Simplification Method Using K-NN Estimation and Fuzzy C-Means Clustering Algorithm

  • Abdelaaziz MahdaouiEmail author
  • A. Bouazi
  • A. Hsaini Marhraoui
  • E. H. Sbai
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 858)


Acquisition systems, such as 3D scanners, are developing in resolution and accuracy. They give set of points that contain huge number of 3D data set. These acquisition systems are incapable to give the optimum number of points. For this reason, it is preferable to go through the simplification phase to optimize the number of 3D points constituting a point cloud and to consider only the relevant points. In this work, we will present a hybrid simplification method using two concepts. This technique is based on Shannon entropy and on the fuzzy c-means clustering algorithm.


Simplification Clustering Fuzzy C-Means Entropy 3D point cloud 


  1. 1.
    Pauly, M., Gross, M., Kobbelt, L.P.: Efficient simplification of point-sampled surfaces. In: IEEE Visualization, 2002, pp. 163–170. VIS (2002)Google Scholar
  2. 2.
    Wu, J., Kobbelt, L.: Optimized sub-sampling of point sets for surface splatting. Comput. Graph. Forum 23(3), 643–652 (2004)CrossRefGoogle Scholar
  3. 3.
    Ohtake, Y., Belyaev, A., Seidel, H.-P.: An integrating approach to meshing scattered point data. In: Proceedings of the 2005 ACM Symposium on Solid and Physical Modeling - SPM 2005, pp. 61–69 (2005)Google Scholar
  4. 4.
    Shi, B.-Q., Liang, J., Liu, Q.: Adaptive simplification of point cloud using K-means clustering. Comput. Des. 43(8), 910–922 (2011)Google Scholar
  5. 5.
    Linsen, L.: Point cloud representation. Univ. Karlsruhe, Ger. Technical report, Fac. Informatics, pp. 1–18 (2001)Google Scholar
  6. 6.
    Dey, T.K., Giesen, J., Hudson, J.: Decimating samples for mesh simplification. In: Proceedings of 13th Canadian Conference on Computational Geometry, pp. 85–88 (2001)Google Scholar
  7. 7.
    Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. In: Proceedings of the Sixteenth Annual Symposium on Computational Geometry - SCG 2000, pp. 213–222 (2000)Google Scholar
  8. 8.
    Alexa, M., Behr, J., Cohen-Or, D., Fleishman, S., Levin, D., Silva, C.T.: Point set surfaces. In: Proceedings Visualization 2001, VIS 2001, pp. 21–28 (2001)Google Scholar
  9. 9.
    Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH 1997, pp. 209–216 (1997)Google Scholar
  10. 10.
    Allegre, R., Chaine, R., Akkouche, S.: Convection-driven dynamic surface reconstruction. In: International Conference on Shape Modeling and Applications (SMI 2005), pp. 33–42 (2005)Google Scholar
  11. 11.
    Mahdaoui, A., Bouazi, A., Hsaini, M., Sbai, E.H.: Entropic Method for 3D Point Cloud Simplification, pp. 613–621. Springer, Cham (2018)CrossRefGoogle Scholar
  12. 12.
    Boissonnat, J., Oudot, S.: An effective condition for sampling surfaces with guarantees. Symp. A Q. J. Mod. Foreign Lit. 101–112 (2004)Google Scholar
  13. 13.
    Boissonnat, J.D., Oudot, S.: Provably good surface sampling and approximation. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 9–18 (2003)Google Scholar
  14. 14.
    Boissonnat, J.-D., Oudot, S.: An effective condition for sampling surfaces with guarantees. In: Proceedings of the Ninth ACM Symposium on Solid Modeling and Applications, pp. 101–112 (2004)Google Scholar
  15. 15.
    Boissonnat, J.-D., Oudot, S.: Provably good sampling and meshing of surfaces. Graph. Models 67(5), 405–451 (2005)CrossRefGoogle Scholar
  16. 16.
    Chew, L.P., Paul, L.: Guaranteed-quality mesh generation for curved surfaces. In: Proceedings of the Ninth Annual Symposium on Computational Geometry - SCG 1993, pp. 274–280 (1993)Google Scholar
