“The Vector Heat Method” by Sharp, Soliman and Crane

  • ©Nicholas Sharp, Yousuf Soliman, and Keenan Crane

Conference:


Type(s):


Title:

    The Vector Heat Method

Session/Category Title:   Transport: Parallel and Optimal

Presenter(s)/Author(s):


Abstract:


    This article describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method, we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations so that they can be used in the context of tangent vector field processing.

References:


    1. D. Adalsteinsson and J. Sethian. 1999. The fast construction of extension velocities in level set methods. J. Comput. Phys. 148, 1 (1999), 2–22.
    2. M. Alexa and M. Wardetzky. 2011. Discrete Laplacians on general polygonal meshes. ACM Trans. Graph. 30, 4 (2011), Article 102.
    3. D. Arnold and L. Li. 2017. Finite element exterior calculus with lower-order terms. Math. Comput. 86, 307 (2017), 2193–2212.
    4. O. Azencot, M. Ovsjanikov, F. Chazal, and M. Ben-Chen. 2015. Discrete derivatives of vector fields on surfaces—An operator approach. ACM Trans. Graph. 34, 3 (2015), Article 29.
    5. A. Belyaev and P. A. Fayolle. 2015. On variational and PDE-based distance function approximations. Comp. Graph. Forum 34, 8 (2015), 104–118.
    6. N. Berline, E. Getzler, and M. Vergne. 1992. Heat Kernels and Dirac Operators. Springer.
    7. A. Bobenko and B. Springborn. 2007. A discrete Laplace-Beltrami operator for simplicial surfaces. Disc. Comp. Geom. 38, 4 (2007), 740–756.
    8. F. Bogo, J. Romero, M. Loper, and M. Black. 2014. FAUST: Dataset and evaluation for 3D mesh registration. In Proceedings of the 2014 IEEE Conference on Computer Vision and Pattern Recognition.
    9. D. Bommes, B. Lévy, N. Pietroni, E. Puppo, C. Silva, M. Tarini, and D. Zorin. 2013. Quad-mesh generation and processing: A survey. Comp. Graph. Forum 32, 6 (2013), 51–76.
    10. R. Bridson. 2007. Fast Poisson disk sampling in arbitrary dimensions. In Proceedings of ACM SIGGRAPH 2007 Sketches.
    11. A. Brun. 2007. Manifolds in Image Science and Visualization. Ph.D. Dissertation. Institutionen för Medicinsk Teknik.
    12. S. Buss and J. Fillmore. 2001. Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20, 2 (2001), 95–126.
    13. T. Caissard, D. Coeurjolly, J. O. Lachaud, and T. Roussillon. 2017. Heat kernel Laplace-Beltrami operator on digital surfaces. In Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, Vol. 10502. Springer, 241–253.
    14. J. Chen and Y. Han. 1990. Shortest paths on a polyhedron. In Proceedings of the 6th Annual Symposium on Computational Geometry (SCG’90).
    15. Y. Chen, T. Davis, W. Hager, and S. Rajamanickam. 2008. Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35, 3 (2008), Article 22.
    16. K. Crane, F. de Goes, M. Desbrun, and P. Schröder. 2013a. Digital geometry processing with discrete exterior calculus. In Proceedings of ACM SIGGRAPH 2013 Courses (SIGGRAPH’13).
    17. K. Crane, M. Desbrun, and P. Schröder. 2010. Trivial connections on discrete surfaces. Comp. Graph. Forum 29, 5 (2010), 1525–1533.
    18. K. Crane, C. Weischedel, and M. Wardetzky. 2013b. Geodesics in heat: A new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 5 (2013).
    19. T. A. Davis. 2004. Algorithm 832: UMFPACK V4.3—An unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 2 (2004), Article 152.
    20. F. de Goes, M. Desbrun, M. Meyer, and T. DeRose. 2016. Subdivision exterior calculus for geometry processing. ACM Trans. Graph. 35, 4 (2016), Article 133.
    21. M. Desbrun, E. Kanso, and Y. Tong. 2006. Discrete differential forms for computational modeling. In Proceedings of ACM SIGGRAPH 2006 Courses (SIGGRAPH’06).
    22. Q. Du, M. Gunzburger, and L. Ju. 2003. Constrained centroidal Voronoi tessellations for surfaces. SIAM J. Sci. Comp. 24, 5 (2003), 1488–1506.
    23. N. El Karoui and H. Wu. 2015. Graph connection Laplacian and random matrices with random blocks. Inf. Inference 4, 1 (03 2015), 1–44.
    24. M. Fisher, B. Springborn, A. Bobenko, and P. Schroder. 2006. An algorithm for the construction of intrinsic Delaunay triangulations with applications to digital geometry processing. In Proceedings of ACM SIGGRAPH 2006 Courses.
    25. P. Fletcher, S. Venkatasubramanian, and S. Joshi. 2009. The geometric median on Riemannian manifolds with application to robust atlas estimation. NeuroImage 45, 1 (2009), S142–S152.
    26. A. Gentle. 2002. Regge calculus: A unique tool for numerical relativity. Gen. Rel. and Grav. 34, 10 (2002), 1701–1718.
    27. A. Gillman and P. G. Martinsson. 2014. A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method. SIAM J. Sci. Comp. 36, 4 (2014), A2023–A2046.
    28. A. Grigor’yan. 2009. Heat Kernel and Analysis on Manifolds. American Mathematical Society.
    29. P. Herholz, T. Davis, and M. Alexa. 2017a. Localized solutions of sparse linear systems for geometry processing. ACM Trans. Graph. 36, 6 (2017), Article 183.
    30. P. Herholz, F. Haase, and M. Alexa. 2017b. Diffusion diagrams: Voronoi cells and centroids from diffusion. Comp. Graph. Forum 36, 2 (2017), 163–175.
    31. J. Itoh and R. Sinclair. 2004. Thaw: A tool for approximating cut loci on a triangulation of a surface. Experiment. Math. 13, 3 (2004), 309–325.
    32. I. Kezurer, S. Kovalsky, R. Basri, and Y. Lipman. 2015. Tight relaxation of quadratic matching. In Proceedings of the Eurographics Symposium on Geometry Processing (SGP’15). 115–128.
    33. R. Kimmel and J. Sethian. 1998. Fast marching methods on triangulated domains. Proc. Nat. Acad. Sci. 95, 15 (1998), 8431–8435.
    34. F. Knöppel, K. Crane, U. Pinkall, and P. Schröder. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4 (2013), Article 59.
    35. F. Knöppel, K. Crane, U. Pinkall, and P. Schröder. 2015. Stripe patterns on surfaces. ACM Trans. Graph. 34, 4 (2015), Article 39.
    36. R. Kyng, Y. T. Lee, R. Peng, S. Sachdeva, and D. Spielman. 2016. Sparsified Cholesky and multigrid solvers for connection Laplacians. In Proceedings of the Symposium on Theory of Computing (STOC’16).
    37. B. Lin, J. Yang, X. He, and J. Ye. 2014. Geodesic distance function learning via heat flow on vector fields. In Proceedings of the International Conference on Machine Learning.
    38. Y. Liu, B. Prabhakaran, and X. Guo. 2012. Point-based manifold harmonics. IEEE Trans. Vis. Comp. Graph. 18, 10 (2012), 1693–1703.
    39. Y. J. Liu, C. X. Xu, R. Yi, D. Fan, and Y. He. 2016. Manifold differential evolution (MDE): A global optimization method for geodesic centroidal Voronoi tessellations on meshes. ACM Trans. Graph. 35, 6 (2016), Article 243.
    40. M. Lorenzi and X. Pennec. 2014. Efficient parallel transport of deformations in time series of images: From Schild’s to pole ladder. J. Math. Imaging Vis. 50, 1 (2014), 5–17.
