Xavier Pennec. Statistical Computing on Manifolds for Computational Anatomy 3: Metric and Affine Geometric Settings for Lie Groups

Transcription

1 Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterranée, France With contributions from Vincent Arsigny, Pierre Fillard, Marco Lorenzi, Christof Seiler, Jonathan Boisvert, Nicholas Ayache, etc Statistical Computing on Manifolds for Computational Anatomy 3: Metric and Affine Geometric Settings for Lie Groups Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis

2 Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb

3 Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Riemannian frameworks on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection The SVF framework for diffeomorphisms Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb

4 Statistical Analysis of the Scoliotic Spine Data Statistics 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 Main translation variability is axial (growth?) Main rot. var. around anterior-posterior axis 4 first variation modes related to King s classes X. Pennec - ESI - Shapes, Feb [ J. Boisvert et al. ISBI 06, AMDO 06 and IEEE TMI 27(4), 2008 ] 4

5 Bronze Standard Rigid Registration Validation Best explanation of the observations (ML) : LSQ criterion Robust Fréchet mean Robust initialization and Newton gradient descent d C ij d 2 ( T ij, Tˆ ij ) ( T1, T2 ) min ( T1, T2 ), Result Ti, j,, rot trans [ T. Glatard & al, MICCAI 2006, Int. Journal of HPC Apps, 2006 ] Derive tests on transformations for accuracy / consistency X. Pennec - ESI - Shapes, Feb

6 Morphometry through Deformations 1 Atlas 5 Patient 1 Patient Patient 2 Patient 3 Patient 4 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] X. Pennec - ESI - Shapes, Feb

7 Natural Riemannian Metrics on Transformations Transformation are Lie groups: Smooth manifold G compatible with group structure Composition g o h and inversion g -1 are smooth Left and Right translation L g (f) = g f R g (f) = f g Conjugation Conj g (f) = g f g -1 Natural Riemannian metric choices Chose a metric at Id: <x,y> Id Propagate at each point g using left (or right) translation <x,y> g = < DL (-1) g.x, DLg (-1).y >Id Implementation Exp Practical computations using left (or right) translations ( 1) x f Exp (DL (.x) fg Log (g) DL.Log (f g) f Id 1) f X. Pennec - ESI - Shapes, Feb f f Id

8 Space of rotations SO(3): Example on 3D rotations Manifold: R T.R=Id and det(r)=+1 Lie group ( R 1 o R 2 = R 1.R 2 & Inversion: R (-1) = R T ) Metrics on SO(3): compact space, there exists a bi-invariant metric Left / right invariant / induced by ambient space <X, Y> = Tr(X T 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 Log R (U) = R log(r T U) Exp R (X) = R exp(r T X) Bi-invariant Riemannian distance d(r,u) = log(r T U) = q( R T U ) X. Pennec - ESI - Shapes, Feb

9 General Non-Compact and Non-Commutative case No Bi-invariant Mean for 2D Rigid Body Transformations Metric at Identity: dist(id, θ; t 1 ; t 2 ) 2 = θ 2 + t t 2 2 T 1 = π ; 2 ; T 2 = 0; 2; 0 T 3 = π ; 2 ; Left-invariant Fréchet mean: 0; 0; 0 Right-invariant Fréchet mean: 0; 3 2 ; 0 (0; ; 0) Questions for this talk: Can we design a mean compatible with the group operations? Is there a more convenient structure for statistics on Lie groups? X. Pennec - ESI - Shapes, Feb

10 Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Riemannian frameworks on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection The SVF framework for diffeomorphisms Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb

11 Basics of Lie groups Flow of a left invariant vector field X = DL. x from identity γ x t exists for all time One parameter subgroup: γ x s + t = γ x s. γ x t Lie group exponential Definition: x g Exp x = γ x 1 ε G Diffeomorphism from a neighborhood of 0 in g to a neighborhood of e in G (not true in general for inf. dim) 3 curves parameterized by the same tangent vector Left / Right-invariant geodesics, one-parameter subgroups Question: Can one-parameter subgroups be geodesics? X. Pennec - ESI - Shapes, Feb

12 Affine connection spaces Affine Connection (infinitesimal parallel transport) Acceleration = derivative of the tangent vector along a curve Projection of a tangent space on a neighboring tangent space Geodesics = straight lines Null acceleration: γ γ = 0 2 nd order differential equation: Normal coordinate system Adapted from Lê Nguyên Hoang, science4all.org Local exp and log maps (Strong form of Whitehead theorem: In an affine connection space, each point has a normal convex neighborhood (unique geodesic between any two points included in the NCN) X. Pennec - ESI - Shapes, Feb

13 Canonical connections on Lie groups Left invariant connections DL.X DL. Y = DL. X Y Characteric bilinear form on the Lie algebra a x, y = X Y e g Symmetric part 1 (a x, y + a y, x ) specifies geodesics 2 Skew symmetric part 1 (a x, y a y, x ) specifies torsion along them 2 Bi-invariant connections a Ad g. x, Ad g. y = Ad g. a x, y a [z, x], y + a x, [z, y] = z, a x, y, x, y, z in g Cartan Schouten connections (def. of Postnikov) LInv connections for which one-parameter subgroups are geodesics Matrices : M(t) = exp(t.v) Diffeos : translations of Stationary Velocity Fields (SVFs) Uniquely determined by a x, x = 0 (skew symmetry) X. Pennec - ESI - Shapes, Feb

14 Canonical Cartan connections on Lie groups Bi-invariant Cartan Schouten connections Family a x, y = λ x, y (-,0, + connections for l=0,1/2,1) Turner Laquer 1992: exhaust all of them for compact simple Lie groups except SU(n) (2- dimensional family) Same group geodesics (a x, y + a y, x = 0): one-parameter subgroups and their left and right translations Curvature: R x, y = λ λ 1 [ x, y, z] Torsion: T x, y = 2a x, y x, y Left/Right Cartan-Schouten Connection (l=0/l=1) Flat space with torsion (absolute parallelism) Left (resp. Right)-invariant vector fields are covariantly constant Parallel transport is left (resp. right) translation Unique symmetric bi-invariant Cartan connection (l=1/2) a x, y = 1 2 x, y Curvature R x, y z = 1 4 x, y, z Parallel transport along geodesics: Π exp (y) x = DL exp ( y 2 ). DL exp ( y 2 ).x X. Pennec - ESI - Shapes, Feb

15 Cartan Connections are generally not metric Levi-Civita Connection of a left-invariant (pseudo) metric is left-invariant Metric dual of the bracket < ad x, y, z > = < x, z, y > a x, y = 1 2 x, y 1 2 ad x, y + ad y, x Bi-invariant (pseudo) metric => Symmetric Cartan connection A left-invariant (pseudo) metric is right-invariant if it is Ad-invariant < x, y > = < Ad g x, Ad g y > Infinitesimaly: < x, z, y > + < x, y, z > = 0 or ad x, y + ad y, x = 0 Existence of bi-invariant (pseudo) metrics A Lie group admits a bi-invariant metric iff Ad(G) is relatively compact Ad G O g GL(g) No bi-invariant metrics for rigid-body transformations Bi-invariant pseudo metric (Quadratic Lie groups): Medina decomposition [Miolane MaxEnt 2014] Bi-inv. pseudo metric for SE(n) for n=1 or 3 only X. Pennec - ESI - Shapes, Feb

16 Group Geodesics Group geodesic convexity Take an NCN V at Id There exists a NCN V g = g V V g at each g s.t.: g Exp x = Exp(Ad g. x) g Log g h g 1 = Ad g. Log h Group geodesics in V g Exp g v = g Exp DL g 1. v = Exp DR g 1. v g Log g h = DL g. Log g 1 h = DR g. Log h g 1 X. Pennec - ESI - Shapes, Feb

17 Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Riemannian frameworks on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection The SVF framework for diffeomorphisms Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb

18 Mean value on an affine connection space Fréchet / Karcher means not usable (no distance) but: argmin Edist(, ) 2 y x Ε x Exponential barycenters [Emery & Mokobodzki 91, Corcuera & Kendall 99] ym E xx xx. p ( z). dm( z) 0 P( C) 0 Log x y μ(dy) = 0 or Existence? Uniqueness? i Log x y i = 0 OK for convex affine manifolds with semi-local convex geometry [Arnaudon & Li, Ann. Prob. 33-4, 2005] Use a separating function (convex function separating points) instead of a distance Algorithm to compute the mean: fixed point iteration (stability?) M x X. Pennec - ESI - Shapes, Feb

19 Bi-invariant Mean on Lie Groups Exponential barycenter of the symmetric Cartan connection Locus of points where Log m 1. g i = 0 (whenever defined) Iterative algorithm: m t+1 = m t Exp 1 Log m 1 n t. g i First step corresponds to the Log-Euclidean mean Corresponds to the first definition of bi-invariant mean of [V. Arsigny, X. Pennec, and N. Ayache. Research Report RR-5885, INRIA, April 2006.] Mean is stable by left / right composition and inversion If m is a mean of g i and h is any group element, then h m is a mean of h g i, m h is a mean of the points g i h and m ( 1) is a mean of g i ( 1) [Pennec & Arsigny, Ch.7 p , Matrix Information Geometry, Springer, 2012] X. Pennec - ESI - Shapes, Feb

20 Bi-invariant Mean on Lie Groups Fine existence If the data points belong to a sufficiently small normal convex neighborhood of some point, then there exists a unique solution in this NCN. Moreover, the iterated point strategy converges at least at a linear rate towards this unique solution, provided the initialization is close enough. Proof: using an auxiliary metric, the iteration is a contraction. Closed-form for 2 points m(t) = x Exp (t. Log x 1 y ) X. Pennec - ESI - Shapes, Feb

21 Special Matrix Groups Scaling and translations ST(n) No bi-invariant metric Group geodesics defined globally, all points are reachable Existence and uniqueness of bi-invariant mean (closed form) Group / left-invariant / right-invariant geodesics X. Pennec - ESI - Shapes, Feb

22 Special matrix groups Heisenberg Group (resp. Scaled Upper Unitriangular Matrix Group) No bi-invariant metric Group geodesics defined globally, all points are reachable Existence and uniqueness of bi-invariant mean (closed form resp. solvable) Rigid-body transformations Logarithm well defined iff log of rotation part is well defined, i.e. if the 2D rotation have angles θ i < π Existence and uniqueness with same criterion as for rotation parts (same as Riemannian) Invertible linear transformations Logarithm unique if no complex eigenvalue on the negative real line Generalization of geometric mean (as in LE case but different) X. Pennec - ESI - Shapes, Feb

23 Example mean of 2D rigid-body transformation T 1 = π 4 ; 2 2 ; 2 2 T 2 = 0; 2; 0 T 3 = π 4 ; 2 2 ; 2 2 Metric at Identity: dist(id, θ; t 1 ; t 2 ) 2 = θ 2 + t t 2 2 Left-invariant Fréchet mean: 0; 0; 0 Log-Euclidean mean: 0; 2 π/4 3 ; 0 (0; ; 0) Bi-invariant mean: 0; 2 π/4 1+π/4( 2+1) ; 0 (0; ; 0) Right-invariant Fréchet mean: 0; 3 2 ; 0 (0; ; 0) X. Pennec - ESI - Shapes, Feb

24 Generalization of the Statistical Framework Covariance matrix & higher order moments Defined as tensors in tangent space Σ = Log x y Log x y μ(dy) Matrix expression changes according to the basis Other statistical tools Mahalanobis distance well defined and bi-invariant μ m,σ (g) = Log m g i ( 1) Σ ij Logm g j μ(dy) Tangent Principal Component Analysis (t-pca) Principal Geodesic Analysis (PGA), provided a data likelihood Independent Component Analysis (ICA) X. Pennec - ESI - Shapes, Feb

