omogeneity
els already
neasurable)
tion:
(2)
omogeneity
value is go
edicate 71,
(#, =1),
(3)
ding to this
cate H, is:
(zo, yo)
ire comple-
omogeneity
rable. Ac-
re. Hence,
ider the a-
…, K. We
the calcu-
eds a given
belongs to
rY
int random
obability of
:twork (see
;ontain the
nfluence of
one, an arc
able to the
pendencies
'edicate for
"on R, =
5. The cor-
The proba-
true given
homogene-
iven in the
yesian net-
region has
ar Bayesian
ef
Pn (4)
Figure 1: Bayesian network for a particular decision situation
The nominator Pz of equation (4) is calculated by margina-
lizing the joint probability distribution:
Pz = S > P(H, =], {af}, Io, {9x}, go)
aC? Io
3
with j = 1,...,J, k — 1,.,K. Considering the de-
pendencies between the random variables in the Bayesian
network in Figure 1, the joint probability distribution
P (31. (a£), Io, (94), go) of the random variables in the
network results to:
P(Hr, {af}, Io, {9x}, go) = Pu(Hr | {a}, Io) x
Pis | 10) Po) ([] Poe | 07) (IP)
k=1 j=1
Using the probability density functions as given in section 2
and observing that for the pixels (zx, yx), k = 1,..., K al-
ready assigned to region &, equation (2) is fulfilled, the ex-
pression for Pz becomes to:
1
Pz= ei Pu(H,=1 | {a}, 10) — x
a m Ju } u MO AI
1 J
1
TT
exp _ (go m) ] yx
2x0 20 ( 270)
(o. = fe aj) $$ (zs, »)))
N
II exp| — 203 x
k=1
J (r) 2
1 (aj —m;)
en ua) )
j
=1
Because Py(H, = 1 | (af?), 10) = 1 only for Iy =
i at? 9 (zo, yo), after calculating the integral with re-
spect to Jo, the expression for Pz results to:
Po j / zh 1 1
a{" at AI (V2xo0) (V2xo;)”
x (o. m (32. a (æn,4e))
II exp| — 92 x
k=0
J (r) 2
(af) - mj)
[I exe (Sum da; (5)
sl 5
The results of the integrals in equation (5) can be expressed
in a closed form. The detailed calculation is given in (Quint,
X
665
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
1994). The integrals which appear in the denominator of
equation (4) are calculated in a similar way. One finally ob-
tains for the probability of homogeneity:
1 NË /det C
P Hr = 1 ; =
| Hoch 90) faoV2r0 (N -- 1)8 det C* i
exp (5 det Cext (N + 1) det C t ) (6)
20? det C 202 det C*
Being a probability, the values taken by expression (6) are in
the domain: P(H, = 1 | {gr}, go) € [0, 1].
In equation (6), the factor f, is defined as:
1 go — Imin 1 Imaz — go
=—erf| =——— | + = orf | ———=
het ( V2o ) 2 ( v20 )
where erf(x) is the Gaussian error function:
erf(z) = = [ $c? dt.
0
Using images with eight bits per pixel, the minimal and max-
imal intensity values are: [min = 0 and Imaz = 255.
The elements of the matrices appearing in equation (6) are
given in Table 1. The matrix C = (c;;) is J x J and com-
posed of the elements c;;, i,j — 1,...,J given in Table 1.
The matrix Cext = (ci;) is (J +1) x (J +1). The upper
left J x J submatrix of Cext is identical with the matrix C.
Column and row J 4- 1 respectively are composed of the el-
ements c;,5+1 given in Table 1. The matrices C*and Cz,
are constructed in a similar way, but now the elements c7;
from Table 1 are used. All matrices are symmetrical. For
computing the matrix elements c;;, the summation has to be
done over the product of the functions 90 and 9? at all
pixel locations (z&, yx), k — 1,..., N already marked as be-
longing to region &,. In addition to this, for calculating c7;
the summation is extended over the pixel (zo, yo) for which
predicate testing is under way.
4 COMPUTATIONAL ASPECTS
For calculating the probability of homogeneity for a region R.
and a pixel (zo, yo), partial knowledge of the regions model
is necessary: the functions 9 have to be known for the re-
gion. However, this does not assume, that these functions are
the same for all possible regions of the image. The complete
model of a region is given if one also knows the coefficients
aC? in the linear combination (2). These coefficients could
be estimated from the gray values of the image. Since we are
only interested in the segmentation of the image and not in
the actual values of these coefficients, in our segmentation
