International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
determine the parameters of the geometric object 
modelling the data of the seed region 
2. Increase the threshold of region growing. 
3. Apply region growing again and recalculate the 
geometric object parameters (Figure 3b). 
4. If the calculated geometric object parameters differ 
largely from the geometric object parameters of the 
previous iteration step, or if they do not fulfil certain 
conditions, stop the iteration process, else go to step 1. 
The geometric object parameters of the newly grown region can 
be compactness (U*/(4Ar), where U is the perimeter and A the 
area); the ratio of bounding box-area and object area; semi- 
major and semi-minor axis or the ratio of them etc. If these 
parameters vary a lot between two iteration steps, one can 
assume that the newly grown region went over the building 
edge as shown in Figure 3c. The conditions mentioned in step 4 
can be used as additional information to stop the iteration 
process (For example the maximal area of a building in our 
study area, or maximal number of corners). 
Additionally, after every region growing process an open-close 
(blow-shrink) function is applied onto the found regions (Figure 
3d). This is necessary to fill holes that might exist within the 
building regions, or to smooth frayed out areas. The kernel size 
of the open-close procedure must be chosen manually and it is 
recommended to choose a sequence of opening and closing 
steps with individual kernel sizes (e.g. close 3x3 — open 5x5). 
  
  
  
Figure 3. Example for building extraction 
The found region of interest is now vectorized (Figure 3e). The 
result is a very long coordinate list, since every single corner is 
included. Many of these corner points do not really represent 
the building geometry, but were created due to inaccuracies of 
the region-growing process. The task is to minimize this point 
list without impairing the geometric properties of the building 
(Figure 3f). 
One solution would be to calculate regression lines through 
points of the vectorized data (Figure 3e) that possibly belong to 
the same edge of the building. 
Another approach is applying a Hough-Transformation on the 
vectorized data in order to find the building boundaries. 
The second solution has been implemented and should be 
briefly described in the following (Gonzalez and Woods, 1992; 
Jain, 1989). 
The Hough transform is defined for a function F(x, y) as shown 
in Equation |. 
H(0,p)= | [FG vo -X cos(0)— ysin(0))dxdy (1) 
MO 
where à is the Dirac delta-function. With F(x. y). each point (x. 
y) in the original image, F, is transformed into a sinusoid p = 
xcos0 - ysin0, where p is the perpendicular distance from the 
origin of a line at an angle 0 (see Figure 4). 
  
Hough Domain 
Image Domain 
Figure 4. Hough Transformation 
Points that lie on the same line in the image will produce 
sinusoids that all cross at a single point in the Hough transform. 
For the inverse transform, or back-projection, each point in the 
Hough domain is transformed into a straight line in the image. 
Usually, the Hough function is used with binary images, in 
which case H(0, p) gives the total number of sinusoids that 
cross at point (0, p), and hence, the total number of points 
making up the line in the original image. By choosing a 
threshold 7 for H(0, p), and using the inverse Hough function, 
the original image is filtered to keep only lines that contain at 
least T points. 
The problem we face is that the points representing a building 
edge do not lie on one single line, but close to it. This means 
that the sinusoids in Hough domain will not intersect in exactly 
one single point, but a local maximum of the overlapping 
sinusoids will appear. This local maximum has to be 
determined and then we can perform the back transformation to 
obtain the according, representative line in image space. 
Unfortunately, the finding of local maxima is in Hough space is 
not easy since the sinusoids are periodic function which leads to 
double-identifications at the close to the beginning (0 = 0) and 
end (0 = 2z) of the period. 
The implemented solution to this problem carries out an 
iterative step-by-step Hough transformation (Figure 5). 
— 
— First step: apply the Hough-Transformation to all 
points lying on the boundary of the found building. 
— Start of iteration loop: Determine the global 
maximum in the Hough domain. This maximum 
location delivers exactly one 0 and one p value. These 
two values represent one line in image space. (Figure 
5, first row) 
— Calculate the back-transform of the global maximum. 
— Eliminate all points in the vectorized image that have 
a smaller perpendicular distance than the tolerance 7 
from the back-transformed line. (Figure 5, second, 
third and forth row) 
— Apply the Hough Transformation again to the 
remaining image points and start next iteration. 
This iterative process is terminated either if a requested number 
of lines has been found (7 number of iteration steps), or when 
no more points lie on/near to a line. This would mean that the 
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