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Estimation of differential quantities using Hermite RBF interpolation

  • M. Prantl
  • L. Váša
Original Article
  • 113 Downloads

Abstract

Curvature is an important geometric property in computer graphics that provides information about the character of object surfaces. The exact curvature can only be calculated for a limited set of surface descriptions. Most of the time, we deal with triangles, point sets or some other discrete representation of the surface. For those, curvature can only be estimated. However, surfaces can be fitted by some kind of interpolation function and from it, curvature can be calculated directly. This paper proposes a method for curvature estimation and normal vector re-estimation based on surface fitting using Hermite Radial Basis Function interpolation. Hermite variation uses not only control points, but normal vectors at those points as well. This leads to a better and more robust interpolation than if only control points are used. Once the interpolant is obtained, the curvature and other possible properties can be directly computed using known approaches. The proposed algorithm was tested on several explicit and implicit functions, and it outperforms current state-of-the-art methods if exact normals are available. For normals calculated directly from a triangle mesh, the proposed algorithm works on par with existing state-of-the-art methods.

Keywords

Curvature RBF HRBF Discrete Differential Geometry Computer Graphics 

Notes

Acknowledgements

This work was supported by the UWB grant SGS-2016-013 Advanced Graphical and Computing Systems.

References

  1. 1.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38(1), 91–109 (2001). doi: 10.1006/jagm.2000.1127. http://www.sciencedirect.com/science/article/pii/S0196677400911271
  2. 2.
    Batagelo, H.C., Wu, S.T.: Estimating curvatures and their derivatives on meshes of arbitrary topology from sampling directions. Vis. Comput. 23(9), 803–812 (2007). doi: 10.1007/s00371-007-0133-8 CrossRefGoogle Scholar
  3. 3.
    Bridson, R.: Fast poisson disk sampling in arbitrary dimensions. In: ACM SIGGRAPH 2007 Sketches, SIGGRAPH ’07. ACM, New York, NY, USA (2007). doi: 10.1145/1278780.1278807
  4. 4.
    Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3d objects with radial basis functions. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’01, pp. 67–76. ACM, New York, NY, USA (2001). doi: 10.1145/383259.383266
  5. 5.
    Cazals, F., Pouget, M.: Estimating differential quantities using polynomial fitting of osculating jets. Comput. Aided Geom. Des. 22(2), 121–146 (2005). doi: 10.1016/j.cagd.2004.09.004. http://www.sciencedirect.com/science/article/pii/S016783960400113X
  6. 6.
    Cohen-Steiner, D., Morvan, J.M.: Restricted delaunay triangulations and normal cycle. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, SCG ’03, pp. 312–321. ACM, New York, NY, USA (2003). doi: 10.1145/777792.777839
  7. 7.
    Goldfeather, J., Interrante, V.: A novel cubic-order algorithm for approximating principal direction vectors. ACM Trans. Graph. 23(1), 45–63 (2004). doi: 10.1145/966131.966134 CrossRefGoogle Scholar
  8. 8.
    Goldman, R.: Curvature formulas for implicit curves and surfaces. Comput. Aided Geom. Des. 22(7), 632 – 658 (2005). doi: 10.1016/j.cagd.2005.06.005. http://www.sciencedirect.com/science/article/pii/S0167839605000737. Geometric Modelling and Differential Geometry
  9. 9.
    Gray, A.: Surfaces in 3-dimensional space via mathematica. In: Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn., chap. 17, pp. 394–401. CRC Press, Inc., Taylor & Francis group. Boca Raton, FL, USA (1997)Google Scholar
  10. 10.
    Guennebaud, G., Jacob, B., et al.: Eigen v3. http://eigen.tuxfamily.org (2010)
  11. 11.
    Hildebrandt, K., Polthier, K.: Generalized shape operators on polyhedral surfaces. Comput. Aided Geom. Des. 28(5), 321–343 (2011). doi: 10.1016/j.cagd.2011.05.001. http://www.sciencedirect.com/science/article/pii/S0167839611000628
  12. 12.
    Ju, T., Losasso, F., Schaefer, S., Warren, J.: Dual contouring of hermite data. ACM Trans. Graph. 21(3), 339–346 (2002). doi: 10.1145/566654.566586 CrossRefGoogle Scholar
  13. 13.
