“Steklov Spectral Geometry for Extrinsic Shape Analysis” by Wang, Ben-Chen, Polterovich and Solomon

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    Steklov Spectral Geometry for Extrinsic Shape Analysis

Session/Category Title:   Maps and Operators

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Abstract:


    We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace–Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace–Beltrami operator with the Dirichlet-to-Neumann operator.

References:


    1. Yonathan Aflalo, Haim Brezis, and Ron Kimmel. 2015. On the optimality of shape and data representation in the spectral domain. SIAM Journal on Imaging Sciences 8, 2 (2015), 1141–1160.
    2. Hartwig Anzt, Stanimire Tomov, and Jack Dongarra. 2015. Accelerating the LOBPCG method on GPUs using a blocked sparse matrix vector product. In Proc. Symposium on High Performance Computing. 75–82.
    3. Mathieu Aubry, Ulrich Schlickewei, and Daniel Cremers. 2011. The wave kernel signature: A quantum mechanical approach to shape analysis. In Proceedings of the International Conference on Computer Vision (ICCV’11). 1626–1633.
    4. Omri Azencot, Mirela Ben-Chen, Frédéric Chazal, and Maks Ovsjanikov. 2013. An operator approach to tangent vector field processing. In Computer Graphics Forum, Vol. 32. 73–82.
    5. Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of vector fields on surfaces–An operator approach. Transactions on Graphics 34, 3 (2015), 29.
    6. Omri Azencot, Orestis Vantzos, and Mirela Ben-Chen. 2016. Advection-based function matching on surfaces. In Computer Graphics Forum, Vol. 35. 55–64.
    7. Omri Azencot, Steffen Weißmann, Maks Ovsjanikov, Max Wardetzky, and Mirela Ben-Chen. 2014. Functional fluids on surfaces. In Computer Graphics Forum, Vol. 33. 237–246.
    8. Mikhail Belkin, Jian Sun, and Yusu Wang. 2008. Discrete Laplace operator on meshed surfaces. In Proceedings of the Symposium on Discrete Algorithms (SODA’08). 278–287.
    9. Mikhail Belkin, Jian Sun, and Yusu Wang. 2009. Constructing Laplace operator from point clouds in R<sup>d</sup>. In Proceedings of the Symposium on Discrete Algorithms (SODA’09). 1031–1040.
    10. Michele Benzi, Gene H. Golub, and Jörg Liesen. 2005. Numerical solution of saddle point problems. Acta Numerica 14 (2005), 1–137.
    11. Silvia Bertoluzza. 2003. Non conforming domain decomposition: The Steklov–Poincaré operator point of view. In Domain Decomposition Methods in Science and Engineering. 15–26.
    12. Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch. 2003. Hierarchical Matrices. Technical Report. Max Planck Institute.
    13. Davide Boscaini, Davide Eynard, Drosos Kourounis, and Michael M. Bronstein. 2015. Shape-from-operator: Recovering shapes from intrinsic operators. In Computer Graphics Forum, Vol. 34. 265–274.
    14. James H. Bramble and Joseph E. Pasciak. 1988. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation 50, 181 (1988), 1–17.
    15. Haim Brezis. 2010. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science 8 Business Media.
    16. Alexander M. Bronstein, Michael M. Bronstein, Leonidas J. Guibas, and Maks Ovsjanikov. 2011. Shape Google: Geometric words and expressions for invariant shape retrieval. Transactions on Graphics 30, 1 (2011), 1.
    17. Alexander M. Bronstein, Michael M. Bronstein, and Ron Kimmel. 2008. Numerical Geometry of Non-rigid Shapes. Springer Science 8 Business Media.
    18. Oscar P. Bruno and Stéphane K. Lintner. 2013. A high-order integral solver for scalar problems of diffraction by screens and apertures in three-dimensional space. Journal of Computational Physics 252 (2013), 250–274.
    19. John Canny and John Reif. 1987. New lower bound techniques for robot motion planning problems. In Annual Symposium on Foundations of Computer Science. 49–60.
    20. Xiaobai Chen, Aleksey Golovinskiy, and Thomas Funkhouser. 2009. A benchmark for 3D mesh segmentation. In Transactions on Graphics, Vol. 28. ACM, 73.
    21. Ming Chuang, Linjie Luo, Benedict J. Brown, Szymon Rusinkiewicz, and Michael Kazhdan. 2009. Estimating the laplace–Beltrami operator by restricting 3D functions. In Computer Graphics Forum, Vol. 28. 1475–1484.
    22. Ronald R. Coifman and Stéphane Lafon. 2006. Diffusion maps. Applied and Computational Harmonic Analysis 21, 1 (2006), 5–30.
    23. Bruno Colbois, Ahmad El Soufi, and Alexandre Girouard. 2011. Isoperimetric control of the Steklov spectrum. Journal of Functional Analysis 261, 5 (2011), 1384–1399.
    24. Etienne Corman, Justin Solomon, Mirela Ben-Chen, Leonidas Guibas, and Maks Ovsjanikov. 2017. Functional characterization of intrinsic and extrinsic geometry. Transactions on Graphics 36, 2 (March 2017), 14:1–14:17.
    25. Keenan Crane, Clarisse Weischedel, and Max Wardetzky. 2013. Geodesics in heat: A new approach to computing distance based on heat flow. Transactions on Graphics 32, 5 (2013), 152.
    26. Johannes J. Duistermaat and Victor W. Guillemin. 1975. The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Mathematicae 29 (1975), 39–80.
    27. Nader Engheta, William D. Murphy, Vladimir Rokhlin, and Marius S. Vassiliou. 1992. The fast multipole method (FMM) for electromagnetic scattering problems. Transactions on Antennas and Propagation 40, 6 (1992), 634–641.
    28. Ming Gao, Nathan Mitchell, and Eftychios Sifakis. 2014. Steklov-Poincaré skinning. In Proc. Symposium on Computer Animation. 139–148.
    29. Alexandre Girouard, Jean Lagacé, Iosif Polterovich, and Alessandro Savo. 2017. The Steklov spectrum of cuboids. arXiv preprint arXiv:1711.03075 (2017).
    30. Alexandre Girouard and Iosif Polterovich. 2017. Spectral geometry of the Steklov problem. Journal of Spectral Theory 7, 2 (2017), 321–359.
    31. Gerd Grubb. 2009. Pseudodifferential methods for boundary value problems. Distributions and operators. Springer Science and Business Media, 305–333.
    32. Wolfgang Hackbusch. 1999. A sparse matrix arithmetic based on H -matrices. Part I: Introduction to H-matrices. Computing 62, 2 (1999), 89–108.
    33. Shoji Hamada. 2011. GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method. Computer Physics Communications 182, 5 (2011), 1162–1168.
    34. Klaus Hildebrandt, Christian Schulz, Christoph von Tycowicz, and Konrad Polthier. 2010. Eigenmodes of surface energies for shape analysis.. In Proceedings of Geometric Modeling and Processing (GMP’10). Springer, 296–314.
    35. Hugues Hoppe. 1996. Progressive meshes. In Proc. SIGGRAPH. 99–108.
    36. Hajime Igarashi and Toshihisa Honma. 1996. A boundary element method for potential fields with corner singularities. Applied Mathematical Modelling 20, 11 (1996), 847–852.
    37. John David Jackson. 1998. Classical electrodynamics. In Classical Electrodynamics, 3rd ed., John David Jackson (Ed.). Wiley-VCH, 832.
    38. Alec Jacobson. 2013. Algorithms and Interfaces for Real-time Deformation of 2d and 3d Shapes. Ph.D. Dissertation. ETH Zurich.
    39. Alec Jacobson, Ladislav Kavan, and Olga Sorkine-Hornung. 2013. Robust inside-outside segmentation using generalized winding numbers. ACM Transactions on Graphics (TOG) 32, 4 (2013), 33.
    40. Alexander Kachalov, Yaroslav Kurylev, and Matti Lassas. 2001. Inverse Boundary Spectral Problems. CRC Press.
    41. Ram P. Kanwal. 2013. Linear Integral Equations. Springer.
