Time-Discrete Geodesics in the Space of Shells
Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation...
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| Veröffentlicht in: | Computer graphics forum Jg. 31; H. 5; S. 1755 - 1764 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2012
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so‐called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. |
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| AbstractList | Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so‐called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so-called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. [PUBLICATION ABSTRACT] |
| Author | Wardetzky, M. Rumpf, M. 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: M. surname: Wardetzky fullname: Wardetzky, M. organization: Institute of Num. and Appl. Math, University of Göttingen, Germany – sequence: 4 givenname: B. surname: Wirth fullname: Wirth, B. organization: Courant Institute, New York University, USA |
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| Cites_doi | 10.1109/TPAMI.2010.184 10.1090/qam/1632326 10.1111/j.1467-8659.2011.01974.x 10.1007/s11263-006-0635-2 10.1007/s11263-010-0323-0 10.1145/1276377.1276457 10.1145/166117.166119 10.1137/090781139 10.1002/cpa.10048 10.1007/s11263-010-0416-9 10.4171/JEMS/37 10.1007/s10851-009-0156-z 10.1007/s10208-003-0094-x 10.1007/s11263-008-0172-2 10.1109/CVPR.2010.5539778 10.1016/S0097-8493(97)00081-2 10.1111/j.1467-8659.2009.01600.x 10.1007/978-3-7643-8621-4_16 10.1016/S1631-073X(03)00028-1 10.1145/258734.258849 |
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| References_xml | – reference: Friesecke G., James R. D., Müller S.: Rigorous derivation of nonlinear plate theory and geometric rigidity. Tech. rep., Max-Planck-Institut, Leipzig , 2001. – reference: Sundaramoorthi G., Mennucci A., Soatto S., Yezzi A.: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM Journal on Imaging Sciences 4, 1 (2011), 109-145. – reference: Grinspun E., Gingold Y., Reisman J., Zorin D.: Computing discrete shape operators on general meshes. Eurographics 25 (2006). – reference: Kilian M., Mitra N. J., Pottmann H.: Geometric modeling in shape space. In ACM Transactions on Graphics (2007), vol. 26, pp. #64, 1-8. – reference: Dupuis D., Grenander U., Miller M.: Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics 56 (1998), 587-600. – reference: Friesecke G., James R., Mora M. G., Müller S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336 8 (2003), 697-702. – reference: Michor P. W., Mumford D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8 (2006), 1-48. – reference: Bronstein A. M., Bronstein M. M., Kimmel R.: Topology-invariant similarity of nonrigid shapes. International Journal of Computer Vision 81, 3 (2009), 281-301. – reference: Fröhlich S., Botsch M.: Example-driven deformations based on discrete shells. Computer Graphics Forum 30, 8 (2011), 2246-âĂŞ2257. – reference: Wirth B., Bar L., Rumpf M., Sapiro G.: A continuum mechanical approach to geodesics in shape space. IJCV 93, 3 (2011), 293-318. – reference: Desbrun M., Kanso E., Tong Y.: Discrete differential forms for computational modeling. Oberwolfach Seminars: Discrete differential geometry 38 (2008), 287-324. – reference: Sundaramoorthi G., Yezzi A., Mennucci A.: Sobolev active contours. International Journal of Computer Vision. 73, 3 (2007), 345-366. – reference: Friesecke G., James R. D., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Communications on Pure and Applied Mathematics 55, 11 (2002), 1461-1506. – reference: LeDret H., Raoult A.: The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 73 (1995), 549-578. – reference: Fuchs M., Jüttler B., Scherzer O., Yang H.: Shape metrics based on elastic deformations. J. Math. Imaging Vis. 35, 1 (2009), 86-102. – reference: Strutt J. W.: Theory of Sound, vol. 2. Dover Publications, 1945. – reference: Charpiat G., Faugeras O., Keriven R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations of Computational Mathematics 5, 1 (2005), 1-58. – reference: Winkler T., Drieseberg J., Alexa M., Hormann K.: Multi-scale geometry interpolation. Computer Graphics Forum 29, 2 (2010), 309-318. – reference: Srivastava A., Klassen E., Joshi S. H., Jermyn I. H.: Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33, 7 (july 2011), 1415-1428. – year: 2011 – volume: 25 year: 2006 article-title: Computing discrete shape operators on general meshes publication-title: Eurographics – volume: 30 issue: 8 year: 2011 article-title: Example‐driven deformations based on discrete shells publication-title: Computer Graphics Forum – volume: 73 start-page: 549 year: 1995 end-page: 578 article-title: The nonlinear membrane model as a variational limit of nonlinear three‐dimensional elasticity publication-title: J. Math. Pures Appl. – start-page: 77 year: 2008 end-page: 86 – year: 