    41. M. Louis, B. Charlier, P. Jusselin, S. Pal, and S. Durrleman. 2018. A fanning scheme for the parallel transport along geodesics on Riemannian manifolds. SIAM J. Numer. Anal. 56, 4 (2018), 2563–2584.
    42. M. Ludewig. 2018. Heat kernel asymptotics, path integrals and infinite-dimensional determinants. J. Geom. Phys. 131 (2018), 66–88.
    43. E. Melvær and M. Reimers. 2012. Geodesic polar coordinates on polygonal meshes. Comp. Graph. Forum 31, 8 (2012), 2423–2435.
    44. J. S. B. Mitchell, D. M. Mount, and C. H. Papadimitriou. 1987. The discrete geodesic problem. SIAM J. Comput. 16, 4 (1987), 647–668.
    45. M. Mohamed, A. Hirani, and R. Samtaney. 2016. Comparison of discrete Hodge star operators for surfaces. Comput. Aided Des. 78, C (2016), 118–125.
    46. A. Myles, N. Pietroni, and D. Zorin. 2014. Robust field-aligned global parametrization. ACM Trans. Graph. 33, 4 (2014), Article 135.
    47. T. Nguyen, K. Karciauskas, and J. Peters. 2016. C<sup>1</sup> finite elements on non-tensor-product 2d and 3d manifolds. Appl. Math. Comput. 272 (2016), 148–158.
    48. D. Panozzo, I. Baran, O. Diamanti, and O. Sorkine-Hornung. 2013. Weighted averages on surfaces. ACM Trans. Graph. 32, 4 (2013), 1–11.
    49. K. Polthier and M. Schmies. 1998. Straightest Geodesics on Polyhedral Surfaces. Springer-Verlag.
    50. Y. Qin, X. Han, H. Yu, Y. Yu, and J. Zhang. 2016. Fast and exact discrete geodesic computation based on triangle-oriented wavefront propagation. ACM Trans. Graph. 35, 4 (2016), Article 125.
    51. N. Ray, W. C. Li, B. Lévy, A. Sheffer, and P. Alliez. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4 (2006), 1460–1485.
    52. T. Regge. 1961. General relativity without coordinates. Il Nuovo Cimento 19, 3 (1961), 558–571.
    53. R. Schmidt. 2013. Stroke parameterization. Comp. Graph. Forum 32, 2 (2013), 1–9.
    54. R. Schmidt, C. Grimm, and B. Wyvill. 2006. Interactive decal compositing with discrete exponential maps. ACM Trans. Graph. 25, 3 (2006), 605–613.
    55. R. Schmidt and K. Singh. 2010. Meshmixer: An interface for rapid mesh composition. In Proceedings of SIGGRAPH 2010 Talks.
    56. N. Sharp and K. Crane. 2018. Variational surface cutting. ACM Trans. Graph. 37, 4 (2018), Article 156.
    57. A. Singer and H.-T. Wu. 2012. Vector diffusion maps and the connection Laplacian. Commun. Pure Appl. Math. 65, 8 (2012), 1–72.
    58. D. Spielman and S.-H. Teng. 2004. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC’04). 81–90.
    59. V. Surazhsky, T. Surazhsky, D. Kirsanov, S. Gortler, and H. Hoppe. 2005. Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24, 3 (2005), 553–560.
    60. X. Tricoche, G. Scheuermann, and H. Hagen. 2000. Higher order singularities in piecewise linear vector fields. In Proceedings of the 9th IMA Conference on the Mathematics of Surfaces. 99–113.
    61. A. Vaxman, M. Campen, O. Diamanti, D. Panozzo, D. Bommes, K. Hildebrandt, and M. Ben-Chen. 2016. Directional field synthesis, design, and processing. Comp. Graph. Forum 35, 2 (2016), 1–28.
    62. E. Weiszfeld. 1937. Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Math. Journal 43 (1937), 355–386.
    63. X. Ying, X. Wang, and Y. He. 2013. Saddle vertex graph (SVG): A novel solution to the discrete geodesic problem. ACM Trans. Graph. 32, 6 (2013), Article 170.
    64. E. Zhang, K. Mischaikow, and G. Turk. 2006. Vector field design on surfaces. ACM Trans. Graph. 25, 4 (2006), 1294–1326.

ACM Digital Library Publication:


Overview Page: