0 
Gp = pp 08,8) = -Aexp(-(@* +4") — (n 
We can find a derivative in an arbitrary 0 direction using follow- 
ing filter function, Gpe = cos(0)Gpo + sin(0)G,z 
We convolve our DSM with this derivative filter in 0 € [0, 7/12, 
…, 23/12] directions as follows, Je = D(x, y) * Gpe 
If 0 angle is perpendicular to the building ridge-line orientation, 
then one side of the building rooftop gives positive response, and 
the other side of the building rooftop gives negative response 
to filter. Assuming B’(x,y) is the jth connected component 
in B(z, y) binary building segment matrix, we assume that two 
sides of the building rooftop (RP; and RN}) can be extracted as 
follows, 
RP) — B'(z,y) x (Je » 0) (8) 
RN] — B'(z,y) x (Ja « 0) (9) 
Since we have no pre-information about building orientations, 
we should do the derivative filtering in all possible orientations. 
Therefore, we calculate >, RL; for 0 € [0,7/12, ..., 237/12] 
directions. Building ridge-line will be extracted in 0; filtering 
angle which is almost perpendicular to the ridge-line orientation. 
However the ridge-line will be also detected in 0; — 7/12 and 
6; + 7/12 neighbor filtering directions. Therefore, the ridge-line 
will have a value of higher than 2 in the ) , RL} result. As a 
result, ridge-line of jth building rooftop can be obtained by cal- 
culating R (z,y) = Y_, RL), > 2 and eliminating connected 
components which are less than 10 pixels in order to eliminate 
redundant information coming from small objects on rooftop. 
Next, we use detected ridge-lines for roof-type classification and 
three-dimensional rooftop model reconstruction purposes. We 
benefit from detected roof ridge-lines to classify rooftops as "flat 
roof’ or ’gable roof’ type. If there is no ridge-line detection re- 
sult on a building segment, we assume that building as flat roof. 
If there is a ridge-line on the jth building segment, then we calcu- 
late mean of DSM height values on ridge-line location by calcu- 
lating > RI (z,y) x D(x,y)/M, where M is the total number 
(zy) 
of ridge-line pixels in R? (z, y) binary matrix. We also calculate 
mean of DSM height values on building border by calculating 
S sv) B?(z,y) x D(z,y) /N, where N holds the total num- 
ber of building border pixels in B?(z, y) binary matrix. If the 
difference between these two mean values is lower than 2 me- 
ters, then we assume the rooftop as a flat rooftop. Otherwise, it 
is assumed as gable rooftop. 'This 2 meters criteria 1s obtained by 
observing gable rooftop characteristics over test area. 
3.1.3 Obtaining Three-Dimensional Representation We start 
with generating a zero matrix D»(z, y) with the same size with 
D(z,y) matrix. New height values belonging to objects in the 
city will be stored in D»(x, y) matrix. In order to eliminate noise 
in non-built regions, we apply median filtering to the original 
DSM (D(z, y)) by using a 3 x 3 pixel size window obtaining 
a filtered DSM (D;(z, y)). For non-builtup areas, or in other 
areas for (x, y) coordinates which satisfy B(x,y) = 0, we ap- 
ply Do(z,y) — Dy(x,y). That means: we assign smoothed 
ground height values for non-built regions. As building wall, 
we insert single height value to each building boundary which is 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
stored in B(z, y) binary building shape matrix. For each build- 
ing, the building height value is assigned by calculating the mean 
of D(z, y) values on all building boundary pixels. After smooth- 
ing the noise on the ground and inserting sharp building walls, 
finally rooftop height values are assigned to D»(zx,y) matrix, 
Rooftop height assignment is done by considering the roof-type. 
If the roof is classified as a flat roof, then only a single height 
value is assigned to all building area which is equal to building 
wall heights. If the rooftop is classified as a gable-roof, we fol- 
low different approaches depending on the building complexity. 
If the building is not detected as a complex building as struc- 
ture type (if the Euler number is equal to 1), then we can find 
polygons which defines the rooftop in three-dimensional space. 
To this end, we detect corners of building segment using Har- 
ris corner detection algorithm (Harris and Stephens, 1988) over 
panchromatic image of the test region. Besides, we also detect 
endpoints of the building ridge-line. We pick each building cor- 
ner one by one. Then, we find the closest ridge-line endpoint. 
A line between building corner and the closest ridge-line end- 
point can divide the rooftop into polygons. A detailed demon- 
stration of this approach is illustrated in (Sirmacek et al., 2011). 
Height values of rooftop polygons are assigned to corresponding 
pixels in D(z, y) matrix. If the building is detected as a com- 
plex structure or if the building ridge-line could not be extracted 
properly, unfortunately we cannot use the same idea for building 
rooftop reconstruction and more advanced rooftop reconstruction 
approaches are needed. Herein we leave the building rooftop re- 
construction at this level. For complex building rooftops and for 
rooftops for which we cannot extract the ridge-line properly, we 
only insert corresponding pixel values from D ; (x, y) matrix. 
3.2 Hierarchical Generation of 3D Prismatic Models of Build- 
ings 
An approach is presented to extract 3D model of buildings ac- 
cording to LODI (Level-of-Detail 1) based on CityGML standard 
(Kolbe et al., 2005). The approach starts with approximation and 
regularization of the building boundaries. Two algorithms are 
proposed for approximation of the building polygons based on 
the main orientation of the buildings (Arefi et al., 2007). The 
algorithms are selected according to the number of main orienta- 
tions of the buildings and implemented as follows: 
e If the building is formed by a rectilinear polygon, i.e., sides 
are perpendicular to each others from the top view, a method 
based on Minimum Bounding Rectangle (MBR) is applied 
for approximation. This method is a top-down, and model- 
based approach that hierarchically optimizes the initial rec- 
tilinear model by fitting MBR to all details of the data set. 
Principles of MBR based polygon approximation is presented 
in Figure 5. 
Accordingly, after determination of the main orientation, the 
building polygon is rotated to the main orientation, as shown 
in Figure 5(a). In the next step the MBR image (Fig. 5(b))is 
subtracted from the rotated building region. After subtrac- 
tion, new regions will be produced (Fig. 5(c)). For any of 
those regions a MBR will be calculated (Fig. 5(d)). They are 
again subtracted from their corresponding regions produced 
in the previous step (Fig. 5(e)). As illustrated in Figure 5(e) 
some small regions are created. The process is followed 
by computing new MBR regions and subtracts them from 
their corresponding regions. This hierarchical procedure i5 
continued until "no regions" are produced any more. That 
means the progress stops when either no new regions cre: 
ated any more or the size of produced regions is less than 
predefined threshold. After convergence the final polygons 
  
    
   
       
     
     
    
   
   
    
    
    
  
  
  
  
    
   
   
   
   
  
  
    
   
   
   
    
   
    
    
  
  
  
  
     
    
    
    
    
    
   
   
   
   
   
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