Journal of Scientific Computing

, Volume 54, Issue 2–3, pp 577–602 | Cite as

Robust and Efficient Implicit Surface Reconstruction for Point Clouds Based on Convexified Image Segmentation

  • Jian Liang
  • Frederick Park
  • Hongkai Zhao


We present an implicit surface reconstruction algorithm for point clouds. We view the implicit surface reconstruction as a three dimensional binary image segmentation problem that segments the entire space \(\mathbb R ^3\) or the computational domain into an interior region and an exterior region while the boundary between these two regions fits the data points properly. The key points with using an image segmentation formulation are: (1) an edge indicator function that gives a sharp indicator of the surface location, and (2) an initial image function that provides a good initial guess of the interior and exterior regions. In this work we propose novel ways to build both functions directly from the point cloud data. We then adopt recent convexified image segmentation models and fast computational algorithms to achieve efficient and robust implicit surface reconstruction for point clouds. We test our methods on various data sets that are noisy, non-uniform, and with holes or with open boundaries. Moreover, comparisons are also made to current state of the art point cloud surface reconstruction techniques.


Principal component analysis (PCA) Distance function  Anisotropic Gaussian Edge indicator Normal Image segmentation Total variation 



This work is partially supported by ARO/MURI grant W911NF-07-1-0185, ONR grant N00014-11-1-0602 and NGA NURI HM1582-10-1-0012. The authors would like to thank Edward Castillo for graciously providing us with his code for computing surface normals to PC data. The author would like to express their thanks to the Stanford 3D scanning Repository for their generosity in distributing their 3D data. The authors would also like to thank Professor Michael Kazhdan for graciously distributing his source code for Poisson surface reconstruction.


