## An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

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Venue: | Applied and Computational Harmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URL http://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf |

Citations: | 47 - 22 self |

### BibTeX

@INPROCEEDINGS{Michor_anoverview,

author = {Peter W. Michor and David Mumford},

title = {An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach},

booktitle = {Applied and Computational Harmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URL http://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf},

year = {}

}

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### Abstract

Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

### Citations

216 | The convenient setting of global analysis
- Kriegl, Michor
- 1997
(Show Context)
Citation Context ...tangent bundlesRIEMANNIAN METRICS ON SPACES OF CURVES, HAMILTONIAN APPROACH 9 have G-gradients. This requirement has only to be satisfied for the first derivative, for the higher ones it follows (see =-=[9]-=-). We shall denote by C ∞ G (Imm(S1 , R 2 )) the space of such smooth functions. We shall always assume that G is invariant under the reparametrization group Diff(S 1 ), hence each such metric induces... |

168 |
Computing large deformation metric mappings via geodesic flows of diffeomorphisms
- Beg, Miller, et al.
- 2005
(Show Context)
Citation Context ... i − j)! R2 〈∂ i x∂ j yX,∂ i x∂ j yY 〉dx 1 dx 2 〈LX,Y 〉dx.dy, where L = (1 − A∆) n ,∆ = ∂ 2 x + ∂ 2 y. These metrics have been extensively studied by Miller, Younes and Trouvé and their collaborators =-=[5, 15, 16, 21]-=-. Since these metrics are right invariant, all maps to coset spaces Diff(R2 ) → Diff(R2 )/H are submersions. In particular, this metric gives a quotient metric on Emb(S1 , R2 ) and Be which we will de... |

123 | Computable elastic distance between shapes
- Younes
(Show Context)
Citation Context ...mannian metric leads to geodesics, curvature and diffusion and, hopefully, to an understanding of the global geometry of the space. Much work has been done in this direction recently (see for example =-=[10, 13, 15, 16, 24]-=-). The purpose of the present paper is two-fold. On the one hand, we want to survey the spectrum of Riemannian metrics which have been proposed (omitting, however, the Weil-Peterson metric). On the ot... |

117 |
On the metrics and Euler-Lagrange equations of computational anatomy. Annual review of biomedical engineering
- Miller, Trouve, et al.
- 2002
(Show Context)
Citation Context ...mannian metric leads to geodesics, curvature and diffusion and, hopefully, to an understanding of the global geometry of the space. Much work has been done in this direction recently (see for example =-=[10, 13, 15, 16, 24]-=-). The purpose of the present paper is two-fold. On the one hand, we want to survey the spectrum of Riemannian metrics which have been proposed (omitting, however, the Weil-Peterson metric). On the ot... |

112 | Riemannian Geometries on Spaces of Plane Curves
- Michor
- 2004
(Show Context)
Citation Context ...mannian metric leads to geodesics, curvature and diffusion and, hopefully, to an understanding of the global geometry of the space. Much work has been done in this direction recently (see for example =-=[10, 13, 15, 16, 24]-=-). The purpose of the present paper is two-fold. On the one hand, we want to survey the spectrum of Riemannian metrics which have been proposed (omitting, however, the Weil-Peterson metric). On the ot... |

108 | Group actions, homeomorphisms, and matching: a general framework
- Miller, Younes
- 2001
(Show Context)
Citation Context |

53 |
Gradient Flows
- Ambrosio, Gigli, et al.
- 2005
(Show Context)
Citation Context ....9. The Wasserstein metric and a related GΦ-metric. The Wasserstein metric (also known as the Monge-Kantorovich metric) is a metric between probability measures on a common metric space, see [1], and =-=[2]-=- for more details. It has been studied for many years globally and is defined as follows: let µ and ν be 2 probability measures on a metric space (X,d). Consider all measures ρ on X × X whose marginal... |

41 | Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms - Michor, Mumford - 2005 |

38 | On shape of plane elastic curves
- Mio, Srivastava, et al.
(Show Context)
Citation Context ...Ln(h),k〉ds where (2) S1 Ln(h) or Ln,c(h) = (I + (−1) n A.D 2n s )(h) The interesting special case n = 1 and A → ∞ has been studied by Trouvé and Younes in [21, 24] and by Mio, Srivastava and Joshi in =-=[17, 18]-=-. In this case, the metric reduces to: S 1 G imm,1,∞ � c (h,k) = S1 〈Ds(h),Ds(k)〉.ds which ignores translations, i.e. it is a metric on Imm(S 1 , R 2 ) modulo translations. Now identify R 2 with C, so... |

