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Estimation of curvatures in point sets based on geometric algebra

: Seibert, Helmut; Hildenbrand, Dietmar; Becker, Meike; Kuijper, Arjan

Institute for Systems and Technologies of Information, Control and Communication -INSTICC-, Setubal:
VISIGRAPP 2010, International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications. Proceedings. CD-ROM : Angers, France May 17 - 21, 2010
Porto: INSTICC Press, 2010
ISBN: 978-989-674-028-3
International Conference on Computer Vision Theory and Applications (VISAPP) <5, 2010, Angers>
Conference Paper
Fraunhofer IGD ()
geometric algebra; curvature; point set surface; Forschungsgruppe Geometric Algebra Computing (GACO)

For applications like segmentation, feature extraction and classification of point sets it is essential to know the principal curvatures and the corresponding principal directions. For the purpose of curvature estimation conformal geometric algebra promises to be a natural mathematical language: Local curvatures can be described with the help of osculating circles or spheres. On one hand, conformal geometric algebra is able to directly compute with these geometric objects, as well as with lines and planes needed for the description of vanishing curvature. On the other hand, distance measures for fitting these objects into point sets can be handled in a linear way, leading to efficient algorithms.
In this paper we use conformal geometric algebra advantageously in order to locally compute continuous curvatures as well as principal curvatures of point sets without the need of costly pre-processing of raw data. We show results on artificial and real data. Numerical verification on artificial data shows the accuracy of our approach.