    42. Andrew V. Knyazev. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM Journal on Scientific Computing 23, 2 (2001), 517–541.
    43. Iasonas Kokkinos, Michael M. Bronstein, Roee Litman, and Alex M. Bronstein. 2012. Intrinsic shape context descriptors for deformable shapes. In Proceedings of the Computer Vision and Pattern Recognition (CVPR’12). 159–166.
    44. John M. Lee and Gunther Uhlmann. 1989. Determining anisotropic real-analytic conductivities by boundary measurements. Communications on Pure and Applied Mathematics 42, 8 (1989), 1097–1112.
    45. Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher. 2017. Sloshing, Steklov and corners I: Asymptotics of sloshing eigenvalues. arXiv Preprint arXiv:1709.01891 (2017).
    46. Jian Liang, Rongjie Lai, Tsz Wai Wong, and Hongkai Zhao. 2012. Geometric understanding of point clouds using Laplace–Beltrami operator. In Proceedings of the Computer Vision and Pattern Recognition (CVPR’12). 214–221.
    47. Stéphane K. Lintner and Oscar P. Bruno. 2015. A generalized Calderón formula for open-arc diffraction problems: Theoretical considerations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 145, 2 (2015), 331–364.
    48. Yaron Lipman, Raif M. Rustamov, and Thomas A. Funkhouser. 2010. Biharmonic distance. Transactions on Graphics 29, 3 (2010), 27.
    49. Roee Litman, Alexander M. Bronstein, and Michael M. Bronstein. 2012. Stable volumetric features in deformable shapes. Computers 8 Graphics 36, 5 (2012), 569–576.
    50. Haixiang Liu, Nathan Mitchell, Mridul Aanjaneya, and Eftychios Sifakis. 2016. A scalable Schur-complement fluids solver for heterogeneous compute platforms. Transactions on Graphics 35, 6 (2016), 201.
    51. Hsueh-Ti Derek Liu, Alec Jacobson, and Keenan Crane. 2017. A Dirac operator for extrinsic shape analysis. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 139–149.
    52. Robert Osada, Thomas Funkhouser, Bernard Chazelle, and David Dobkin. 2002. Shape distributions. Transactions on Graphics 21, 4 (2002), 807–832.
    53. Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas. 2012. Functional maps: A flexible representation of maps between shapes. Transactions on Graphics 31, 4 (2012), 30.
    54. Giuseppe Patané. 2015. Volumetric heat kernel: Padé-Chebyshev approximation, convergence, and computation. Computers 8 Graphics 46 (2015), 64–71.
    55. Giuseppe Patané. 2016. Laplacian spectral kernels and distances for geometry processing and shape analysis. In Computer Graphics Forum, Vol. 35. 599–624.
    56. Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 1 (1993), 15–36.
    57. Iosif Polterovich and David A. Sher. 2015. Heat invariants of the Steklov problem. Journal of Geometric Analysis 25, 2 (2015), 924–950.
    58. Alfio Quarteroni and Alberto Valli. 1999. Domain Decomposition Methods for Partial Differential Equations. Oxford.
    59. Dan Raviv, Michael M. Bronstein, Alexander M. Bronstein, and Ron Kimmel. 2010. Volumetric heat kernel signatures. In Proceedings of the Workshop on 3D Object Retrieval (3DOR’10). 39–44.
    60. Martin Reuter, Silvia Biasotti, Daniela Giorgi, Giuseppe Patanè, and Michela Spagnuolo. 2009. Discrete Laplace–Beltrami operators for shape analysis and segmentation. Computers 8 Graphics 33, 3 (2009), 381–390.
    61. Martin Reuter, Franz-Erich Wolter, and Niklas Peinecke. 2006. Laplace–Beltrami spectra as “Shape-DNA” of surfaces and solids. Computer-Aided Design 38, 4 (2006), 342–366.
    62. Vladimir Rokhlin. 1985. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics 60, 2 (1985), 187–207.
    63. Raif M. Rustamov. 2011. Interpolated eigenfunctions for volumetric shape processing. Visual Computer 27, 11 (2011), 951–961.
