íi 
V 
x 
  
  
  
  
  
  
  
  
  
  
0 200 400 600 800 1000 
Figure 2: Each additional measurement further bounds the 
elevation estimations 
Refer to appendix 4 for the derivation of this equation. 
Note that this is a very simple model for the propagation of 
the measurements. It should be extended by using the current 
DEM and the surface orientations. 
Thus from each measurement other 'measurements' can be 
derived for arbitrary positions. 
3.2 Combination of the measurements 
If for two measurements at the same position the probability 
densities are given by zi and 2; the variances are given by 
01 and 03) and the measurements are independent, then the 
combination is given by (see e.g. [Papoulis, 1984]): 
2 2 
0921 + 0122 
E i 7 
Te 02-402 () 
and 
2 _0103 (8) 
? elt] 
The propagation step delivers the derived measurements that 
can be used for the combination. 
Figure 3.2 shows the results for the same examples as were 
shown for the simpler min-max approach earlier. The 20 
boundary corresponds to the z.,,,, of fig.3.1 and the variance 
of the measurements is 1m. 
A disadvantage of the proposed method is that near a mea- 
sured elevation of a feature point the gradient of the surface 
is estimated badly. This is because close to a measurement 
821 
  
  
  
  
  
  
  
  
  
0 200 400 600 800 1000 
Figure 3: Bounding of the elevations by using estimations 
with a Normal distribution. Shown are the o and 20 bound- 
aries 
the propagated measurements are almost solely depending on 
the original measurement. This can be improved if surface 
orientation measurements are used as well. 
3.3 Including surface orientation measurements 
Including surface orientation measurements is straightfor- 
ward. In eq.5 the expectation of z'(x) is not 0 now, but 
the measured orientation. Like the point feature elevation 
measurements the surface orientation measurements can be 
propagated as well. 
3.4 Extension to 3 dimensional DEM's 
The extension to the third dimension is straight forward. For 
each position (x, y) in the DEM, now the following properties 
are kept: 
e elevation estimation: z 
e variance of elevation estimation: o? 
e surface gradient: (2, 2%} 2 (27.25) 
e variance of surface gradient: 0°, , 0°, 
æ y 
An example of a 3 dimensional DEM and its estimation is 
shown in fig.3.4. The reconstruction of the DEM was done 
using 1000 measurements of the elevation for arbitrary chosen 
positions and with different accuracies. 
Unfortunately the feature point tracking and surface orienta- 
tion measurement systems were not available yet, hence only 
results on simulated surfaces are shown. The complete sys- 
tem will be evaluated using a landscape model with a movable 
camera setup. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996