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