Exploring the Geometry of the Space of Shells
We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time‐ and space‐discrete geodesic calculus. In particular we show how to...
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| Veröffentlicht in: | Computer graphics forum Jg. 33; H. 5; S. 247 - 256 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Oxford
Blackwell Publishing Ltd
01.08.2014
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time‐ and space‐discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space. |
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| AbstractList | We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time- and space-discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space. [PUBLICATION ABSTRACT] We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time‐ and space‐discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space. |
| Author | Wardetzky, M. Rumpf, M. Schröder, P. Heeren, B. Wirth, B. |
| Author_xml | – sequence: 1 givenname: B. surname: Heeren fullname: Heeren, B. organization: Institute for Numerical Simulation, University of Bonn, Germany – sequence: 2 givenname: M. surname: Rumpf fullname: Rumpf, M. organization: Institute for Numerical Simulation, University of Bonn, Germany – sequence: 3 givenname: P. surname: Schröder fullname: Schröder, P. organization: Caltech, USA – sequence: 4 givenname: M. surname: Wardetzky fullname: Wardetzky, M. organization: Institute of Num. and Appl. Math, University of Göttingen, Germany – sequence: 5 givenname: B. surname: Wirth fullname: Wirth, B. organization: Institute for Computational and Applied Mathematics, University of Münster, Germany |
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| Cites_doi | 10.1111/j.1467-8659.2011.01974.x 10.1109/TPAMI.2011.233 10.4171/JEMS/37 10.1002/nme.958 10.1109/ISBI.2002.1029263 10.1515/9781400881802 10.1111/j.1467-8659.2006.00974.x 10.1145/1276377.1276457 10.1023/A:1026473418439 10.1007/s11263-006-0635-2 10.1109/TVCG.2007.1054 10.1145/2019627.2019638 10.3934/jgm.2011.3.389 10.1111/j.1467-8659.2012.03180.x 10.1145/37402.37427 10.1111/j.1467-8659.2009.01600.x 10.1145/1015706.1015759 10.1109/TMI.2004.831793 10.1007/s11263-010-0416-9 10.1109/TVCG.2008.103 10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P 10.1007/s10851-009-0156-z |
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| References_xml | – reference: Kheyfets A., Miller W.A., Newton G.A.: Schild's ladder parallel transport procedure for an arbitrary connection. Internat. J. Theoret. Phys. 39, 12 (2000), 2891-2898. 5 – reference: Michor P.W., Mumford D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8 (2006), 1-48. 2 – reference: Zhang L., Snavely N., Curless B., Seitz S.M.: Spacetime faces: High resolution capture for modeling and animation. In ACM Tran. Graph. (2004), vol. 23, pp. 548-558. 8 – reference: Fletcher P., Lu C., Pizer S., Joshi S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imag. 23, 8 (2004), 995-1005. 2 – reference: Heeren B., Rumpf M., Wardetzky M., Wirth B.: Time-discrete geodesics in the space of shells. Comp. Graph. Forum 31, 5 (2012), 1755-1764. 2, 3, 4 – reference: Kilian M., Mitra N.J., Pottmann H.: Geometric modeling in shape space. In ACM Tran. Graph. (2007), vol. 26, pp. 1-8. 2, 5 – reference: Strutt J.: Theory of sound: Vol. 2. Dover Publications, 1945. 4 – reference: Zienkiewicz O.C., Taylor R.L.: The finite element method: The basis, 5th ed., vol. 1. Butterworth and Heinemann, 2000. 2, 4 – reference: Hildebrandt K., Schulz C., Tycowicz C. v., Polthier K.: Interactive surface modeling using modal analysis. ACM Trans. Graph. 30, 5 (2011), 119:1-11. 2, 7 – reference: Lew A., Marsden M., Oritz M., West M.: Variational time integrators. Int. J. Numer. Meth. Engng. 60 (2004), 153-212. 3 – reference: Grinspun E., Gingold Y., Reisman J., Zorin D.: Computing discrete shape operators on general meshes. Comp. Graph. Forum 25, 3 (2006), 547-556. 2 – reference: Fröhlich S., Botsch M.: Example-driven deformations based on discrete shells. Comp. Graph. Forum 30, 8 (2011), 2246-2257. 2, 5, 6 – reference: Winkler T., Drieseberg J., Alexa M., Hormann K.: Multi-scale geometry interpolation. Comp. Graph. Forum 29, 2 (2010), 309-318. 2, 6 – reference: Kurtek S., Klassen E., Gore J., Ding Z., Srivastava A.: Elastic geodesic paths in shape space of parametrized surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 34, 9 (2012), 1717-1730. 3 – reference: Wirth B., Bar L., Rumpf M., Sapiro G.: A continuum mechanical approach to geodesics in shape space. Inter. J. Comp. Vision 93, 3 (2011), 293-318. 2 – reference: Milnor J.: Morse Theory. Princeton University Press, 1963. 3 – reference: Ciarlet P.G.: Mathematical Elasticity, Vol III: Theory of Shells. North-Holland, 2000. 