ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002 
  
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Figure 3: Plücker representation for lines. 
the 3D model of cylinders onto their images for pose es- 
timation. However, there was no attempt for defining an 
appropriate respresentation of cylinder as a geometrical 
entity. Motion estimation from generalized cylinders has 
been studied in (Sayd et al.,1996, Sayd et al., 1997). How- 
ever, in the same way that, in computer vision, lines are not 
simply considered as particular case of general curves and 
require their own representation and mathematical frame- 
work, straight cylinders need to be studied as a geomertical 
entity and not as a particular case of generalized cylinders. 
In this paper we introduce a canonical representation for 
cylinders. This is closely related to the line Plücker rep- 
resentation. This representation is suitable for computer 
vision and photogrammetry applications. We use this rep- 
resentation to define the general equation of a cylinder. 
We then use this respresentation to describe the motion of 
cylinders and their viewing geometry. We show that a pair 
of non parallel cylinders observed by two cameras provide 
enough constraint for recovery of the motion and structure 
trough a non-linear minimization. Finally, we introduce 
the main equations dominating the three view geometry 
of cylinders. We show the relationship between cylinders’ 
three-view geometry and that of lines and points defined by 
the trilinear tensor (Hartley, 1997). We also develop a lin- 
ear method, which uses the correspondences between six 
cylinders over three views in order to recover the motion 
and structure. 
2 CANONICAL REPRESENTATION OF CYLINDERS 
In this section we first review the Pliicker representation 
for lines. Then we introduce a new cylinder representation 
and its relationship to the Pliicker coordinates of lines. 
2.1 Pliicker representation of 3D lines 
A 3D line is represented by two vectors 1 and N. 1 is the 
unit vector representing the orientation of the line. The 
vector N is orthogonal to the plane defined by the 3D line 
and the origin of the coordinate system. The lenght of IN, 
h — |[N ||, is the distance from origin to the 3D line. Figure 
3 illustrates these entities. 
A point M belongs to the 3D line if and only if its Pliicker 
coordinates (1, N) satisfy the following equation: 
Mx1=N (1) 
  
  
  
  
Figure 4: Distance of a point M from a line defined by its 
Plücker coordinates: illustration for the proof of the Theo- 
rem. 
The nearest point of a line to the origin, H — N xlLisa 
particular one and we use it often in this paper. The dis- 
tance of the line from origin is defined by ^ — ||H ||. 
2.2 Motion of lines 
After a coordinate transformation defined by the rotation 
matrix R and the translation vector T, a 3D line (11, Nı) 
is represented by (lo, No) such that (Navab and Faugeras, 
1997): 
lo = R 0O3x3 l; 
[S1 [e HE e 
Where E — TR is known as the essential matrix. 
2.3 Image ofa line 
The 2D image of the 3D line (1, N) is presented by n — 
TNT in the camera coordinate system. There are three more 
parameters to estimate in order to recover the 3D line, two 
for the orientation 1 of the line and one for the depth A of 
the line from the camera. 
2.4 Canonical representation for Cylinders 
In this section we first introduce a canonical representation 
for cylinders. We then define the equation of the cylinder 
using this representation. 
We represent a cylinder by (1, N, 7), where (1, N) is the 
Plücker representation of the axis of the cylinder and r is 
the radius of the cylinder. 
Theorem: A point belongs to the surface of a cylinder 
represented by its canonical representation (1, N, r) if and 
only if: 
IM x1-Nl||27 (3) 
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