  1. 1.
    Adamson, A., Alexa, M.: Anisotropic point set surfaces. Comput. Graph. Forum 25, 717–724 (2006). doi: 10.1111/j.1467-8659.2006.00994.x CrossRefGoogle Scholar
  2. 2.
    Alexa, M., Adamson, A.: Interpolatotory point set surfaces—convexity and hermite data. ACM Trans. Graph. 28 (2009). doi: 10.1145/1516522.1516531
  3. 3.
    Alexa, M., Adamson, A.: On normals and projection operators for surfaces defined by point sets. In: Eurographics Symposium on Point-Based Graphics, pp. 149–155, 2009. doi:
  4. 4.
    Alliez, P., Cohen-Steiner, D., Tong, Y., Desbrun, M.: Voronoi-based variational reconstruction of unoriented point sets. In: Proceedings of 5th Eurographics Symposium on Geometry Process (SGP ’05), pp. 39–48, July (2007). doi: 10.1145/1281991.1281997
  5. 5.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 32, 387–438 (1993). doi: 10.1137/0331020 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Amenta, N., Yong, J.K.: Defining point-set surfaces. ACM Trans. Graph. 23 (2004). doi: 10.1145/1015706.1015713
  7. 7.
    Amenta, N, Bern, M., Kamvysselis, M.: A new Voronoi-based surface reconstruction algorithm. In: Proceedings of 25th Annual Conference Computer Graphics and Interactive Techniques (SIGGRAPH ’98), pp. 415–421, (1998). doi: 10.1145/280814.280947
  8. 8.
    Amenta, N., Bern, M., Eppstein, D.: The crust and the beta-skeleton: combinatorial curve reconstruction. Graph. Models Image Process. 60, 125–135 (1998). doi: 10.1006/gmip.1998.0465 CrossRefGoogle Scholar
  9. 9.
    Aujol, J.F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67, 111–136 (2006). doi: 10.1007/s11263-006-4331-z CrossRefGoogle Scholar
  10. 10.
    Bae, E., Yuan, J., Tai, X.C.: Global minimization for continuous multiphase partitioning problems using a dual approach. Int. J. Comput. Vis. 92, 112–119 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bak, S., McLaughlin, J., Renzi, D.: Some improvements for the fast sweeping method. SIAM J. Sci. Comput 32, 2853–2874 (2010). doi: 10.1137/090749645 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Boissonnat, J.D.: Geometric structures for three dimensional shape reconstruction. ACM Trans. Graph. 3, 266–286 (1984). doi: 10.1145/357346.357349 CrossRefGoogle Scholar
  13. 13.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: IEEE International Conference on Computer Vision (ICCV ’03), vol. 1, pp. 26–33, October (2003). doi: 10.1109/ICCV.2003.1238310
  14. 14.
    Bresson, X., Chan, T.: Active contours based on Chambolle’s mean curvature motion. In: IEEE International Conference on Image Processing (ICIP ’07), vol. I, pp. 33–36, (2007). doi: 10.1109/ICIP.2007.4378884
  15. 15.
    Bresson, X., Esedoglu, S., Vanderheynst, P., Thiran, J.P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007). doi: 10.1007/s10851-007-0002-0 CrossRefGoogle Scholar
  16. 16.
    Burger, M., Osher, S., Xu, J., Gilboa, G.: Nonlinear inverse scale space methods for image restoration. Commun. Math. Sci. 3752, 25–36 (2005). doi: 10.1007/11567646_3 Google Scholar
  17. 17.
    Carr, J.C., Fright, W.R., Beatson, R.K.: Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Med. Imag. 16, 96–107 (1997). doi: 10.1109/42.552059 CrossRefGoogle Scholar
  18. 18.
    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 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’01), pp. 67–76, (2001). doi: 10.1145/383259.383266
  19. 19.
    Carter, J.: Dual methods for total variation-based image restoration. Ph.D. disserattion, Department of Mathematics, UCLA (2001)Google Scholar
  20. 20.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proceedings of 5th International Conference on Computer Vision (ICCV ’97), pp. 694–699, (1997). doi: 10.1109/ICCV.1995.466871
  21. 21.
    Castillo, E., Liang, J., Zhao, H.K.: Point cloud segmentation via constrained nonlinear least squares surface normal estimates, Chapter 13. In: Breuss, M., Bruckstein, A., Maragos, P. (eds.) Innovations for Shape Analysis: Models and Algorithms, Springer-Verlag, New York (2013)Google Scholar
  22. 22.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004). doi: 10.1023/B:JMIV.0000011321.19549.88 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 20(1), 120–145 (2011). doi: 10.1007/s10851-010-0251-1 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chambolle, A., De Vore, R.A., Lee, N.T., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7, 319–335 (1998). doi: 10.1109/83.661182 MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \({L}^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005). doi:  10.1137/040604297 MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001). doi: 10.1109/83.902291 zbMATHCrossRefGoogle Scholar
  27. 27.
