International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
LIDAR-data points. We know that they lie besides the out- 
liers on the surface of the wanted object. By assigning a 
little standard field to every particle, the points will prop- 
agate it into a certain neighborhood around them and in- 
fluence the neighbored points. In this way they adapt the 
form of the field of the respresented object. Afterwards we 
can interpolate the field in the space between the particles 
and extract meaningful points like edges and corner points 
by searching for maxima. 
2.2 Tensor encoding 
The above mentioned field is a tensor field, that means each 
point in space has an associated tensor. In our case this ten- 
sor is a second order symmetrical tensor. If we formulate 
the tensor like in (1) it encodes a 3D-ellipsoid rotated in 
space. (2) is an equivalent writing for this. In fig. (1) we 
can see the geometrical meaning of (1). The normalized 
vectors ej, eo, es are the main axis of the ellipsoid. They 
build a local right-handed coordinate system. 
App "9 ef 
Tu 1. €), €3 0 Ao 0 el 
ei So el 
(1) 
T=); eye! + Agesel + Aseael (2) 
This Ellipsoid has the dimensions Ai, Ay and A; in the 
main axis directions. We define that A; > Ay > Az i.e. 
we claim that the ellipsoid is always oriented in the direc- 
tion of e4. With this definition we can rewrite (2) into (3) 
by saying that A; is the basic part in all three directions 
where the differences of 49 — A3 and A, — A» are added in 
the directions e»,e, and ej. The geometrical interpretation 
is shown in fig. (2). The Ellipsoid is decomposed into the 
As-part, which constructs a 3D sphere, encodes the likeli- 
hood of this location to be a point also called point-ness. 
The (A? — Aa)-part which defines a 2D disk in the e»- 
es-plane here called the surface-ness of the location and 
the (A; — À» )-part which defines a one-dimensional stick- 
portion and which is the curve-ness. 
T = (Mrz Ag)eiel "(T As)(ere] = exe; ) 
+A3(e; el = eses == ese) (3) 
  
Figure 1: An ellipsoid with its local coordinatesystem and 
the dimenions A1, Ag. À3 
  
Point Surface Curve 
Figure 2: The decomposition of a Tensor 
2.3 Voting as communication 
Every initial location sends out a tensor field and propa- 
gates it into the space in a certain neighborhood. Every 
other location in this neighborhood is then influenced by 
this field. To calculate the total influence on a certain lo- 
cation we simply have to summarize the tensor fields of all 
neighbors in a given radius. therefore we have to look how 
the tensor field propagates in space. 
As we have seen above, the tensor field represents three 
vectorfields with different meanings. Thus these vector 
fields behave differently while propagating, we have to han- 
dle each part by itself and assemble them again afterwards 
(4). The reason why we formulate the three fields in a sin- 
gle tensor is the comunication between these fields. This 
implicit communication is shown in fig. (3) where two 
votes of two different voting sites accumulate at the reciev- 
ing site to a surface portion. 
Figure 3: Communication between the curve field 
A £ A rp Bj mB; Bi 
Ti IG 19 + x | (TE, * T Lite x va) (4) 
i 
The design of the voting fields is derived by considering 
the analogy with the flow of force in particle physics 0. 
The three voting fields are shown in fig. (4). The length 
and orientation of the sticks depicted in fig. (4) is the 
strength and orientation of the field sent out by the vot- 
ing location (which is located in the center) and received 
at the site position relative to the voting location in a local 
coordinate system. 
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