# American Institute of Mathematical Sciences

September  2016, 8(3): 273-304. doi: 10.3934/jgm.2016008

## 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

Received  June 2015 Revised  May 2016 Published  September 2016

Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects.
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.
Citation: Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008
##### References:
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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.  Google Scholar [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.  Google Scholar [14] R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics,, 2nd edition, (1989).   Google Scholar [15] M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds,, The Visual Computer, 31 (2015), 1179.  doi: 10.1007/s00371-014-1001-y.  Google Scholar [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.  Google Scholar [17] H. 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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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] K. Shoemake, Animating rotation with quaternion curves,, SIGGRAPH Comput. Graph., 19 (1985), 245.  doi: 10.1145/325334.325242.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] L. Younes, Computable elastic distances between shapes,, SIAM Journal on Applied Mathematics, 58 (1998), 565.  doi: 10.1137/S0036139995287685.  Google Scholar [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.  Google Scholar

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##### References:
 [1] A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie},, J. Analyse Math., 13 (1964), 1.  doi: 10.1007/BF02786619.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [11] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry,, North-Holland Publishing Co., (1975).   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics,, 2nd edition, (1989).   Google Scholar [15] M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds,, The Visual Computer, 31 (2015), 1179.  doi: 10.1007/s00371-014-1001-y.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] K. Shoemake, Animating rotation with quaternion curves,, SIGGRAPH Comput. Graph., 19 (1985), 245.  doi: 10.1145/325334.325242.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] L. Younes, Computable elastic distances between shapes,, SIAM Journal on Applied Mathematics, 58 (1998), 565.  doi: 10.1137/S0036139995287685.  Google Scholar [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.  Google Scholar
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