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(8) 
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(10) 
(11) 
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(12) 
International Archives of the Photogrammetry, Remote Sensin 
3.3.2 Translation In case of determining the transla- 
tion vector by using planes at least three corresponding 
pairs are required, because of restrictions to the position 
of planes in space. If the planes are corresponding, the 
normal vectors of the planes are approximately equal. The 
difference between the planes is the translation, the plane 
is shifted by the values Az, Ay and Az. The equations of 
corresponding planes can be written as: 
a(z — Az) 4 b(y — Ay) -- c(z — Az) - d?? = 0 (13) 
az -b5y-cz-pq? z0 (14) 
If the equations are equated one obtains: 
aÂx +bAy 4- cAz — d?? — q* (15) 
Generally, in matrix notation equation (15) can be written 
as: 
AT 
Cl 3 dil andata et) A (16) 
Each pair of corresponding planes yields one equation. A 
least squares solution is used to calculate the translation 
vector 1: 
l+v= A2 (17) 
Be (AT A LAT] (18) 
4 EXAMPLE AND RESULTS 
The method proposed in the previous chapter was applied 
to scan data gathered with a Riegl LMS 360i scanner. In 
a first test, a demo data set was recorded. A corner in a 
room was selected and scanned from two different scan 
positions. A corner provides three planar surfaces, which 
are perpendicular and thus provide enough information to 
determine the rotation as well as the translation parame- 
ters. The transformation parameters have also been deter- 
mined using traditional methods. Retro-reflective targets 
were distributed, identified in the scans and the transfor- 
mation matrix was calculated. 
Figure 5 shows the measurement setup. The scan positions 
are visualized by the scanners, the two scans are displayed 
in different colours and are already registered. The cal- 
culated transformation matrix can be treated as reference. 
The matrix is structured as described in equation (7) and 
reads as follows: 
0.4590 —0.8880 —0.0297 3.5393 
T = 0.8879 0.4596 | —0.0210 —1.9763 
1.0.0823 —0.0167 0.9993 — —0.5283 
0 0 0 1 
(19) 
For the registration process without the targets, first pla- 
nar surfaces are extracted from the scan data. The distance 
threshold (see chapter 3.2.3) for the region growing has to 
be selected greater than the accuracy of the scanner. In this 
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g and Spatial Information Sciences, Vol XXXV , Part B3. Istanbul 2004 
  
Figure 5: Registered scans and scan positions 
case the noise of the scanner does not affect the segmenta- 
tion. In the example the threshold is selected to 2 cm. The 
region growing process results in three extracted planes for 
each scan, if the small regions are neglected. The next step 
is to assign corresponding planes. At first for the test data 
set this is done manually. 
Scan 1 Scan 2 
Measured point cloud: 
  
Segmented planes: 
   
Figure 6: Test data 
Figure 6 illustrates both, scans and the result of segmenting 
the points to planes. The numbers in the figure indicate the 
corresponding planes. The plane parameters were calcu- 
lated using all segmented points by the method described 
in chapter 3.2.1 and are estimated to: 
The plane parameters in table 1 are used to compute the ro- 
tation and the translation between the two scan positions. 
For the rotation matrix three different combinations with 
pairs of corresponding planes are possible, whereas all the 
planes are required to calculate the translation component. 
The method described in section 3.3 applied to all combi- 
nations (2 and 1, 3 and 1, 2 and 3) yields an averaged trans- 
formation matrix including rotation and translation of: