Shape analysis on Lie groups with applications in computer animation

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

Shape analysis on Lie groups with applications in computer animation

Elena Celledoni ^1, , Markus Eslitzbichler ^1, and Alexander Schmeding ^1,

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

Abstract
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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.

Keywords: Shape analysis, infinite dimensional manifolds, computer animation., curve matching, Lie groups.

Mathematics Subject Classification: 65D18, 58J90, 58D10, 49Q1.

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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[2]	M. Bauer and M. Bruveris, A new riemannian setting for surface registration, 2011, 182-193, URL http://hal.inria.fr/inria-00624210.
[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), 139-165. 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, 1-38.
[5]	M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-Guided Elastic Shape Analysis of Human Character Motions, arXiv:1502.07666 [cs], URL http://arxiv.org/abs/1502.07666.
[6]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438.
[7]	M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian computing in computer vision, Springer, Cham, 2016, 233-255.
[8]	Carnegie-Mellon, Carnegie-Mellon Mocap Database, 2003, URL http://mocap.cs.cmu.edu/.
[9]	E. Celledoni, H. Marthinsen and B. Owren, An introduction to lie group integrators - basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061. 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), 421-438. doi: 10.1016/S0045-7825(02)00520-0.
[11]	J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 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), 103-121. 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), 350-362. doi: 10.1006/jfan.1995.1149.
[14]	R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics, 2nd edition, Heldermann Verlag, Berlin, 1989.
[15]	M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190. 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), 86-102. doi: 10.1007/s10851-009-0156-z.
[17]	H. Glöckner, Regularity properties of infinite-dimensional Lie groups, and semiregularity, 2012, URL http://arxiv.org/abs/1208.0715, arXiv: 1208.0715 [math].
[18]	H. Glöckner, Fundamentals of submersions and immersions between infinite-dimensional manifolds, 2015, URL http://arxiv.org/abs/1208.0715, arXiv:1502.05795v3 [math].
[19]	G. González Castro, M. Athanasopoulos and H. Ugail, Cyclic animation using partial differential equations, The Visual Computer, 26 (2010), 325-338.
[20]	F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipz. Ber., 58 (1906), 19-48.
[21]	J. Hilgert and K. H. Neeb, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer, New York, 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), 215-365. 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, URL http://stat.fsu.edu/~anuj/pdf/papers/3DCurves.pdf.
[24]	L. Kovar and M. Gleicher, Flexible Automatic Motion Blending with Registration Curves, in Proceedings of the 2003 ACMSIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 2003, 214-224.
[25]	L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, in ACM Transactions on Graphics (TOG), ACM, 23 (2004), 559-568. doi: 10.1145/1186562.1015760.
[26]	L. Kovar, M. Gleicher and F. Pighin, Motion graphs, ACM Trans. Graph., 21 (2002), 473-482.
[27]	A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups, Journal of Lie Theory, 7 (1997), 61-99.
[28]	A. Kriegl and P. W. Michor, The convenient Setting of Global Analysis, vol. 53 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 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, 1625-1632. 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, 33-38. doi: 10.1109/MMBIA.2012.6164739.
[31]	S. Lahiri, D. Robinson and E. Klassen, Precise matching of PL curves in $\mathbb{R}^N2$ in the square root velocity framework, Geom. Imaging Comput., 2 (2015), 133-186, \urlprefixhttp://arxiv.org/abs/1501.00577, arXiv:1501.00577 [math]. doi: 10.4310/GIC.2015.v2.n3.a1.
[32]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 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), 74-113. 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), 307-324. doi: 10.1007/s11263-006-9968-0.
[35]	K.-H. Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math., 1 (2006), 291-468. 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), 202-226. 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), 87-123. 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), 116-125. 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), 55-75. doi: 10.1109/CVPR.2004.1315185.
[40]	K. Shoemake, Animating rotation with quaternion curves, SIGGRAPH Comput. Graph., 19 (1985), 245-254. 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), 590-602. 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), 1415 -1428. 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), 530-552. 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, 620-627. doi: 10.1109/CVPR.2014.86.
[45]	L. Younes, Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58 (1998), 565-586. doi: 10.1137/S0036139995287685.
[46]	L. Younes, Spaces and manifolds of shapes in computer vision: An overview, Image and Vision Computing, 30 (2012), 389-397. 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), 1-114. doi: 10.1007/BF02786619.
[2]	M. Bauer and M. Bruveris, A new riemannian setting for surface registration, 2011, 182-193, URL http://hal.inria.fr/inria-00624210.
[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), 139-165. 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, 1-38.
[5]	M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-Guided Elastic Shape Analysis of Human Character Motions, arXiv:1502.07666 [cs], URL http://arxiv.org/abs/1502.07666.
[6]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438.
[7]	M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian computing in computer vision, Springer, Cham, 2016, 233-255.
[8]	Carnegie-Mellon, Carnegie-Mellon Mocap Database, 2003, URL http://mocap.cs.cmu.edu/.
[9]	E. Celledoni, H. Marthinsen and B. Owren, An introduction to lie group integrators - basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061. 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), 421-438. doi: 10.1016/S0045-7825(02)00520-0.
[11]	J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 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), 103-121. 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), 350-362. doi: 10.1006/jfan.1995.1149.
[14]	R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics, 2nd edition, Heldermann Verlag, Berlin, 1989.
[15]	M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190. 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), 86-102. doi: 10.1007/s10851-009-0156-z.
[17]	H. Glöckner, Regularity properties of infinite-dimensional Lie groups, and semiregularity, 2012, URL http://arxiv.org/abs/1208.0715, arXiv: 1208.0715 [math].
[18]	H. Glöckner, Fundamentals of submersions and immersions between infinite-dimensional manifolds, 2015, URL http://arxiv.org/abs/1208.0715, arXiv:1502.05795v3 [math].
[19]	G. González Castro, M. Athanasopoulos and H. Ugail, Cyclic animation using partial differential equations, The Visual Computer, 26 (2010), 325-338.
[20]	F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipz. Ber., 58 (1906), 19-48.
[21]	J. Hilgert and K. H. Neeb, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer, New York, 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), 215-365. 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, URL http://stat.fsu.edu/~anuj/pdf/papers/3DCurves.pdf.
[24]	L. Kovar and M. Gleicher, Flexible Automatic Motion Blending with Registration Curves, in Proceedings of the 2003 ACMSIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 2003, 214-224.
[25]	L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, in ACM Transactions on Graphics (TOG), ACM, 23 (2004), 559-568. doi: 10.1145/1186562.1015760.
[26]	L. Kovar, M. Gleicher and F. Pighin, Motion graphs, ACM Trans. Graph., 21 (2002), 473-482.
[27]	A. Kriegl and P. W. Michor, Regular infinite dimensional Lie groups, Journal of Lie Theory, 7 (1997), 61-99.
[28]	A. Kriegl and P. W. Michor, The convenient Setting of Global Analysis, vol. 53 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 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, 1625-1632. 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, 33-38. doi: 10.1109/MMBIA.2012.6164739.
[31]	S. Lahiri, D. Robinson and E. Klassen, Precise matching of PL curves in $\mathbb{R}^N2$ in the square root velocity framework, Geom. Imaging Comput., 2 (2015), 133-186, \urlprefixhttp://arxiv.org/abs/1501.00577, arXiv:1501.00577 [math]. doi: 10.4310/GIC.2015.v2.n3.a1.
[32]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 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), 74-113. 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), 307-324. doi: 10.1007/s11263-006-9968-0.
[35]	K.-H. Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math., 1 (2006), 291-468. 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), 202-226. 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), 87-123. 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), 116-125. 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), 55-75. doi: 10.1109/CVPR.2004.1315185.
[40]	K. Shoemake, Animating rotation with quaternion curves, SIGGRAPH Comput. Graph., 19 (1985), 245-254. 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), 590-602. 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), 1415 -1428. 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), 530-552. 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, 620-627. doi: 10.1109/CVPR.2014.86.
[45]	L. Younes, Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58 (1998), 565-586. doi: 10.1137/S0036139995287685.
[46]	L. Younes, Spaces and manifolds of shapes in computer vision: An overview, Image and Vision Computing, 30 (2012), 389-397. doi: 10.1016/j.imavis.2011.09.009.

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