The greatest difference in the values of so and 5, oc- 
curs, if different numbers of pictures are used for surface 
reconstruction with FAST Vision. The values are the 
lower, the more pictures are used. Fig. 5.1 shows sy for 
all stripes and all numbers of pictures for regulariza- 
tion by curvature minimization and A=2000. Fig. 5.2 
shows 5 for the same case. 
The number of iterations using adaptive regularization is 
almost always higher than that using regularization by 
curvature minimization. This effect results from the hig- 
her degree of freedom of adaptive regularization. 
A comparison of the heights resulting from automatic 
surface reconstruction with FAST Vision and the values 
measured by a human operator will be carried out in 
the near future. 
standard deviation 
  
of unit weight 
    
  
    
“lo [] 2 pictures 
3 pictures 
4 pictures 
  
    
  
      
  
    
{ 
stripe 1 
1 
stripe 2 
  
I i 
stripe 3 stripe 4 / 
Fig. 5.1: Standard deviation of unit weight dependent on 
number of pictures used for FAST Vision 
  
mean standard deviation 
of heights (metres) 
  
  
  
  
0,05 
  
1 2 pictures 
3 pictures 
2 4 pictures 
Fig. 5.2: Mean standard deviation of heights dependent 
on number of pictures used for FAST Vision 
  
0,00 w* I I Ï I 
  
  
stripe 1 stripe 2 stipe 3 stripe 4 
  
6. Conclusions 
Two aspects were important in the experiments carried 
out in this paper: How will the results of the recon- 
struction of different surfaces differ, if both methods of 
regularization, regularization by curvature minimization 
or by adaptive regularization, are used? And will there 
be a noticable improvement in the reconstruction of 
surfaces, if the input data consists of more than two 
pictures? 
The answer to the latter question is clearly positive. The 
addition of a third picture not only lowers the values of 
standard deviation of unit weight and the mean stan- 
dard deviations of heights, but also cuts down the 
number of iterations necessary to meet the break-off 
criterion. Using four instead of three pictures does not 
result in such a big improvement, but one has to keep 
in mind, that one of three pictures or its orientation data 
might be of poorer quality. Then, the addition of a 
fourth picture would increase the reliability of the 
recontruction results. 
816 
The comparison of the two regularization methods de- 
pends on several prameters. Surfaces containing edges 
are better reconstructed using adaptive regularization, 
whereas the assumption of smoothness implied in regu- 
larization by curvature minimization yields better re- 
construction of smooth surfaces as long as the regulari- 
zation parameter À is chosen not too high. Surface 1 
(rfpar) and 2 (rfrot) are composed of Z-facets with 
zero curvature almost everywhere. On surfaces 3 (cyl. 
par.) and 4 (cylrot) one of the principal curvatures is 
zero everywhere and the other varies from almost zero 
to low values only. So, there are good conditions for 
applying regularization by curvature minimization. The 
secorid method - adaptive regularization - is practically 
independent of the type of surface curvatures, but as it 
offers much more degrees of freedom to the object sur- 
face model the results show up more roughness than 
with the first method. Adaptive regularization was 
introduced in order to yield recontruction results not so 
dependent on that choice of A, this property is confir- 
med by the experiments: The reconstruction results deri- 
ved from regularization by curvature minimization get 
worse with increasing A in general, which is not true if 
adaptive regularization is used. The price to be paid for 
using adaptive regularization is a higher number of 
iterations. 
T. References 
Tikhonov, A.N./Arsenin, V.Y.: Solutions of Ill-Posed Pro- 
blems. V.H. Winston & Sons, Washington D.C. 
1977 
Weisensee, M.: Modelle und Algorithmen für das Facet- 
ten-Stereosehen. Deutsche Geodätische Kom- 
mission, Reihe C, Nr. 374, München, 1992 
Wrobel, B.: Digital Image Matching by Facets Using Ob- 
ject Space Models. 4th International Symposi- 
um on Optical and Optoelectr. Appl. Science 
and Engineering, March 30th-April 3rd 1987, 
The Hague, Netherlands, SPIE 804, p.p. 
325-333 
Wrobel, B.: The Evolution Of Digital Photogrammetry 
From Analytical Photogrammetry. Photogram- 
metric Record, 13(TT), p.p. T65-TT6,. April 
1991 R 
Wrobel, B./ Kaiser, B./ Hausladen, ]: Adaptive Regulari- 
zation of Surface Reconstruction By Image 
Inversion. In: Fórstner, W./ Ruwiedel, St. (eds) : 
Robust Computer Vision. Wichmann Verlag. 
Karlsruhe 1992, (Wrobel et al. 1992a) 
Wrobel, B./ Kaiser, B./ Hausladen, J.: Adaptive Regulari- 
zation - a New Method for Stabilization of 
Surface Reconstruction from Images. Presented 
Paper, XVIIth Congress of ISPRS, Comm. III/2, 
Washington 1992, (Wrobel et al., 1992b) 
These investigations are supported by Deutsche 
Forschungsgemeinschaft.