Functional Characterization of Deformation Fields

Abstract

In this article, we present a novel representation for deformation fields of 3D shapes, by considering the induced changes in the underlying metric. In particular, our approach allows one to represent a deformation field in a coordinate-free way as a linear operator acting on real-valued functions defined on the shape. Such a representation provides both a way to relate deformation fields to other classical functional operators and enables analysis and processing of deformation fields using standard linear-algebraic tools. This opens the door to a wide variety of applications such as explicitly adding extrinsic information into the computation of functional maps, intrinsic shape symmetrization, joint deformation design through precise control of metric distortion, and coordinate-free deformation transfer without requiring pointwise correspondences. Our method is applicable to both surface and volumetric shape representations and we guarantee the equivalence between the operator-based and standard deformation field representation under mild genericity conditions in the discrete setting. We demonstrate the utility of our approach by comparing it with existing techniques and show how our representation provides a powerful toolbox for a wide variety of challenging problems.

References

1. Bart Adams, Maks Ovsjanikov, Michael Wand, Hans-Peter Seidel, and Leonidas J. Guibas. 2008. Meshless modeling of deformable shapes and their motion. In Proceedings of the 2008 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Eurographics Association, 77--86.
2. 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.
3. Omri Azencot, Etienne Corman, Mirela Ben-Chen, and Maks Ovsjanikov. 2017. Consistent functional cross field design for mesh quadrangulation. ACM Transactions on Graphics (TOG) 36, 4 (2017), 92.
4. Omri Azencot, Maks Ovsjanikov, Frédéric Chazal, and Mirela Ben-Chen. 2015. Discrete derivatives of vector fields on surfaces—An operator approach. ACM Transactions on Graphics (TOG) 34, 3 (2015), 29.
5. Ilya Baran, Daniel Vlasic, Eitan Grinspun, and Jovan Popović. 2009. Semantic deformation transfer. In ACM Transactions on Graphics (TOG) 28, 3 (August 2009), Article 36.
6. Alexander I. Bobenko, Helmut Pottmann, and Johannes Wallner. 2010. A curvature theory for discrete surfaces based on mesh parallelity. Mathematische Annalen 348, 1 (2010), 1--24.
7. Federica Bogo, Javier Romero, Matthew Loper, and Michael J. Black. 2014. FAUST: Dataset and evaluation for 3D mesh registration. In Proc. CVPR. IEEE.
8. David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-integer quadrangulation. In ACM Transactions on Graphics (TOG) 28, 3 (August 2009), Article 77.
9. 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. Wiley Online Library, 265--274.
10. Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Lévy. 2010. Polygon Mesh Processing. CRC Press.
11. Mario Botsch and Olga Sorkine. 2008. On linear variational surface deformation methods. IEEE Transactions on Visualization and Computer Graphics 14, 1 (2008), 213--230.
12. A. Bronstein, M. Bronstein, and R. Kimmel. 2007. Numerical Geometry of Non-rigid Shapes. Springer-Verlag.
13. George Celniker and Dave Gossard. 1991. Deformable curve and surface finite-elements for free-form shape design. In ACM SIGGRAPH Computer Graphics, Vol. 25. ACM, 257--266.
14. Philippe G. Ciarlet. 2000. Theory of Shells. Vol. 3. Elsevier.
15. Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2011. Spin transformations of discrete surfaces. In ACM Transactions on Graphics (TOG) 30, 4 (July 2011), Article 104.
16. Tamal K. Dey, Pawas Ranjan, and Yusu Wang. 2012. Eigen deformation of 3D models. The Visual Computer 28, 6--8 (2012), 585--595.
17. M. P. do Carmo. 2013. Riemannian Geometry. Birkhäuser, Boston.
18. Hans-Christian Ebke, David Bommes, Marcel Campen, and Leif Kobbelt. 2013. QEx: Robust quad mesh extraction. ACM Transactions on Graphics (TOG) 32, 6 (2013), 168.
