0 0 0 0 
0.000,00 0 0,60 005010 0 0.0 D 0 
(c) A expanded by H (d) Ashrunken by H 
(A ®H) (A ® H) 
Fig.9 Special cases of dilation and erosion 
expansion and shrink 
Another two very important operators are opening and closing. 
They are defined as follows: 
Opening: AoB=(AOB)®B (3) 
Closing: AeB=(A®B) © B 4) 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
where, A is the original image and B is called the structuring 
element. 
Based on the two basic operators, i.e. dilation and erosion, a 
number of other new operators have also been developed, such 
as thinning, thickening, hit or miss, conditional dilation, 
conditional erosion, conditional thinning, conditional 
thickening, sequential dilation, and conditional sequential 
dilation, and so on. However, it is not the purpose of this paper 
to discuss all of them. More detailed information can be found 
from the book by Serra (1982). 
Structuring element is the key element in a morphological 
operator. Structuring elements could take any shape. Fig.10 
shows some of commonly used structuring elements. Indeed, it 
is through the proper manipulation of structuring elements that 
the morphological operators alter the shape, form and structure 
of spatial objects. 
B x = 
  
(a) Circular (b) Diagnonal (c) Linear 
(d) Squared (e) Cross 
Fig.10 Some possible structuring elements 
6. DISCUSSION 
After the introduction of so many new concepts such as scale 
dimension, natural principle and morphological operators, it 
seems appropriate to usev an example to illustrate how 
morphological operators can be used to depict the 
transformation of spatial representation in scale dimension. 
Fig.11 is one of those example (Su et al, 1996). In this 
diagram, the size of the structuring element B is determined by 
the natural principle. After applying some morphological 
operators, the representation shown in Fig.11(a) is transformed 
into that shown in Fig.11(c). Features are smoothed and 
combined. The reduced image is shown in Fig.11(d). 
8* . 
(a) Original features A (b) Structuring element B 
  
de 
(c) Areas combined (c)Combined area reduced 
C=AeB C=AeB 
Fig.11 Transformation of the representation of area features in 
scale dimension: area aggregation (Su et al, 1996) 
457