ATA 
jstruction 
industrial envionments. 
n to be able to model the 
sses he semi automatic 
ans taken from different 
djustments. The method 
leters are estimated from 
re needed. A method to 
t scans the registration is 
we needed. 
nbination of range data 
> to be registered. In this 
; capable of doing this. 
ts measured in different 
on parameters. In case 
possible to register the 
or cylinders are fitted in 
an operator assigns 
d objects. The last step 
the registration of laser 
it (ICP) algorithm (Besl, 
responding points from 
oresponding points a 
s of this method are that 
e data sets and that one 
of the other. That is it 
in both scenes, which 1s 
rmore occlusions cannot 
ne idea as ICP but can 
ets of one another. This 
ences by dynamically 
en points throughout the 
t designed to establish 
tablish the relationship 
ions on the surfaces 
ier data set (Chen and 
he techniques described 
ages whereas (Eggert et 
scans simultaneously. 
/e, however, is designed 
imultaneously, although 
istration of 2D data and 
Hanson, 1994) 
The following section elaborates on the formulation of the 
observation equations used to find the pose and shape 
parameters of planes and circles in point clouds. Furthermore 
the parameterisation of the objects is discussed. The third 
section describes the mathematical model used to registerlaser 
scan. In the fifth section initial results are presented. 
2. FITTING OBJECT MODELS TO POINT CLOUDS 
Before the registration can be started corresponding objects 
need to be modelled in the different scans. Up till now 
algorithms have been developed to fit planes (modelled as a 
flat box) and cylinders to point clouds. The algorithms are 
designed in such a way that the same ideas can be applied to 
more complex CSG models. The principal idea is to minimise 
distances from the points to the CSG model to be found by 
adjusting the shape and pose parameters. The distance is 
defined as the perpendicular distance from the point to the 
object. For more complex CSG models this is not trivial to 
calculate, however advanced 3D modelling software packages 
offer this feature. 
The specific changes between the observation equations of 
different models lie in the fact that other partial derivatives 
with respect to the shape and pose parameters are needed. 
Section 2.1 gives observation equations for the fitting of a 
plane. As this is a non-linear problem initial values are needed 
for the pose parameters of the plane, which is described in this 
section as well. Section 2.2. mentions the observation 
equations to be used for cylinder fitting and describes a 
method to find initial values for the shape and pose 
parameters. 
2. Plane 
In close range laser scanning one has to take into account the 
fact that planes can have any orientation when setting a certain 
parameterisation whereas in aerial laser altimetry the data can 
be seen as 2.5D in which vertical planes are not present. 
Therefore choosing the right parameterisation 
is important. Describing a plane by the normal vector and the 
perpendicular distance from the origin to the plane (Figure 1): 
provides a singularity free representation (Heuvel, 1999). A 
plane can be written as a function of the normal vectorn and a 
signed distance /,: 
n-x—/ =0 (1) 
  
Figure 1. Parameterisation of a plane by a normal vectorn and 
a distance /, 
Using this parameterisation the distances from a point j to a 
plane defined by its normal vector n and the perpendicular 
distance /,, can be calculated using: 
d; Tp, ny, tnmx, Ex (2) 
Where elements 7, n; and n; are the elements of the plane's 
normal vector and elements x;, x, and x; are coordinates of the 
point from which the distance is to be calculated. 
The observation equations used are based on minimising the 
distance from the points to the plane. The observation 
equations that describe the relation between a change in the 
distance and a change in the parameters is: 
n 
od 
AP; 
where Ad is the perpendicular distance from a point to the 
object, p; are the object parameters, Ap; are the unknown 
changes to the parameters and n is the number of parameters. 
The partial derivative of the distance point p-plane with 
respect to the rotation elements is: 
b 4 
On, tij (e 
I 
The partial derivative that describes the connection between 
the perpendicular distance from the origin to the plane and the 
measured distance is: 
  
Before the iteration process can start approximate values for 
the normal vector n and the distance /, of the plane have to be 
available. As mentioned earlier the technique to find 
approximate values is designed specifically for the use of 
planes and will be different for other objects. The strategy to 
find approximate values consists of two straightforward steps: 
first an approximation of the normal vector is computed, 
second the distance from the origin to the plane along the 
normal vector is derived. 
The normal vector of the plane is determined by calculating a 
normal vector in each point of the plane. À normal vector in a 
point is calculated by performing a nearest neighbour search 
for the point p. In the next step vectors are calculated from 
point p to the neighbouring points. In case the angles between 
the vectors are sufficiently big the cross products are taken 
between those vectors. The constraint on the angles is used to 
reduce the effects of noise on the calculated normals. Finally 
the average normal vector is taken as the normal vector at 
point p. The normal of a plane is +n or —n. This is not a 
problem when solving equations (3) as the plane’s distance 
will have a different sign whether + or —n is used. 
After the normal vector n of the plane has been 
found the average distance is calculated from the distances 
using: 
1 N 
L =y 2" x, (6) 
j=1 
in which N is the number of points used. 
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