“Computing minimal surfaces with differential forms” by Wang and Chern

  • ©Stephanie Wang and Albert Chern




    Computing minimal surfaces with differential forms



    We describe a new algorithm that solves a classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve. Existing numerical methods face challenges due to the non-convexity of the problem. Using a representation of curves and surfaces via differential forms on the ambient space, we reformulate this problem as a convex optimization. This change of variables overcomes many difficulties in previous numerical attempts and allows us to find the global minimum across all possible surface topologies. The new algorithm is based on differential forms on the ambient space and does not require handling meshes. We adopt the Alternating Direction Method of Multiplier (ADMM) to find global minimal surfaces. The resulting algorithm is simple and efficient: it boils down to an alternation between a Fast Fourier Transform (FFT) and a pointwise shrinkage operation. We also show other applications of our solver in geometry processing such as surface reconstruction.


    1. Andreas Arnez, Konrad Polthier, Martin Steffens, and Christian Teitzel. 2007. Touching Soap Films (Springer VideoMATH). Springer-Verlag.Google Scholar
    2. Moheb Sabry Aziz et al. 2016. Biomimicry as an approach for bio-inspired structure with the aid of computation. Alexandria Engineering Journal 55, 1 (2016), 707–714.Google ScholarCross Ref
    3. Marcel Berger. 2012. A panoramic view of Riemannian geometry. Springer Science & Business Media.Google Scholar
    4. Alexander I Bobenko, Tim Hoffmann, and Boris A Springborn. 2006. Minimal surfaces from circle patterns: Geometry from combinatorics. Annals of Mathematics (2006), 231–264.Google Scholar
    5. Alexander I Bobenko and Ulrich Pinkall. 1994. Discrete isothermic surfaces. Technical Report. SCAN-9412128.Google Scholar
    6. Kenneth A Brakke. 1992. The surface evolver. Experimental mathematics 1, 2 (1992), 141–165.Google Scholar
    7. Haïm Brezis and Petru Mironescu. 2019. The Plateau problem from the perspective of optimal transport. Comptes Rendus Mathematique 357, 7 (2019), 597–612.Google ScholarCross Ref
    8. Blanche Buet, Gian Paolo Leonardi, and Simon Masnou. 2018. Discretization and approximation of surfaces using varifolds. Geometric Flows 3, 1 (2018), 28–56.Google ScholarCross Ref
    9. Nicolas Charon and Alain Trouvé. 2014. Functional currents: a new mathematical tool to model and analyse functional shapes. Journal of mathematical imaging and vision 48, 3 (2014), 413–431.Google ScholarDigital Library
    10. David Cohen-Steiner, André Lieutier, and Julien Vuillamy. 2020. Lexicographic optimal homologous chains and applications to point cloud triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik.Google Scholar
    11. Paul Concus. 1967. Numerical solution of the minimal surface equation. Math. Comp. 21, 99 (1967), 340–350.Google ScholarCross Ref
    12. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin transformations of discrete surfaces. In ACM SIGGRAPH 2011 papers. 1–10.Google ScholarDigital Library
    13. Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H Barr. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques. 317–324.Google ScholarDigital Library
    14. Tamal K Dey, Anil N Hirani, and Bala Krishnamoorthy. 2011. Optimal homologous cycles, total unimodularity, and linear programming. SIAM J. Comput. 40, 4 (2011), 1026–1044.Google ScholarDigital Library
    15. Jesse Douglas. 1927. A method of numerical solution of the problem of Plateau. Annals of Mathematics (1927), 180–188.Google Scholar
    16. Tevian Dray. 1999. The hodge dual operator. Oregon State University report (1999), 1–6.Google Scholar
    17. Nathan M Dunfield and Anil N Hirani. 2011. The least spanning area of a knot and the optimal bounding chain problem. In Proceedings of the twenty-seventh annual symposium on Computational geometry. 135–144.Google ScholarDigital Library
    18. Stanley Durrleman, Pierre Fillard, Xavier Pennec, Alain Trouvé, and Nicholas Ayache. 2011. Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage 55, 3 (2011), 1073–1090. Google ScholarCross Ref
