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December  2011, 3(4): 363-387. doi: 10.3934/jgm.2011.3.363

Un-reduction

Martins Bruveris 1, , David C. P. Ellis 1, , Darryl D. Holm 2, and  François Gay-Balmaz 3,

1. 

Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom, United Kingdom
2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
3. 

Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure de Paris, 75005 Paris, France

Received  December 2010 Revised  November 2011 Published  February 2012

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally conceived for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.
Keywords: image matching., Lagrangian reduction, un-reduction, force fields.
Mathematics Subject Classification: 37K05, 37K6.
Citation: Martins Bruveris, David C. P. Ellis, Darryl D. Holm, François Gay-Balmaz. Un-reduction. Journal of Geometric Mechanics, 2011, 3 (4) : 363-387. doi: 10.3934/jgm.2011.3.363
References:
[1]

M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, to appear in: SIAM Journal on Imaging Sciences, ().   Google Scholar

[2]

A. Clark, C. Cotter and J. Peiro, A reparameterisation-based approach to geodesic shooting for 2D curve matching,, work in progress., ().   Google Scholar

[3]

H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, in "Geometry of Differential Equations" (eds. A. Khovanskii, A. Varchenko and V. Vassiliev), Amer. Math. Soc. Trans. Ser. 2, 186, Amer. Math. Soc., Providence, RI, (1998), 1-25.  Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp.  Google Scholar

[5]

C. J. Cotter, The variational particle-mesh method for matching curves, J. Phys. A, 41 (2008), 344003, 18 pp.  Google Scholar

[6]

C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry, J. Geom. Mech., 2 (2010), 51-68.  Google Scholar

[7]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Rational Mech. Anal., 197 (2010), 811-902. doi: 10.1007/s00205-010-0305-y.  Google Scholar

[8]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys., 61 (2011), 2120-2146.  Google Scholar

[9]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Comm. Math. Phys. \textbf{309}(2), 309 (): 413.   Google Scholar

[10]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, Journal of the Brazilian mathematical society, 42 (): 579.   Google Scholar

[11]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[12]

A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Mathematical Surveys and Monographs, 53, AMS, Providence, RI, 1997.  Google Scholar

[13]

J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Note Series, 174, Cambridge University Press, Cambridge, 1992.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[15]

P. W. Michor, Manifolds of smooth maps. III: The principal bundle of embeddings of a noncompact smooth manifold, Cah. Top. Géom. Diff., 21 (1980), 325-337.  Google Scholar

[16]

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.  Google Scholar

[17]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc., 8 (2006), 1-48. doi: 10.4171/JEMS/37.  Google Scholar

[18]

R. Montgomery, Gauge theory of the falling cat, in "Dynamics and Control of Mechanical Systems" (Waterloo, ON, 1992), Fields Inst. Comm., 1, Amer. Math. Soc., Providence, RI, (1993), 193-218.  Google Scholar

[19]

F.-X. Vialard, "Hamiltonian Approach to Shape Spaces in a Diffeomorphic Framework: From the Discontinuous Image Matching Problem to a Stochastic Growth Model," Ph.D Thesis, École Normale Supérieure de Cachan, 2009. Google Scholar

[20]

L. Younes, "Shapes and Diffeomorphisms," Applied Mathematical Sciences, 171, Springer-Verlag, Berlin, 2010.  Google Scholar

[21]

L. Younes, F. Arrate and M. I. Miller, Evolutions equations in computational anatomy, NeuroImage, 45 (2009), S40-S50. doi: 10.1016/j.neuroimage.2008.10.050.  Google Scholar

show all references

References:
[1]

M. Bauer, P. Harms and P. W. Michor, Almost local metrics on shape space of hypersurfaces in n-space,, to appear in: SIAM Journal on Imaging Sciences, ().   Google Scholar

[2]

A. Clark, C. Cotter and J. Peiro, A reparameterisation-based approach to geodesic shooting for 2D curve matching,, work in progress., ().   Google Scholar

[3]

H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, in "Geometry of Differential Equations" (eds. A. Khovanskii, A. Varchenko and V. Vassiliev), Amer. Math. Soc. Trans. Ser. 2, 186, Amer. Math. Soc., Providence, RI, (1998), 1-25.  Google Scholar

[4]

H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), x+108 pp.  Google Scholar

[5]

C. J. Cotter, The variational particle-mesh method for matching curves, J. Phys. A, 41 (2008), 344003, 18 pp.  Google Scholar

[6]

C. J. Cotter and D. D. Holm, Geodesic boundary value problems with symmetry, J. Geom. Mech., 2 (2010), 51-68.  Google Scholar

[7]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze and T. S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Rational Mech. Anal., 197 (2010), 811-902. doi: 10.1007/s00205-010-0305-y.  Google Scholar

[8]

D. C. P. Ellis, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys., 61 (2011), 2120-2146.  Google Scholar

[9]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard, Invariant higher-order variational problems,, Comm. Math. Phys. \textbf{309}(2), 309 (): 413.   Google Scholar

[10]

F. Gay-Balmaz, D. D. Holm and T. S. Ratiu, Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, Journal of the Brazilian mathematical society, 42 (): 579.   Google Scholar

[11]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[12]

A. Kriegl and P. W. Michor, "The Convenient Setting of Global Analysis," Mathematical Surveys and Monographs, 53, AMS, Providence, RI, 1997.  Google Scholar

[13]

J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Note Series, 174, Cambridge University Press, Cambridge, 1992.  Google Scholar

[14]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[15]

P. W. Michor, Manifolds of smooth maps. III: The principal bundle of embeddings of a noncompact smooth manifold, Cah. Top. Géom. Diff., 21 (1980), 325-337.  Google Scholar

[16]

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.  Google Scholar

[17]

P. W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc., 8 (2006), 1-48. doi: 10.4171/JEMS/37.  Google Scholar

[18]

R. Montgomery, Gauge theory of the falling cat, in "Dynamics and Control of Mechanical Systems" (Waterloo, ON, 1992), Fields Inst. Comm., 1, Amer. Math. Soc., Providence, RI, (1993), 193-218.  Google Scholar

[19]

F.-X. Vialard, "Hamiltonian Approach to Shape Spaces in a Diffeomorphic Framework: From the Discontinuous Image Matching Problem to a Stochastic Growth Model," Ph.D Thesis, École Normale Supérieure de Cachan, 2009. Google Scholar

[20]

L. Younes, "Shapes and Diffeomorphisms," Applied Mathematical Sciences, 171, Springer-Verlag, Berlin, 2010.  Google Scholar

[21]

L. Younes, F. Arrate and M. I. Miller, Evolutions equations in computational anatomy, NeuroImage, 45 (2009), S40-S50. doi: 10.1016/j.neuroimage.2008.10.050.  Google Scholar

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