moO
kt
e computed
tions in the
0.
ct to degra-
ff the model
n the upper
background
per column
ation of the
th the same
ay values in
age, violate
arabolic rule
e bottom of
in both row
te gray value
ge degraded
and for dif-
, 1995). In
ic image de-
nce o2 — 30
Figure 2: Segmentation result of the synthetic image de-
graded with Gaussian white noise with a2 — 30
is given. As model parameters we have used in this case:
mi = m = 0, m3 = 128, 01 = 02 = 03 = 3 and 0? = 30.
We have used the value § = 0.8 as the decision threshold in
the homogeneity predicate testing.
Experiments have shown that the segmentation results for
regions for which the correct model was chosen are good up
to values for the standard deviation of the added noise which
are three times higher than the gradient of the gray values
within the region. At the left border of region 1 and the
right border of region 2 there appear inaccuracies which are
expected since the gray values of the two regions reach at
these borders background level. Model violations, as shown
with region 3 of the synthetic image, are partly tolerated.
It is mainly the parameter c? which controls the amount
of noise or model violation tolerated by the segmentation
algorithm. For optimality, this parameter should be chosen
equal to the actual noise variance in the image. Choosing this
parameter smaller than the variance of the actual noise results
in a segmented image containing many single points rejected
by the algorithm. However, these points could be eliminated
in a following stage by morphological operations. Choosing
this parameter bigger than the variance of the actual noise
is more critical since in this case different regions could be
merged in the segmented image.
6 AERIAL IMAGE SEGMENTATION
We are using the homogeneity predicate described in this arti-
cle for the segmentation of colour aerial images. The regions
gained this way are used together with line segments as prim-
itives in our model based aerial image understanding system
Moses (Quint and Sties, 1995).
As a control algorithm for the segmentation process a region
growing scheme is used. The process starts with a set of
initial seed regions. For all regions, pixels are sought which are
neighbour to at least one region and which are not yet marked
as belonging to a region. The probability of homogeneity
is calculated for these pixels and each of their neighbouring
regions. If this probability exceeds the decision threshold 6,
the pixel is marked as belonging to the corresponding region.
If the initial regions cannot be extended any longer new seed
regions are chosen in areas with small gray value differences.
The digital images used in our project are acquired by scan-
ning aerial colour photographies and have a raster size of
30 cm x 30 cm on the ground. The image in Figure 3 shows
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
a part of the campus of University of Karlsruhe. The German
Topographic Base Map 1:5000 which is available in digital
form is used to gain estimates for the positions of initial seed
regions. In order to obtain stable values in the calculation
of the probability of homogeneity, seed regions should have
a minimum size. Experiments have shown, that an initial
region size of 5 x 5 pixels is suitable.
Map knowledge is also used to choose the model for a given
region according to the known class of the objects. For the
segmentation of the image in Figure 3 we have used two
types of models: the planar model presented in section 5 for
regions corresponding to buildings, parking areas and streets,
and a Markov Random Field (MRF) model for wood and grass
regions. MRF approaches already have been used in previous
work (see e.g. (Cohen and Fan, 1992), (Herlin et al., 1994))
for the segmentation. of textured surfaces.
In our approach we use a second order MRF model:
1 1
> S. Alm (I(x%® — l, yk — m) — ux) = 0.
I=-1 m=-1
Since the light intensities I” are unmeasurable they are re-
placed with the gray values at the corresponding pixel loca-
tion. Hence, the model functions $;(z&, yx) in equation (1)
are:
$; (xk, yk) 7 g(za — l, yk — m) — px
with I,m € {-1,0,1} excepting the pair (I,m) = (0,0).
There are eight model functions and thus for the probability
of homogeneity determinants of 8 x 8 and 9 x 9 matrices have
to be calculated. For the parameter pj, we use the local mean
of the gray values in the neighbourhood. The variance c? of
the noise in the three channels of the images is estimated
using the method described in (Brügelmann and Fórstner,
1992).
For each channel we calculate the corresponding probability
of homogeneity. The value used for testing the homogeneity
predicate is obtained in analogy to the law of total probability
as a linear combination of the three probabilities of homo-
geneity. The factors in this linear combination are chosen
inverse proportional to the variance of the noise in the corre-
sponding channel.
Figure 4 gives the segmentation result of the aerial image of
Figure 3. Pixels belonging to the same region are marked in
Figure 4 with the same gray value. As a decision threshold the
value à = 0.8 was used. A number of 14 initial seed regions
were extracted from the map. After our segmentation the
image was divided in 86 regions. As one can observe, man
made objects like buildings, streets and walking ways, for
which the planar model was used, are segmented with good
accuracy. The MRF model provided good results in the area
with regular planted trees in the lower left corner of the image,
but difficulties arise in the wood area in the upper part of the
image. The gray values in this area are very inhomogeneous
and cannot be represented by the used model. As a result,
the wood area was splitten into several regions.
7 SUMMARY AND CONCLUSION
Our approach for a homogeneity predicate is based on the
a-posteriori probability for a pixel to fulfill the model assump-
tions for a region. For some practical relevant models (poly-
gonal surfaces, MRF models) we have derived a closed for-
mula to calculate the probability of homogeneity. Using this