  17. 17.
    Adamson, A., Alexa, M.: Approximating and intersecting surfaces from points. In: Proceedings of Symposium on Geometry Processing, pp. 230–239 (2003)Google Scholar
  18. 18.
    Pauly, M., Gross, M.: Spectral processing of point-sampled geometry. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH 2001, pp. 379–386 (2001)Google Scholar
  19. 19.
    Witkin, P., Heckbert, P.S.: Using particles to sample and control implicit surfaces. In: Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH 1994, pp. 269–277 (1994)Google Scholar
  20. 20.
    Wang, J., Li, X., Ni, J.: Probability density function estimation based on representative data samples. In: IET International Conference on Communication Technology and Application (ICCTA 2011), pp. 694–698 (2011)Google Scholar
  21. 21.
    Dunn, J.C.: A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J. Cybern. 3(3), 32–57 (1973)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shannon, E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Parzen: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27(3), 832–837 (1956)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall (1986)Google Scholar
  26. 26.
    Müller, H.-G., Petersen, A.: Density estimation including examples. In: Wiley StatsRef: Statistics Reference Online, pp. 1–12. Wiley, Chichester (2016)Google Scholar
  27. 27.
    Cignoni, P., Rocchini, C., Scopigno, R.: Metro: measuring error on simplified surfaces. Comput. Graph. Forum 17(2), 167–174 (1998)CrossRefGoogle Scholar
  28. 28.
    Wang, J., Wu, J., Wang, J., Zhang, H., Chen, X.: Multi-criteria decision-making methods based on the Hausdorff distance of hesitant fuzzy linguistic numbers. Soft. Comput. 20(4), 1621–1633 (2016)CrossRefGoogle Scholar
  29. 29.
    Mahdaoui, A., Hsaini, M., Bouazi, A., Sbai, E.H.: Comparative study of combinatorial 3D reconstruction algorithms. Int. J. Eng. Trends Technol. 48(5), 247–251 (2017)CrossRefGoogle Scholar
  30. 30.
    Bernardini, F., Mittleman, J., Rushmeier, H., Silva, C., Taubin, G.: The ball-pivoting algorithm for surface reconstruction. IEEE Trans. Vis. Comput. Graph. 5(4), 349–359 (1999)CrossRefGoogle Scholar
  31. 31.
    Jain, K.: Data clustering: 50 years beyond K-means. Patt. Recognit. Lett. 31(8), 651–666 (2010)CrossRefGoogle Scholar
  32. 32.
    Mahdaoui, A., Bouazi, A., Hsaini, M., Sbai, E.H.: Comparison of K-means and fuzzy C-Means algorithms on simplification of 3D point cloud based on entropy estimation. Adv. Sci. Technol. Eng. Syst. J. 2(5), 38–44 (2017)CrossRefGoogle Scholar
  33. 33.
    Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Springer US, Boston (1981)CrossRefGoogle Scholar
  34. 34.
    Hsaini, M., Bouazi, A., Mahdaoui, A., Sbai, E.H., Bernstein-bezier, A.R.: Reconstruction and adjustment of surfaces from a 3-D point cloud. Int. J. Comput. Trends Technol. 37(2), 105–109 (2016)CrossRefGoogle Scholar
  35. 35.
    Gueziec: Locally toleranced surface simplification. IEEE Trans. Vis. Comput. Graph, 5(2), 168–189 (1999)CrossRefGoogle Scholar
  36. 36.
    Bank, R.E.: PLTMG: a software package for solving elliptic partial differential equations. Society for Industrial and Applied Mathematics (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Abdelaaziz Mahdaoui
    • 1
    Email author
  • A. Bouazi
    • 2
  • A. Hsaini Marhraoui
    • 2
  • E. H. Sbai
    • 2
  1. 1.Department of Physics, Faculty of ScienceMoulay Ismail UniversityMeknèsMorocco
  2. 2.Technology High SchoolMoulay Ismail UniversityMeknèsMorocco

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