    64. Raif M. Rustamov, Maks Ovsjanikov, Omri Azencot, Mirela Ben-Chen, Frédéric Chazal, and Leonidas Guibas. 2013. Map-based exploration of intrinsic shape differences and variability. Transactions on Graphics 32, 4 (2013), 72.
    65. Rohan Sawhney and Keenan Crane. 2017. Boundary first flattening. ACM Transactions on Graphics (TOG) 37, 1 (2017), 5.
    66. Wojciech Śmigaj, Timo Betcke, Simon Arridge, Joel Phillips, and Martin Schweiger. 2015. Solving boundary integral problems with BEM++. Transactions on Mathematical Software 41, 2 (2015), 6.
    67. Barry Smith, Petter Bjorstad, and William Gropp. 2004. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge.
    68. Olga Sorkine. 2005. Laplacian mesh processing. In Eurographics State of the Art Report.
    69. Olaf Steinbach. 2007. Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer.
    70. Olaf Steinbach and Wolfgang L. Wendland. 1998. The construction of some efficient preconditioners in the boundary element method. Advances in Computational Mathematics 9, 1–2 (1998), 191–216.
    71. Mark J. Stock and Adrin Gharakhani. 2010. A GPU-accelerated boundary element method and vortex particle method. In AIAA 40th Fluid Dynamics Conference and Exhibit, Vol. 1.
    72. Martin Stoll and A. J. Wathen. 2007. The Bramble–Pasciak Preconditioner for Saddle Point Problems. Technical Report. University of Oxford.
    73. Jian Sun, Maks Ovsjanikov, and Leonidas Guibas. 2009. A concise and provably informative multi-scale signature based on heat diffusion. In Computer Graphics Forum, Vol. 28. 1383–1392.
    74. Toru Takahashi and Tsuyoshi Hamada. 2009. GPU-accelerated boundary element method for Helmholtz equation in three dimensions. International Journal for Numerical Methods in Engineering 80, 10 (2009), 1295–1321.
    75. Federico Tombari, Samuele Salti, and Luigi Di Stefano. 2010. Unique signatures of histograms for local surface description. In Proceedings of the European Conference on Computer Vision (ECCV’10). 356–369.
    76. Andrea Toselli and Olof B. Widlund. 2005. Domain Decomposition Methods: Algorithms and Theory. Vol. 34. Springer.
    77. Bruno Vallet and Bruno Lévy. 2008. Spectral geometry processing with manifold harmonics. In Computer Graphics Forum, Vol. 27. 251–260.
    78. Amir Vaxman, Mirela Ben-Chen, and Craig Gotsman. 2010. A multi-resolution approach to heat kernels on discrete surfaces. Transactions on Graphics 29, 4 (2010), 121.
    79. Gang Wang and Yalin Wang. 2015. Multi-scale heat kernel based volumetric morphology signature. In International Conference on Medical Image Computing and Computer-Assisted Intervention. 751–759.
    80. Hao Wang, Tong Lu, Oscar Kin-Chung Au, and Chiew-Lan Tai. 2014. Spectral 3D mesh segmentation with a novel single segmentation field. Graphical Models 76, 5 (2014), 440–456.
    81. Kenong Wu and Martin D. Levine. 1997. 3D part segmentation using simulated electrical charge distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 11 (1997), 1223–1235.
    82. Rio Yokota, Jaydeep P. Bardhan, Matthew G. Knepley, Lorena A. Barba, and Tsuyoshi Hamada. 2011. Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUs and a billion unknowns. Computer Physics Communications 182, 6 (2011), 1272–1283.
    83. Hao Zhang, Oliver Van Kaick, and Ramsay Dyer. 2010. Spectral mesh processing. In Computer Graphics Forum, Vol. 29. 1865–1894.
    84. Qingnan Zhou, Eitan Grinspun, Denis Zorin, and Alec Jacobson. 2016. Mesh arrangements for solid geometry. Transactions on Graphics 35, 4 (July 2016), 39:1–39:15.

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