19. Ilya Eckstein, J.-P. Pons, Yiying Tong, C.-C. J. Kuo, and Mathieu Desbrun. 2007. Generalized surface flows for mesh processing. In Proc. SGP. Eurographics Association, 183--192.
20. Michael Eigensatz and Mark Pauly. 2009. Positional, metric, and curvature control for constraint-based surface deformation. In Computer Graphics Forum, Vol. 28. Wiley Online Library, 551--558.
21. Michael Grant and Stephen Boyd. 2014. CVX: Matlab Software for Disciplined Convex Programming, version 2.1. http://cvxr.com/cvx.
22. Igor Guskov, Wim Sweldens, and Peter Schröder. 1999. Multiresolution signal processing for meshes. In Proc. SIGGRAPH. 325--334.
23. Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky. 2016. A discrete parametrized surface theory in R³ . International Mathematics Research Notices (2016).
24. Martin Kilian, Niloy J. Mitra, and Helmut Pottmann. 2007. Geometric modeling in shape space. ACM Transactions on Graphics (TOG) 26, 3 (2007), 64.
25. Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally optimal direction fields. ACM Transactions on Graphics (TOG) 32, 4 (2013), 59.
26. Leif Kobbelt, Swen Campagna, Jens Vorsatz, and Hans-Peter Seidel. 1998. Interactive multi-resolution modeling on arbitrary meshes. In Proc. SIGGRAPH. 105--114.
27. Artiom Kovnatsky, Michael M. Bronstein, Alexander M. Bronstein, Klaus Glashoff, and Ron Kimmel. 2013. Coupled quasi-harmonic bases. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 439--448.
28. Bruno Lévy. 2006. Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In Proceedings of the IEEE International Conference on Shape Modeling and Applications.
29. Yaron Lipman, Olga Sorkine, Daniel Cohen-Or, David Levin, Claudio Rossi, and Hans-Peter Seidel. 2004. Differential coordinates for interactive mesh editing. In Shape Modeling Applications. 181--190.
30. Or Litany, Emanuele Rodolà, Alex M. Bronstein, and Michael M. Bronstein. 2017. Fully spectral partial shape matching. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 247--258.
31. Beibei Liu, Yiying Tong, Fernando De Goes, and Mathieu Desbrun. 2016. Discrete connection and covariant derivative for vector field analysis and design. ACM Transactions on Graphics (TOG) 35, 3 (2016), 23.
32. J. Martinez Esturo, C. Rössl, and H. Theisel. 2013. Generalized metric energies for continuous shape deformation. Springer LNCS (Proc. Curves and Surfaces 2012) 8177, 1 (2013), 135--157.
33. Matthias Müller, Nuttapong Chentanez, Tae-Yong Kim, and Miles Macklin. 2014. Strain based dynamics. In Proc. SCA. 149--157.
34. Andrew Nealen, Matthias Müller, Richard Keiser, Eddy Boxerman, and Mark Carlson. 2006. Physically based deformable models in computer graphics. In Computer Graphics Forum, Vol. 25. Wiley Online Library, 809--836.
35. Andrew Nealen, Olga Sorkine, Marc Alexa, and Daniel Cohen-Or. 2005. A sketch-based interface for detail-preserving mesh editing. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH) 24, 3 (2005), 1142--1147.
36. Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas. 2012. Functional maps: A flexible representation of maps between shapes. ACM Transactions on Graphics (TOG) 31, 4 (2012), 30.
37. Maks Ovsjanikov, Etienne Corman, Michael Bronstein, Emanuele Rodolà, Mirela Ben-Chen, Leonidas Guibas, Frederic Chazal, and Alex Bronstein. 2016. Computing and processing correspondences with functional maps. In SIGGRAPH ASIA 2016 Courses. ACM, 9.
38. Daniele Panozzo, Yaron Lipman, Enrico Puppo, and Denis Zorin. 2012. Fields on symmetric surfaces. ACM Transactions on Graphics (TOG) 31, 4 (2012), 111.
39. Daniele Panozzo, Enrico Puppo, Marco Tarini, and Olga Sorkine-Hornung. 2014. Frame fields: Anisotropic and non-orthogonal cross fields. Proc. ACM SIGGRAPH) 33, 4 (2014), 134:1--134:11.
40. Nikolas Paries, Patrick Degener, and Reinhard Klein. 2007. Simple and efficient mesh editing with consistent local frames. In Proceedings of the 15th Pacific Conference on Computer Graphics and Applications (PG’07). IEEE, 461--464.
41. Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E. Marsden, and Mathieu Desbrun. 2011. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena 240, 6 (2011), 443--458.
42. Jonathan Pokrass, Alexander M. Bronstein, Michael M. Bronstein, Pablo Sprechmann, and Guillermo Sapiro. 2013. Sparse modeling of intrinsic correspondences. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 459--468.
43. Martin Rumpf and Max Wardetzky. 2014. Geometry processing from an elastic perspective. GAMM-Mitteilungen 37, 2 (2014), 184--216.
44. 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. ACM Trans. Graph. 32, 4 (2013), 72:1--72:12.
45. Matan Sela, Yonathan Aflalo, and Ron Kimmel. 2015. Computational caricaturization of surfaces. CVIU 141 (2015), 1--17.
46. Justin Solomon, Mirela Ben-Chen, Adrian Butscher, and Leonidas Guibas. 2011. As-killing-as-possible vector fields for planar deformation. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 1543--1552.
47. Olga Sorkine and Marc Alexa. 2007. As-rigid-as-possible surface modeling. In Proc. SGP. 109--116.
48. Olga Sorkine, Daniel Cohen-Or, Yaron Lipman, Marc Alexa, Christian Rössl, and H.-P. Seidel. 2004. Laplacian surface editing. In Proc. SGP. ACM, 175--184.
49. Robert W. Sumner and Jovan Popović. 2004. Deformation transfer for triangle meshes. In ACM Transactions on Graphics (TOG) 23, 3 (August 2004). 399--405.
50. Demetri Terzopoulos, John Platt, Alan Barr, and Kurt Fleischer. 1987. Elastically deformable models. In ACM Siggraph Computer Graphics 21, 4 (July 1987), 205--214.
51. Bernhard Thomaszewski, Simon Pabst, and Wolfgang Strasser. 2009. Continuum-based strain limiting. In Computer Graphics Forum, Vol. 28. Wiley Online Library, 569--576.
52. Federico Tombari, Samuele Salti, and Luigi Di Stefano. 2010. Unique signatures of histograms for local surface description. In European Conference on Computer Vision. Springer, 356--369.
53. Bruno Vallet and Bruno Lévy. 2008. Spectral geometry processing with manifold harmonics. Computer Graphics Forum 27, 2 (2008), 251--260.
54. Amir Vaxman, Christian Müller, and Ofir Weber. 2015. Conformal mesh deformations with Möbius transformations. ACM Transactions on Graphics (TOG) 34, 4 (2015), 55.
55. Wolfram von Funck, Holger Theisel, and Hans-Peter Seidel. 2006. Vector field based shape deformations. In Proc. SIGGRAPH, Vol. 25. 1118--1125.
56. Christoph Von-Tycowicz, Christian Schulz, Hans-Peter Seidel, and Klaus Hildebrandt. 2015. Real-time nonlinear shape interpolation. ACM Transactions on Graphics (TOG) 34, 3 (2015), 34.
57. William Welch and Andrew Witkin. 1992. Variational surface modeling. In ACM SIGGRAPH Computer Graphics, Vol. 26. ACM, 157--166.
58. Mengxiao Yin, Guiqing Li, Huina Lu, Yaobin Ouyang, Zhibang Zhang, and Chuhua Xian. 2015. Spectral pose transfer. Computer Aided Geometric Design 35 (2015), 82--94.
59. Yizhou Yu, Kun Zhou, Dong Xu, Xiaohan Shi, Hujun Bao, Baining Guo, and Heung-Yeung Shum. 2004. Mesh editing with poisson-based gradient field manipulation. ACM Transactions on Graphics (TOG) 23, 3 (2004), 644--651.
60. Rhaleb Zayer, Christian Rössl, Zachi Karni, and Hans-Peter Seidel. 2005. Harmonic guidance for surface deformation. In Computer Graphics Forum, Vol. 24. Wiley Online Library, 601--609.
61. Zhibang Zhang, Guiqing Li, Huina Lu, Yaobin Ouyang, Mengxiao Yin, and Chuhua Xian. 2015. Fast as-isometric-as-possible shape interpolation. Computers 8 Graphics 46 (2015), 244--256.
62. Qian Zheng, Zhuming Hao, Hui Huang, Kai Xu, Hao Zhang, Daniel Cohen-Or, and Baoquan Chen. 2015. Skeleton-intrinsic symmetrization of shapes. In Computer Graphics Forum, Vol. 34. Wiley Online Library, 275--286.
63. Denis Zorin, Peter Schröder, and Wim Sweldens. 1997. Interactive multiresolution mesh editing. In Proc. SIGGRAPH. 259--268.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Shape analysis

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Discretization