Reparameterization invariant metric on the space of curves

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14330

Résumé

This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions =Imm([0,1],M) by pullback of a metric on the tangent bundle T derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on T induces a first-order Sobolev metric on with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.

Reparameterization invariant metric on the space of curves

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application/pdf Reparameterization invariant metric on the space of curves Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco

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Optimal matching between curves in a manifold
Drone Tracking Using an Innovative UKF
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
Session Geometrical Structures of Thermodynamics (chaired by Frédéric Barbaresco, François Gay-Balmaz)
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GSI'17-Closing session
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Démonstrateur franco-britannique "IRM" : gestion intelligente et homéostatique des radars multifonctions
Principes & applications de la conjugaison de phase en radar : vers les antennes autodirectives
Nouvelles formes d'ondes agiles en imagerie, localisation et communication
Compréhension et maîtrise des tourbillons de sillage
Wake vortex detection, prediction and decision support tools
Ordonnancement des tâches pour radar multifonction avec contrainte en temps dur et priorité
Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics
Reparameterization invariant metric on the space of curves
Probability density estimation on the hyperbolic space applied to radar processing
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Opening Session (chaired by Frédéric Barbaresco)
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Koszul Information Geometry & Souriau Lie Group 4Thermodynamics
MaxEnt’14, The 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
Koszul Information Geometry & Souriau Lie Group Thermodynamics
Robust Burg Estimation of stationary autoregressive mixtures covariance
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Koszul Information Geometry and Souriau Lie Group Thermodynamics
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Information/Contact Geometries and Koszul Entropy
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This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions =Imm([0,1],M) by pullback of a metric on the tangent bundle T derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on T induces a first-order Sobolev metric on with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.