    19. Stanley Durrleman, Xavier Pennec, Alain Trouvé, Paul Thompson, and Nicholas Ayache. 2008. Inferring brain variability from diffeomorphic deformations of currents: an integrative approach. Medical image analysis 12, 5 (2008), 626–637.Google Scholar
    20. Stanley Durrleman, Xavier Pennec, Alain Trouvé, and Nicholas Ayache. 2009. Statistical models of sets of curves and surfaces based on currents. Medical Image Analysis 13, 5 (2009), 793–808. Includes Special Section on the 12th International Conference on Medical Imaging and Computer Assisted Intervention. Google ScholarCross Ref
    21. Gerhard Dziuk. 1990. An algorithm for evolutionary surfaces. Numer. Math. 58, 1 (1990), 603–611.Google ScholarDigital Library
    22. Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun. 2007. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics (TOG) 26, 1 (2007), 4–es.Google ScholarDigital Library
    23. Michele Emmer. 2013. Minimal surfaces and architecture: New forms. Nexus Network Journal 15, 2 (2013), 227.Google ScholarCross Ref
    24. Myfanwy E Evans and Gerd E Schröder-Turk. 2015. In a material world: Hyperbolic geometry in biological materials. Asia Pacific Mathematics Newsletter 5, 2 (2015), 21–30.Google Scholar
    25. Herbert Federer and Wendell H Fleming. 1960. Normal and integral currents. Annals of Mathematics (1960), 458–520.Google Scholar
    26. Harley Flanders. 1963. Differential forms with applications to the physical sciences by harley flanders. Elsevier.Google Scholar
    27. Wendell H Fleming. 2015. Geometric measure theory at brown in the 1960s.Google Scholar
    28. Joan Glaunes, Alain Trouvé, and Laurent Younes. 2004. Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching. In Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004., Vol. 2. IEEE, II–II.Google ScholarCross Ref
    29. Tom Goldstein, Brendan O’Donoghue, Simon Setzer, and Richard Baraniuk. 2014. Fast alternating direction optimization methods. SIAM Journal on Imaging Sciences 7, 3 (2014), 1588–1623.Google ScholarDigital Library
    30. Allen Hatcher. 2002. Algebraic topology. Cambridge University Press.Google Scholar
    31. Masahiro Hinata, Masaaki Shimasaki, and Takeshi Kiyono. 1974. Numerical solution of Plateau’s problem by a finite element method. Math. Comp. 28, 125 (1974), 45–60.Google Scholar
    32. Anil Nirmal Hirani. 2003. Discrete exterior calculus. Ph.D. Dissertation. California Institute of Technology.Google ScholarDigital Library
    33. David Hoffman and Henri Matisse. 1987. The computer-aided discovery of new embedded minimal surfaces. The Mathematical Intelligencer 9, 3 (1987), 8–21.Google ScholarCross Ref
    34. Cyril Isenberg. 1978. The science of soap films and soap bubbles. Tieto Cleveton, UK.Google Scholar
    35. Dominic Joyce. 2003. Riemannian holonomy groups and calibrated geometry. In Calabi-Yau Manifolds and Related Geometries. Springer, 1–68.Google Scholar
    36. Hermann Karcher and Konrad Polthier. 1996. An introduction to minimal surfaces. http://page.mi.fu-berlin.de/polthier/booklet/architecture.htmlGoogle Scholar
    37. Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe. 2006. Poisson surface reconstruction. In Proceedings of the fourth Eurographics symposium on Geometry processing, Vol. 7.Google ScholarDigital Library
    38. Michael Kazhdan, Jake Solomon, and Mirela Ben-Chen. 2012. Can mean-curvature flow be modified to be non-singular?. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1745–1754.Google Scholar
    39. Jacob JK Kirkensgaard, Myfanwy E Evans, Liliana De Campo, and Stephen T Hyde. 2014. Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks. Proceedings of the National Academy of Sciences 111, 4 (2014), 1271–1276.Google ScholarCross Ref
    40. Antoni A Kosinski. 2013. Differential manifolds. Courier Corporation.Google Scholar
    41. JL Lagrange. 1762. Essai d’une nouvelle méthode pour déterminer les maxima et les formules des intégrales minimum indéfinies. Œuvres de Lagrange (1867–1892) 1 (1762), 335–362.Google Scholar
    42. Wai Yeung Lam. 2018. Discrete minimal surfaces: critical points of the area functional from integrable systems. International Mathematics Research Notices 2018, 6 (2018), 1808–1845.Google ScholarCross Ref
    43. Linhu Li, Ching Hua Lee, and Jiangbin Gong. 2019. Emergence and full 3D-imaging of nodal boundary Seifert surfaces in 4D topological matter. Communications physics 2, 1 (2019), 1–11.Google Scholar