2005 – volume: 29 start-page: 309 issue: 2 year: 2010 end-page: 318 article-title: Multi‐scale geometry interpolation publication-title: Computer Graphics Forum – volume: 73 start-page: 345 issue: 3 year: 2007 end-page: 366 article-title: Sobolev active contours publication-title: International Journal of Computer Vision. – year: 2007 – year: 2001 – volume: 8 start-page: 697 year: 2003 end-page: 702 article-title: Derivation of nonlinear bending theory for shells from three‐dimensional nonlinear elasticity by Gamma‐convergence publication-title: C. R. Math. Acad. Sci. Paris 336 – year: 2003 – start-page: 182 year: 2011 end-page: 193 – 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. – volume: 5 start-page: 1 issue: 1 year: 2005 end-page: 58 article-title: Approximations of shape metrics and application to shape warping and empirical shape statistics publication-title: Foundations of Computational Mathematics – start-page: 209 year: 1997 end-page: 216 – volume: 55 start-page: 1461 issue: 11 year: 2002 end-page: 1506 article-title: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity publication-title: Communications on Pure and Applied Mathematics – volume: 81 start-page: 281 issue: 3 year: 2009 end-page: 301 article-title: Topology‐invariant similarity of nonrigid shapes publication-title: International Journal of Computer Vision – year: 2010 – start-page: 27 year: 1998 end-page: 36 – volume: 33 start-page: 1415 issue: 7 year: july 2011 end-page: 1428 article-title: Shape analysis of elastic curves in euclidean spaces publication-title: Pattern Analysis and Machine Intelligence, IEEE Transactions on – volume: 26 start-page: 1 year: 2007 end-page: 8 article-title: Geometric modeling in shape space publication-title: ACM Transactions on Graphics – volume: 56 start-page: 587 year: 1998 end-page: 600 article-title: Variational problems on flows of diffeomorphisms for image matching publication-title: Quarterly of Applied Mathematics – volume: 4 start-page: 109 issue: 1 year: 2011 end-page: 145 article-title: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering publication-title: SIAM Journal on Imaging Sciences – year: 2006 – 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: IJCV – volume: 2 year: 1945 – volume: 38 start-page: 287 year: 2008 end-page: 324 article-title: Discrete differential forms for computational modeling publication-title: Oberwolfach Seminars: Discrete differential geometry – year: 1993 – 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. – year: 1999 – start-page: 77 volume-title: ACM SIGGRAPH/Eurographics Symposium on Computer Animation year: 2008 ident: e_1_2_11_2_2 – ident: e_1_2_11_28_2 doi: 10.1109/TPAMI.2010.184 – ident: e_1_2_11_6_2 doi: 10.1090/qam/1632326 – ident: e_1_2_11_15_2 – ident: e_1_2_11_9_2 – ident: e_1_2_11_10_2 doi: 10.1111/j.1467-8659.2011.01974.x – ident: e_1_2_11_18_2 – ident: e_1_2_11_31_2 doi: 10.1007/s11263-006-0635-2 – ident: e_1_2_11_25_2 doi: 10.1007/s11263-010-0323-0 – ident: e_1_2_11_7_2 – ident: e_1_2_11_23_2 doi: 10.1145/1276377.1276457 – ident: e_1_2_11_19_2 doi: 10.1145/166117.166119 – volume: 73 start-page: 549 year: 1995 ident: e_1_2_11_24_2 article-title: The nonlinear membrane model as a variational limit of nonlinear three‐dimensional elasticity publication-title: J. Math. Pures Appl. – ident: e_1_2_11_3_2 – ident: e_1_2_11_22_2 – volume-title: Theory of Sound year: 1945 ident: e_1_2_11_30_2 – ident: e_1_2_11_29_2 doi: 10.1137/090781139 – ident: e_1_2_11_12_2 doi: 10.1002/cpa.10048 – ident: e_1_2_11_32_2 doi: 10.1007/s11263-010-0416-9 – ident: e_1_2_11_26_2 doi: 10.4171/JEMS/37 – ident: e_1_2_11_14_2 doi: 10.1007/s10851-009-0156-z – ident: e_1_2_11_5_2 doi: 10.1007/s10208-003-0094-x – ident: e_1_2_11_4_2 doi: 10.1007/s11263-008-0172-2 – volume-title: Rigorous derivation of nonlinear plate theory and geometric rigidity year: 2001 ident: e_1_2_11_11_2 – volume: 25 year: 2006 ident: e_1_2_11_16_2 article-title: Computing discrete shape operators on general meshes publication-title: Eurographics – ident: e_1_2_11_21_2 doi: 10.1109/CVPR.2010.5539778 – ident: e_1_2_11_20_2 doi: 10.1016/S0097-8493(97)00081-2 – ident: e_1_2_11_27_2 – ident: e_1_2_11_33_2 doi: 10.1111/j.1467-8659.2009.01600.x – ident: e_1_2_11_8_2 doi: 10.1007/978-3-7643-8621-4_16 – ident: e_1_2_11_13_2 doi: 10.1016/S1631-073X(03)00028-1 – ident: e_1_2_11_17_2 doi: 10.1145/258734.258849 |
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| SubjectTerms | Bending Computation Computational mathematics Computer graphics Deformation Distortion Evolution Geodetics I.3.5 [Computer Graphics]: Computational geometry and object modeling-Physically based modeling Mathematical models Shells Studies Thin walled shells |
| Title | Time-Discrete Geodesics in the Space of Shells |
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| Volume | 31 |
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