2 
  
  
  
Ke K 
1e 1 (r) 2 T 1 (r) 2 o T 
cji = N X (a 2009) + No? C N+1 S («| (TE; y) + +a? j=1,...,J 
k=1 J k=0 
ré fx 
Cij = N S. 9 (z,, Yk) $C (vs, yk) Ci = N+1 S (nn, uk) 9 (a, yk) 4j = 1, ...z J 
k=1 k=0 
izj 
1 N c? 7n; 1 X (r) c? LU 
T * SE T 2 enis 
C+ = Ww > (zi, yk) gk * No? GH = Fg »»D (zx, yk) gk (N31)? j=1,..,J 
CJ+1,J+1 = N > ai CJ+1,J+1 = Nil > où 
k=1 k=0 
2 
  
  
Table 1: Definition of the matrix elements used in the calculation of the probability of homogeneity 
procedure the estimation is not done explicitly. The calcu- 
. lation of the probability of homogeneity can be performed 
without knowing the actual values of the coefficients of for 
the current region. 
It is possible to calculate the matrix elements given in Table 1 
iteratively. One observes that after a successful assignment of 
a pixel to a region, the matrix elements cj; of the current step 
will become the matrix elements ci; of the next step. Thus, 
for testing the homogeneity predicate, only the elements c;; 
have to be calculated. These elements can be calculated 
iteratively using the elements ci; from the previous step. 
The computational complexity depends from the size of the 
model used, i.e. from the number J of functions used in the 
linear combination (2). The main effort spent in the calcu- 
lation of the probability of homogeneity is for the calculation 
of the determinant of a (J + 1) x (J + 1) matrix. Due to 
iterative calculation the effort is independent from the size 
(number of pixels N) of a region. 
5 ASIMPLE EXAMPLE 
To illustrate the usage of our approach with a simple example, 
we assume that the image is composed part by part of planar 
surfaces. Although this is a limitation, it can be used with 
good approximation for range images of scenes with mostly 
planar surfaces or even for light intensity images of objects 
without textured surfaces. lt is a reduction to the simplest 
case of a polygonal approximation of surfaces, which is often 
used in the segmentation of range images, see e.g. (Besl, 
1988), (Silverman and Cooper, 1988). 
In this case, the model functions for the regions to be seg- 
mented are: 
di(zk,yk) — Tk 
P2(Xk, Yk) = Yk 
$s(zk, yx) — 1 
and J — 3. According to the definitions given in Table 1 the 
matrices C and Cex: take the form: 
Sz lzy mz 
C= ley Sy my 
m. my lcg 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
ley Sy My lyg 
Cext = m m 1+ Foy ma 4 mao? 
€ y Nog g Nos 
2 
m 
lag lyg mg + > Sg 
3 
and the matrix elements are defined as follows: 
  
  
K 
1 2 T 1 2 ag 
s2= 5 DT + 32 = RL NS 
k=1 k=1 
} i Ae 
89 = Ww 9 ly = 3 e 
k=l k=1 
N N 
1 2 1 mac? 
lzg E N PR N 2 lag = N 2 na No? 
The elements of the matrices C* and Cz,4 are computed 
in a similar way. For these matrices, the summations in the 
definition of their elements start with index k — 0. 
To test the sensitivity of our approach with respect to degra- 
dation with noise and with respect to violations of the model 
assumptions, we have used a synthetic image. In the upper 
left area of the image, the gray values rise from background 
level with a constant slope of two gray values per column 
until the middle column of the image. In continuation of the 
first region, the gray values fall in region 2 with the same 
slope until they reach background level. The gray values in 
region 3, which is situated in the middle of the image, violate 
the model assumptions: they depend upon a parabolic rule 
from their position. In region 4, situated in the bottom of 
the image, the gray values have a slope of 1.5 in both row 
and column direction. Within each region absolute gray value 
differences up to 200 occur. 
The segmentation results for the synthetic image degraded 
with Gaussian white noise of different variances and for dif- 
ferent parameter settings are given in (Landes, 1995). In 
Figure 2, the segmentation result for the synthetic image de- 
graded with Gaussian white noise with the variance o2 — 30 
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