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74 
Figure 2: The interpretation plane 
3.2 Vanishing point detection and image orientation 
The extracted straight lines are input to a vanishing point 
detection procedure. This procedure starts with the selection of 
the longest extracted line which is assumed to intersect with 
other image lines in one of the three main vanishing points. In 
fact, the procedure is based on the analysis of intersection 
constraints on interpretation planes and thus does not require 
the lines to actually intersect in image space. Image lines are 
grouped, based on accepted statistical tests on the intersection 
constraint of three interpretation planes: 
0=(n, xn,)-n; (3) 
A complete set of independent constraints is adjusted. The 
object orientations (v) that relates to the vanishing points are 
computed from the adjusted observations: 
v -n,xn, (4) 
Here we concentrate on the results, as the procedure has been 
described in detail in (van den Heuvel, 1998). 
As a by-product of the vanishing point detection, the orientation 
of the image relative to the object is found. The rotation matrix 
can be constructed when at least two of the three vanishing 
points — associated with orthogonal object orientations — have 
been detected (Fórstner and Gülch, 1999): 
R - (v, v5, V) (5) 
However, this rotation matrix is ambiguous because there is no 
unique relation between the object orientations (v) from the 
vanishing points and the object coordinate system. To reduce 
the ambiguity in this rotation matrix it is assumed that the 
object orientation that is closest to the y-axis of the camera 
  
  
  
  
  
N = 
5 epipolei ——— 
      
system corresponds to the Z-axis of the object system. Now 
four options for the rotation matrix remain, corresponding to 
four 90 degree rotations of the object system around the Z-axis. 
Note that — up to this ambiguity — the orientation of the images 
is found by the vanishing point detection, and thus only relative 
position remains to be determined in order to complete the 
relative orientation. 
In the sequel of the procedure only those image lines are used 
that have been uniquely grouped to one vanishing point. 
Especially lines on or near the connecting line between two 
vanishing points (a so-called horizon) cannot be uniquely 
grouped and therefore these lines are not used for the final and 
main step of the procedure for automatic relative orientation 
described in the next section. 
3.3 Correspondence and relative position 
With the orientation of the images relative to the building 
known, the relative position and correspondence problem shows 
many similarities with the vanishing point detection problem. 
The line in space that connects the two projection centres 
intersects the images in a point that is called the epipole (Figure 
3). The spatial orientation associated with a vanishing point is 
found as the intersection of interpretation planes, while the 
relative position vector is found as the intersection of epipolar 
planes. An interpretation plane is constructed from the two 
endpoints of an image line. An epipolar plane also needs two 
(corresponding) image points, one from each image. Because of 
these similarities the procedures for the detection of the epipole 
is also similar to the one for the vanishing point detection. 
However, there are some important differences. First, 
correspondence between the two images is unknown, while in 
the vanishing point detection an image line links the two 
endpoints. On the other hand there exist only one epipole (per 
image), while an image of a building usually shows two or 
more vanishing points. 
Like for the vanishing point detection, statistical tests on the 
intersection of planes (now epipolar instead of interpretation 
planes) can be grouped for detecting correspondences that 
support the same epipole. However, the number of possible 
correspondences, and consequently the number of statistical 
tests to evaluate would explode without the use of additional 
object knowledge. With n points per image, there are n 
correspondence hypotheses, and thus n^ possible epipolar 
planes. As an intersection constraint involves three planes, the 
number of tests is of the order n°. 
  
  
  
   
vanishing — ^ 
point 
Figure 3 : Two epipolar planes and the epipoles (left), two interpretation planes and the vanishing point (right) 
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