ESTIMATION OF CURVATURES IN POINT SETS BASED ON GEOMETRIC ALGEBRA

Helmut Seibert, Dietmar Hildenbrand, Meike Becker, Arjan Kuijper

Abstract

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. Furthermore, the results are obtained in a fast manner and are also visually satisfactory.

References

  1. Adamson, A. and Alexa, M. (2003). Approximating and Intersecting Surfaces from Points . In Eurographics Symposium on Geometry Processing (SGP), pages 230-239.
  2. Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., and Silva, C. T. (2001). Point set surfaces. In Conference on Visualization (VIS), pages 21-28.
  3. Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. (2003). Anisotropic polygonal remeshing. ACM Transactions on Graphics, 22(3):485-493.
  4. Chen, X. and Schmitt, F. (1992). Intrinsic surface properties from surface triangulation. In European Conference on Computer Vision (ECCV), pages 739-743, London, UK. Springer-Verlag.
  5. Gatzke, T., Grimm, C., Garland, M., and Zelinka, S. (2005). Curvature Maps For Local Shape Comparison. In Shape Modeling and Applications, pages 244-253.
  6. Gois, J., Tejada, E., Etiene, T., Nonato, L., Castelo, A., and Ertl, T. (2006). Curvature-driven modeling and rendering of point-based surfaces. In Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI), pages 27-36.
  7. Gross, M. and Pfister, H. (2007). Point Based Graphics. Morgan Kaufmann.
  8. Guennebaud, G., Germann, M., and Gross, M. (2008). Dynamic sampling and rendering of algebraic point set surfaces. In Eurographics, pages 653-662.
  9. Guennebaud, G. and Gross, M. H. (2007). Algebraic point set surfaces. ACM Transactions on Graphics, 26(3):23.
  10. Hameiri, E. and Shimshoni, I. (2003). Estimating the principal curvatures and the darboux frame from real 3-d range data. IEEE Systems, Man, and Cybernetics B (SMC-B), 33(4):626-637.
  11. Hildenbrand, D. (2005). Geometric computing in computer graphics using conformal geometric algebra. Computers & Graphics, 29(5):802-810.
  12. Hornung, A. and Kobbelt, L. (2006). Robust Reconstruction of Watertight 3D Models from Non-uniformly Sampled Point Clouds Without Normal Information . In Eurographics Symposium on Geometry Processing (SGP), pages 41-50.
  13. IEEE (2001). Std. 1241-2000 IEEE standard for terminology and test methods foranalog-to-digital converters, chapter 3, pages 26-27.
  14. Kalogerakis, E., Simari, P., Nowrouzezahrai, D., and Singh, K. (2007). Robust Statistical Estimation of Curvature on Discretized Surfaces. In Eurographics Symposium on Geometry Processing (SGP), pages 13-22.
  15. Kolluri, R., Shewchuk, J. R., and O'Brien, J. F. (2004). Spectral Surface Reconstruction From Noisy Point Clouds . In Eurographics Symposium on Geometry Processing (SGP), pages 11-22.
  16. Lavoue, G. (2007). A Roughness Measure for 3D Mesh Visual Masking. In ACM SIGGRAPH Symposium on Applied Perception in Graphics and Visualization (APGV), pages 57 - 60.
  17. Mederos, B., Amenta, N., Velho, L., and de Figueiredo, L. H. (2005). Surface Reconstruction for Noisy Point Clouds. In Eurographics Symposium on Geometry Processing (SGP), pages 53-62.
  18. Mitra, N. J., Gelfand, N., Pottmann, H., and Guibas, L. (2004). Registration of Point Cloud Data from a Geometric Optimization Perspective . In Symposium on Geometry Processing, pages 23-32.
  19. Taubin, G. (1995). Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation. In IEEE International Conference on Computer Vision (ICCV), pages 902-907.
  20. Vince, J. (2008). Geometric Algebra for Computer Graphics. Springer.
  21. Yang, Y.-L., Lai, Y.-K., Hu, S.-M., and Pottmann, H. (2006). Robust Principal Curvatures on Multiple Scales. In Eurographics Symposium on Geometry Processing (SGP), pages 223-226.
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Paper Citation


in Harvard Style

Seibert H., Hildenbrand D., Becker M. and Kuijper A. (2010). ESTIMATION OF CURVATURES IN POINT SETS BASED ON GEOMETRIC ALGEBRA . In Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2010) ISBN 978-989-674-028-3, pages 12-19. DOI: 10.5220/0002817800120019


in Bibtex Style

@conference{visapp10,
author={Helmut Seibert and Dietmar Hildenbrand and Meike Becker and Arjan Kuijper},
title={ESTIMATION OF CURVATURES IN POINT SETS BASED ON GEOMETRIC ALGEBRA},
booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2010)},
year={2010},
pages={12-19},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002817800120019},
isbn={978-989-674-028-3},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2010)
TI - ESTIMATION OF CURVATURES IN POINT SETS BASED ON GEOMETRIC ALGEBRA
SN - 978-989-674-028-3
AU - Seibert H.
AU - Hildenbrand D.
AU - Becker M.
AU - Kuijper A.
PY - 2010
SP - 12
EP - 19
DO - 10.5220/0002817800120019