ame origin 
ned by lin- 
> restricted 
its can be 
(man and 
  
i) 
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nd a least 
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| is added 
de, 1997] 
has nega- 
t A. This 
has over- 
nd Kanade, 
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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
  
Figure 6: Section of the Eremitage se- 
quence showing the tracked features 
HS 
Figure 7: Section of the Eremitage se- 
quence showing the tracked features. It 
is the frame following the frame shown in 
Figure 6. Note the change in feature loca- 
tions. 
To solve this problem we propose to parameterize, Q as: 
220 0 
Q(e, À) = R(e) | 042 0 | R(e)” (11) 
00) 
where R(e) is a rotation matrix with the three Euler angles 
denoted by e. The term aZ Qa, — bf Qb, = 1 is replaced 
by det(A) — 1, such that the over all scale of A is much 
more robust and less sensitive to the noise in a particular 
frame. 
Hence the estimation of Q is a nonlinear optimization prob- 
lem in six variables, with a guaranteed symmetric positive 
definite Q. Our experience shows that this approach to the 
problem is well behaved with a quasi-Newton optimiza- 
tion method. 
5 EXPERIMENTAL RESULTS 
We illustrate the capabilities of the proposed algorithm via 
three sets of experiments. The first demonstrate the capa- 
bility of using different error functions. This is followed by 
a more systematic test of the tolerance for different kinds 
of possible errors. Finally we show an example of why 
the proposed method for Eucledian reconstruction is to be 
preferred. 
5.1 Different Error Functions 
To demonstrate the capability of using different error func- 
tions, we used an image sequence of the Eremitage castle, 
see Figure 4. The 2D features were extracted via the Har- 
ris corner-detector [Harris and Stephens, 1988], where- 
upon the epipolar geometry was used for regularization via 
RANSAC/MLESAC [Torr, 2000] followed by a non-linear 
optimization [Hartley and Zisserman, 2000]. 
  
Figure 8: Section of the Eremitage sequence showing the 
tracked features and residuals from the roof using the trun- 
cated quadratic with the proposed method. The residuals 
are denoted by the dark lines. 
This enforcement of the epipolar geometry enhanced the 
quality of the data, but it did not yield perfectly tracked 
data. There are two main reasons for this. 
First, the trees around the castle yield a myriad of pottential 
matches since the branches look pretty much alike. The 
restriction of correspondances to the epipolar line is not 
sufficient to amend the situation as is shown in Figures 6 
and 7. 
Second, when filming a castle, one moves approximately 
in a plane — both feet on the ground. This plane is paral- 
lel to many of the repeating structures in the image, e.g. 
windows are usually located at the same horizontal level. 
Hence the epipolar lines are approximately located ’along” 
these repeating structures and errors here cannot be cor- 
rected by enforcing the epipolar geometry. In general the 
sequence containes plenty of missing features, mismatched 
featuress and noise. 
The truncated quadratic [Black and Rangarajan, 1996], the 
Hubers M-estimator [Huber, 1981] and the 2-norm were 
tested as error functions. The reason the proposed method 
was used with the 2-norm and not the original Christy- 
Horaud method is, that there were missing features in the 
data-set. These missing features are incompatible with the 
Christy-Horaud approach, but the approach presented here 
deal with them by modeling them as located in the middle 
of the image with a weight 109 times smaller than the 'nor- 
mal' data. It is noted that this approach for dealing with 
missing features can not be expressed in the framework of 
[Irani and Anandan, 2000]. 
In order to evaluate the performance the residuals between 
the 2D features and the corresponding reprojected 3D fea- 
tures were calculated. The desired effect is that the resid- 
uals of the 'good' 2D features should be small, hereby in- 
dicating that they were not ’disturbed’ by the erroneous 
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