    Kalogerakis, E., Simari, P., Nowrouzezahrai, D., Singh, K.: Robust statistical estimation of curvature on discretized surfaces. In: Proceedings of the Fifth Eurographics Symposium on Geometry Processing, SGP ’07, pp. 13–22. Eurographics Association, Aire-la-Ville, Switzerland, Switzerland (2007). http://dl.acm.org/citation.cfm?id=1281991.1281993
  14. 14.
    Lorensen, W.E., Cline, H.E.: Marching cubes: A high resolution 3d surface construction algorithm. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’87, pp. 163–169. ACM, New York, NY, USA (1987). doi: 10.1145/37401.37422
  15. 15.
    Macedo, I., Gois, J., Velho, L.: Hermite interpolation of implicit surfaces with radial basis functions. In: Computer Graphics and Image Processing (SIBGRAPI), 2009 XXII Brazilian Symposium on, pp. 1–8 (2009). doi: 10.1109/SIBGRAPI.2009.11
  16. 16.
    Macedo, I., Gois, J.P., Velho, L.: Hermite radial basis functions implicits. Comput. Graph. Forum 30(1), 27–42 (2011). doi: 10.1111/j.1467-8659.2010.01785.x CrossRefGoogle Scholar
  17. 17.
    Max, N.: Weights for computing vertex normals from facet normals. J. Graph. Tools 4(2), 1–6 (1999). doi: 10.1080/10867651.1999.10487501 CrossRefGoogle Scholar
  18. 18.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics III, Mathematics and Visualization, pp. 35–57. Springer, Berlin (2003). doi: 10.1007/978-3-662-05105-4_2
  19. 19.
    Pottmann, H., Wallner, J., Huang, Q.X., Yang, Y.L.: Integral invariants for robust geometry processing. Comput.Aided Geom. Des. 26(1), 37–60 (2009). doi: 10.1016/j.cagd.2008.01.002. http://www.sciencedirect.com/science/article/pii/S0167839608000095
  20. 20.
    Pouget, M., Cazals, F.: Estimation of local differential properties of point-sampled surfaces. In: CGAL User and Reference Manual, 4.8 edn. CGAL Editorial Board (2016). http://doc.cgal.org/4.8/Manual/packages.html#PkgJet_fitting_3Summary
  21. 21.
    Razdan, A., Bae, M.: Curvature estimation scheme for triangle meshes using biquadratic bzier patches. Comput. Aided Des. 37(14), 1481–1491 (2005). doi: 10.1016/j.cad.2005.03.003. http://www.sciencedirect.com/science/article/pii/S0010448505000825
  22. 22.
    Rusinkiewicz, S.: Estimating curvatures and their derivatives on triangle meshes. In: Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium, 3DPVT ’04, pp. 486–493. IEEE Computer Society, Washington, DC, USA (2004). doi: 10.1109/3DPVT.2004.54
  23. 23.
    The stanford 3D scanning repository. Electronic. http://graphics.stanford.edu/data/3Dscanrep/
  24. 24.
    Taubin, G.: Estimating the tensor of curvature of a surface from a polyhedral approximation. In: Proceedings of the Fifth International Conference on Computer Vision, 1995, pp. 902–907 (1995). doi: 10.1109/ICCV.1995.466840
  25. 25.
    Theisel, H., Rossi, C., Zayer, R., Seidel, H.P.: Normal based estimation of the curvature tensor for triangular meshes. In: Proceedings of 12th Pacific Conference on Computer Graphics and Applications, 2004, PG 2004, pp. 288–297 (2004). doi: 10.1109/PCCGA.2004.1348359
  26. 26.
    Vaillant, R.: Recipe for implicit surface reconstruction with hrbf. Electronic (2013). http://rodolphe-vaillant.fr/?e=12
  27. 27.
    Váša, L., Vaněček, P., Prantl, M., Skorkovská, V., Martínek, P., Kolingerová, I.: Mesh statistics for robust curvature estimation. Comput. Graph. Forum (2016). doi: 10.1111/cgf.12982 Google Scholar
  28. 28.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(1), 389–396 (1995). doi: 10.1007/BF02123482 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yang, P., Qian, X.: Direct computing of surface curvatures for point-set surfaces. In: Botsch, M., Pajarola, R., Chen, B., Zwicker M. (eds.) Eurographics Symposium on Point-Based Graphics. The Eurographics Association (2007). doi: 10.2312/SPBG/SPBG07/029-036
  30. 30.
    Zhihong, M., Guo, C., Yanzhao, M., Lee, K.: Curvature estimation for meshes based on vertex normal triangles. Comput. Aided Des. 43(12), 1561–1566 (2011). doi: 10.1016/j.cad.2011.06.006. http://www.sciencedirect.com/science/article/pii/S0010448511001448

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.NTIS - New Technologies for Information SocietyUniversity of West BohemiaPlzeňCzech Republic

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