Estimating principal curvatures and directions of a surface from a polyhedral
approximation with a large number of faces has become a basic step in many
computer vision algorithms, particularly those targeted at medical
applications.
The author wishes to design an algorithm for accurately and efficiently
estimating the
principal curvature and direction at each point of an underlying unknown
smooth surface from a polyhedral approximation.
This paper presents a simple new efficient algorithm for estimating the
principal curvature and principal direction at each vertex of a polyhedral
approximation of a smooth surface. These are computed using certain
3 x 3
mathend000# symmetric matrices defined by integral formulas. These matrices
are closely related to the matrix representation of the tensor of
curvature, which is the map
p 1#1 2#2
mathend000# that assigns to each point
pmathend000# on a surface Smathend000# the function
2#2(T)
mathend000# which is the directional
curvature of Smathend000# at pmathend000# in the direction of the unit length vector Tmathend000#.
The computations in the curvature computation algorithm presented in this
paper are simple and direct. No expensive iterative numerical computations
were needed even for computing eigenvalues and eigenvectors since closed form
expressions are used instead. The experiments presented in this paper
suggest that the accuracy of this algorithm is not worse than existing
algorithms, and is perhaps much better.