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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