    Chan, T.F., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20, 1964–1977 (1999). doi: 10.1137/S1064827596299767 MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66, 1632–1648 (2006). doi: 10.1137/040615286 MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11, 215–234 (1967). doi: 10.1147/rd.112.0215 MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Curless, B., Levoy, M.: A volumetric method for building complex models from range images. In: Proceedings of 23rd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’96), pp. 303–312, (1996). doi: 10.1145/237170.237269
  31. 31.
    Digne, J., Morel, J.M., Souzani, C.M., Lartigue, C.: Scale space meshing of raw data point sets. Comput. Graph. Forum 30, 1630–1642 (2011). doi: 10.1111/j.1467-8659.2011.01848.x CrossRefGoogle Scholar
  32. 32.
    Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995). doi: 10.2307/2291512 MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Duguet, F., Durand, F., Drettakis, G.: Robust higher-order filtering of points. Tech. Rep. INRIA, April (2004), doi:
  34. 34.
    Edelsbrunner, H.: Shape reconstruction with Delaunay complex. In: Proceedings of 3rd Latin American Symposium on Theoretical Information (LATIN ’98), vol. 1380, pp. 119–132, 1998. doi: 10.1007/BFb0054315
  35. 35.
    Edelsbrunner, H., Mucke, E.P.: Three dimensional alpha shapes. ACM Trans. Graph. 13, 43–72 (1994). doi: 10.1145/174462.156635 zbMATHCrossRefGoogle Scholar
  36. 36.
    Federer, H.: Geometric Measure Theory. Springer, New York (1969)zbMATHGoogle Scholar
  37. 37.
    Fleishman, S., Drori, I., Cohen-Or, D.: Bilateral mesh denoising. In: ACM Transactions on Graphics, Proceedings of ACM SIGGRAPH 2003, vol. 22, pp. 950–953, July (2003). doi: 10.1145/882262.882368
  38. 38.
    Fleishman, S., Cohen-Or, D., Silva, C.T.: Robust moving least-squares fitting with sharp features. In: ACM Transactions on Graphics, Proceedings of ACM SIGGRAPH 2005, vol. 24, July (2005). doi: 10.1145/1186822.1073227
  39. 39.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Boston (1984)zbMATHGoogle Scholar
  40. 40.
    Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009). doi: 10.1137/080725891 MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Goldstein, T., Bresson, X., Osher, S.: Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction. In: CAM, LA, CA, Rep. 09-06, (2009)Google Scholar
  42. 42.
    Guennebaud, G., Gross, M.: Algebraic point set surfaces. ACM Trans. Graph. 26 (2007). doi: 10.1145/1276377.1276406
  43. 43.
    Hoppe, H., Derose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. In: Proceedings of 19th Annual Conference on Computer Graphics and Interactive Techniques, ACM SIGGRAPH. Computer Graphics, vol. 26, pp. 71–78, July (1992). doi: 10.1145/142920.134011
  44. 44.
    Hornung, A., Kobbelt, L.: Robust reconstruction of watertight 3D models from non-uniformly sampled point clouds without normal information. In: Proceedings of 4th Eurographics Symposium on Geometry Processing (SGP ’06), June (2006). doi: 10.1145/1281957.1281963
  45. 45.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vision 1, 321–331 (1988). doi: 10.1007/BF00133570 CrossRefGoogle Scholar
  46. 46.
    Kazhdan, M., Bolitho, M., Hoppe, H.: Poisson surface reconstruction. In: Proceedings of 4th Eurographics Symposium on Geometry Processing (SGP ’06), pp. 61–70, (2006)Google Scholar
  47. 47.
    Kolluri, R., Shewchuk, J.R., O’Brien, J.F.: Spectral surface reconstruction from noisy point clouds. In: Proceedings of 2004 Eurographics/ACM SIGGRAPG Symposium on Geometry Processing (SGP ’04), pp. 11–21, (2004). doi: 10.1145/1057432.1057434
  48. 48.
    Lempitsky, V., Boykov, V.: Global optimization for shape fitting. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’07), pp. 1–8, June (2007). doi: 10.1109/CVPR.2007.383293
  49. 49.
    Luo, S., Guibas, L.J., Zhao, H.K.: Euclidean skeletons using closest points. Inverse Probl Imaging 5, 95–113 (2011). doi: 10.3934/ipi.2011.5.95 MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Mederos, B., Velho, L., de Figueiredo, L.H.: Point cloud denoising. In: Proceeding of SIAM Conference on Geometric Design and Computing, (2003)Google Scholar
  51. 51.
    Medioni, G., Lee, M.S., Tang, C.K.: A Computational Framework for Segmentation and Grouping. Elsevier, New York (2000)zbMATHGoogle Scholar
  52. 52.
    Morse, B.S., Yoo, T.S., Chen, D.T., Rheingans, P., Subramanian, K.R.: Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. In: Proceedings of International Conference on Shape Modeling & Application (SMI ’01), (2001)Google Scholar
  53. 53.
    Mullen, P., de Goes, F., Desbrun, M., Cohen-Steiner, D., Alliez, P.: Signing the unsigned: robust surface reconstruction from raw pointsets. Comput. Graph. Forum 29(5), 1733–1741 (2010). doi: 10.1111/j.1467-8659.2010.01782.x CrossRefGoogle Scholar
  54. 54.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989). doi: 10.1002/cpa.3160420503 MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Ohtake, Y., Belyaev, A., Seidel, H.P.: A multi-scale approach to 3D scattered data interpolation with compactly supported basis function. In: Shape Modeling International, pp. 153–161, May (2003), doi: 10.1109/SMI.2003.1199611