27 | Elastic-String Models for Representation and Analysis
- Mio, Srivastava
- 2004
(Show Context)
Citation Context ...Ln(h),k〉ds where (2) S1 Ln(h) or Ln,c(h) = (I + (−1) n A.D 2n s )(h) The interesting special case n = 1 and A → ∞ has been studied by Trouvé and Younes in [21, 24] and by Mio, Srivastava and Joshi in =-=[17, 18]-=-. In this case, the metric reduces to: S 1 G imm,1,∞ � c (h,k) = S1 〈Ds(h),Ds(k)〉.ds which ignores translations, i.e. it is a metric on Imm(S 1 , R 2 ) modulo translations. Now identify R 2 with C, so... |

24 | H0-type Riemannian metrics on spaces of planar curves
- Shah
- 2005
(Show Context)
Citation Context ...by scaled arc length. We show that it is sandwiched between the conformal metric Gℓ−1 Weights of the form Φ(ℓ,κ) = f(ℓ) and G ΦW where ΦW = ℓ −1 + 1 12 ℓκ2 . were studied in [10] and independently by =-=[19]-=-. The latter are attractive because they give metrics which are conformally equivalent to G 0 . These metrics are a borderline case between really stable metrics on Be and the metric G 0 for which pat... |

22 | Some Geometric Evolution Equations Arising as Geodesic Equations on Groups of Diffeomorphism, Including the Hamiltonian Approach
- Michor
- 2006
(Show Context)
Citation Context ...plectic form from the contantent bundle to T Imm(S 1 , R 2 ). We use the basics of symplectic geometry and momentum mappings on cotangent bundles in infinite dimensions, and we explain each step. See =-=[12]-=-, section 2, for a detailed exposition in similar notation as used here. 2.1. The setting. Consider as above the smooth Fréchet manifold Imm(S 1 , R 2 ) of all immersions S 1 → R 2 which is an open su... |

21 | Local geometry of deformable templates
- Trouvé, Younes
- 2005
(Show Context)
Citation Context ...s in Emb(S1 , R2 ).sRIEMANNIAN METRICS ON SPACES OF CURVES, HAMILTONIAN APPROACH 37 An even stronger theorem, proving global existence on the level of H k -diffeomorphisms on R 2 , has been proved by =-=[20, 22, 23]-=-. Proof. Let c ∈ H k . We begin by checking that F ′ c is a pseudo differential operator kernel of order −2n + 2 as we did for Fc in 5.3. c(θ1) − c(θ2) =: ˜c(θ1,θ2)(θ1 − θ2) gradF(x) = 1 � e 2π i〈x,ξ〉... |

15 |
Infinite Dimensional Group Action and Pattern Recognition. Technical report, DMI, École Nationale Supérieure
- Trouvé
- 1995
(Show Context)
Citation Context ...s in Emb(S1 , R2 ).sRIEMANNIAN METRICS ON SPACES OF CURVES, HAMILTONIAN APPROACH 37 An even stronger theorem, proving global existence on the level of H k -diffeomorphisms on R 2 , has been proved by =-=[20, 22, 23]-=-. Proof. Let c ∈ H k . We begin by checking that F ′ c is a pseudo differential operator kernel of order −2n + 2 as we did for Fc in 5.3. c(θ1) − c(θ2) =: ˜c(θ1,θ2)(θ1 − θ2) gradF(x) = 1 � e 2π i〈x,ξ〉... |

12 |
Metrics in the space of curves
- Yezzi, Mennucci
- 2005
(Show Context)
Citation Context ...mannian metric leads to geodesics, curvature and diffusion and, hopefully, to an understanding of the global geometry of the space. Much work has been done in this direction recently (see for example =-=[10, 12, 14, 15, 23]-=-). The purpose of the present paper is two-fold. On the one hand, we want to survey the spectrum of Riemannian metrics which have been proposed (omitting, however, the Weil-Peterson metric). On the ot... |

