Triangular Curvature Approximation of Surfaces - Filtering the Spurious Mode

Paavo Nevalainen, Ivan Jambor, Jonne Pohjankukka, Jukka Heikkonen, Tapio Pahikkala

Abstract

Curvature spectrum is a useful feature in surface classification but is difficult to apply to cases with high noise typical e.g. to natural resource point clouds. We propose two methods to estimate the mean and the Gaussian curvature with filtering properties specific to triangulated surfaces. Methods completely filter a highest shape mode away but leave single vertical pikes only partially dampened. Also an elaborate computation of nodal dual areas used by the Laplace-Beltrami mean curvature can be avoided. All computation is based on triangular setting, and a weighted summation procedure using projected tip angles sums up the vertex values. A simplified principal curvature direction definition is given to avoid computation of the full second fundamental form. Qualitative evaluation is based on numerical experiments over two synthetical examples and a prostata tumor example. Results indicate the proposed methods are more robust to presence of noise than other four reference formulations.

References

  1. Crane, K., de Goes, F., Desbrun, M., and Schröder, P. (2013). Digital geometry processing with discrete exterior calculus. In ACM SIGGRAPH 2013 Courses, SIGGRAPH 7813, pages 7:1-7:126, New York, NY, USA. ACM.
  2. Dorst, L., Fontijne, D., and Mann, S. (2007). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition.
  3. Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations (3rd Ed.). Johns Hopkins University Press, Baltimore, MD, USA.
  4. Jin, S., Lewis, R., and West, D. (2005). A comparison of algorithms for vertex normal computation. The Visual Computer, 21:71-82.
  5. Lee, J., Nishikawa, R. M., Reiser, I., Boone, J. M., and Lindfors, K. K. (2015). Local curvature analysis for classifying breast tumors: Preliminary analysis in dedicated breast ct. Medical Physics, 42(9).
  6. Max, N. (1999). Weights for computing vertex normals from facet normals. Journal of Graphics Tools, 4(2).
  7. Mesmoudi, M. M., De Floriani, L., and Magillo, P. (2012). Discrete curvature estimation methods for triangulated surfaces. In Applications of Discrete Geometry and Mathematical Morphology, pages 28-42. Springer.
  8. Meyer, M., Desbrun, M., Schr öder, P., and Barr, A. H. (2003). Visualization and Mathematics III, chapter Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, pages 35-57. Springer Berlin Heidelberg, Berlin, Heidelberg.
  9. Mitra, N. J. and Nguyen, A. (2003). Estimating surface normals in noisy point cloud data. In Proceedings of the Nineteenth Annual Symposium on Computational Geometry, SCG03, pages 322-328, New York, NY, USA. ACM.
  10. Nevalainen, P., Middleton, M., Kaate, I., Pahikkala, T., Sutinen, R., and Heikkonen, J. (2015). Detecting stony areas based on ground surface curvature distribution. In 2015 International Conference on Image Processing Theory, Tools and Applications, IPTA 2015, Orleans, France, November 10-13, 2015, pages 581-587.
  11. Nevalainen, P., Middleton, M., Sutinen, R., Heikkonen, J., and Pahikkala, T. (2016). Detecting terrain stoniness from airborne laser scanning data . Remote Sensing, 8(9):720.
  12. Pierzchala, M., Talbot, B., and Astrup, R. (2016). Measuring wheel ruts with close-range photogrammetry. Forestry, 89(4):383-391.
  13. Pressley, A. (2010). Elementary Differential Geometry. Springer Undergraduate Mathematics Series. Springer London.
  14. Rusinkiewicz, S. (2004). Estimating curvatures and their derivatives on triangle meshes. In Symposium on 3D Data Processing, Visualization, and Transmission.
  15. Schaer, P., Skaloud, J., Landtwing, S., and Legat, K. (2007). Accuracy Estimation for Laser Point Cloud Including Scanning Geometry. In Mobile Mapping Symposium 2007, Padova.
  16. Theisel, H., Rössl, C., Zayer, R., and Seidel, H. P. (2004). Normal based estimation of the curvature tensor for triangular meshes. In In PG04: Proceedings of the Computer Graphics and Applications, 12th Pacific Conference on (PG2004), pages 288-297. IEEE Computer Society.
  17. van Oosterom, A. and Strackee, J. (1983). A solid angle of a plane triangle. IEEE Trans. Biomed. Eng., 30(2):125- 126.
  18. Vranic, D. V. and Saupe, D. (2001). 3d shape descriptor based on 3d fourier transform. In Fazekas, K., editor, 3D Shape Descriptor Based on 3D Fourier Transform In Proceedings of the EURASIP Conference on Digital Signal Processing for Multimedia Communications and Services (ECMCS 2001), pages 271-274.
  19. Wardetzky, M., Mathur, S., Kaelberer, F., and Grinspun, E. (2007). Discrete laplace operators: No free lunch. In Belyaev, A. and Garland, M., editors, Geometry Processing. The Eurographics Association.
  20. Yang, P. and Qian, X. (2007). Direct computing of surface curvatures for point-set surfaces. In SPBG'07, pages 29-36.
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Paper Citation


in Harvard Style

Nevalainen P., Jambor I., Pohjankukka J., Heikkonen J. and Pahikkala T. (2017). Triangular Curvature Approximation of Surfaces - Filtering the Spurious Mode . In Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM, ISBN 978-989-758-222-6, pages 684-692. DOI: 10.5220/0006249206840692


in Bibtex Style

@conference{icpram17,
author={Paavo Nevalainen and Ivan Jambor and Jonne Pohjankukka and Jukka Heikkonen and Tapio Pahikkala},
title={Triangular Curvature Approximation of Surfaces - Filtering the Spurious Mode},
booktitle={Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},
year={2017},
pages={684-692},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006249206840692},
isbn={978-989-758-222-6},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,
TI - Triangular Curvature Approximation of Surfaces - Filtering the Spurious Mode
SN - 978-989-758-222-6
AU - Nevalainen P.
AU - Jambor I.
AU - Pohjankukka J.
AU - Heikkonen J.
AU - Pahikkala T.
PY - 2017
SP - 684
EP - 692
DO - 10.5220/0006249206840692