PROCESS 
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3.2 - Two First Sides Detection 
As soon as a seed point is chosen we proceed to 
monocular detection of building sides. For each seed 
potential position we apply a line detection process based 
on a criteria of radiometry discontinuity (i.c. gradient) 
and sign continuity of this discontinuity. 
Sclection criteria of the first side is based on biggest 
gradient along a line and on sign continuity of this 
gradient along the same line. Thus, for each line D, (its 
equation being Y-A,X--B,) passing by the seed point we 
compute a cost function Gp, which we try to maximize. 
This cost function takes the form of 
izn 
Gp, = M aradiX 11a; +B, S00) (1) 
iz 
SG) a 1 
if sign(grad|Xo][A,Xq B,]) 7 sign(graa[X ;] [A,X;-- B,]) 
ifhot S() « 0 
Index i limits computation insidc area of interest. S(i) 
express sign continuity along D, line. When we have 
extracted the first side, it is very easy to find the second 
one because it is perpendicular to the first onc. We used 
the same function cost to detect perpendicular side. 
We used two types of gradient in order to maximize our 
function cost, the classical and the declivity ones. Results 
show that the second one provides best localization of the 
two sides detected. In effect, some detected sides are not 
lines with real building sides (sec figure 5) when we used 
classical gradient, so wc will keep declivity gradient (see 
figure 6) in the following (for more details about 
declivity operator sec [Quiguer 91). 
  
    
Figure 6: Use of Declivity Gradient on Same Bdg 
659 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
3.3 - Parallelogram Closing 
We used criterion of parallelism in order to close 
parallelogram which constitutes a building. So, we apply 
a set of parallel lines of the two first sides we have 
already detected. We realize closing by minimizing a cost 
function F,. This function integrates homogeneity and 
discontinuity notions. Homogeneity appears inside roof 
of buildings and discontinuity on their sides. 
Homogeneity expresses likeness between grey levels 
inside building along two parallel sides (see figure 7). It 
has to be low ; it is computed by a difference of two 
means. 
  
Homogeneity = | m1 - m2| 
mean ml 
  
  
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| mean m2 
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Figure 7: Computation of Homogeneity 
Fr takes the form of : 
I, = Homogeneity | Gradient (2) 
In effect, using only gradient is insufficient because 
urban zones are complex scenes and include several 
parallel sides belonging to different buildings. So, we can 
separate buildings using luminance criterion. 
Nevertheless, using only this criterion is insufficient too 
because it can’t exist local minimum of function F, 
inside building. So we compute F, with ranked gradient 
into a decreasing order. Optimum corresponds to the first 
local minimum (see figure 8). 
  
Fr 
  
{ 
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[f > 
Gradient in decreasing order 
| Optimum of Function Fr 
  
Figure 8: Optimum of Cost Function F,