Bi-invariant means on Lie groups with Cartan-Schouten connections

28/08/2013
Auteurs : Xavier Pennec
OAI : oai:www.see.asso.fr:2552:5082
DOI :

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Bi-invariant means on Lie groups with Cartan-Schouten connections

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Xavier Pennec Asclepios team, INRIA Sophia- Antipolis – Mediterranée, France Bi-invariant Means on Lie groups with Cartan-Schouten connections GSI, August 2013 X. Pennec - GSI, Aug. 30, 2013 2 Design Mathematical Methods and Algorithms to Model and Analyze the Anatomy  Statistics of organ shapes across subjects in species, populations, diseases…  Mean shape  Shape variability (Covariance)  Model organ development across time (heart-beat, growth, ageing, ages…)  Predictive (vs descriptive) models of evolution  Correlation with clinical variables Computational Anatomy Statistical Analysis of Geometric Features Noisy Geometric Measures  Tensors, covariance matrices  Curves, tracts  Surfaces  Transformations  Rigid, affine, locally affine, diffeomorphisms Goal:  Deal with noise consistently on these non-Euclidean manifolds  A consistent statistical (and computing) framework X. Pennec - GSI, Aug. 30, 2013 3 X. Pennec - GSI, Aug. 30, 2013 4 Statistical Analysis of the Scoliotic Spine Data  307 Scoliotic patients from the Montreal’s St-Justine Hosp  3D Geometry from multi-planar X-rays  Articulated model:17 relative pose of successive vertebras Statistics  Main translation variability is axial (growth?)  Main rot. var. around anterior-posterior axis  4 first variation modes related to King’s classes [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ] Morphometry through Deformations 5X. Pennec - GSI, Aug. 30, 2013 Measure of deformation [D’Arcy Thompson 1917, Grenander & Miller]  Observation = “random” deformation of a reference template  Deterministic template = anatomical invariants [Atlas ~ mean]  Random deformations = geometrical variability [Covariance matrix] Patient 3 Atlas Patient 1 Patient 2 Patient 4 Patient 5 1 2 3 4 5 Hierarchical Deformation model Varying deformation atoms for each subject M3 M4 M5 M6 M1 M2 M0 K M3 M4 M5 M6 M1 M2 M0 1 … Subject level: 6 Spatial structure of the anatomy common to all subjects w0 w1 w2 w3 w4 w5 w6 Population level: Aff(3) valued trees X. Pennec - GSI, Aug. 30, 2013[Seiler, Pennec, Reyes, Medical Image Analysis 16(7):1371-1384, 2012] X. Pennec - GSI, Aug. 30, 2013 7 Outline Riemannian frameworks on Lie groups Lie groups as affine connection spaces A glimpse of applications in infinite dimensions Conclusion and challenges X. Pennec - GSI, Aug. 30, 2013 8 Riemannian geometry is a powerful structure to build consistent statistical computing algorithms Shape spaces & directional statistics  [Kendall StatSci 89, Small 96, Dryden & Mardia 98] Numerical integration, dynamical systems & optimization  [Helmke & Moore 1994, Hairer et al 2002]  Matrix Lie groups [Owren BIT 2000, Mahony JGO 2002]  Optimization on Matrix Manifolds [Absil, Mahony, Sepulchre, 2008] Information geometry (statistical manifolds)  [Amari 1990 & 2000, Kass & Vos 1997]  [Oller & Corcuera Ann. Stat. 1995, Battacharya & Patrangenaru, Ann. Stat. 2003 & 2005] Statistics for image analysis  Rigid body transformations [Pennec PhD96]  General Riemannian manifolds [Pennec JMIV98, NSIP99, JMIV06]  PGA for M-Reps [Fletcher IPMI03, TMI04]  Planar curves [Klassen & Srivastava PAMI 2003] Geometric computing  Subdivision scheme [Rahman,…Donoho, Schroder SIAM MMS 2005] X. Pennec - GSI, Aug. 30, 2013 9 The geometric framework: Riemannian Manifolds Riemannian metric :  Dot product on tangent space  Speed, length of a curve  Geodesics are length minimizing curves  Riemannian Distance Operator Euclidean space Riemannian manifold Subtraction Addition Distance Gradient descent )( ttt xCxx   )(yLogxy x xyxy  xyyx ),(dist x xyyx ),(dist )(xyExpy x ))(( txt xCExpx t   xyxy  Unfolding (Logx), folding (Expx)  Vector -> Bipoint (no more equivalent class) Exponential map (Normal coord. syst.) :  Geodesic shooting: Expx(v) = g(x,v)(1)  Log: find vector to shoot right (geodesic completeness!) 10 Statistical tools: Moments Frechet / Karcher mean minimize the variance Existence and uniqueness : Karcher / Kendall / Le / Afsari Gauss-Newton Geodesic marching Covariance (PCA) [higher moments]  xyEwith)(expx x1  vvtt        M M )().(.x.xx.xE TT zdzpzz xxx xx         0)(0)().(.xxE),dist(Eargmin 2   CPzdzpy y MM MxxxxxΕ X. Pennec - GSI, Aug. 30, 2013 [Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ] 11 Distributions for parametric tests Generalization of the Gaussian density:  Stochastic heat kernel p(x,y,t) [complex time dependency]  Wrapped Gaussian [Infinite series difficult to compute]  Maximal entropy knowing the mean and the covariance Mahalanobis D2 distance / test:  Any distribution:  Gaussian:           2/x..xexp.)( T xΓxkyN       rOk n /1.)det(.2 32/12/    Σ    rO /Ric3 1)1(    ΣΓ yx..yx)y( )1(2   xxx t    n)(E 2 xx  rOn /)()( 322  xx [ Pennec, NSIP’99, JMIV 2006 ] X. Pennec - GSI, Aug. 30, 2013 Natural Riemannian Metrics on Lie Groups Lie groups: Smooth manifold G compatible with group structure  Composition g o h and inversion g-1 are smooth  Left and Right translation Lg(f) = g o f Rg (f) = f o g Natural Riemannian metric choices  Chose a metric at Id: Id  Propagate at each point g using left (or right) translation g = < DLg (-1) .x , DLg (-1) .y >Id  Practical computations using left (or right) translations X. Pennec - GSI, Aug. 30, 2013 12   g)(f.LogDL(g)Logfgx).DL(ExpfxExp 1)( Idffff 1)(    Id 13 Example on 3D rotations Space of rotations SO(3):  Manifold: RT.R=Id and det(R)=+1  Lie group ( R1 o R2 = R1.R2 & Inversion: R(-1) = RT ) Metrics on SO(3): compact space, there exists a bi-invariant metric  Left / right invariant / induced by ambient space = Tr(XT Y) Group exponential  One parameter subgroups = bi-invariant Geodesic starting at Id  Matrix exponential and Rodrigue’s formula: R=exp(X) and X = log(R)  Geodesic everywhere by left (or right) translation LogR(U) = R log(RT U) / ExpR(X) = R exp(RT X) / Bi-invariant Riemannian distance  d(R,U) = ||log(RT U)|| = q( RT U ) X. Pennec - GSI, Aug. 30, 2013 14 General Non-Compact and Non-Commutative case No Bi-invariant Mean for 2D Rigid Body Transformations  Metric at Identity: