where Mi is correlation matrix, | M*l -
matrix determinant, n ; - Pj vector dimen
sion equal to region pixels number,
Ij - unit vector of rii dimension.
Since echo intensity fluctuations in
neighbouring pixels are statistically
independent (Sullivan,1980;Wang,1984),
Pi vector may be presented as f\~sXpo i >
where o t - is frequency measurement error
vector in Q t - region pixels, and pixels
frequency correlation matrix in Q f regi
on may be presented as follows
(7)
where 6t - standart deviation of Qj re
gion 1th pixel Doppler frequency.
So far as p(P)= const for a specific
image, we come to the problem
.fjwith the proviso
isl (2)
Substituting (4),(5) and (6) into (8)
and taking a logarithm result in
Mp(FjQ)ffi
-~>vax (9 )
where E xs designates exponential functi
on index in (9). To determine unknown
values l&t entering in (9) we utilize
maximum likelihood estimation and choose
g t paramètres values which maximize a
posteriori probability with specific im
age partitioning into regions. To achi
eve this goal it is sufficient to calcu
late the partial derivatives in (9) of
all g^ paramétrés and equate them to ze
ro
This equation differentiation and res
pective equations solution result in un-
knoun paramétrés optimal estimates
o - F
9r TwTi ", ^ i/5*
(10)
(10) follows from (7)«
Por background DOI regions, combining
zero state pixels, g- t =0, hence maximum
likelihood estimation should be applied
only for unit state pixels regions.
Proceeding from above observations we
maximize criterion (9) under condition
718
that in Qi unit pixels image regions un
known parameter g,- is set equal to g t -
while in zero pixels image regions - eq
ual to zero.
6. SEGMENTATION ALGORITHM CONSTRUCTION
The acceptable problem solution method
implies the application of sequential
iteration algorithm of sliding window
type (Therrien,1986; Pratt,1978). Let
some initial DOI partitioning is given
and all pixels are assigned specific
state values (zero or unit). We set win
dow size equal to Markovian random field
model definitional domain according to
(5). After that we choose an arbitrary
(x,y) pixel and superpose the window
centre on it. Then we vary the central
pixel state without other pixels state
change and calculate the corresponding
values of a posteriori probability (with
an accuracy of up to 1/p(P))
p(FlQ}P(Q) z p(Fls x;f =L,S lx;f) ) P(s x:f ~-L, £?
at L=1 and 0
where £2 - all pixels states set with
out (x,y) pixel. Note that
Since the probability P(S^ i<i() ) doesn’t
change with a pixel state variation so
as to determine a posteriori probability
maximum P(s) which according to
the adopted Markovian model of pixels
states relation depends only on S(x,y),
i.e. it is a conditional probability of
Markovian field (5)
P(s^llS ( J=P(s,,-Lin t , 3> )
Similary, it may be shown that
Pft kxi L. S(* s))~p(f x Jff*»), S [x ^j
where P(x,y) - DOI without (x,y) pixel,
", ( x,y) frequency. The second factor
of the left part of the equation doesn't
depend on s xy , so later on only the
conditional probability densiti may be
taken into account
pfaxvlF«*),s k;j ~L ,¿(*3))
Thus the search of a posteriori probabi
lity maximum for (x,y) pixel state vari
ation is reduced to the mequality vali
dity check
p({«H II&xd) <C
^ P Q S(xi))P(s v / o 15 (x J
and the assignment of a new state to a