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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
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Figure 4. TIN model of Figure 2 that is condensed with fractals 
H=0.8, units are in meter. 
    
   
  
  
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Figure 5. TIN model of Figure 3 that is 
condensed with fractals H=0.3, units are in 
meter. 
In order to perform a quantitative analysis of the surfaces 
rendered by the fractal interpolator approaches, we compared 
the results with the real surfaces. So 15-20 check points was 
selected for each region. Then for each point error function that 
defined as the difference between the real elevation and the 
estimated elevation was computed. Table 1 shows the residuals 
on the check points for the rough and smoothed region. 
In addition for H=0.1-0.9 with increment 0.1, standard 
deviation of the error function was computed that results have 
shown that the best H for irregular and regular surfaces are 0.3 
and 0.8 respectively (Table 2) .It means that irregular surface 
has fractal dimension 2.7 while regular surface has fractal 
dimension 2.2. 
553 
Table 1, Residuals on check points for smooth and rough region 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
(m) 
Point number Residuals on check | Residuals on check 
points for smooth points for rough 
region (m) region (m) 
| -0.090 0.340 
2 0.100 0.280 
3 0.060 0.210 
4 -0.332 0.726 
5 -0.381 -1.196 
6 1.159 0.141 
7 0.066 0.267 
8 -0.016 6.262 
9 0.037 -1.364 
10 0.558 6.763 
11 0.164 1.726 
12 0.121 -0.056 
13 -0.423 0.570 
14 -0.045 -2.015 
15 -0.890 -3.635 
16 0.041 -0.088 
17 -0.357 
18 0.010 
19 0.427 
20 0.022 
  
  
  
Table 2, standard deviation of the error function for smooth and 
rough region (m) 
  
  
  
  
  
  
  
  
  
  
  
standard deviation standard deviation 
of the error of the error 
H function for function for rough 
smooth region region 
0.1 2.675 0.419 
0.2 2.680 0.425 
0.3 2.682 0.404 
0.4 2.713 0.435 
0.5 2.676 0.465 
0.6 2.666 0.550 
0.7 2.789 0.598 
0.8 2.618 0.626 
0.9 2.888 1.283 
  
  
  
4. CONCLUSION 
Fractal methods can be successfully used when the real surface 
represents a natural phenomenon like elevation. The major 
problem seems to be the definition. of the appropriate 
parameters H and S (H is relative smoothness at different scales 
and S is scale or roughness factor) to best represent the 
variations of the real surface.