    44. Thomas Mollenhoff and Daniel Cremers. 2019. Lifting vectorial variational problems: a natural formulation based on geometric measure theory and discrete exterior calculus. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 11117–11126.Google ScholarCross Ref
    45. Frank Morgan. 2016. Geometric measure theory: a beginner’s guide. Academic press.Google ScholarCross Ref
    46. Patrick Mullen, Alexander McKenzie, Yiying Tong, and Mathieu Desbrun. 2007. A variational approach to Eulerian geometry processing. ACM Transactions on Graphics (TOG) 26, 3 (2007), 66–es.Google ScholarDigital Library
    47. Hao Pan, Yang Liu, Alla Sheffer, Nicholas Vining, Chang-Jian Li, and Wenping Wang. 2015. Flow aligned surfacing of curve networks. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1–10.Google ScholarDigital Library
    48. Harold Parks. 1977. Explicit determination of area minimizing hypersurfaces. Duke Mathematical Journal 44, 3 (1977), 519–534.Google ScholarCross Ref
    49. Harold R Parks. 1992. Numerical approximation of parametric oriented area-minimizing hypersurfaces. SIAM journal on scientific and statistical computing 13, 2 (1992), 499–511.Google Scholar
    50. Harold R Parks and Jon T Pitts. 1997. Computing least area hypersurfaces spanning arbitrary boundaries. SIAM Journal on Scientific Computing 18, 3 (1997), 886–917.Google ScholarDigital Library
    51. Harold R Parks and Jon T Pitts. 2020. Explicit determination in RN of (N – 1)-dimensional area minimizing surfaces with arbitrary boundaries. The Journal of Geometric Analysis 30, 1 (2020), 601–616.Google ScholarCross Ref
    52. Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experimental mathematics 2, 1 (1993), 15–36.Google Scholar
    53. Joseph Plateau. 1873. Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires. Vol. 1. Gauthier-Villars.Google Scholar
    54. EV Popov. 1996. On some variational formulations for minimum surface. Transactions of the Canadian Society for Mechanical Engineering 20, 4 (1996), 391–400.Google ScholarCross Ref
    55. Filippo Santambrogio. 2015. Optimal transport for applied mathematicians. Birkäuser, NY 55, 58-63 (2015), 94.Google Scholar
    56. Henrik Schumacher and Max Wardetzky. 2019. Variational convergence of discrete minimal surfaces. Numer. Math. 141, 1 (2019), 173–213.Google ScholarDigital Library
    57. Günter Schwarz. 2006. Hodge Decomposition-A method for solving boundary value problems. Springer.Google Scholar
    58. Hermann Amandus Schwarz. 1890. Gesammelte mathematische abhandlungen. Berlin J. Springer.Google Scholar
    59. Stefan Sechelmann and Alexander I Bobenko. 2007. Discrete minimal surfaces, Koebe polyhedra, and Alexandrov’s theorem. variational principles, algorithms, and implementation. Ph.D. Dissertation. Citeseer.Google Scholar
    60. Yousuf Soliman, Dejan Slepčev, and Keenan Crane. 2018. Optimal cone singularities for conformal flattening. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1–17.Google ScholarDigital Library
    61. Justin Solomon, Raif Rustamov, Leonidas Guibas, and Adrian Butscher. 2014. Earth mover’s distances on discrete surfaces. ACM Transactions on Graphics (TOG) 33, 4 (2014), 1–12.Google ScholarDigital Library
    62. John M Sullivan. 1990. A crystalline approximation theorem for hypersurfaces. ProQuest LLC. Ann Arbor, Thesis (Ph. D.), Princeton University (1990).Google Scholar
    63. Marc Vaillant and Joan Glaunes. 2005. Surface matching via currents. In Biennial International Conference on Information Processing in Medical Imaging. Springer, 381–392.Google ScholarDigital Library
    64. H-J Wagner. 1977. A contribution to the numerical approximation of minimal surfaces. Computing 19, 1 (1977), 35–58.Google ScholarCross Ref
    65. Matthias Weber. 2013. Minimal surface archive. https://minimal.sitehost.iu.edu/archive/index.htmlGoogle Scholar
    66. Walter L Wilson. 1961. On discrete Dirichlet and Plateau problems. Numer. Math. 3, 1 (1961), 359–373.Google ScholarDigital Library
    67. Rundong Zhao, Mathieu Desbrun, Guo-Wei Wei, and Yiying Tong. 2019. 3D Hodge decompositions of edge-and face-based vector fields. ACM Transactions on Graphics (TOG) 38, 6 (2019), 1–13.Google ScholarDigital Library

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