not limited only to the segmentation of optical images, it can 
also be applied to the segmentation of range images or other 
types of images. 
In an ideal image formation process the light intensities of 
points in the scene form the intensity I(xx,9æ) at pixel 
(zk, yr) in the image. Because of degradation, the , true” 
intensities J(z&, yx) are not observable, accessible are only 
the gray values g(x, yx) of the image. For simplicity we will 
denote a location (zx, yx) only with its index k, e.g. instead 
of g(zx, yx) we write gx. 
We assume that the degradation is due to additive white noise 
with a Gaussian probability density function (pdf) and zero 
mean value. The noise is statistically independent from the 
light intensities I(x, yx). Nonlinearities due to saturation, 
aliasing and quantization effects are neglected. Accordingly, 
we have for the gray values in the image: 
gk = Ir +n, 
where n is a realization of the Gaussian white noise. This 
leads for the a-posteriori pdf of the gray values in the image 
to: 
a 
fa (94 | It) = m e (- , 
where c? is the variance of the Gaussian noise. 
  
We also need a prior model for the light intensity Ip of the 
pixel (zo, yo), for which the homogeneity condition is tested. 
The prior model reflects our expectations in the value of the 
intensity Io before the pixel was assigned to a particular re- 
gion. Since a-priori we have no reason to believe that some 
intensities are preferred, we assume a uniform density on the 
bounded definition space D;. With AT = Imaz — Imin, we 
have: 
1 
A; : lo€Di 
JA] 
filo) - { 0 otherwise. 
2.2 Region model 
Our model for a region R is a parametric model. The „true” 
light intensities of the pixels belonging to the same region 
satisfy the equation: 
J 
HORDE (we, yr) (1) 
j=1 
with (k | (zx, yx) € R}, a; E R. 
The functions ¢;(z,y) are arbitrary, real-valued functions, 
which are supposed to be known for a given region. How- 
ever, it is not necessary that these functions are the same 
for all regions in the image. In our task of map based seg- 
mentation of aerial images, we choose the model of a region 
(i.e. the functions ¢;(z,y)) according to knowledge gained 
from maps. 
The parameters a;, j — 1,..., J in equation (1) are unknown 
and have to be estimated. However, as we will show later, if 
we are interested only in the segmentation of the image and 
not in the parametric description of the regions, the explicit 
calculation of their values is not necessary. We assume that 
these parameters are random variables over the set of regions 
in the image and have an a-priori Gaussian pdf with mean m; 
and standard deviation o;: 
Sui iust (a; — mj)? 
ied sr 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
2.3 Homogeneity predicate 
We now define the predicate used for testing the homogeneity 
condition. Let (zx,yx), k — 1,..., K be the pixels already 
marked as belonging to region R,. Their (unmeasurable) 
light intensities Z(z&, yx) (or short 7,) fulfill equation: 
J 
ho Ya on) (2) 
j=1 
We denote with (xo, yo) the pixel for which the homogeneity 
predicate is tested in the current step. Its gray value is go 
and its light intensity is Jo. The homogeneity predicate H. 
for pixel (zo, yo) and region R, evaluates to true (H, = 1), 
if 
J 
Io =} a4" @o, yo). (3) 
j=1 
Otherwise, H, evaluates to false (H, = 0). According to this 
definition, the conditional probability of the predicate #, is: 
j=1 
0 : otherwise. 
J 
Io = Sag 
Pui. =1]aI)=4 L5 lo 2 ja 4$. (zo. o) 
Pul. 0 af^), Io) and P4 (71. —1 | af"), Io) are comple- 
mentary. 
The random variables needed for testing the homogeneity 
predicate according to equation (3) are unmeasurable. Ac- 
cessible are only the gray values gi of the image. Hence, 
we redefine our homogeneity predicate and consider the a- 
posteriori probability Py(H, = 1 | gx), k — 0,..,K. We 
call this expression probability of homogeneity. If the calcu- 
lated value for the probability of homogeneity exceeds a given 
threshold we take the decision, that pixel (xo, yo) belongs to 
the region R, 
3 PROBABILITY OF HOMOGENEITY 
To illustrate the dependencies between the different random 
variables which appear in the calculation of the probability of 
homogeneity, we represent them in a Bayesian network (see 
e.g. (Pearl, 1986)). The nodes of the network contain the 
random variables. If there exists a direct causal influence of 
one random variable on the behavior of a second one, an arc 
of the graph leads from the node of the first variable to the 
node of the second one. The strengths of the dependencies 
are quantified by conditional probabilities. 
Consider the situation, where the homogeneity predicate for 
pixel (xo,ÿo) and region R, is tested. The region R, = 
((zx, yx) | k = 1... K} already contains K pixels. The cor- 
responding Bayesian network is given in Figure 1. The proba- 
bility for the homogeneity predicate to evaluate to true given 
the gray values of the image (i.e. the probability of homogene- 
ity) is calculated considering the dependencies given in the 
network. After successful predicate testing the Bayesian net- 
work is updated since the number of pixels in the region has 
increased. Each decision situation has its particular Bayesian 
network. 
The probability of homogeneity can be written as: 
Pr zi. {gx}, go) le: P, 
PERPE Pw (4) 
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