lane in x and y direction 
ing (7) gives: 
F2bzi)yo -* (8) 
(9) 
-2bz; —1Y) x, 
: | Vo (10) 
263, —1K 7 
n 
ubstituting the calculated 
  
ylinder are shown. On the 
jected on a plane with the 
lane 
TION 
rameters to use for the 
ical. Take for example the 
at the lengths of a cylinder 
f occlusion in one or both 
the centre of gravity. It is 
centre of gravity for the 
; registration results. Only 
point set they are derived 
1 with planes the invariants 
1 infinite plane, which are: 
? origin to the plane. 
yistration with cylindrical 
finite cylinder. They are: 
is of the cylinder 
ndicular to the symmetry 
hrough the origin 
e origin to the cylinder 
ion 2.2 is chosen such that 
| easily from cylinders with 
ders are being used for the 
erent objects have been 
, the following observation 
1 be obtained. 
n is chosen in such a way 
ations of the linearised 
observation equations at a singular point results in no 
convergence). 
A singularity free representation is the  quaternion 
parameterisation (Shih, 1990). This representation has one 
redundant parameter. In the observation equations this 
parameter is removed by demanding that the quaternion vector 
is a unit vector. 
The conversion from the four quaternion parameter, also 
known as the Euler-Rodrigues symmetric parameters, ql, q2, 
q3 and q4, to the 3*3 rotation matrix R is given by: 
2(g193 ^ 4294) la 1) 
2(q293 * 4194) 
2(q192 * 4394) 
5 
1 
44414 
RE Had: 4:94) 9 
299: * 4294) 2(9.9: ~ 9194) 
Toot 
pq 0 tg ty. 
The partial derivatives of R, with respect to the quatemion 
elements OR/0q,, are calculated by taking the partial 
derivatives of each element in the matrix individually. The 
partial derivative of a normal vector in scan s,, with respect to 
one of the rotation parameters is then calculated as, 
n. (12) 
oq; oq; 
1 1 
in which the vector s,, is expressed in the coordinate system of 
that scan no. 
The linearised observation equation for the rotation becomes: 
  
4 0 
OR : 
As, = ) oq, So Aq; (13) 
iz i 
Where As, is the difference between the normal vector sy in 
the base scan (the scan that defines the coordinate system) and 
the rotated normal vector s,,, R° is the rotation matrix with 
approximate values for the four quaternion elements. 
Translation 
The translation parameters position the scans in the world. As 
scale is not an issue of modelling (see later in this section) 
they can be found by entering equations, which state that the 
measured corresponding distances in different scans should be 
the same. The distance measured in one scan can be expressed 
in terms of translations along the x-, y-, z-axis in the other 
scan by: 
T 
ds, = ds, + (i. t (14) 
Where ds,, is the distance measured in one scan, ds, the 
corresponding distance in the another scan and t is the 
translation (x, y, z) vector. R^! can be calculated by: 
R'-R' (15) 
= T ; 
(R no) t can be written as nf, 4 njt; 4 nt, 
From this it follows that: 
Ods 
no = n. 1 6 
raie (16) 
1 
With this the linearised observation equation for the distance 
becomes: 
Ads, = Ads, + nA + n,At, + AL, (17) 
no 
in which Ads,, is the difference between the actual measured 
distances in the scan to be registered and the distance 
calculated with approximate values for the rotation and 
translation parameters as well as the observed distance in the 
base scan. 
Scale 
A scale difference is not assumed between different scans and 
is therefore not modelled. This is a valid assumption assuming 
the laser scanner(s) used to be calibrated before taking the 
measurements. Furthermore introducing scale factors would 
lead big to correlations between those factors and the 
translation parameters in many cases. 
Based on a model assuming a rotation and translation between 
the scans, the total set of equations becomes: 
  
  
  
As, 
As... As, 
Ads, : 
: AS. 
Ads, | 4 Ads, (18) 
AS oi A, s 
Ads, , , 
AS nien Aq 
Ads, At 
Ads oisn 
  
in which Aq are the unknown changes to the four quaternions 
and At consists unknown changes to the three translation 
elements. Al in this equation simply states that the elements 
on the left side of equation 6 are the same as the ones on the 
right side. A2 consists of the partial derivatives given by (13) 
and (17). 
From this set of equations it is obvious that the scans are 
registered with respect to the coordinate system defined by w. 
Furthermore the data measured in all scans is assumed to have 
noise as the observable of the base scan w are added as 
stochastic observable as well. The benefit of this is that in case 
multiple scans are being registered it is possible to test the data 
acquired from all scans, even from the base scan w. This can 
be done by analysing the observation residuals after the 
adjustment is performed. This is not implemented in the 
software yet. 
The registration can be done with cylinders and planes. For 
each plane added in the registration the set of equations can be 
expanded by: 
1. One equation to get the normal vectors of the planes to 
point in the same direction 
2. One equation that states that the perpendicular distances 
from the origin to the plane should be the same. 
In case a cylinder is being used for registration the same 
equations as used for registration based on planes can be 
added, additionally an extra equation can be added of the same 
-15-