“Surface Reconstruction Based on Modified Gauss Formula” by Lu, Shi, Sun and Wang

  • ©Wenjia Lu, Zuoqiang Shi, Jian Sun, and Bin Wang




    Surface Reconstruction Based on Modified Gauss Formula

Session/Category Title: Scene and Object Reconstruction



    In this article, we introduce a surface reconstruction method that has excellent performance despite nonuniformly distributed, noisy, and sparse data. We reconstruct the surface by estimating an implicit function and then obtain a triangle mesh by extracting an iso-surface. Our implicit function takes advantage of both the indicator function and the signed distance function. The implicit function is dominated by the indicator function at the regions away from the surface and is approximated (up to scaling) by the signed distance function near the surface. On one hand, the implicit function is well defined over the entire space for the extracted iso-surface to remain near the underlying true surface. On the other hand, a smooth iso-surface can be extracted using the marching cubes algorithm with simple linear interpolations due to the properties of the signed distance function. Moreover, our implicit function can be estimated directly from an explicit integral formula without solving any linear system. An approach called disk integration is also incorporated to improve the accuracy of the implicit function. Our method can be parallelized with small overhead and shows compelling performance in a GPU version by implementing this direct and simple approach. We apply our method to synthetic and real-world scanned data to demonstrate the accuracy, noise resilience, and efficiency of this method. The performance of the proposed method is also compared with several state-of-the-art methods.


