Sobolev metrics on shape space of surfaces

Martin Bauer; Philipp Harms; Peter W. Michor

doi:10.3934/jgm.2011.3.389

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December 2011, 3(4): 389-438. doi: 10.3934/jgm.2011.3.389

Sobolev metrics on shape space of surfaces

Martin Bauer ^1, , Philipp Harms ^2, and Peter W. Michor ^1,

1.	Fakultät f¨ur Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria, Austria
2.	EdLabs, Harvard University, 44 Brattle Street, Cambridge, MA 02138, United States

Received September 2010 Revised August 2011 Published February 2012

Abstract
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Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \overline{g}( P^fh, k) vol (f^*\overline{g})$$ where $\overline{g}$ is some fixed metric on $N$, $f^*\overline{g}$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\overline{g}$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.

Keywords: shape space, well-posedness, Sobolev type metric, geodesic equation., Surface matching.

Mathematics Subject Classification: Primary: 58B20, 58D15, 58E1.

Citation: Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

References:

[1]	V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233. Google Scholar
[2]	M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space, preprint, 2010, arXiv:math/1001.0717. Google Scholar
[3]	Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'' Ph.D thesis, University of Vienna, 2010. Google Scholar
[4]	Arthur L. Besse, "Einstein Manifolds,'' Reprint of the 1987 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Google Scholar
[5]	V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions, Differential Geom. Appl., 1 (1991), 391-401. doi: 10.1016/0926-2245(91)90015-2. Google Scholar
[6]	Adrian Constantin and Boris Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. Google Scholar
[7]	Jürgen Eichhorn, "Global Analysis on Open Manifolds,'' Nova Science Publishers, Inc., New York, 2007. Google Scholar
[8]	Jürgen Eichhorn and Jan Fricke, The module structure theorem for Sobolev spaces on open manifolds, Math. Nachr., 194 (1998), 35-47. doi: 10.1002/mana.19981940105. Google Scholar
[9]	François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55-58. Google Scholar
[10]	Philipp Harms, "Sobolev Metrics on Shape Space of Surfaces,'' Ph.D thesis, University of Vienna, 2010. Google Scholar
[11]	Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Google Scholar
[12]	I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,'' Springer-Verlag, Berlin, 1993. Google Scholar
[13]	Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'' Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997. Google Scholar
[14]	A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound., 10 (2008), 423-445. doi: 10.4171/IFB/196. Google Scholar
[15]	Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, in "Phase Space Analysis of Partial Differential Equations," Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, (2006), 133-215. Google Scholar
[16]	Peter W. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, 93, American Mathematical Society, Providence, RI, 2008. Google Scholar
[17]	Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic). Google Scholar
[18]	Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. Google Scholar
[19]	Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004. Google Scholar
[20]	M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'' Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Google Scholar
[21]	Alain Trouvé and Laurent Younes, "Diffeomorphic Matching Problems in One Dimension: Designing and Minimizing Matching Functionals," Computer Vision, Vol. 1842, ECCV, 2000. Google Scholar
[22]	Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'' Ph.D thesis, Katholieke Universiteit Leuven, 2008. Google Scholar
[23]	L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57. Google Scholar
[24]	Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic). doi: 10.1137/S0036139995287685. Google Scholar

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References:

[1]	V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233. Google Scholar
[2]	M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space, preprint, 2010, arXiv:math/1001.0717. Google Scholar
[3]	Martin Bauer, "Almost Local Metrics on Shape Space of Surfaces,'' Ph.D thesis, University of Vienna, 2010. Google Scholar
[4]	Arthur L. Besse, "Einstein Manifolds,'' Reprint of the 1987 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Google Scholar
[5]	V. Cervera, F. Mascaró and P. W. Michor, The action of the diffeomorphism group on the space of immersions, Differential Geom. Appl., 1 (1991), 391-401. doi: 10.1016/0926-2245(91)90015-2. Google Scholar
[6]	Adrian Constantin and Boris Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. Google Scholar
[7]	Jürgen Eichhorn, "Global Analysis on Open Manifolds,'' Nova Science Publishers, Inc., New York, 2007. Google Scholar
[8]	Jürgen Eichhorn and Jan Fricke, The module structure theorem for Sobolev spaces on open manifolds, Math. Nachr., 194 (1998), 35-47. doi: 10.1002/mana.19981940105. Google Scholar
[9]	François Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilv. Univ. Braşov Ser. III, 2(51) (2009), 55-58. Google Scholar
[10]	Philipp Harms, "Sobolev Metrics on Shape Space of Surfaces,'' Ph.D thesis, University of Vienna, 2010. Google Scholar
[11]	Shoshichi Kobayashi and Katsumi Nomizu, "Foundations of Differential Geometry," Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Google Scholar
[12]	I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,'' Springer-Verlag, Berlin, 1993. Google Scholar
[13]	Andreas Kriegl and Peter W. Michor, "The Convenient Setting of Global Analysis,'' Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997. Google Scholar
[14]	A. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound., 10 (2008), 423-445. doi: 10.4171/IFB/196. Google Scholar
[15]	Peter W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach, in "Phase Space Analysis of Partial Differential Equations," Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, (2006), 133-215. Google Scholar
[16]	Peter W. Michor, "Topics in Differential Geometry,'' Graduate Studies in Mathematics, 93, American Mathematical Society, Providence, RI, 2008. Google Scholar
[17]	Peter W. Michor and David Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic). Google Scholar
[18]	Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. Google Scholar
[19]	Peter W. Michor and David Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113. doi: 10.1016/j.acha.2006.07.004. Google Scholar
[20]	M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,'' Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Google Scholar
[21]	Alain Trouvé and Laurent Younes, "Diffeomorphic Matching Problems in One Dimension: Designing and Minimizing Matching Functionals," Computer Vision, Vol. 1842, ECCV, 2000. Google Scholar
[22]	Steven Verpoort, "The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects,'' Ph.D thesis, Katholieke Universiteit Leuven, 2008. Google Scholar
[23]	L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57. Google Scholar
[24]	Laurent Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic). doi: 10.1137/S0036139995287685. Google Scholar

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