717 
assume thet D01 segmentation problem 
involves Q set partitioning into Q 4 * 
subsets through identical state pixels 
grouping so that 
U Ql~Q. and QiHdj-0 at ¥i^J (2) 
whore Qrlix^y.Sxy-Li], R~i 
Our goal is to i'ind through pixel s sta 
tes variation the partitioning static- 
fying the following condition 
QiQif-tQkyF)-* 
where < t > (Q l ,...,Q k ,F) is some segiaenv.a¡.Ion 
quality criterion, k is a number of 
image regions being segmented. The seg 
mentation is implemented according to a 
posteriori probability maximum criterion 
P(i,.. (3) 
where P(a Q.IF) is the probability of 
QQ* regions presence in image on 
condition that F DOI is observed, 
is joint probability den 
sity of all pixels Doppler frequencies 
on condition that image is partitioned 
into regions , P(QQ„) is a 
priori probability of Q t ,. .. , Q* regions 
presence in image; p(F) is uncondition 
al joint probability density of all pix 
els Doppler frequencies, 
4«MODELS SELECTION FOR IMAGE DESCRIPTION 
For formulated problem solution we must 
specify respective models adequately 
describing real DOI. In a number of pa 
pers (Hanson,1982; Besag,1974) Markovian 
random field models are used for pm, o.) 
description. Tnis approach is called 
the dual stochastic model. The essence 
of this model is that each region is 
described by its own stationary random 
process and the transition from one .im 
age region to another is modeled by Mar 
kovian process. 
It is obvious that possible image par 
titionings set is mutually-definitively 
related to pixels possible states set 
and the probability of different image 
regions presence may be substituted by 
joint pixels possible states probabili 
ty. 
ground itself are coupled regions of so 
me extension, each pixels state is rela 
ted to neighbouring pixels state in pro 
babili tic manner. Thus, for small-size 
objects the analysis may be limited to 
eight neighbouring pixels. Let denote 
the set of (x,y) pixel's eight neighbo 
uring pixels states by S(x</)» For concre 
te definition of (3) we must predetermi 
ne neighbouring pixels states probabili 
stic relationship so that the following 
equation is satisfied 
where F(s | ) is the (x,y) pixels 
state probability on conditiun that 
neighbouring pixels have S (w states, 
corresponding to specified image parti 
tioning. 
We utilise the presentation form of pix 
els state dependency on its neighbours 
states proposed in (Therrien,1983) 
" D i?o+Pi fex+/ t y + ‘Sx-J,i) + 
wherefraxe arbitrary constants, 
D is a normalization factor. 
Let us consider the form of multiplier 
in (3). We adopt the plane-parallel mo 
tion model for objects being observed. 
Then all the pixels in Q t - image region, 
corresponding to a moving object, must 
have equal conditional frequency expec 
tation g/ , which takes nonzero random 
value. Assuming that the background is 
stationary relative to a sensor we adopt 
conditional expectation, g,- , equal to 
zero for Q,- regions corresponding to the 
background. 
3. PROBLEM FORMALIZATION BASED 
ON ADOPTED MODELS 
Let us denote the set of F matrix ele 
ments corresponding to Q ¡_ , by F / 
Since all F/ are independent, frequency 
values of pixels belonging to different 
regions, are independent too. And there 
fore, for image model being considered 
p(FlQ,,..,Q,)=p(F,IQ l }p(FM)-‘p(i : *IQ«) 
For many DOI processing we may regard 
the presence of regions of only two 
types - moving objects and background 
generated by random processes of two 
types. Then pixel state 3«^ may take on 
ly two values - 0 or 1 and the problem 
of image partitioning is reduced to bi 
nary segmentation. For certainly we 
shall assume that pixels of image .regi 
ons having unit state (3,., =1) are con 
sidered as belonging to moving objects 
and pixels with zero state (S K y =0) as 
belonging to background. 
Sins moving objects patterns and back 
Within the scope of general statistical 
approach to DOI segmentation problem let 
us consider the case when Doppler freq 
uency mesurement errors are normally dis 
tributed keeping in mind that real echoes 
are described by narrowband normal ran 
dom processes (Papurt,1981) with instan 
taneous frequency g t - asymptotically nor 
mal distribution law for large signal- 
to-noise ratio ( Левин, 1974 ) • Then con 
sidering the set of Fj pixels, corres 
ponding to Qi , as a vector, the condi 
tional joint probability density о Г Q; 
region pixels Doppler frequencies may be 
expressed as follows