3 – reference: Bauer M., Harms P., Michor P.W.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3, 4 (2011), 389-438. 3 – reference: Fuchs M., Jüttler B., Scherzer O., Yang H.: Shape metrics based on elastic deformations. J. Math. Imaging Vis. 35, 1 (2009), 86-102. 2 – reference: Sundaramoorthi G., Yezzi A., Mennucci A.: Sobolev active contours. Inter. J. Comp. Vision 73, 3 (2007), 345-366. 2 – reference: Tycowicz C. v., Schulz C., Seidel H.-P, Hildebrandt K.: An efficient construction of reduced deformable objects. ACM Trans. Graph. 32, 6 (2013). 2, 7 – reference: Jin M., Zeng W., Luo F., Gu X.: Computing Teichmüller shape space. IEEE Trans. Vis. Comp. Graph. 15, 3 (2009), 504-517. 3 – reference: Botsch M., Sorkine O.: On linear variational surface deformation methods. IEEE Trans. Vis. Comp. Graph. 14 (2008), 213-230. 2 – reference: Hughes T. J. R.: Finite Element Method - Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, 1987. 2, 4 – start-page: 11 year: 2006 end-page: 20 – volume: 32 issue: 6 year: 2013 article-title: An efficient construction of reduced deformable objects publication-title: ACM Trans. Graph – volume: 35 start-page: 86 issue: 1 year: 2009 end-page: 102 article-title: Shape metrics based on elastic deformations publication-title: J. Math. Imaging Vis – volume: 31 start-page: 1755 issue: 5 year: 2012 end-page: 1764 article-title: Time‐discrete geodesics in the space of shells publication-title: Comp. Graph. Forum – volume: 2 start-page: 309 year: 2010 end-page: 318 article-title: Multi‐scale geometry interpolation publication-title: Comp. Graph. Forum 29 – volume: 73 start-page: 345 issue: 3 year: 2007 end-page: 366 article-title: Sobolev active contours publication-title: Inter. J. Comp. Vision – volume: 26 start-page: 1 year: 2007 end-page: 8 article-title: Geometric modeling in shape space publication-title: ACM Tran. Graph – volume: 23 start-page: 995 issue: 8 year: 2004 end-page: 1005 article-title: Principal geodesic analysis for the study of nonlinear statistics of shape publication-title: IEEE Trans. Med. Imag – year: 1987 – volume: 8 start-page: 1 year: 2006 end-page: 48 article-title: Riemannian geometries on spaces of plane curves publication-title: J. Eur. Math. Soc – start-page: 341 year: 2002 end-page: 344 – year: 2000 – volume: 60 start-page: 153 year: 2004 end-page: 212 article-title: Variational time integrators publication-title: Int. J. Numer. Meth. Engng – volume: 171 year: 2010 – volume: 25 start-page: 547 issue: 3 year: 2006 end-page: 556 article-title: Computing discrete shape operators on general meshes publication-title: Comp. Graph. Forum – volume: 31 year: 2006 – start-page: 62 year: 2003 end-page: 67 – volume: 15 start-page: 504 issue: 3 year: 2009 end-page: 517 article-title: Computing Teichmüller shape space publication-title: IEEE Trans. Vis. Comp. Graph – volume: 3851 start-page: 612 year: 2006 end-page: 621 – year: 1963 – start-page: 63 year: 1972 end-page: 84 – volume: 23 start-page: 548 year: 2004 end-page: 558 article-title: Spacetime faces: High resolution capture for modeling and animation publication-title: ACM Tran. Graph – volume: 30 start-page: 2246 issue: 8 year: 2011 end-page: 2257 article-title: Example‐driven deformations based on discrete shells publication-title: Comp. Graph. Forum – volume: 34 start-page: 1717 issue: 9 year: 2012 end-page: 1730 article-title: Elastic geodesic paths in shape space of parametrized surfaces publication-title: IEEE Trans. Pattern Anal. Mach. Intell – volume: 1 year: 2000 – volume: 4174 start-page: 142 year: 2006 end-page: 151 – year: 1945 – volume: 93 start-page: 293 issue: 3 year: 2011 end-page: 318 article-title: A continuum mechanical approach to geodesics in shape space publication-title: Inter. J. Comp. Vision – volume: 14 start-page: 213 year: 2008 end-page: 230 article-title: On linear variational surface deformation methods publication-title: IEEE Trans. Vis. Comp. Graph – volume: 30 start-page: 119:1 issue: 5 year: 2011 end-page: 11 article-title: Interactive surface modeling using modal analysis publication-title: ACM Trans. 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| Snippet | We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells... |
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| SubjectTerms | Analysis Animation Calculus Categories and Subject Descriptors (according to ACM CCS) Computer animation Computer graphics Deformation Elastic deformation Extrapolation Foundations I.3.5 [Computer Graphics]: Computational geometry and object modeling-Physically based modeling Mathematical analysis Rigid-body dynamics Shells Studies Transport |
| Title | Exploring the Geometry of the Space of Shells |
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