Cl PA 2003 XIX 11 ' International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
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Figure 4. The first four images of a sequence taken with the Rollei d30 metric 5 camera and the points tracked through it. 
Figure 5. Two views of the VRML model generated fully automatically from the sequence. The cameras are given in green. 
4.2 (Auto-)Calibration of the Sequence 
Basically, to obtain a metric, i.e., calibrated sequence, the 
orientations as well as the 3D points have to be upgraded to 
metric space. Here, we first assume, that the calibration matrix 
is already given. In a second step we explain, how we obtain 
an approximation for the calibration. 
To calibrate a sequence given a calibration matrix containing 
the ratios of the principal distance to the image width and 
height, the principal coordinates in terms of the image width 
and height, as well as the sheer of the sensor (Hartley and 
Zisserman, 2000), we use a linear algorithm for the image pair. 
It results in orientations for the two images as well as in 
(metric) 3D points. The 3D points are used to calculate the 
orientations for succeeding images which in turn can be used 
to compute 3D coordinates for points not visible in the first 
two images. (Metric) bundle adjustment is employed to 
determine a highly precise and reliable result. 
To obtain a solution also if no calibration information is 
available, we have developed a solution based on sampling, 
which gives reliable results in most cases, as long as the 
principal point is not too far from the center of the image. 
Though we are therefore not able to deal with parts cut out of 
larger images, the solution is from our experience (Hao and 
Mayer, 2003) more stable, especially for triplets or short 
sequences, than those based for instance on the absolute conic 
(Hartley and Zisserman, 2000). The basic idea is to set the 
sheer as well as the principal point to zero. Then, the size of a 
pixel is assumed to be a square and ratios of principal distance 
to image width from 2.5 to 0.5 are sampled in 30 logarithmic 
steps. For all steps the first triplet is calibrated using the 
resulting calibration matrices. The criterion for choosing the 
ratio of principal distance to image width is the average 
standard deviation of the image coordinates. After obtaining 
the principal distance for the width, the ratio of the width and 
the height of the pixel is varied from 1.2 to 0.8, i.e., the 
principal distance for the image height is computed. Finally, 
more precise values for the principal distances are determined 
by a bundle adjustment with additional parameters. 
For the above sequence we first used the given calibration 
information. The ratio of principal distance to the image width 
is known to be 1.7219, the ratio to the image height is 2.2887, 
and the principal point is close to the center of the image (- 
0.0112, -0.0298). The skew is assumed to be zero. With this 
information we have obtained a standard deviation of the 
calibrated sequence of 0.39 pixels. We have produced a 
VRML (the virtual reality modeling language; ISO / IEC 
14772-1:1997 standard) model from the calibrated 3D points. 
Two views with the cameras in green and the points in red are 
given in Figure 5. Our procedure for auto-calibration resulted 
in 1.84 for the ratio of the principal distance to the image 
width and 2.34 for the ratio to the image height. The standard 
deviation in image space was estimated to be 0.48 pixels. This 
is not an extremely good result, but for visualization the 
differences were minor. One also has to consider that this is a 
relatively challenging scene, as the basis between the single 
cameras is rather small compared to the distance to the object.