=> = 
|. (24) 
=~ 
7 6 7. The 
the analysis 
Matrix B - 
ngular. The 
Section 2). 
\ is equal to 
OGICAL 
: 
problem of 
developed 
sred cross- 
non-linear 
ased on the 
problem of 
adiometrical 
orphological 
ct set on a 
unction f(x)) 
lid 
(25) 
element of 
nd the norm 
(26) 
sed shape» 
ith constant 
as 
(27) 
on Ci, - 
(28) 
n - number of regions with constant intensity on a field of 
view X. 
Let F - class of one-dimensional functions. Let's consider 
the images f(x)eL?, and F(f(x))eL^, xeX, FeF. The 
«shape» of the image f(x) is said to be not more 
complex than the «shape» of the image f(x)) (denoted 
f,<f) if there exists FeF that fi(x)=F(f(x)). 
Consider a set of the images which shape is not more 
complex than the shape of f(x)). We denote this set by Vr. 
V is obtained from f with the help of various functions F 
from a class F. Let the class of functions F the following 
restrictions are imposed on: 
1. {F(2)=z,zeR4}eF 
2. ifFıe F, F2e F ‚then Fı(F2(zZ)) € F 
3. ifF,e F, F2e F , then ayFı(z)+ a2F2(z) e F, a4,02 >0. 
The set V: formed with the help of the various functions 
from the class F satisfying 1-3 is the closed convex set in 
iiu Then for anyone gel“, there exists a unique element 
from V; nearest to q. This determines the projection 
operator P: on the set V;. It is defined from a condition: 
Ie e|- inf le - 7l eS 
The shape of f is defined as the projection operator P: on 
the set Vi. 
The projection operator P; satisfies to the following 
properties: 
1. Vr*(o: o7P: ©} ; 
2. || Proll s |lol], oe L T || Pr oll 7 lol] © o<f 
3. || Pre- o|[20, oe L^, 
In the case of preliminary image segmentation (27) the 
explicit form of operator Ps ¢ is 
(9. 2.) 
P= 2 Ga e 
From properties of the projection operator follows, that 
the size | Pro - o || is a measure of distinction of images 
f and o, and the following characteristic 
"uL (31) 
can be considered as a similarity measure of two images. 
The result of segmentation s (15) can be considered as a 
shape of the template f(x,y). Therefore morphological 
correlation coefficient between f(x,y) and g(x,y) is 
calculated as follows 
lel 
We denote N; - number of pixels in the region xi, g, - 
average intensity of the image g(x,y) in the region xi. By 
definition of the projection operator 
Q 
loaf EN EA Xs) 
i (x,); 
2 
1 2 
N S if 28; " = em 
i ij NQGy)i 
i 1 
- Ap! raf x > eg Jap 
i j N(QG)i (x, )); 
ll - 3 3:8 G5) 53 Y grab) = Ap" X Ye elon 
i (xy), i (xy) i (xy) 
Thus, the expression for a square of morphological 
correlation coefficient is 
Ap! AAp 
= EL (34) 
Ap! BAp 
k (Ap) 
where 
A-XnW. n-——XYg B-XEXas 
N; (x) T G3» 
The matrices A and B have the same properties as 
corresponding matrices from Section 5. 
7. CORRELATION COEFFICIENT MAXIMIZATION 
The correlation coefficient maximization problem (22), 
(34) has a kind of a problem (10) which is reduced to the 
generalized eigenvalues problem (11). The block 
structure of matrices (24) allows to write down the 
generalized eigenvalues problem for each submatrix 
AixX*ABx, i=1...n, (35) 
Where A; = ni". For the problem (35) the statement from 
the Section 2 is valid. The non-zero eigenvalue Aj, i=1...n 
is under the formula 
199