International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
    
   
  
  
configuration 
number of points 
  
  
  
  
  
5 [ qe 15 
"Tetra’: camera distance approx. 3 m 83% | 28% | 09% 
"Airl: camera distance approx. 1500 m 21594 41.7955 | 80.6 96 
'Air2: camera distance approx. 1500 m 5 90 7% | 06% 
  
  
  
  
number of points 
  
  
  
  
  
configuration T. 
Bay » ^35 
'"Strect1': camera distance approx. 10 m 50 96 50 96 25 96 
'Street2': reference distance approx. 2.85 m 50 96 25 % | 125% 
  
  
  
  
  
  
  
  
Table 1: Minimum thickness in percent of the camera distance for which the computation of the trifocal tensor was still 
successful. For the configurations Tetra’, ’Airl’ and ’Air2’ the given values hold for any computation method, whereas 
for the two ’Street’ configurations they hold only for the constrained method minimizing reprojection error (CR). 
k points, (ii) retrieve the exterior orientation parameters 
for the relative orientation and since the camera is cali- 
brated use the known calibration (2061.0, -1339.0, 2292.8) 
for this purpose. The extracted orientation parameters 
shall serve as approximate values for a subsequent bundle 
bock adjustment of the three images (with fixed interior 
orientation and fixed distortion) using the subset of the k 
points. After the block adjustment all 121 points in the 
three images are computed in object space by spatial in- 
tersection. This Euclidian reconstruction differs from the 
known positions of the retro reflecting targets by an ab- 
solute orientation A, i.e. shift, rotation and scale. This 
transformation A is computed using all 121 points. The 
remaining discrepancies after the absolute orientation are 
used to judge the quality of the initialization of the block 
adjustment. 
The aim is to find the smallest possible subset of points 
for the computation of the tensor, which still provides 
good enough approximate values for the initialization of 
the bundle block adjustment. This task can already be 
solved with the minimum number of 7 points, which are 
shown in figure 3 right part. And as it turned out, for 
the creation of the approximate values it is not relevant 
whether the tensor is computed in the simple way "UCA’ 
(without the constraints and minimizing algebraic error) 
or in the rigorous way ‘CR’ (with the constraints and min- 
imizing reprojection error). The remaining errors after 
finishing the bundle block adjustment, using the known 
calibrated interior orientation and the known non-linear 
distortion parameters, and the absolute orientation are: 
  
  
reconstruction errors [m] 
mean max 
x. 0.009 -0.030 
y | 0.009 -0.033 
z | 0.006 0.030 
  
  
  
  
  
For another task with the real data we could also ne- 
glect the known interior orientation of the camera and use 
Kruppa's equations to derive a common interior orienta- 
tion for the three images. In this case, however, the depth 
range of the used subset of object points has to be ex- 
panded a lot, see figure 3 right part, and at least 15 points 
must be used: 11 points from within the facade, 2 points 
from the roof (lying 3.3 m behind the facade) and 2 points 
3.1 m in front of it. For this point sample the rigorous 
computation of the tensor in the Gauss-Helmert model 
by minimizing reprojection error and considering the in- 
ternal constraints (method 'CR/) is necessary, because for 
the direct linear solution (method 'UCA") no valid inte- 
rior orientation can be obtained using Kruppa's equations. 
The remaining errors after finishing the bundle block ad- 
justment, using the determined interior orientation (2059.7, 
-1043.8, 2726.6) fixed and without any non-linear distortion 
parameters, and the absolute orientation are: 
  
  
reconstruction errors [m] 
mean max 
x1 3115 0.326 
vii 0.113 0.370 
z | 0:100 -0.429 
  
  
  
  
  
5 SUMMARY 
Concerning the computation of the trifocal tensor we dis- 
covered the following using different synthetic examples: 
e the difference between minimizing algebraic and re- 
projection error in the computation is negligible the 
more point correspondences are used and the more 
the respective object points deviate from a common 
plane, 
e minimization of reprojection error is more important 
than considering the internal constraints 
e if the image geometry is not too bad and at least 10 
point correspondences are used, a minimum thickness 
of the object points of about 1 % of the camera dis- 
tance is already enough to allow a proper solution for 
the trifocal tensor. 
International Arch 
JHCTRIHCHUT 13 0H 
  
Radial 
Canon 
207 
15 —— 
  
-B5 de 
Distortion [pixel] 
-10 A—— 
  
-15 d 
Radial 
  
Figure 3: Left ; 
Right part: Th 
mark the points 
on the roof signi, 
the trifocal tensc 
Guided by this 
using real image: 
we see: 
if the interio 
the exterior 
sor is sufficic 
even in the 
and even if 1 
number of " 
mum thickn 
1 96 of the 
away from t 
e ifthe interio 
computed w 
icantly away 
values for th 
ing Kruppa’ 
Although for mo 
ple direct linear 
rigorous solution 
turn similar resul 
one, because for 
to yield a usable 
Therefore the re 
tion is to first est 
Gauss-Helmert n 
mon interior oriei 
and finally to ini 
à bundle block a 
by additionally n