  56. 56.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model. Simul. 4, 460–489 (2005). doi: 10.1137/040605412 MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Oztireli, C., Guennebaud, G., Gross, M.: Feature preserving point set surfaces based on non-linear kernel regression. Comput. Graph. Forum 28(2), 493–501 (2009). doi: 10.1111/j.1467-8659.2009.01388.x CrossRefGoogle Scholar
  58. 58.
    Pauly, M., Gross, M., Kobbelt, L.P.: Efficient simplification of point-sampled surfaces. In: Proceedings of Conference on Visualization (VIS ’02), (2002)Google Scholar
  59. 59.
    Pauly, M., Keiser, R., Kobbelt, L.P., Gross, M.: Shape modeling with point-sampled geometry. ACM Trans. Graph. 22(3), 641–650 (2003)CrossRefGoogle Scholar
  60. 60.
    Pauly, M., Mitra, N.J., Guibas, L.: Uncertainty and variability in point cloud surface data. In: Symposium on Point-Based Graphics, pp. 77–84, (2004)Google Scholar
  61. 61.
    Peng, D., Marriman, B., Osher, S., Zhao, H.K., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999). doi: 10.1006/jcph.1999.6345 Google Scholar
  62. 62.
    Piegl, L., Tiller, M.: The NURBS Book, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar
  63. 63.
    Rogers, D.F.: An Introduction to NURBS: With Historical Perspective, 1st edn. Morgan Kaufmann, San Francisco (2001)Google Scholar
  64. 64.
    Rogers, D.F.: An Introduction to NURBS. Morgan Kaufmann, San Francisco (2003)Google Scholar
  65. 65.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992). doi: 10.1016/0167-2789(92)90242-F zbMATHCrossRefGoogle Scholar
  66. 66.
    Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl Acad. Sci. USA 93, 1591–1595 (1996). doi: 10.1073/pnas.93.4.1591 MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Sethian, J.A.: Level Set Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 2nd edn. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  68. 68.
    Sharf, A., Lewiner, T., Shamir, A., Kobbelt, L., Cohen-or, D.: Competing fronts for coarse-to-fine surface reconstruction. Comput. Graph. Forum 25, 389–398 (2006). doi: 10.1111/j.1467-8659.2006.00958.x CrossRefGoogle Scholar
  69. 69.
    Shen, C., Brien, J.F., Shewchuk, J.R.: Interpolation and approximating implicit surfaces from polygonal soup. ACM Trans. Graph. 23 (2004). doi: 10.1145/1015706.1015816
  70. 70.
    Takeda, H., Farsiu, S., Milanfar, P.: Kernel regression for image processing and reconstruction. IEEE Trans. Image Process. 16, 349–366 (2007). doi: 10.1109/TIP.2006.888330 MathSciNetCrossRefGoogle Scholar
  71. 71.
    Tsai, Y.R.: Rapid and accurate computation of the distance function using grids. J. Comput. Phys. 178, 175–195 (2002). doi: 10.1006/jcph.2002.7028 MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Wan, M., Wang, Y., Bae, E., Tai, X.C., Wang, V.: Reconstructing open surfaces via graph-cuts. In: CAM, LA, CA, Rep. 10-29, (2010)Google Scholar
  73. 73.
    Wendland, H.: Fast evaluation of radial basis functions: methods based on partition of unity. Approx. Theory X 473, 473–483 (2002). doi: MathSciNetGoogle Scholar
  74. 74.
    Ye, J., Bresson, X., Goldstein, T., Osher, S.: A fast variational method for surface reconstruction from sets of scattered points. In: CAM, LA, CA, Rep. 01-10, (2001)Google Scholar
  75. 75.
    Zach, C.: Fast and high quality fusion of depth maps. In: International Symposium on 3D Data Processing, Visualization and Transmission (3DPVT), June (2008)Google Scholar
  76. 76.
    Zach, C., Pock, T., Bischof, H.: A globally optimal algorithm for robust TV-L1 range image integration. In: Proceedings of 11th IEEE International Conference on Computer Vision (ICCV ’07), vol. 1, pp. 1–8, October (2007). doi: 10.1109/ICCV.2007.4408983
  77. 77.
    Zhao, H.K.: A fast sweeping method for Eikonal equations. Math. Comput. 74, 603–627 (2005). doi: zbMATHGoogle Scholar
  78. 78.
    Zhao, H.K., Osher, S., Merriman, B., Kang, M.: Implicit, nonparametric shape reconstruction from unorganized points using a variational level set method. Comput. Vis. Image Underst. 80, 295–314 (2000). doi: 10.1006/cviu.2000.0875 zbMATHCrossRefGoogle Scholar
  79. 79.
    Zhao, H.K., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: Proceedings of IEEE Workshop Variational and Level Set Methods Computer Vision, pp. 194–201, August (2002). doi: 10.1109/VLSM.2001.938900
  80. 80.
    Zhu, M., Chan, T.F.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. In: CAM, LA, CA, Rep. 08-34, (2008)Google Scholar

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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, IrvineIrvineUSA
  2. 2.Department of MathematicsWhittier CollegeWhittierUSA

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