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Shape analysis on Lie groups with applications in computer animation
1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, Norway, Norway |
  In this paper, we develop a framework for shape analysis of curves in Lie groups for problems of computer animations. In particular, we will use these methods to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones.
References:
[1] |
A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie},, J. Analyse Math., 13 (1964), 1.
doi: 10.1007/BF02786619. |
[2] |
M. Bauer and M. Bruveris, A new riemannian setting for surface registration, 2011,, 182-193, (): 182. Google Scholar |
[3] |
M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves,, Differential Geometry and its Applications, 34 (2014), 139.
doi: 10.1016/j.difgeo.2014.04.008. |
[4] |
M. Bauer, M. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, Journal of Mathematical Imaging and Vision, (): 1. Google Scholar |
[5] |
M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-Guided Elastic Shape Analysis of Human Character Motions,, arXiv:1502.07666 [cs], (). Google Scholar |
[6] |
M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, Journal of Geometric Mechanics, 3 (2011), 389.
|
[7] |
M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves,, in Riemannian computing in computer vision, (2016), 233.
|
[8] |
Carnegie-Mellon, Carnegie-Mellon Mocap Database, 2003,, URL , (). Google Scholar |
[9] |
E. Celledoni, H. Marthinsen and B. Owren, An introduction to lie group integrators - basics, new developments and applications,, J. Comput. Phys., 257 (2014), 1040.
doi: 10.1016/j.jcp.2012.12.031. |
[10] |
E. Celledoni and B. Owren, Lie group methods for rigid body dynamics and time integration on manifolds,, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 421.
doi: 10.1016/S0045-7825(02)00520-0. |
[11] |
J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry,, North-Holland Publishing Co., (1975).
|
[12] |
C. J. Cotter, A. Clark and J. Peiró, A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching,, International Journal of Computer Vision, 99 (2012), 103.
doi: 10.1007/s11263-012-0520-0. |
[13] |
T. Dobrowolski, Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere,, J. Funct. Anal., 134 (1995), 350.
doi: 10.1006/jfan.1995.1149. |
[14] |
R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics,, 2nd edition, (1989).
|
[15] |
M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds,, The Visual Computer, 31 (2015), 1179.
doi: 10.1007/s00371-014-1001-y. |
[16] |
M. Fuchs, B. Jüttler, O. Scherzer and H. Yang, Shape Metrics Based on Elastic Deformations,, Journal of Mathematical Imaging and Vision, 35 (2009), 86.
doi: 10.1007/s10851-009-0156-z. |
[17] |
H. Glöckner, Regularity properties of infinite-dimensional Lie groups, and semiregularity, 2012,, URL , (1208). Google Scholar |
[18] |
H. Glöckner, Fundamentals of submersions and immersions between infinite-dimensional manifolds, 2015,, URL , (). Google Scholar |
[19] |
G. González Castro, M. Athanasopoulos and H. Ugail, Cyclic animation using partial differential equations,, The Visual Computer, 26 (2010), 325. Google Scholar |
[20] |
F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie,, Leipz. Ber., 58 (1906), 19. Google Scholar |
[21] |
J. Hilgert and K. H. Neeb, Structure and Geometry of Lie Groups,, Springer Monographs in Mathematics, (2012).
doi: 10.1007/978-0-387-84794-8. |
[22] |
A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215.
doi: 10.1017/S0962492900002154. |
[23] |
E. Klassen and A. Srivastava, A path-straightening method for finding geodesics in shape spaces of closed curves in R3,, SIAM Journal of Applied Mathematics, (). Google Scholar |
[24] |
L. Kovar and M. Gleicher, Flexible Automatic Motion Blending with Registration Curves,, in Proceedings of the 2003 ACMSIGGRAPH/Eurographics Symposium on Computer Animation, (2003), 214. Google Scholar |
[25] |
L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets,, in ACM Transactions on Graphics (TOG), 23 (2004), 559.
doi: 10.1145/1186562.1015760. |
[26] |
L. Kovar, M. Gleicher and F. Pighin, Motion graphs,, ACM Trans. Graph., 21 (2002), 473. Google Scholar |
[27] |
A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups,, Journal of Lie Theory, 7 (1997), 61.
|
[28] |
A. Kriegl and P. W. Michor, The convenient Setting of Global Analysis, vol. 53 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).
doi: 10.1090/surv/053. |
[29] |
S. Kurtek, E. Klassen, Z. Ding and A. Srivastava, A novel riemannian framework for shape analysis of 3d objects,, in 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2010), 1625.
doi: 10.1109/CVPR.2010.5539778. |
[30] |
S. Kurtek and A. Srivastava, Elastic symmetry analysis of anatomical structures,, in 2012 IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA), (2012), 33.
doi: 10.1109/MMBIA.2012.6164739. |
[31] |
S. Lahiri, D. Robinson and E. Klassen, Precise matching of PL curves in $\mathbbR^N$ in the square root velocity framework,, Geom. Imaging Comput., 2 (2015), 133.
doi: 10.4310/GIC.2015.v2.n3.a1. |
[32] |
S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999).
doi: 10.1007/978-1-4612-0541-8. |
[33] |
P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004. |
[34] |
W. Mio, A. Srivastava and S. Joshi, On shape of plane elastic curves,, Int. J. Comput. Vision, 73 (2007), 307.
doi: 10.1007/s11263-006-9968-0. |
[35] |
K.-H. Neeb, Towards a Lie theory of locally convex groups,, Jpn. J. Math., 1 (2006), 291.
doi: 10.1007/s11537-006-0606-y. |
[36] |
T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications,, Computer Graphics Forum, 29 (2010), 202.
doi: 10.1111/j.1467-8659.2009.01591.x. |
[37] |
A. Schmeding and C. Wockel, The Lie group of bisections of a Lie groupoid,, Ann. Global Anal. Geom., 48 (2015), 87.
doi: 10.1007/s10455-015-9459-z. |
[38] |
T. Sebastian, P. Klein and B. Kimia, On aligning curves,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 116.
doi: 10.1109/TPAMI.2003.1159951. |
[39] |
E. Sharon and D. Mumford, 2d-shape analysis using conformal mapping,, International Journal of Computer Vision, 70 (2006), 55.
doi: 10.1109/CVPR.2004.1315185. |
[40] |
K. Shoemake, Animating rotation with quaternion curves,, SIGGRAPH Comput. Graph., 19 (1985), 245.
doi: 10.1145/325334.325242. |
[41] |
A. Srivastava, S. Joshi, W. Mio and X. Liu, Statistical shape analysis: Clustering, learning, and testing,, IEEE Trans. Pattern Anal. Mach. Intell, 27 (2005), 590.
doi: 10.1109/TPAMI.2005.86. |
[42] |
A. Srivastava, E. Klassen, S. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011).
doi: 10.1109/TPAMI.2010.184. |
[43] |
J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance,, The Annals of Applied Statistics, 8 (2014), 530.
doi: 10.1214/13-AOAS701. |
[44] |
J. Su, A. Srivastava, F. de Souza and S. Sarkar, Rate-invariant analysis of trajectories on riemannian manifolds with application in visual speech recognition,, in 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2014), 620.
doi: 10.1109/CVPR.2014.86. |
[45] |
L. Younes, Computable elastic distances between shapes,, SIAM Journal on Applied Mathematics, 58 (1998), 565.
doi: 10.1137/S0036139995287685. |
[46] |
L. Younes, Spaces and manifolds of shapes in computer vision: An overview,, Image and Vision Computing, 30 (2012), 389.
doi: 10.1016/j.imavis.2011.09.009. |
show all references
References:
[1] |
A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie},, J. Analyse Math., 13 (1964), 1.
doi: 10.1007/BF02786619. |
[2] |
M. Bauer and M. Bruveris, A new riemannian setting for surface registration, 2011,, 182-193, (): 182. Google Scholar |
[3] |
M. Bauer, M. Bruveris, S. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves,, Differential Geometry and its Applications, 34 (2014), 139.
doi: 10.1016/j.difgeo.2014.04.008. |
[4] |
M. Bauer, M. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups,, Journal of Mathematical Imaging and Vision, (): 1. Google Scholar |
[5] |
M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-Guided Elastic Shape Analysis of Human Character Motions,, arXiv:1502.07666 [cs], (). Google Scholar |
[6] |
M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces,, Journal of Geometric Mechanics, 3 (2011), 389.
|
[7] |
M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves,, in Riemannian computing in computer vision, (2016), 233.
|
[8] |
Carnegie-Mellon, Carnegie-Mellon Mocap Database, 2003,, URL , (). Google Scholar |
[9] |
E. Celledoni, H. Marthinsen and B. Owren, An introduction to lie group integrators - basics, new developments and applications,, J. Comput. Phys., 257 (2014), 1040.
doi: 10.1016/j.jcp.2012.12.031. |
[10] |
E. Celledoni and B. Owren, Lie group methods for rigid body dynamics and time integration on manifolds,, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 421.
doi: 10.1016/S0045-7825(02)00520-0. |
[11] |
J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry,, North-Holland Publishing Co., (1975).
|
[12] |
C. J. Cotter, A. Clark and J. Peiró, A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching,, International Journal of Computer Vision, 99 (2012), 103.
doi: 10.1007/s11263-012-0520-0. |
[13] |
T. Dobrowolski, Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere,, J. Funct. Anal., 134 (1995), 350.
doi: 10.1006/jfan.1995.1149. |
[14] |
R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics,, 2nd edition, (1989).
|
[15] |
M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds,, The Visual Computer, 31 (2015), 1179.
doi: 10.1007/s00371-014-1001-y. |
[16] |
M. Fuchs, B. Jüttler, O. Scherzer and H. Yang, Shape Metrics Based on Elastic Deformations,, Journal of Mathematical Imaging and Vision, 35 (2009), 86.
doi: 10.1007/s10851-009-0156-z. |
[17] |
H. Glöckner, Regularity properties of infinite-dimensional Lie groups, and semiregularity, 2012,, URL , (1208). Google Scholar |
[18] |
H. Glöckner, Fundamentals of submersions and immersions between infinite-dimensional manifolds, 2015,, URL , (). Google Scholar |
[19] |
G. González Castro, M. Athanasopoulos and H. Ugail, Cyclic animation using partial differential equations,, The Visual Computer, 26 (2010), 325. Google Scholar |
[20] |
F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie,, Leipz. Ber., 58 (1906), 19. Google Scholar |
[21] |
J. Hilgert and K. H. Neeb, Structure and Geometry of Lie Groups,, Springer Monographs in Mathematics, (2012).
doi: 10.1007/978-0-387-84794-8. |
[22] |
A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett and A. Zanna, Lie-group methods,, Acta Numerica, 9 (2000), 215.
doi: 10.1017/S0962492900002154. |
[23] |
E. Klassen and A. Srivastava, A path-straightening method for finding geodesics in shape spaces of closed curves in R3,, SIAM Journal of Applied Mathematics, (). Google Scholar |
[24] |
L. Kovar and M. Gleicher, Flexible Automatic Motion Blending with Registration Curves,, in Proceedings of the 2003 ACMSIGGRAPH/Eurographics Symposium on Computer Animation, (2003), 214. Google Scholar |
[25] |
L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets,, in ACM Transactions on Graphics (TOG), 23 (2004), 559.
doi: 10.1145/1186562.1015760. |
[26] |
L. Kovar, M. Gleicher and F. Pighin, Motion graphs,, ACM Trans. Graph., 21 (2002), 473. Google Scholar |
[27] |
A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups,, Journal of Lie Theory, 7 (1997), 61.
|
[28] |
A. Kriegl and P. W. Michor, The convenient Setting of Global Analysis, vol. 53 of Mathematical Surveys and Monographs,, American Mathematical Society, (1997).
doi: 10.1090/surv/053. |
[29] |
S. Kurtek, E. Klassen, Z. Ding and A. Srivastava, A novel riemannian framework for shape analysis of 3d objects,, in 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2010), 1625.
doi: 10.1109/CVPR.2010.5539778. |
[30] |
S. Kurtek and A. Srivastava, Elastic symmetry analysis of anatomical structures,, in 2012 IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA), (2012), 33.
doi: 10.1109/MMBIA.2012.6164739. |
[31] |
S. Lahiri, D. Robinson and E. Klassen, Precise matching of PL curves in $\mathbbR^N$ in the square root velocity framework,, Geom. Imaging Comput., 2 (2015), 133.
doi: 10.4310/GIC.2015.v2.n3.a1. |
[32] |
S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999).
doi: 10.1007/978-1-4612-0541-8. |
[33] |
P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach,, Applied and Computational Harmonic Analysis, 23 (2007), 74.
doi: 10.1016/j.acha.2006.07.004. |
[34] |
W. Mio, A. Srivastava and S. Joshi, On shape of plane elastic curves,, Int. J. Comput. Vision, 73 (2007), 307.
doi: 10.1007/s11263-006-9968-0. |
[35] |
K.-H. Neeb, Towards a Lie theory of locally convex groups,, Jpn. J. Math., 1 (2006), 291.
doi: 10.1007/s11537-006-0606-y. |
[36] |
T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications,, Computer Graphics Forum, 29 (2010), 202.
doi: 10.1111/j.1467-8659.2009.01591.x. |
[37] |
A. Schmeding and C. Wockel, The Lie group of bisections of a Lie groupoid,, Ann. Global Anal. Geom., 48 (2015), 87.
doi: 10.1007/s10455-015-9459-z. |
[38] |
T. Sebastian, P. Klein and B. Kimia, On aligning curves,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 116.
doi: 10.1109/TPAMI.2003.1159951. |
[39] |
E. Sharon and D. Mumford, 2d-shape analysis using conformal mapping,, International Journal of Computer Vision, 70 (2006), 55.
doi: 10.1109/CVPR.2004.1315185. |
[40] |
K. Shoemake, Animating rotation with quaternion curves,, SIGGRAPH Comput. Graph., 19 (1985), 245.
doi: 10.1145/325334.325242. |
[41] |
A. Srivastava, S. Joshi, W. Mio and X. Liu, Statistical shape analysis: Clustering, learning, and testing,, IEEE Trans. Pattern Anal. Mach. Intell, 27 (2005), 590.
doi: 10.1109/TPAMI.2005.86. |
[42] |
A. Srivastava, E. Klassen, S. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011).
doi: 10.1109/TPAMI.2010.184. |
[43] |
J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance,, The Annals of Applied Statistics, 8 (2014), 530.
doi: 10.1214/13-AOAS701. |
[44] |
J. Su, A. Srivastava, F. de Souza and S. Sarkar, Rate-invariant analysis of trajectories on riemannian manifolds with application in visual speech recognition,, in 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2014), 620.
doi: 10.1109/CVPR.2014.86. |
[45] |
L. Younes, Computable elastic distances between shapes,, SIAM Journal on Applied Mathematics, 58 (1998), 565.
doi: 10.1137/S0036139995287685. |
[46] |
L. Younes, Spaces and manifolds of shapes in computer vision: An overview,, Image and Vision Computing, 30 (2012), 389.
doi: 10.1016/j.imavis.2011.09.009. |
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