threshold t. 
B. Solve the average v of region 0. 
C. Computer the difference between the gray value 
If the 
absolute difference is less than t, this point is 
of any neighbor in region 0 and v. 
considered as the domain of region 1. 
D. If new region is no longer extended, loop stop. 
Otherwise, repeat like from step A to Step C. 
Suppose po be a starting point, and py; be one of the 
new growed pixels. According to the step C, pi and 
po form a path. pj and pi; then establish a path 
through po, i. e. any two points in the region 1 can 
form a path. Similarly, p, is a new growed pixel 
from region k —1 to region k. Suppose region k— 1 
be connected, a path is formed from p, to any pixel 
of region k— 1. So, new paths are formed from py; to 
pg through pixels of region k — 1, i. e. any two 
points in the region k can form a path . According to 
the definition of connectivity, such a set of pixels 
posses connectivity and is called to be connected 
region. 
In fact, the above procedure is region growing. We 
say here that the connectivity of digital topology 
constrains region growing. 
1. 2 Matrix Structure 
Matrix is well known by people who are engaged in 
each different kind of the field. But, using matrix as 
representation and calculation structure in image 
processing is just at the first step (Dougherty & 
Giardina 19877. A digital image is similar to a matrix 
or array of numbers. As a result,it is a useful tool for 
standarizing representation of the pictures and the 
parallel algorithms. In this paper, we introduce a 
mathematical structure, the “bound matrix”, for the 
representation of digital images. This structure is used 
in conjunction with block diagrams to serve as a 
concise expression of digital image processing 
operation. 
Definition 1. A array — type structure consisting of 
m by n entities; 
10 
3. dis“ Gin 
d21 022 * dan 
lai 0,2 . Gr) T, 
where, 
1) each ay is a real number or a x (star) 
2) ISp<M, I Sq<n 
3) r and t are integers 
is called a BOUND MATRIX, or an m x n bound 
matrix, and the stars denote values that are not 
known. The location of the a; entry is (r,t). The 
direction from aj; to aj makes the value of q 
increased, and the direction from a; to a; makes the 
value of p decreased. This * (star) representation 
here is helpful to visualize all values outside a bound 
matrix to be stars. Unusually, the origin is changable 
and available in the location of any entry. 
In many disciplines, such as computer architecture 
and communications, block diagram is a useful 
device. Therein, they serve as a language for 
describing and performing operations with and 
between images and bound matries. When this bound 
matrix is used for representing the gray value 
distribution on the two — dimensional plane, it is 
called gray — matrix structure. The procedure of 
above region growing can be denoted by this gray — 
matrix structure. 
2. IMAGE THINNING AND 
MATRIX REPRESENTATION 
2. ] Image Thinning 
Image thinning is a common preprocessing operation 
in computer aided design, automated cartography, 
facsimile transmission, and pattern recognition. 
Many algorithms of image thinning do not preserve 
topology. This is not permissive in the fields needing 
to utilize the topological properties. Therefore , Acrelli 
proposed thinning algorithms should be obedient to 
the following criterian for North : 
WU GS UE AM GW) UNWUECGN) 
f\ &N) U NE U C-E) () CE) U SE U CS)