9 |
Gradient flows with metric and differentiable structures, and applications to the Wasserstein space
- Ambrosio, Gigli, et al.
(Show Context)
Citation Context ...several special cases. The weights Φ(ℓ,κ) = 1 + Aκ 2 were introduced and studied in [13]. As we shall see, this metric is also closely connected to the Wasserstein metric on probability measures (see =-=[1]-=-), if we assign to a curve C the probability measure given by scaled arc length. We show that it is sandwiched between the conformal metric Gℓ−1 Weights of the form Φ(ℓ,κ) = f(ℓ) and G ΦW where ΦW = ℓ... |

7 |
A Lie group structure for pseudo differential operators
- Adams, Ratiu, et al.
- 1986
(Show Context)
Citation Context ...on, along the lines used above. 5.5. The geodesic equation on Emb(S 1 , R 2 ), direct approach. The space of invertible pseudo differential operators on a compact manifold is a regular Lie group (see =-=[3]-=-), so we can use the usual formula d(A −1 ) = −A −1 .dA.A −1 for computing the derivative of Lc with respect to c. Note that we have a simple expression for Dc,hFc, namely Dc,hFc(θ1,θ2) = dF(c(θ1) − c... |

7 |
A computational fluid mechanics solution to the Monge– Kantorovich mass transfer problem
- Benamou, Brenier
- 1987
(Show Context)
Citation Context ... all measures ρ on X × X whose marginals under the 2 projections are µ and ν. Then: �� dwass(µ,ν) = inf d(x,y)dρ(x,y). ρ:p1,∗(ρ)=µ,p2,∗(ρ)=ν X×X It was discovered only recently by Benamou and Brenier =-=[6]-=- that, if X = R n , this is, in fact, path length for a Riemannian metric on the space of probability measures P. In their theory, the tangent space at µ to the space of probability measures and the i... |

5 | Local analysis on a shape manifold
- Trouvé, Younes
- 2002
(Show Context)
Citation Context ...s in Emb(S1 , R2 ).sRIEMANNIAN METRICS ON SPACES OF CURVES, HAMILTONIAN APPROACH 37 An even stronger theorem, proving global existence on the level of H k -diffeomorphisms on R 2 , has been proved by =-=[20, 22, 23]-=-. Proof. Let c ∈ H k . We begin by checking that F ′ c is a pseudo differential operator kernel of order −2n + 2 as we did for Fc in 5.3. c(θ1) − c(θ2) =: ˜c(θ1,θ2)(θ1 − θ2) gradF(x) = 1 � e 2π i〈x,ξ〉... |

4 |
The homotopy type of the space of degree 0 immersed curves
- Kodama, Michor
(Show Context)
Citation Context ...s we have introduced can be combined in one commutative diagram: Diff(R 2 ) ↓ Diff(R 2 )/Diff 0 (R 2 ,∆) ≈ −→ Emb(S 1 , R 2 ) ⊂ Imm(S 1 , R 2 ) ↓ ↓ ↓ ≈ −→ Be ⊂ Bi Diff(R 2 )/Diff(R 2 ,∆) See [13] and =-=[8]-=- for the homotopy type of the spaces Imm(S 1 , R 2 ) and Bi. What is the infinitesimal version of this? We will use the notation X(R 2 ) to denote the Lie algebra of Diff(R 2 ), i.e., either the space... |

2 |
la gomtrie diffrentielle des groupes de Lie de dimension infinie et ses applications l’hydrodynamique des fluides parfaits. Annales de l’institut Fourier
- Sur
- 1966
(Show Context)
Citation Context ...ive the geodesic equations: they are all in the same family as fluid flow equations. We prove well posedness of the geodesic equation on Emb(S1 , R2 ) and on Be. Although there is a formula of Arnold =-=[4]-=- for the sectional curvature of any right-invariant metric on a Lie group, we have not computed sectional curvatures for the quotient spaces. In the final section 6, we study two examples to make clea... |

1 |
Sobolev–type metrics in the space of curves. arXiv:math.DG/0605017
- Mennucci, Yezzi, et al.
(Show Context)
Citation Context ...nvariant momentum along a geodesic turns out to be the time derivative of log(ℓ). At this time, we do not know the sectional curvature for this metric. Sobolev-metrics of type H n are also studied in =-=[11]-=-, in particular in view of the completion of the space of curves. In the next section 5, we start with the basic right invariant metrics on Diff(R 2 ) which are given by the Sobolev H n -inner product... |