M) €-S) U SE U -W) = 0 (1) 
where, “YY”, “I” and *- " "denote boolean 
disjunction, conjunction, and negation , respectively ; 
w-—0 or 1 according as the west neighbor of p is a 
signal or background, and similarly for the other 
compass directions. In fact, N=1, since this rule 
applies only to north border points. If “east”, “ 
south”, and “west” substitute for “north”, it is 
applicable to delete points in parallel from each side in 
turn (e. g. ,in the N,S,E,W). 
2. 2 Matrix Representation 
Before we deal with the representation method for 
equation (1), we expand some basic concepts 
CDougherty &. Giardina 1987] used for equation (1) 
with their block diagrams. Suppose f and g be the 
input images . 
CADD(f,g, +++) Gi, j) 
£(i,j)+g(i,j) +--+. if all inputs are 
== defined at(i,j) 
* elsewhere 
Here, we have extended the above concept from two 
images to infinite number images. This entension will 
efficiently cut down the computational complexity for 
more than 2 images. This situation is similar to most 
of the matrix operations. The block diagram for 
ADD is 
  
f— 
g—> 
—ADD(f,g) 
— 
  
  
  
CMULT(f,g)26,j) 
fü,DXgG,D X, if all inputs are 
defined at(i,j) 
if either input is 
undefined at(i,j) 
The block diagram corresponding to multiplication is 
  
fo 
g—| MULT 
- MULT  (f,g,:) 
  
  
— 
  
11 
GGUB(D2G,D-— —fG,j) 
the negation of * — x . The block diagram for SUB 
is defined as 
t--[SUB|-- SUB (f) 
TRAN(f;i. Dur 
The block diagram for TRAN is given by 
  
f= (ann 
I> 
TRAN|--TRAN(f,i,j) — (Qn) r+i,t+i 
  
  
j* 
  
(EXTADD(f,g,...)J, D— 
fü,D--gü,.D-d:, as long as any image 
is defined at (i,j), add here. 
if all inputs 
are undefined at (i,j) 
We can here absorb these concepts into boolean 
operations. 
CBN(COJ G, j) 
a (a if the pixel is undefined at(i, j) 
A0 
if the pixel is defined at (i,j) 
CAND(É,g,+++)I Gi, j) 
ii , if all inputs are 1 at(i,j) 
=. 
COR(f,g,°:))G,j) 
= , if at least 1 image is ture at(i,j) 
== a 
For equation (1) N can be gotten by translating the 
origin from (0,0) to (0, 4-1) , and similarly for the 
other compass directions. (- N) can be obtained from 
elsewhere 
elsewhere 
N by negation BN, and similarly for W, E,and S. 
* (1"and *U" can be denoted by OR and AND, 
respectively. For a non — isolated and non — end 
point, it is solved by employing EXTADD after 
translating for 8 compass directions. Obviously ,if the 
accumulative value of the neighbors whose are ] is