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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
Once p,, p,. p, and p; are determined, the j-th rectangle 
is formed as By; = {p, Pos Ps Pu} - 
If the two lines starting from p, and p, cannot join 
together at the same point p;, we temporarily create an 
auxiliary point. p,(x,, y,) for rectangulation (Figure 2d). 
685 
680 + 
675 
670 + 
665 - 
660 - 
655 + 
650 + 
645 + 
  
640 J À À L 1 1 1 1 1 1 J 
315. 320 325 330. ;:338 340. 345 3560 388 360: 2365: 370 
Figure 2b). Roof points. 
645 - 
  
640 1 1 I L 1 1 1 L J L 1 
15-320 325 330 .335.. 34D. 345. 3590 .355. 360. 365 370 
Figure 2c). Intermediate points. 
685 
[ 
  
680 + 
675 + 
Pn Pj 
670 + 
  
660 - 
B55 F 
  
  
  
  
d & 
B] . 
) 
p D; Pe 
645} 
  
J | 1 L 1 1 am] 
315 30 35 33) 335 340 345 350 355 200 365 370 
  
Figure 2d). Rectangulation. 
If there is no auxiliary point created, repeat step 1 to step 5 
555 
to form the (j+/)-th rectangle on this level by 
P uy 7 B; - B 
i i 
7. Once auxiliary points are created, repeat step 1 to step 5 by 
Pan =F; —B; + Pag where F,; denotes auxiliary points 
in the j-th round. 
8. After the rectangulation process continually repeats, if the 
number of the rest points is less than four, or if there are 
points that cannot form a rectangle, what remains can be 
classified as roof points. 
2.2 Forming 3-D primitives 
After finding rectangular bases of primitive roofs, the next step 
is to reconstruct 3-D polyhedral roof primitives. This process is 
working on the 3-D space. However, to reconstruct these 
surfaces, we first apply a 2-D range query to find roof points for 
each rectangular base. This is a 2-D check across different 
levels. The range query is to check the x, y coordinates of roof 
points P and base points. If the roof points lie inside the base in 
the nearest level above the base, they can form a roof primitive. 
The ^ rectangular bac can be represented by 
B, =[x:x"]x[y:y"] and P. lying inside B, if 
x, e[x:x'] and y, e[y: y]. 3) 
The primitive is determined according to| P. |. If | P. |=1and 2, 
it is a one-ridge point roof and two-ridge points roof, 
respectively (Figure 2e). If an upper rectangular base B, ,., in 
the nearest level exists, and satisfies 
B, C B;. (4) 
a four-ridge points roof can be formed. Then the 3-D polyhedral 
surfaces of a roof primitive can be formed by connecting roof 
points and base points (Figure 2f). 
688r 
  
  
B45 
  
  
640 1 À LL. 1 1 1 À 1 1 L J 
315.5320 325 5330. 3358. 340. 345 350. 385. 300 305 370 
Figure 2e). Roof points inside bases. 
2.3 Combining 3-D primitives 
Since the primitive roof models have been reconstructed in the 
previous step, the following steps are combining primitive 
models by Boolean operators in order to obtain a complete 
building. This process is first working on 2-D horizontal level 
for rectangular bases and roof points separately.