    1. Pierre Alliez, David Cohen-Steiner, Yiying Tong, and Mathieu Desbrun. 2007. Voronoi-based variational reconstruction of unoriented point sets. In Symposium on Geometry processing, Vol. 7. 39–48.
    2. Nina Amenta, Marshall Bern, and Manolis Kamvysselis. 1998. A new Voronoi-based surface reconstruction algorithm. In Proceedings of SIGGRAPH’98. ACM, 415–421.
    3. N. Amenta, S. Choi, T. K. Dey, and N. Leekha. 2002. A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geom. Appl. 12 (2002), 125–141.
    4. Nina Amenta, Sunghee Choi, and Ravi Krishna Kolluri. 2001. The power crust, unions of balls, and the medial axis transform. Comput. Geom. 19, 2 (2001), 127–153.
    5. Nina Amenta and Yong Joo Kil. 2004a. Defining Point-set Surfaces. ACM Trans. Graph. 23, 3 (Aug. 2004), 264–270.
    6. Nina Amenta and Yong Joo Kil. 2004b. The domain of a point set surface. In SPBG. 139–147.
    7. Matthew Berger, Joshua A. Levine, Luis Gustavo Nonato, Gabriel Taubin, and Claudio T. Silva. 2013. A benchmark for surface reconstruction. ACM Trans. Graph. 32, 2, Article 20 (April 2013), 17 pages.
    8. Matthew Berger, Andrea Tagliasacchi, Lee M. Seversky, Pierre Alliez, Gael Guennebaud, Joshua A. Levine, Andrei Sharf, and Claudio T. Silva. 2017. A survey of surface reconstruction from point clouds. Comput. Graphics Forum 36, 1 (2017), 301–329.
    9. Jean-Daniel Boissonnat and Steve Oudot. 2005. Provably good sampling and meshing of surfaces. Graph. Models 67, 5 (Sept. 2005), 405–451.
    10. Martin Burtscher and Keshav Pingali. 2011. An efficient CUDA implementation of the tree-based barnes hut n-body algorithm. In GPU Computing Gems Emerald Edition.75.
    11. Fatih Calakli and Gabriel Taubin. 2011. SSD: Smooth signed distance surface reconstruction. Comput. Graphics Forum 30, 7 (2011), 1993–2002.
    12. J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, and T. R. Evans. 2001. Reconstruction and representation of 3D objects with radial basis functions. In Proceedings of SIGGRAPH’01. ACM, 67–76.
    13. Jiazhou Chen, Gael Guennebaud, Pascal Barla, and Xavier Granier. 2013. Non-oriented MLS gradient fields. Comput. Graphics Forum 32, 98–109.
    14. Paolo Cignoni, Claudio Rocchini, and Roberto Scopigno. 1998. Metro: measuring error on simplified surfaces. Comput. Graphics Forum 17, 2 (1998), 167–174.
    15. Brian Curless and Marc Levoy. 1996. A volumetric method for building complex models from range images. In Proceedings of SIGGRAPH’96. ACM, 303–312.
    16. Tamal K. Dey and Samrat Goswami. 2004. Provable surface reconstruction from noisy samples. In Proceedings of SCG’04. ACM, 330–339.
    17. Tamal K. Dey and Jian Sun. 2005. An adaptive MLS surface for reconstruction with guarantees. In Proceedings of SGP’05. Eurographics Association, Article 43, 43–52.
    18. Herbert Edelsbrunner. 2003. Surface reconstruction by wrapping finite sets in space. In Discrete and Computational Geometry. Algorithms and Combinatorics, Vol. 25. Springer, Berlin, 379–404.
    19. Simon Fuhrmann and Michael Goesele. 2014. Floating scale surface reconstruction. ACM Trans. Graph. 33, 4, Article 46 (July 2014), 11 pages.
    20. Simon Fuhrmann, Michael Kazhdan, and Michael Goesele. 2015. Accurate isosurface interpolation with hermite data. In 2015 International Conference on 3D Vision (3DV). IEEE, 256–263.
    21. Joachim Giesen and Matthias John. 2008. The flow complex: A data structure for geometric modeling. Comput. Geom. 39, 3 (2008), 178–190.
    22. Leslie Greengard and Vladimir Rokhlin. 1987. A fast algorithm for particle simulations. J. Comput. Physics 73, 2 (1987), 325–348.
    23. Hugues Hoppe, Tony DeRose, Tom Duchamp, John McDonald, and Werner Stuetzle. 1992. Surface reconstruction from unorganized points. In Proceedings of SIGGRAPH’92. ACM, 71–78.
    24. Evelyne Hubert. 2012. Convolution surfaces based on polygons for infinite and compact support kernels. Graphical Models 74, 1 (2012), 1–13.
    25. Alec Jacobson, Ladislav Kavan, and Olga Sorkine-Hornung. 2013. Robust inside-outside segmentation using generalized winding numbers. ACM Trans. Graph. 32, 4, Article 33 (July 2013), 12 pages.
    26. Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe. 2006. Poisson surface reconstruction. In Proceedings of SGP’06. Eurographics Association, 61–70.
    27. Michael Kazhdan and Hugues Hoppe. 2013. Screened Poisson surface reconstruction. ACM Trans. Graph. 32, 3, Article 29 (July 2013), 13 pages.
    28. Michael Kazhdan, Allison Klein, Ketan Dalal, and Hugues Hoppe. 2007. Unconstrained isosurface extraction on arbitrary octrees. In Symposium on Geometry Processing, Vol. 7.
    29. Ravikrishna Kolluri, Jonathan Richard Shewchuk, and James F. O’Brien. 2004. Spectral surface reconstruction from noisy point clouds. In Proceedings of SGP’04. ACM, 11–21.
    30. David Levin. 2004. Mesh-independent surface interpolation. In Geometric Modeling for Scientific Visualization. Springer, 37–49.
    31. William E. Lorensen and Harvey E. Cline. 1987. Marching cubes: A high resolution 3D surface construction algorithm. In Proceedings of SIGGRAPH’87. ACM, 163–169.
    32. Shigeru Muraki. 1991. Volumetric shape description of range data using “blobby model.” In Proceedings of SIGGRAPH’91. ACM, 227–235.
    33. A. Cengiz Öztireli, Gael Guennebaud, and Markus Gross. 2009. Feature Preserving Point Set Surfaces based on Non-Linear Kernel Regression. Comput. Graphics Forum 28, 493–501.
    34. Scott Schaefer and Joe Warren. 2004. Dual marching cubes: Primal contouring of dual grids. In Proceedings of the 12th Pacific Conference on Computer Graphics and Applications (PG’04). IEEE, 70–76.
    35. Chen Shen, James F. O’Brien, and Jonathan R. Shewchuk. 2004. Interpolating and approximating implicit surfaces from polygon soup. In ACM SIGGRAPH 2004 Papers (SIGGRAPH’04). ACM, 896–904.
    36. Christian Walder, Olivier Chapelle, and Bernhard Schölkopf. 2005. Implicit Surface Modelling As an Eigenvalue Problem. In Proceedings of ICML’05. ACM, 936–939.
    37. W. L. Wendland. 2009. On the double layer potential. In Analysis, Partial Differential Equations and Applications: The Vladimir Maz¡¯ya Anniversary Volume (2009), 319–334.
    38. Jane Wilhelms and Allen Van Gelder. 1992. Octrees for faster isosurface generation. ACM Trans. Graph. 11, 3 (July 1992), 201–227.
    39. Shiyao Xiong, Juyong Zhang, Jianmin Zheng, Jianfei Cai, and Ligang Liu. 2014. Robust surface reconstruction via dictionary learning. ACM Trans. Graph. 33, 6, Article 201 (Nov. 2014), 12 pages.

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