“Consistent functional cross field design for mesh quadrangulation”

  • ©Omri Azencot, Étienne Corman, Mirela (Miri) Ben-Chen, and Maks Ovsjanikov

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Title:

    Consistent functional cross field design for mesh quadrangulation

Session/Category Title: Global Parameterization


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Abstract:


    We propose a novel technique for computing consistent cross fields on a pair of triangle meshes given an input correspondence, which we use as guiding fields for approximately consistent quadrangulations. Unlike the majority of existing methods our approach does not assume that the meshes share the same connectivity or even have the same number of vertices, and furthermore does not place any restrictions on the topology (genus) of the shapes. Importantly, our method is robust with respect to small perturbations of the given correspondence, as it only relies on the transportation of real-valued functions and thus avoids the costly and error-prone estimation of the map differential. Key to this robustness is a novel formulation, which relies on the previously-proposed notion of power vectors, and we show how consistency can be enforced without pre-alignment of local basis frames, in which these power vectors are computed. We demonstrate that using the same formulation we can both compute a quadrangulation that would respect a given symmetry on the same shape or a map across a pair of shapes. We provide quantitative and qualitative comparison of our method with several baselines and show that it both provides more accurate results and allows to handle more general cases than existing techniques.

References:


    1. Noam Aigerman, Roi Poranne, and Yaron Lipman. 2015. Seamless surface mappings. ACM Transactions on Graphics (TOG) 34, 4 (2015), 72.Google ScholarDigital Library
    2. Dragomir Anguelov, Praveen Srinivasan, Daphne Koller, Sebastian Thrun, Jim Rodgers, and James Davis. 2005. SCAPE: shape completion and animation of people. In ACM Transactions on Graphics (TOG), Vol. 24. ACM, 408–416. Google ScholarDigital Library
    3. 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. Wiley Online Library, 73–82. Google ScholarDigital Library
    4. David Bommes, Bruno Lévy, Nico Pietroni, Enrico Puppo, Claudio Silva, Marco Tarini, and Denis Zorin. 2013. Quad-Mesh Generation and Processing: A Survey. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 51–76. Google ScholarDigital Library
    5. David Bommes, Henrik Zimmer, and Leif Kobbelt. 2009. Mixed-integer quadrangulation. ACM Transactions On Graphics (TOG) 28, 3 (2009), 77.Google ScholarDigital Library
    6. Olga Diamanti, Amir Vaxman, Daniele Panozzo, and Olga Sorkine-Hornung. 2014. Designing N-PolyVector Fields with Complex Polynomials. In Computer Graphics Forum, Vol. 33. Wiley Online Library, 1–11.Google ScholarDigital Library
    7. 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.Google ScholarDigital Library
    8. Theodore Frankel. 2011. The geometry of physics: an introduction. Cambridge University Press. Google ScholarCross Ref
    9. A Jacobson, D Panozzo, C Schüller, O Diamanti, Q Zhou, N Pietroni, and others. 2013. libigl: A simple C++ geometry processing library. (2013).Google Scholar
    10. Vladimir G Kim, Yaron Lipman, and Thomas Funkhouser. 2011. Blended intrinsic maps. In ACM Transactions on Graphics (TOG), Vol. 30. ACM, 79.Google ScholarDigital Library
    11. Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2013. Globally optimal direction fields. ACM Transactions on Graphics (TOG) 32, 4 (2013), 59.Google ScholarDigital Library
    12. O. Litany, E. Rodolà, A. M. Bronstein, M. M. Bronstein, and D. Cremers. 2016. Non-Rigid Puzzles. Computer Graphics Forum (Proc. SGP) 35, 5 (2016), 135–143.Google ScholarDigital Library
    13. Giorgio Marcias, Nico Pietroni, Daniele Panozzo, Enrico Puppo, and Olga Sorkine-Hornung. 2013. Animation-Aware Quadrangulation. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 167–175. Google ScholarDigital Library
    14. Giorgio Marcias, Kenshi Takayama, Nico Pietroni, Daniele Panozzo, Olga Sorkine-Hornung, Enrico Puppo, and Paolo Cignoni. 2015. Data-driven interactive quadrangulation. ACM Transactions on Graphics (TOG) 34, 4 (2015), 65.Google ScholarDigital Library
    15. Min Meng and Ying He. 2016. Consistent quadrangulation for shape collections via feature line co-extraction. Computer-Aided Design 70 (2016), 78 — 88. {SPM} 2015.Google ScholarDigital Library
    16. Shigeyuki Morita. 2001. Geometry of differential forms. Vol. 201. American Mathematical Soc.Google Scholar
    17. 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.Google ScholarDigital Library
    18. 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. Article 9, 60 pages. Google ScholarDigital Library
    19. Daniele Panozzo, Yaron Lipman, Enrico Puppo, and Denis Zorin. 2012. Fields on symmetric surfaces. ACM Trans. Graph. 31, 4 (2012), 111–1. Google ScholarDigital Library
    20. J. Pokrass, A. M. Bronstein, M. M. Bronstein, P. Sprechmann, and G. Sapiro. 2013. Sparse Modeling of Intrinsic Correspondences. Computer Graphics Forum 32, 2pt4 (2013), 459–468. Google ScholarCross Ref
    21. Nicolas Ray, Wan Chiu Li, Bruno Lévy, Alla Sheffer, and Pierre Alliez. 2006. Periodic global parameterization. ACM Transactions on Graphics (TOG) 25, 4 (2006), 1460–1485. Google ScholarDigital Library
    22. Nicolas Ray, Bruno Vallet, Laurent Alonso, and Bruno Levy. 2009. Geometry-aware direction field processing. ACM Transactions on Graphics (TOG) 29, 1 (2009), 1.Google ScholarDigital Library
    23. Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy 2008. N-symmetry direction field design. ACM Transactions on Graphics (TOG) 27, 2 (2008), 10.Google ScholarDigital Library
    24. E. Rodolà, M. Moeller, and D. Cremers. 2015. Point-wise Map Recovery and Refinement from Functional Correspondence. In Proc. Vision, Modeling and Visualization (VMV).Google Scholar
    25. Szymon Rusinkiewicz. 2004. Estimating curvatures and their derivatives on triangle meshes. In 3D Data Processing, Visualization and Transmission, 2004. 3DPVT 2004. Proceedings. 2nd International Symposium on. IEEE, 486–493. Google ScholarCross Ref
    26. Julien Tierny, Joel Daniels II, Luis G. Nonato, Valerio Pascucci, and Claudio T. Silva. 2011. Inspired quadrangulation. Computer-Aided Design 43, 11 (2011), 1516 — 1526. Solid and Physical Modeling 2011.Google ScholarDigital Library
    27. Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David Bommes, Klaus Hildebrandt, and Mirela Ben-Chen. 2016. Directional Field Synthesis, Design, and Processing. Computer Graphics Forum (2016).Google Scholar
    28. Chih-Yuan Yao, Hung-Kuo Chu, Tao Ju, and Tong-Yee Lee. 2009. Compatible quadrangulation by sketching. Computer Animation and Virtual Worlds 20, 2–3 (2009), 101–109.Google ScholarCross Ref


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