2.2 Cylinder 
A cylinder scanned from different positions will often not be 
entirely visible. To be able to use these cylinders for the 
registration of different scans the parameterisation of a 
cylinder should not rely on whether the cylinder is partly 
occluded or not. Even when the cylinder is not occluded the 
end points of a cylinder are not uniquely defined which has to 
be taken into account when choosing a parameterisation. À 
cylinder can be described by (Figure 2): 
1. The orientation of the symmetry axis of the cylinder 
2. The orientation of the vector pointing from the origin to a 
point on the rotation axis closest to the origin 
3. The distance from the origin to the point on the rotation 
axis closest to the origin 
4. Start point and end point 
5. Radius 
One might notice an over parameterisation. In section 3 will 
become clear why this is done. 
  
Figure 2. Parameterisation of a cylinder 
Finding the pose and shape parameters of a cylinder is an 
iterative process. Therefore initial values have to be 
determined beforehand. The parameters are determined in 3 
steps: 
1. Finding the orientation of the rotation axis b of the 
cylinder 
2. Calculating the position of the cylinder closest to the 
origin m.c and simultaneously calculating the radius of 
the cylinder 
3. Deriving the start and end point of the cylinder along it’s 
rotation axis 
To find the direction of the rotation axis of the cylinder normal 
vectors are calculated in all the points that can be assigned to 
the cylinder. Until now assigning points to a specific object is 
done by a human interpreter. The calculation of normal 
vectors of the individual points in a point cloud is described in 
2.1. In the next step the vector product over all those normal 
vectors is calculated and the average is taken as the 
approximation of the rotation axis. 
Once the rotation axis is known a plane is constructed with the 
rotation axis as it’s normal going through the origin. All the 
laser points assigned to the cylinder are projected onto that 
plane. The projected points describe a circle or a fraction of a 
circle as seen in Figure 3. The radius and the centre of the 
circle are found solving: 
07 (x, -x)) t — yo). * (z; - (ax, * by)’ - r (7) 
where x, and y, are the coordinates of the centre of the circle, 
x;, y; and z; are the coordinates of the projected points, a and b 
are the gradients of the projection plane in x and y direction 
and r is the unknown radius. Evaluating (7) gives: 
My. z2 — (2x, 4 2az,)x, * (2y; * 2bz))yo - * (8) 
with 
fzXPp*ye(ax by) -r (9) 
The total set of equation becomes: 
2x, +20z;. Qy, +2bz;, -—lif % 
= ; ; : 1% (10) 
2x, +203, 2v, + 263, Jr 
X HN Zr 
Fons 
An Yu T Zu 
The radius can be obtained by substituting the calculated 
values for xp, Xp, à and b into (9). 
  
Figure 3. On the left points on a cylinder are shown. On the 
right those point are projected on a plane with the 
same orientation as the plane 
3. REGISTRATION 
The selection of what object parameters to use for the 
registration of different scans is critical. Take for example the 
length of a cylinder. It may occur that the lengths of a cylinder 
in two scans are different because of occlusion in one or both 
scan(s), the same holds for using the centre of gravity. It is 
obvious that using length or the centre of gravity for the 
registration will result in erroneous registration results. Only 
parameters that are invariant to the point set they are derived 
from are used. In case of registration with planes the invariants 
parameters are those that describe an infinite plane, which are: 
1. Direction of the normal vector 
2. Perpendicular distance from the origin to the plane. 
The invariants parameters for registration with cylindrical 
objects are those that describe an infinite cylinder. They are: 
1. Orientation of the symmetry axis of the cylinder 
2. Orientation of a vector perpendicular to the symmetry 
axis of the cylinder and going through the origin 
3. Perpendicular distance from the origin to the cylinder 
The parameterisation chosen in section 2.2 is chosen such that 
the infinite cylinders can be derived easily from cylinders with 
a certain length. Those infinite cylinders are being used for the 
registration. 
After the invariants of the different objects have been 
calculated, as described in section 2, the following observation 
equations for the normal vectors can be obtained. 
Rotation 
The parameterisation of the rotation is chosen in such a way 
that no singularities exist (Iterations of the linearised 
i44 
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