id 
nt 
nt 
n 
P 
e o p M 
ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
Since all possible pairs (c, €) actually occur, we can use 
these three cases to classify the type of velocity vector field 
at one instant of an arbitrary smooth motion: Infinitesimal 
translations are characterized by ¢ = o, and infinitesimal 
rotations by c- € = 0. The remaining velocity vector fields 
are said to belong to infinitesimal helical motions. At all 
instants, if the velocity vector field of a smooth motion is 
nonzero, it belongs to one of the three cases. 
If (c, €) represents the velocity vector field of a uniform ro- 
tation or helical motion, then the Plücker coordinates (g, g) 
of the axis A, the pitch p and the angular velocity w are re- 
constructed by 
(£8 —-(ce-pe, p-ce:e/e, w=, G) 
see e.g. (Pottmann Wallner, 2001). 
The Plücker coordinates (g, g) of a straight line A consist 
of a direction vector g and the moment vector g about the 
origin. From the moment vector, we can easily compute a 
point p of the line A, since for all points p on A we have 
the relation g — p x g. 
The above results about infinitesimal motions are a limit 
case of the following fundamental result of 3-dimensional 
kinematics: Any two positions of a rigid body in 3-space 
can be transformed onto each other by a (discrete) helical 
motion (consisting of a rotation about an axis and a trans- 
lation along that axis), including the special cases ofa pure 
rotation and a pure translation. 
Our algorithm actually iteratively computes the velocity 
vector field of a discrete helical motion. Those underly- 
ing helical motions are then used for the displacement. 
4 SIMULTANEOUS REGISTRATION WITH 
KNOWN CORRESPONDENCES 
The first application we have in mind is the simultaneous 
registration of N point clouds which have been obtained by 
stereo photogrammetry. The point clouds partially overlap, 
and in these regions correspondences (plus confidence val- 
ues) between points of different point clouds are known 
from surface texture analysis. 
The N point clouds can be viewed as rigid systems and 
are denoted by X;. An arbitrary number (at least one) 
of the given systems remains fixed. The others shall be 
moved such that after application of the motions the dis- 
tances of corresponding points, weighted with their confi- 
dence value, are as small as possible. Since in our case we 
have N > 2, only an iterative procedure is possible. We 
use a geometric method that involves instantaneous kine- 
matics, and thus it is similar to the approach in (Bourdet 
Clément, 1988). 
Only those points in a cloud are used for the alignment pro- 
cess which belong to an overlapping region with a neigh- 
boring point cloud. For such a given data point pair (x ;, V;) 
we know the index j of the system X; to which x; belongs, 
  
BE 
    
bae ddr 
   
w.... 
Figure 2: Example of multiple registration with known 
correspondences: All given 30 point clouds (top) and detail 
of 4 clouds showing the overlapping areas (bottom). Data 
by courtesy of Gerhard Paar, Joanneum Research. 
and the index K indicating the system X, of the point y;. 
The point pairs have found to be in correspondence with a 
confidence value w; € (0, 1). 
Our goal is to move each system X; by a motion M; ina 
way, such that after application of all these motions Mi, 
the new positions of corresponding points are as close as 
possible to each other in a least squares sense. Thereby we 
have to keep in mind the confidence values of correspon- 
dences. 
4.1 Displacement estimation via instantaneous kine- 
matics 
Since the expected motions are small displacements any- 
way, we replace them by instantaneous motions. The in- 
stantaneous motion of system X; against one fixed system 
(called Hp henceforth) possesses a velocity vector field. It 
is characterized by two vectors c;, €; € R3, and analo- 
gously to Eq. 1, the velocity vector v;o of a point x; € Xj 
is then given by 
Vio(X;) = € + € X Xi. (4) 
For a pair of corresponding points (x;, y;) we would like to 
estimate their distance after the motions have been applied 
to their systems X; and X;, respectively. In first order, we 
A - 267