The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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The 3 parameters of the 3d planes corresponding to the 
segments, i.e. the two angles that define the direction of the 
normal vector of the plane, and the distance between the plane 
and the origin, are computed on-the-fly as the averages of 
values computed in the participating pixels. They are expressed 
in the same polar coordinate system that the range image is 
based on. 
3. SENSOR LOCALIZATION USING EXTRACTED 
PLANES 
The localization of the laser scanner is based on measuring 
distances to a minimum of three non-parallel planar surfaces 
(that intersect in non-parallel lines) in an already mapped 
indoor scene. The first scan maps the scene with respect to the 
scanner coordinate system; i.e., the parameters of the extracted 
planes are computed in the scanner coordinate system at its 
initial position. In the next scan, new ranges to these planes 
with known parameters are measured. The position of the 
scanner with respect to the previous coordinate system is then 
determined by intersecting the measured ranges. 
The determination of the scanner position with respect to the 
previous position requires a minimum of three corresponding 
planes in the two successive scans. Generally a larger number 
of planes are extracted by the segmentation algorithm, and a 
search procedure for finding corresponding planes is needed. 
The following sections describe the plane intersection and the 
search procedure for corresponding planes. 
(3) 
The solution of the set of equations given in (3) provides the 
coordinates of S 2 in the coordinate system of S,. By repeating 
this procedure at each consecutive scan, the position of the 
scanner with respect to the coordinate system of the scanner at 
the previous position can be computed. 
To transform the scanner positions to a single coordinate 
system, e.g. the coordinate system of the first scan, rotations 
between the coordinate systems of the scanners at successive 
scans should also be taken into account. If the positions of the 
scanners are required in a reference coordinate system, then the 
position and orientation of the scanner at the first position with 
respect to the reference coordinate system should be known. 
Assume that a third scan is performed, and the position of the 
scanner at S 3 is to be determined in a reference coordinate 
system O; we have: 
12 13 12 2 12 D T} _,0 
P P ~ ^ -s 3 — n •R.jjRqiSj 
22 23 22 2 12 R R 0 
P P ~ n s 3 — n k 12 k 01 s 3 
32 33 _ „32 _2 _ „12 TJ r» „0 
P P — n • s 3 — n • R 12 R 01 s 3 
(4) 
1.1 Scanner position determination 
The position of a point can be computed by measuring its 
distance to a minimum of three non-parallel planes with known 
parameters. Fig. 4 shows three non-parallel planar surfaces 
scanned from two positions S t and S 2 . As can be seen, if planes 
PI, P2, and P3 are shifted by -p", -p 12 , and -p 13 respectively, 
they intersect at point S 2 . The equation of a plane i in the 
coordinate system j of the scanner at position s 7 is written as: 
p ,J =nfx iJ +4y iJ +n«z ,J 0) 
where Ry denotes a 3D rotation from the coordinate system i to 
the coordinate system j. The rotation matrices can be derived 
from the parameters of corresponding planes in every pair of 
successive scans, except for Roi , which is the orientation of 
the scanner at its initial position with respect to the reference 
coordinate system, and has to be measured in advance (if the 
navigation is to be computed in a reference coordinate system). 
In practice, the scanner can be levelled by using the bubble 
level and levelling screws. In such case, the z axis of the 
scanner will always remain in the upward direction. 
Consequently, the 3D rotation matrices will be simplified to 
rotations around z axis only, which can be easily derived from 
the differences in the parameter 0 of the corresponding planes. 
where p ij is the distance of the plane i from the origin of the 
coordinate system j, ,n‘{, n'j are the coordinates of the 
normal vector n of the plane i in the coordinate system j, and 
x‘ J f y'j 5 z‘ J are the coordinates of a point x on the plane i in the 
coordinate system j. The short form of the equation can be 
written as: 
where . denotes the dot product of the two vectors. To compute 
the position of the scanner at position S 2 we take the 
parameters of three non-parallel planes in the coordinate 
system of Si, shift them according to their distances to S 2 , and 
find the intersection. This can be expressed as: 
Fig. 4: Determining the position of S 2 by measuring distances 
to three planes.