Gatzke and Grimm / Curvature Estimation
curvatures, and vector quantities such as principal curvature
directions. In this paper we focus on the Gaussian curvature
because it captures much of the data in a single number. The
evaluation techniques easily extend to the other properties.
The three main contributions of this work are the devel-
opment of several new curvature calculation techniques, the
construction of a test suite for the evaluation of curvature cal-
culation algorithms, and several insights into how and where
different methods fail. We also discuss how these methods
perform on a segmentation task.
Section 1.1 highlights previous work on curvature calcu-
lations for meshes. Section 2 briefly describes the different
methods. In Section 3, the test cases, mesh parameters, and
noise perturbations are described. Results of the analyses are
presented in Section 4. Section 5 summarizes the conclu-
sions of this study. Finally, Section 6 outlines possible areas
for future work.
1.1. Previous work
A number of researchers
6, 13, 2, 14, 1
have looked at curvature
estimation from 3-D range images for computer vision appli-
cations. While these methods were developed for 3-D range
image data, the methods based on data fitting can also be
applied to meshes. The main difference between curvature
calculation on meshes and curvature calculations based on
range data is that range data provides a rectangular array of
pixel data, while on a mesh, data is available only at discrete
points. Therefore the mesh-based approach requires a pa-
rameterization step before the fitting step. An obvious choice
is to project the local portion of the mesh to a plane, which
may cause folding. A better choice is to use a parameteriza-
tion method such as Desbrun’s
11
(see Section
11
).
Meek and Walton
12
perform asymptotic analysis for sev-
eral methods using both regular data (as in range data) and
irregular data (as in meshes). However, they state that their
asymptotic analysis applies only to discretization and inter-
polation methods, but not to least-squares fitting methods.
Desbrun et. al.
4, 3
defined methods to compute the first
and second order differential attributes (normal vector, mean
and Gaussian curvatures, principal curvatures and principal
directions) for piecewise linear surfaces such as arbitrary tri-
angle meshes. They claim optimality of their discrete curva-
ture operators under mild smoothness conditions. They in-
corporate local operators to denoise arbitrary meshes of vec-
tor fields, while preserving features. However, they also ad-
mit that smoothing techniques do not deal well with large
amounts of noise.
Taubin
15
proposed a method that estimates the tensor of
curvature from the eigenvalues and eigenvectors of a 3 × 3
matrix defined by integral formulas. He also incorporated
a smoothing step for noisy meshes. A key benefit of his
method is its simplicity; the complexity is linear in both time
and space.
Goldfeather
8
compares several methods for calculating
principal curvature directions, looking at the impact of ran-
dom error on the resulting error in the calculated principal
curvature directions. He also compares three methods for
calculating normal directions at vertices. His primary test
cases are a torus and a more complex closed surface, with
random noise added to the vertices. He concludes that small
errors in quantities such as normal curvature can amplify the
error in the principal curvature directions. He describes an
improved method that uses the normal vectors at adjacent
points to generate a third-order fit in the curvature calcula-
tion. He attributes the improved control of the error mag-
nitude for his cubic method to its third-order, rather than
second-order, approximations.
2. Curvature Metrics
Curvature metrics can be grouped as either discrete metrics
or fitting methods. In the following sections we describe both
existing methods and our new or modified methods. The var-
ious methods and the acronyms summarized are summarized
in Table 2.
Curvature Calculation Taxonomy
Discrete Methods
Discrete Curvature Operator DCO
Modified DCO Mod (NEW)
Integral Eigenvalue Method IEM
1-Ring with Normal Fits
Adjacent Normal Cubic ANCE
Exact normals
Adjacent Normal Cubic ANC
Computed normals
Parametric Fitting
Parameterization
Fit Planar Desbrun
Polynomial Fit*P Fit*N (NEW)
Radial Basis RBF*P RBF*N (NEW)
Implicit Functions
Conic Fit Con* (NEW)
* = Number of rings
2.1. Discrete methods
Discrete methods attempt to calculate curvature directly
from a given mesh without an intermediate fitting step. They
derive relations based on a local region (typically the star of
the vertex, or 1-ring) about a point.
c
The Eurographics Association 2003.
