166 
t unknown enables optimization of the positions of the nodes both laterally and along the curve (Forkert, 1994, pp. 
11-15). This might cause a collision of nodes in regions of maximum curvature. In order to prevent this, additional 
observations for the positions of the node points along the curve can be introduced. For a detailed description of the 
algorithm, see (Forkert, 1994). 
As the observation equations are not linear, approximate values for the unknowns are required for bundle block 
adjustment. The approximate position of each curve point is obtained from the intersection of its image ray with the 
polygonal pyramid formed by the rays of another image. As soon as approximations for all curve points are available, 
the nodes are distributed in regular intervals along the polygon of object points. All the algorithms described in this 
section were implemented into the universal bundle block adjustment system ORIENT (Kager, 1989). 
5.2 Reconstruction of our Test Object 
Using the observed image points (see section 4) and the orientation parameters (section 3), we can reconstruct the 
object curves iteratively by spatial intersection as described in the previous section. In our test, the orientation 
parameters were assumed to be constant. The iterative process of curve adjustment consists of two main steps 
alternately applied until the expected accuracy is achieved: 
1) adjustment with a given number of nodes 
2) insertion of additional nodes in the intervals containing points with the greatest mean residuals 
Good approximate values for the 
node points are essential for a nice 
convergence behaviour of curve 
adjustment. Applying the algorithm 
described in the previous section 
results in rather good approxima- 
tions for the object points. Experi- 
ence shows that it is advisable to 
carry out the first iteration of curve 
adjustment using these approxi- 
mated object points as constants in 
order to get a better initial 
distribution of node points (figure 6). 
     
   
     
  
edge points from 
290.0 V . : 
different images 
x adjusted curve 
with node point 
Figure 6 shows the reconstructed 
curve adjusted to five bundles of 
rays. Note that in general there will 
always be images in which only 
parts of the curve are visible. Due 
this fact and due to geometrical 
reasons it is necessary to use more 
than two images to achieve reliable 
results. Dangerous configurations 
are described in (Forkert, 1994, pp. 
101-106). An axonometric plot of 
the reconstructed car can be seen 
in figure 7 (next page). 
approximated curve with 
initial node arrangement 
Figure 6: reconstructed curve (right back door): approximated and final version 
  
  
  
No. images No.edge |r.m.serrorin| max.res.in | No. curves No. nodes mean r.m.s | max. r.m.s. 
points image image on object on object 
12 15704 7 um 60 um 44 501 0.5 mm 2.5 mm 
  
  
  
  
  
  
  
  
  
  
Table 1: results of curve adjustment 
(r.m.s. errors refer to edge points in image and object coordinate systems, respectively) 
The results of curve reconstruction can be seen in table 1. The r.m.s. errors are theoretical values resulting from 
inversion of the normal equation matrix. As mentioned in section 2, check points were also measured geodetically on 
some of the lines. The average distance of these check points from the adjusted curves was 1.1 mm, and the 
maximum distance was 1.7 mm. Another plausibility criterion is symmetry of the adjusted curves. Figure 8 (next 
page) is a plot of curves of the front of the car superimposed by mirrored curves of the opposite side. The greatest 
discrepancies appear at the side window rubber seal. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995