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Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariant metric on the space of curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Geometric Science of Information Workshop 30 octobre 2015 Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Motivations Goal : given n curves in a manifold M, find the Fréchet mean or median curve. cm = argmin c n ∑ i=1 distk (ci ,c) k = 1 : median k = 2 : mean Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Motivations Goal : given n curves in a manifold M, find the Fréchet mean or median curve. cm = argmin c n ∑ i=1 distk (ci ,c) k = 1 : median k = 2 : mean Why compute a mean curve ? – statistical analysis of trajectories (urban or sea traffic, hurricanes...) – shape analysis (medical imagery, movement recognition in video...) Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Motivations Goal : given n curves in a manifold M, find the Fréchet mean or median curve. cm = argmin c n ∑ i=1 distk (ci ,c) k = 1 : median k = 2 : mean Why compute a mean curve ? – statistical analysis of trajectories (urban or sea traffic, hurricanes...) – shape analysis (medical imagery, movement recognition in video...) Why a manifold ? – Organ contour, car trajectory → plane curve – Trajectory of a hurricane or a ship → S2 -valued curve – If the points of the curves do not represent positions in space but more complex objects (covariance matrices) → more complex manifold Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection How does a radar work ? Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection How does a radar work ? Target detection Cell under test ↔ observation zi =t [zi 1,...,zi n] Cells of its environment ↔ mean observation zm Idea : compare zi and zm similar → no target significantly different → target Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection How does a radar work ? Target detection Cell under test ↔ observation zi =t [zi 1,...,zi n] Cells of its environment ↔ mean observation zm Idea : compare zi and zm similar → no target significantly different → target Stationarity hypothesis zi = realization of Zi centered Gaussian stationary process Instead of using directly the observation z, we use the estimation of the covariance matrix Σi =t (zi )zi Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection Information geometry setting Statistical manifold of the covariance matrices endowed with the Fisher Information metric To exploit the form of the matrices (Toeplitz), hyp : autoregressive model of order n −1 There is a bijection Σ ↔ (µ1,µ2,...,µn−1) ∈ Dn−1 reflexion coefficients of the AR model → Equivalent representation in the Poincaré Disk Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection Information geometry setting Statistical manifold of the covariance matrices endowed with the Fisher Information metric To exploit the form of the matrices (Toeplitz), hyp : autoregressive model of order n −1 There is a bijection Σ ↔ (µ1,µ2,...,µn−1) ∈ Dn−1 reflexion coefficients of the AR model → Equivalent representation in the Poincaré Disk Local stationarity hypothesis Each stationary portion is represented by its estimated covariance matrix Each cell is represented by a a path in the space of THDP matrices or several paths in D Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Application to radar detection Information geometry setting Statistical manifold of the covariance matrices endowed with the Fisher Information metric To exploit the form of the matrices (Toeplitz), hyp : autoregressive model of order n −1 There is a bijection Σ ↔ (µ1,µ2,...,µn−1) ∈ Dn−1 reflexion coefficients of the AR model → Equivalent representation in the Poincaré Disk Local stationarity hypothesis Each stationary portion is represented by its estimated covariance matrix Each cell is represented by a a path in the space of THDP matrices or several paths in D What we need – A metric on the space of curves in a general manifold M (space of THPDM or Poincaré disk) – A way to compute a Fréchet mean or median of several curves according to this metric Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Setting and Notations The Riemannian setting – M manifold, M = Imm([0,1],M) space of curves in M – We equip M with a Riemannian metric G (scalar product on TM ) – We get a geodesic distance dist(c0,c1) = inf c(0,·)=c0,c(1,·)=c1 1 0 G ∂c ∂s , ∂c ∂s ds and the equations of the geodesics = (locally) shortest paths between points in the manifold = optimal deformations between curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Setting and Notations The Riemannian setting – M manifold, M = Imm([0,1],M) space of curves in M – We equip M with a Riemannian metric G (scalar product on TM ) – We get a geodesic distance dist(c0,c1) = inf c(0,·)=c0,c(1,·)=c1 1 0 G ∂c ∂s , ∂c ∂s ds and the equations of the geodesics = (locally) shortest paths between points in the manifold = optimal deformations between curves Notations – A point c ∈ M is a curve in the manifold M. – A tangent vetor h ∈ Tc M is a vector field along the curve c in M. – A path s → c(s) in M is a surface (s,t) → c(s,t) in M. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance We want a metric that verifies the equivariance property Gc◦φ(h ◦φ,k ◦φ) = Gc(h,k) ∀φ ∈ Diff([0,1],M) Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance We want a metric that verifies the equivariance property Gc◦φ(h ◦φ,k ◦φ) = Gc(h,k) ∀φ ∈ Diff([0,1],M) Why ? Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance We want a metric that verifies the equivariance property Gc◦φ(h ◦φ,k ◦φ) = Gc(h,k) ∀φ ∈ Diff([0,1],M) Why ? The distance between two curves is the same if we reparameterize them the same way dist(c0 ◦φ,c1 ◦φ) = dist(c0,c1) ∀φ Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance We want a metric that verifies the equivariance property Gc◦φ(h ◦φ,k ◦φ) = Gc(h,k) ∀φ ∈ Diff([0,1],M) Why ? The distance between two curves is the same if we reparameterize them the same way dist(c0 ◦φ,c1 ◦φ) = dist(c0,c1) ∀φ We induce a Riemannian metric on the shape space S = M /Diff([0,1],M) distS (c0 ◦φ,c1 ◦ψ) = distS (c0,c1) ∀φ,ψ A shape ¯c = equivalence class for the relation c1R c2 ssi ∃φ,c2 = c1 ◦φ The geodesics of the shape space S are the horizontal geodesics of the space of curves M ¯Gπ(c) (Tc π(h),Tc π(k)) = Gc (hH ,kH ) Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Reparameterization invariance Illustration : L2 -metric The L2 -metric is probably the most natural metric to think of Gc(h,k) = 1 0 h(t),k(t) dt. To make it reparameterization invariant, we need to take velocity into account Gc(h,k) = 1 0 h(t),k(t) c (t) dt. If we don’t, we can always find a deformation as small as we want that goes from one curve to another [Michor,Mumford,2005] Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives The SRV framework M = Rd [Michor, Mumford, 2005] Even if we include velocity, the distance induced by the L2 -metric on the shape space always vanishes → We introduce higher order derivatives : Sobolev and elastic metrics. [Srivastava, Klassen, Joshi, Jermyn, 2011] Particularly interesting elastic metric Gc(h,k) = D h⊥ ,D k⊥ + 1 4 D h ,D k d pullback of the L2 -metric via the "Square Root Velocity Function" (SRVF) R : c → c / c Gc(h,k) = TcR(h),TcR(k) dt R verifies Tc◦φR(h ◦φ) = |φ |1/2 TcR(h)◦φ, which guaranties the equivariance property. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Our generalization to curves in any manifold M M manifold With the SRVF we send the curves to the tangent space TM equipped with ˜G(ηs(s),ηs(s)) = cs(s,0) 2 + 1 0 ∇sv(s,t) 2 dt where s → η(s,·) = (c(s,·),v(s,·)) is a curve in TM . This gives on M by pullback, just as in the planar case Gc(h,k) = h(0),k(0) + 1 0 ∇ h⊥ ,∇ k⊥ + 1 4 ∇ h ,∇ k d Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Our generalization to curves in any manifold M M manifold With the SRVF we send the curves to the tangent space TM equipped with ˜G(ηs(s),ηs(s)) = cs(s,0) 2 + 1 0 ∇sv(s,t) 2 dt where s → η(s,·) = (c(s,·),v(s,·)) is a curve in TM . This gives on M by pullback, just as in the planar case Gc(h,k) = h(0),k(0) + 1 0 ∇ h⊥ ,∇ k⊥ + 1 4 ∇ h ,∇ k d Induced distance : We get a combination of the distance between the origins and the L2 -distance between the SRVF-images dist(c0,c1) = inf c 1 0 cs(s,0) 2 + 1 0 ∇sq(s,t) 2 dt ds, where q = ct / ct is the the SRV representation of c. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Characterization of geodesics Variational principle We introduce the notion of energy of a curve E(c) := 1 2 Gc(s) (cs(s),cs(s))ds. By Cauchy-Schwarz L2 (c) ≤ 2E(c) → it is enough to minimize the energy. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Characterization of geodesics Variational principle We introduce the notion of energy of a curve E(c) := 1 2 Gc(s) (cs(s),cs(s))ds. By Cauchy-Schwarz L2 (c) ≤ 2E(c) → it is enough to minimize the energy. c0,c1 ∈ M , s → c(s,·) : c(0,·) = c0,c(1,·) = c1. We consider a variation of c ˆc : (−ε,ε) → M a → ˆc(a,·,·) st ˆc(0,·,·) = c. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Characterization of geodesics Variational principle We introduce the notion of energy of a curve E(c) := 1 2 Gc(s) (cs(s),cs(s))ds. By Cauchy-Schwarz L2 (c) ≤ 2E(c) → it is enough to minimize the energy. c0,c1 ∈ M , s → c(s,·) : c(0,·) = c0,c(1,·) = c1. We consider a variation of c ˆc : (−ε,ε) → M a → ˆc(a,·,·) st ˆc(0,·,·) = c. The energy of this variation is ˆE : (−ε,ε) → R+ ˆE(a) := E (ˆc(a,·,·)) = 1 2 Gˆc(a,s,·) (ˆcs(a,s,·),ˆcs(a,s,·))ds and we want s → c(s,·) s.t. d da a=0 ˆE(a) = 0 for any variation ˆc. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Characterization of geodesics In our case, we have ˆE(a) = 1 2 ˆcs(a,s,0),ˆcs(a,s,0) ds + ∇sˆq(s,t),∇sˆq(s,t) dt ds, ˆE (a) = ∇aˆcs(a,s,0),ˆcs(a,s,0) ds + ∇a∇sˆq(a,s,t),∇s ˆq(a,s,t) dt ds, which we can rewrite, for a = 0, in 1 0 ∇scs(s,0)+ 1 0 B(s,t)t,0 dt , ˆca(0,s,0) ds + 1 0 1 0 D(s,t)+ 1 t B(s,u)u,t du , ∇t ˆca(0,s,t) dt ds = 0, with the notations B(s,t) = R (q,∇sq)cs(s,t), D(s,t) = 1√ ct ∇s∇sq(s,t) − 1 2 ct 5/2 ∇s∇sq , ct ct (s,t). Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Characterization of geodesics In our case, we have ˆE(a) = 1 2 ˆcs(a,s,0),ˆcs(a,s,0) ds + ∇sˆq(s,t),∇sˆq(s,t) dt ds, ˆE (a) = ∇aˆcs(a,s,0),ˆcs(a,s,0) ds + ∇a∇sˆq(a,s,t),∇s ˆq(a,s,t) dt ds, which we can rewrite, for a = 0, in 1 0 ∇scs(s,0)+ 1 0 B(s,t)t,0 dt , ˆca(0,s,0) ds + 1 0 1 0 D(s,t)+ 1 t B(s,u)u,t du , ∇t ˆca(0,s,t) dt ds = 0, with the notations B(s,t) = R (q,∇sq)cs(s,t), D(s,t) = 1√ ct ∇s∇sq(s,t) − 1 2 ct 5/2 ∇s∇sq , ct ct (s,t). We obtain equations describing the optimal deformation of one curve into another (∗)    ∇scs(s,0) = r(s,0) ∀s ∇s∇sq(s,t) = q(s,t) r(s,t)+r(s,t) ∀t,s with r(s,t) = − 1 t R (q,∇sq)cs(s,τ)τ,t dτ. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Discretization of our model In practice, we have a series of points that we link to one another Exponential map : to approximate the geodesic starting at point c0 with initial velocity u0, we solve the system (∗) at time s, if we know c(s,·) and cs(s,·), we propagate to time s +ε with c(s +ε,t) = expc(s,t) (εcs(s,ε)) ∀t cs(s +ε,t) = cs(s,t)+ε∇scs(s,t) ∀t, where expc(s,t) (εcs(s,ε)) is the point obtained by following the geodesic of M starting from c(s,t) at velocity εcs(s,ε), and ∇scs is deduced from (∗). Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Discretization of our model In practice, we have a series of points that we link to one another Exponential map : to approximate the geodesic starting at point c0 with initial velocity u0, we solve the system (∗) at time s, if we know c(s,·) and cs(s,·), we propagate to time s +ε with c(s +ε,t) = expc(s,t) (εcs(s,ε)) ∀t cs(s +ε,t) = cs(s,t)+ε∇scs(s,t) ∀t, where expc(s,t) (εcs(s,ε)) is the point obtained by following the geodesic of M starting from c(s,t) at velocity εcs(s,ε), and ∇scs is deduced from (∗). Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Discretization of our model In practice, we have a series of points that we link to one another Exponential map : to approximate the geodesic starting at point c0 with initial velocity u0, we solve the system (∗) at time s, if we know c(s,·) and cs(s,·), we propagate to time s +ε with c(s +ε,t) = expc(s,t) (εcs(s,ε)) ∀t cs(s +ε,t) = cs(s,t)+ε∇scs(s,t) ∀t, where expc(s,t) (εcs(s,ε)) is the point obtained by following the geodesic of M starting from c(s,t) at velocity εcs(s,ε), and ∇scs is deduced from (∗). Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Discretization of our model In practice, we have a series of points that we link to one another Exponential map : to approximate the geodesic starting at point c0 with initial velocity u0, we solve the system (∗) at time s, if we know c(s,·) and cs(s,·), we propagate to time s +ε with c(s +ε,t) = expc(s,t) (εcs(s,ε)) ∀t cs(s +ε,t) = cs(s,t)+ε∇scs(s,t) ∀t, where expc(s,t) (εcs(s,ε)) is the point obtained by following the geodesic of M starting from c(s,t) at velocity εcs(s,ε), and ∇scs is deduced from (∗). Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Towards computations of means and medians Geodesic shooting : computing the optimal deformation of one curve into another Starting from a point c0 ∈ M , "shoot" in the direction u0, then adjust u0 until convergence. Karcher flow : computing a Fréchet mean or median of several curves The mean curve is updated in the direction of the sum of the initial tangent vectors to the geodesics linking it to the curves. Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves Table of contents Introduction Riemannian metric on the space of curves Medians and means of curves Perspectives Thank you ! P. Michor, D. Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10, 217-245, 2005. A. Srivastava, E. Klassen, S. H. Joshi, and I. H. Jermyn. Shape analysis of elastic curves in Euclidean spaces. IEEE T. Pattern Anal., 33(7):1415-1428, 2011. A. L., M. Arnaudon, F. Barbaresco, Reparameterization invariant metric on the space of curves, arXiv:1507.06503. http://arxiv.org/abs/1507.06503 Alice Le Brigant Marc Arnaudon Frédéric Barbaresco Reparameterization invariant metric on the space of curves