International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
split and merge predicates uses one parameter, the last parame- 
ter gives the minimum surface of a region in the final segmenta- 
tion. The adaptation of this algorithm to colour images in RGB 
or L*a*b* spaces, and an evaluation of segmentation strategies 
are presented in (Roux et al., 1997). 
The figure 1 shows the result of the region-based segmentation 
over a small part of an aerial image. 
iro m i 
Figure 1: Part of the original image (©EUROSENSE) and result 
of the segmentation. 
2.2 Planar estimation 
We propose to investigate different approaches for the estimation 
of a 3D plane for each region of the segmentation. The plane 
is estimated with all the laser points whose projection using the 
collinearity equation falls inside the region (see figure 2). 
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/ laser points 
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Figure 2: Projection of the laser points in the image plane, and 
attribution to the regions of the segmentation. 
Given N laser points (zi, yi, zi )i=1..~, the problem is to estimate 
the equation of a plane as: 
axx+bxy+exz+d=0 
or, if vertical and near-vertical faces are disregarded: 
z=aXxz+bXy+c 
Two types of parameter estimation techniques can be evocated: 
linear and non-linear approaches. Among the first ones, the least 
     
squares estimator has been widely adopted because its ease of 
computation. On the other hand, non-linear techniques proved to 
be robust to the presence of outliers: the LMedS and the RANSAC 
estimators have also been tested for this application. An extensive 
tutorial on parameter estimation can be found in (Zhang, 1997). 
Least Squares Estimation (LSE): in the case of the second 
plane equation, the LSE approach consists in the minimization of 
the mean quadratic error: 
  
  
"De n la x x; +b x y: + 6 — zn) 
s > 3 
The solution is given by: 
a Soa] OS eue Si TÉ sun 
So ui Su > vi Sn 
€ Sn > y, N Sm 
e 
Il 
The figure 3 presents 2 views of the 3D reconstruction for the 
black and white regions of the figure 1. These examples show 
clearly the inadequation of the LSE approach when outliers are 
present in the set of 3D points. One way to overcome this be- 
haviour would be to replace the LSE, i.e. minimization of, ei’, 
with one M-estimator, i.e. minimization of 5 ^. p(e;?), where p 
is positive and symmetric function with a unique minimum at 
zero (Rousseeuw and Leroy, 1987). 
   
Figure 3: Two views of the 3D reconstruction of 2 regions using 
the LSE approach. 
Least Median of Squares (LMedS): the LMedS estimates the 
parameters of a shape by solving the non-linear minimization 
problem: 
min median, rj? 
where r; is the residual at the point z. This problem does not have 
an analytical solution. It is solved wifh a search in a large set of 
possible models generated with the data: 
e select n features randomly, and estimate the corresponding 
model, 
e calculate the residual to the model for each feature, 
e sort the square residuals and select the median value as qual- 
ity measure for the estimated model, 
e repeate previous steps £ times and select the model with the 
better quality measure. 
where n is the minimum number of features to calculate a model. 
In the case of 3D planar surface n — 3. The number of iteration 
t should be large enough to have selected at least one set of m | 
features without outliers. The probability of success or failure 
can be calculated according to the maximal proportion of outliers 
a: 
Pfailure = (1 T (1 = œ)")" 
   
    
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