“Gaussian-product subdivision surfaces” by Preiner, Boubekeur and Wimmer

  • ©Reinhold Preiner, Tamy Boubekeur, and Michael Wimmer




    Gaussian-product subdivision surfaces

Session/Category Title: Shape Science



    Probabilistic distribution models like Gaussian mixtures have shown great potential for improving both the quality and speed of several geometric operators. This is largely due to their ability to model large fuzzy data using only a reduced set of atomic distributions, allowing for large compression rates at minimal information loss. We introduce a new surface model that utilizes these qualities of Gaussian mixtures for the definition and control of a parametric smooth surface. Our approach is based on an enriched mesh data structure, which describes the probability distribution of spatial surface locations around each vertex via a Gaussian covariance matrix. By incorporating this additional covariance information, we show how to define a smooth surface via a nonlinear probabilistic subdivision operator based on products of Gaussians, which is able to capture rich details at fixed control mesh resolution. This entails new applications in surface reconstruction, modeling, and geometric compression.


    1. Nina Amenta, Sunghee Choi, and Ravi Krishna Kolluri. 2001. The Power Crust. In Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications (SMA ’01). ACM, New York, NY, USA, 249–266. Google ScholarDigital Library
    2. 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. Google ScholarDigital Library
    3. Henning Biermann, Adi Levin, and Denis Zorin. 2000. Piecewise Smooth Subdivision Surfaces with Normal Control. In Proc. SIGGRAPH. 113–120. Google ScholarDigital Library
    4. Wade Brainerd, Tim Foley, Manuel Kraemer, Henry Moreton, and Matthias Nießner. 2016. Efficient GPU Rendering of Subdivision Surfaces using Adaptive Quadtrees. ACM Transactions on Graphics (TOG) (2016). Google ScholarDigital Library
    5. Max Budninskiy, Beibei Liu, Fernando de Goes, Yiying Tong, Pierre Alliez, and Mathieu Desbrun. 2016. Optimal Voronoi Tessellations with Hessian-based Anisotropy. ACM Trans. Graph. 35, 6, Article 242 (Nov. 2016), 12 pages. Google ScholarDigital Library
    6. Stephane Calderon and Tamy Boubekeur. 2014. Point Morphology. ACM Transactions on Graphics (Proc. SIGGRAPH 2014) (2014). Google ScholarDigital Library
    7. Edwin Catmull and James Clark. 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6 (1978), 350 — 355.Google ScholarCross Ref
    8. David Cohen-Steiner and Frank Da. 2004. A Greedy Delaunay-based Surface Reconstruction Algorithm. Vis. Comput. 20, 1 (April 2004), 4–16. Google ScholarDigital Library
    9. Martin Danelljan, Giulia Meneghetti, Fahad S. Khan, and Michael Felsberg. 2016. A Probabilistic Framework for Color-Based Point Set Registration. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 1818–1826.Google ScholarCross Ref
    10. Arthur P. Dempster, Natalie Laird, and Donald Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. Ser. B 39, 1 (1977), 1–38.Google ScholarCross Ref
    11. Tony DeRose, Michael Kass, and Tien Truong. 1998. Subdivision Surfaces in Character Animation. In Proc. SIGGRAPH. 85–94. Google ScholarDigital Library
    12. Meenakshisundaram Gopi, Shankar Krishnan, and Cláudio T. Silva. 2000. Surface Reconstruction based on Lower Dimensional Localized Delaunay Triangulation. Computer Graphics Forum 19, 3 (2000), 467–478.Google ScholarCross Ref
    13. Philipp Grohs. 2009. Smoothness of interpolatory multivariate subdivision in Lie groups. IMA J. Numer. Anal. 29 (07 2009).Google Scholar
    14. Philipp Grohs. 2010. A General Proximity Analysis of Nonlinear Subdivision Schemes. SIAM J. Math. Analysis 42 (01 2010), 729–750.Google Scholar
    15. Richard D. Hill and Steven R. Waters. 1987. On the cone of positive semidefinite matrices. Linear Algebra Appl. 90 (1987), 81 — 88.Google ScholarCross Ref
    16. Hugues Hoppe, Tony DeRose, Tom Duchamp, Mark Halstead, Hubert Jin, John McDonald, Jean Schweitzer, and Werner Stuetzle. 1994. Piecewise Smooth Surface Reconstruction. In Proc. SIGGRAPH. 295–302. Google ScholarDigital Library
    17. Uri Itai and Nir Sharon. 2013. Subdivision Schemes for Positive Definite Matrices. Foundations of Computational Mathematics 13, 3 (01 Jun 2013), 347–369. Google ScholarDigital Library
    18. Bing Jian and Baba C. Vemuri. 2011. Robust Point Set Registration Using Gaussian Mixture Models. IEEE Trans. Pattern Anal. Mach. Intell. 33, 8 (Aug. 2011), 1633–1645. Google ScholarDigital Library
    19. Thomas Kailath. 1967. The Divergence and Bhattacharyya Distance Measures in Signal Selection. IEEE Trans. on Communication Technology 15, 1 (February 1967), 52–60.Google ScholarCross Ref
    20. Kestutis Karčiauskas and Jörg Peters. 2007. Concentric Tessellation Maps and Curvature Continuous Guided Surfaces. Comput. Aided Geom. Des. 24, 2 (2007), 99–111. Google ScholarDigital Library
    21. Kestutis Karčiauskas and Jörg Peters. 2018. A New Class of Guided C2 Subdivision Surfaces Combining Good Shape with Nested Refinement. Computer Graphics Forum 37, 6 (2018), 84–95.Google ScholarCross Ref
    22. Michael Kazhdan and Hugues Hoppe. 2013. Screened Poisson Surface Reconstruction. ACM Trans. Graph. 32, 3, Article 29 (July 2013), 13 pages. Google ScholarDigital Library
    23. Adi Levin. 2006. Modified Subdivision Surfaces with Continuous Curvature. ACM Trans. Graph. 25, 3 (2006), 1035–1040. Google ScholarDigital Library
    24. Ruosi Li, Lu Liu, Ly Phan, Sasakthi Abeysinghe, Cindy Grimm, and Tao Ju. 2010. Polygonizing Extremal Surfaces with Manifold Guarantees. In Proc. of the 14th ACM Symposium on Solid and Physical Modeling (SPM ’10). ACM, New York, 189–194. Google ScholarDigital Library
    25. Lars Linsen and Hartmut Prautzsch. 2003. Fan Clouds – An Alternative To Meshes. In Geometry, Morphology, and Computational Imaging, Tetsuo Asano, Reinhard Klette, and Christian Ronse (Eds.). Lecture Notes in Computer Science, Vol. 2616. Springer, Berlin, Germany, 39–57. Google ScholarDigital Library
    26. Charles Loop. 1987. Smooth Subdivision Surfaces Based on Triangles. Ph.D. Dissertation.Google Scholar
    27. Matthias Nießner, Charles Loop, Mark Meyer, and Tony DeRose. 2012. Feature-adaptive GPU rendering of Catmull-Clark subdivision surfaces. ACM Trans. Graph. (TOG) 31, 1 (2012), 6. Google ScholarDigital Library
    28. Umut Ozertem and Deniz Erdogmus. 2011. Locally Defined Principal Curves and Surfaces. Journal of Machine Learning Research 12 (2011), 1249–1286. Google ScholarDigital Library
    29. Reinhold Preiner, Oliver Mattausch, Murat Arikan, Renato Pajarola, and Michael Wimmer. 2014. Continuous Projection for Fast L1 Reconstruction. ACM Transactions on Graphics (Proc. of ACM SIGGRAPH 2014) 33, 4 (Aug. 2014), 47:1–47:13. Google ScholarDigital Library
    30. Mael Rouxel-Labbé, Mathijs Wintraecken, and Jean-Daniel Boissonnat. 2016. Discretized Riemannian Delaunay triangulations. In Proceedings 25th International Meshing Roundtable (IMR25). Elsevier, Washington DC, United States.Google ScholarCross Ref
    31. Scott Schaefer, Etienne Vouga, and Ron Goldman. 2008. Nonlinear subdivision through nonlinear averaging. Computer Aided Geometric Design 25, 3 (2008), 162 — 180. Google ScholarDigital Library
    32. Jos Stam. 1998. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values. In Proc. of the 25th Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH ’98). ACM, New York, NY, USA, 395–404. Google ScholarDigital Library
    33. Jos Stam. 1999. Evaluation of Loop Subdivision Surfaces. In SIGGRAPH ’99 Course Notes.Google Scholar
    34. Jochen Süßmuth and Günther Greiner. 2007. Ridge Based Curve and Surface Reconstruction. In Proc. of the Fifth Eurographics Symposium on Geometry Processing (SGP ’07). Eurographics Association, Aire-la-Ville, Switzerland, 243–251. Google ScholarDigital Library
    35. Nuno Vasconcelos and Andrew Lippman. 1999. Learning Mixture Hierarchies. In Advances in Neural Information Processing Systems 11, M. J. Kearns, S. A. Solla, and D. A. Cohn (Eds.). MIT Press, 606–612. Google ScholarDigital Library
    36. Amir Vaxman, Christian Müller, and Ofir Weber. 2018. Canonical Möbius Subdivision. ACM Trans. Graph. 37, 6, Article 227 (Dec. 2018), 15 pages.Google ScholarDigital Library
    37. Johannes Wallner. 2014. On Convergent Interpolatory Subdivision Schemes in Riemannian Geometry. Constructive Approximation 40 (12 2014).Google Scholar
    38. Johannes Wallner and Nira Dyn. 2005. Convergence and C1 analysis of subdivision schemes on manifolds by proximity. Computer Aided Geomtric Design 22, 7 (2005), 593 — 622. Google ScholarDigital Library
    39. Johannes Wallner, Esfandiar Nava Yazdani, and Andreas Weinmann. 2011. Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Advances in Computational Mathematics 34, 2 (01 Feb 2011), 201–218. Google ScholarDigital Library
    40. Andreas Weinmann. 2010. Nonlinear Subdivision Schemes on Irregular Meshes. Constructive Approximation 31, 3 (01 Jun 2010), 395–415.Google Scholar
    41. Xunnian Yang. 2016. Matrix Weighted Rational Curves and Surfaces. Comput. Aided Geom. Des. 42, C (2016), 40–53. Google ScholarDigital Library
    42. Denis Zorin and Peter Schroder. 2000. Subdivision for Modeling and Animation. In ACM SIGGRAPH Course.Google Scholar

ACM Digital Library Publication:

Overview Page: