Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics

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December 2012, 4(4): 365-383. doi: 10.3934/jgm.2012.4.365

Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics

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

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

Received September 2011 Revised May 2012 Published January 2013

Abstract
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In continuation of [7] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i\big(Vol(f)\big)\ \bar{g}\big((P_i)_fh,k\big) vol(f^*\bar{g}) $$ on the space of immersions $Imm(M,N)$ and on shape space $B_i(M,N)=Imm(M,N)/{Diff}(M)$. Here $(N,\bar{g})$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\bar{g}$ is the induced Riemannian metric on $M$, $vol(f^*\bar{g})$ is the induced volume density on $M$, $Vol(f)=\int_M vol(f^*\bar{g})$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\bar{g}}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Phi_i(Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.

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

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

Citation: Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365

References:

[1]	M. Bauer and M. Bruveris, A new Riemannian setting for surface registration, 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (2011), 182-194.
[2]	M. Bauer, M. Bruveris, C. Cotter, S. Marsland and P. W. Michor, Constructing reparametrization invariant metrics on spaces of plane curves,, \arXiv{1207.5965}., ().
[3]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Vanishing geodesic distance for the riemannian metric with geodesic equation the KdV-equation, Ann. Global Analysis Geom., 41 (2012), 461-472. doi: 10.1007/s10455-011-9294-9.
[4]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Glob. Anal. Geom., (). doi: doi:10.1007/s10455-012-9353-x.
[5]	M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space, SIAM J. Imaging Sci., 5 (2012), 244-310. doi: 10.1137/100807983.
[6]	M. Bauer, P. Harms and P. W. Michor, Curvature weighted metrics on shape space of hypersurfaces in n-space, Differential Geometry and its Applications, 30 (2012), 33-41. doi: 10.1016/j.difgeo.2011.10.002.
[7]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438.
[8]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, To appear in, ().
[9]	M. Bauer, "Almost Local Metrics on Shape Space of Surfaces," Ph.D thesis, University of Vienna, 2010.
[10]	A. L. Besse, "Einstein Manifolds," Classics in Mathematics. Springer-Verlag, Berlin, 2008.
[11]	P. Harms, "Sobolev Metrics on Shape Space of Surfaces," Ph.D Thesis, University of Vienna, 2010.
[12]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).
[13]	P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. doi: 10.4171/JEMS/37.
[14]	P. W. Michor and D. 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.
[15]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.
[16]	J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels", Math. Scand., 8 (1960), 116-120.
[17]	J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.
[18]	M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96854-9.
[19]	Jan Slovák, Peetre theorem for nonlinear operators, Ann. Global Anal. Geom., 6 (1988), 273-283. doi: 10.1007/BF00054575.
[20]	A. Yezzi and A. Mennucci, Conformal riemannian metrics in space of curves, EUSIPCO, (2004).
[21]	A. Yezzi and A. Mennucci, Metrics in the space of curves, arXiv:math/0412454, December (2004).
[22]	A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in "Proceedings of the Tenth IEEE International Conference on Computer Vision," 1, 913-919, Washington, (2005). IEEE Computer Society.
[23]	L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Rend. Lincei Mat. Appl., 9 (2008), 25-57. doi: 10.4171/RLM/506.

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References:

[1]	M. Bauer and M. Bruveris, A new Riemannian setting for surface registration, 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (2011), 182-194.
[2]	M. Bauer, M. Bruveris, C. Cotter, S. Marsland and P. W. Michor, Constructing reparametrization invariant metrics on spaces of plane curves,, \arXiv{1207.5965}., ().
[3]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Vanishing geodesic distance for the riemannian metric with geodesic equation the KdV-equation, Ann. Global Analysis Geom., 41 (2012), 461-472. doi: 10.1007/s10455-011-9294-9.
[4]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Glob. Anal. Geom., (). doi: doi:10.1007/s10455-012-9353-x.
[5]	M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space, SIAM J. Imaging Sci., 5 (2012), 244-310. doi: 10.1137/100807983.
[6]	M. Bauer, P. Harms and P. W. Michor, Curvature weighted metrics on shape space of hypersurfaces in n-space, Differential Geometry and its Applications, 30 (2012), 33-41. doi: 10.1016/j.difgeo.2011.10.002.
[7]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438.
[8]	M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on the manifold of all Riemannian metrics,, To appear in, ().
[9]	M. Bauer, "Almost Local Metrics on Shape Space of Surfaces," Ph.D thesis, University of Vienna, 2010.
[10]	A. L. Besse, "Einstein Manifolds," Classics in Mathematics. Springer-Verlag, Berlin, 2008.
[11]	P. Harms, "Sobolev Metrics on Shape Space of Surfaces," Ph.D Thesis, University of Vienna, 2010.
[12]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).
[13]	P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS), 8 (2006), 1-48. doi: 10.4171/JEMS/37.
[14]	P. W. Michor and D. 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.
[15]	J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7 (1959), 211-218.
[16]	J. Peetre, Réctification à l'article "Une caractérisation abstraite des opérateurs différentiels", Math. Scand., 8 (1960), 116-120.
[17]	J. Shah, $H^0$-type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.
[18]	M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96854-9.
[19]	Jan Slovák, Peetre theorem for nonlinear operators, Ann. Global Anal. Geom., 6 (1988), 273-283. doi: 10.1007/BF00054575.
[20]	A. Yezzi and A. Mennucci, Conformal riemannian metrics in space of curves, EUSIPCO, (2004).
[21]	A. Yezzi and A. Mennucci, Metrics in the space of curves, arXiv:math/0412454, December (2004).
[22]	A. Yezzi and A. Mennucci, Conformal metrics and true "gradient flows" for curves, in "Proceedings of the Tenth IEEE International Conference on Computer Vision," 1, 913-919, Washington, (2005). IEEE Computer Society.
[23]	L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Rend. Lincei Mat. Appl., 9 (2008), 25-57. doi: 10.4171/RLM/506.

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