14000
12000
10000
8000
Figure 2: Rendering of a data set acquired with our stripe projection sensor. The object is a precise sphere. Deviations
of the sensed data from the ideal sphere are shown on the right. The number of points is plotted over the deviations in
millimeters.
3.3 Curvature
Surface curvature is derived from the fundamental forms
given above. The principal curvatures are the maximum
curvature kiand minimal curvature k». Alternatively the
mean curvature A and Gaussian curvature K can be used
to describe the surface locally:
kız=H+VH?-K
and
M
K … LN — M
EG — F?
EN --GL—2FM
H a EE ADM
2(EG — F?)
They are translation- and rotation-invariant. While kı, ko
and H, K are both valid pairs for local surface character-
ization, as is noted in (Besl, 1988), there are further con-
siderations which may favor the one over the other. For
one to compute the principal curvatures is computationally
slightly more expensive. Since the expression of which the
Square root is taken can become negative due to numerical
instabilities, additional precautions have to be taken. The
mean curvature is the average of the two principal curva-
tures and is therefore less sensitive to noise. On the other
hand since Gaussian curvature is the product of the two
it is much more sensitive to noise. Using only the signs
of the curvatures six basic surface types can be determined
using principal curvature while eight can be determined us-
ing mean and Gaussian curvature.
Based on principal curvature further local properties of a
surface can be derived. (Koenderink and van Doorn, 1992)
have proposed a shape classification scheme based on two
quantities called S and C:
Cz JcrR
Where 5 describes the shape, and C the strength of cur-
vature. (C is the square root of the deviation from flat-
ness, another derived quantity in differential geometry).
Points of same value for S but differing C, can be seen
as points of same shape with stronger curvature. The main
difference to the description using mean and Gaussian cur-
vature is the possibility to describe surface flatness with
a single quantity C « 0. We will detail the analogy to
our approach below. A study comparing both description
schemes (Cantzler and Fisher, 2001) found no significant
difference of the two. Other authors have extended the SC
scheme and have given different formulas for the shape pa-
rameter. For our studies we have decided to use the HK
scheme.
4 CLASSIFICATION AND SEGMENTATION
In the previous chapter we have given the mathematical
quantities used to describe a surface locally. In order to
apply these quantities in a classification we have to com-
pute them from range data. Due to the nature of the data
described earlier reliable curvature estimation becomes a
difficult task crucial to the success of the segmentation pro-
cess. We have tested the algorithms described below on a
dataset of a test scene consisting of a planar, a cylindrical
and a spherical region, which was acquired with our sensor
(see figure 3).
4.1 Curvature Estimation
Several methods for curvature estimation have been pre-
sented in the past. An overview of the most prominent
methods has been given in (Flynn and Jain, 1989). For
simple approximation the curvature can be computed from
the change of orientation from the point of interest to its
neighbor. Some methods based on this idea have been pre-
sented especially for triangulated surfaces, where surface
normals are computed per mesh. These simple techniques
are often used for edge pixel detection for example in mesh
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