25 Generalizing the statistical setting Covariance matrix & higher order moments Can be defined as a 2-covariant tensor Σ = Log x y Log x y μ(dy) Cannot be defined as S ij = E( <x e i ><x e j >) (no dot product) Diagonalization depends on the basis: change PCA to ICA? Mahalanobis distance is well defined and bi-invariant μ m,σ (g) = Log m g i ( 1) Σ ij Logm g j μ(dy) X. Pennec - ESI - Shapes, Feb

26 Cartan Connections vs Riemannian What is similar Standard differentiable geometric structure [curved space without torsion] Normal coordinate system with Exp x et Log x [finite dimension] Limitations of the affine framework No metric (but no choice of metric to justify) The exponential does always not cover the full group Pathological examples close to identity in finite dimension In practice, similar limitations for the discrete Riemannian framework Global existence and uniqueness of bi-invariant mean? Use a bi-invariant pseudo-riemannian metric? [Miolane MaxEnt 2014] What we gain A globally invariant structure invariant by composition & inversion Simple geodesics, efficient computations (stationarity, group exponential) The simplest linearization of transformations for statistics? X. Pennec - ESI - Shapes, Feb

27 Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Riemannian frameworks on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection The SVF framework for diffeomorphisms Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb

28 Riemannian Metrics on diffeomorphisms Space of deformations Transformation y= (x) Curves in transformation spaces: (x,t) Tangent vector = speed vector field d( x, t) v t ( x) dt Right invariant metric Eulerian scheme Sobolev Norm H k or H (RKHS) in LDDMM diffeomorphisms [Miller, Trouve, Younes, Holm, Dupuis, Beg ] Geodesics determined by optimization of a time-varying vector field Distance d 2 v t v t Geodesics characterized by initial velocity / momentum Optimization for images is quite tricky (and lenghty) (, ) arg min( 0 1 t v t t v t 2 Id t dt) X. Pennec - ESI - Shapes, Feb

29 The SVF framework for Diffeomorphisms Idea: [Arsigny MICCAI 2006, Bossa MICCAI 2007, Ashburner Neuroimage 2007] Exponential of a smooth vector field is a diffeomorphism Parameterize deformation by time-varying Stationary Velocity Fields exp Direct generalization of numerical matrix algorithms Computing the deformation: Scaling and squaring [Arsigny MICCAI 2006] recursive use of exp(v)=exp(v/2) o exp(v/2) Stationary velocity field Diffeomorphism Computing the Jacobian :Dexp(v) = Dexp(v/2) o exp(v/2). Dexp(v/2) Updating the deformation parameters: BCH formula [Bossa MICCAI 2007] exp(v) exp(εu) = exp( v + εu + [v,εu]/2 + [v,[v,εu]]/12 + ) Lie bracket [v,u](p) = Jac(v)(p).u(p) - Jac(u)(p).v(p) X. Pennec - ESI - Shapes, Feb

30 Measuring Temporal Evolution with deformations Optimize LCC with deformation parameterized by SVF φ t x = exp(t. v x ) [ Lorenzi, Ayache, Frisoni, Pennec, Neuroimage 81, 1 (2013) ] X. Pennec - ESI - Shapes, Feb

31 The Stationnary Velocity Fields (SVF) framework for diffeomorphisms SVF framework for diffeomorphisms is algorithmically simple Compatible with inverse-consistency Vector statistics directly generalized to diffeomorphisms. Registration algorithms using log-demons: Log-demons: Open-source ITK implementation (Vercauteren MICCAI 2008) [MICCAI Young Scientist Impact award 2013] Tensor (DTI) Log-demons (Sweet WBIR 2010): LCC log-demons for AD (Lorenzi, Neuroimage. 2013) 3D myocardium strain / incompressible deformations (Mansi MICCAI 10) Hierarchichal multiscale polyaffine log-demons (Seiler, Media 2012) [MICCAI 2011 Young Scientist award] X. Pennec - ESI - Shapes, Feb

32 A powerful framework for statistics Parametric diffeomorphisms [Arsigny et al., MICCAI 06, JMIV 09] One affine transformation per region (polyaffines transformations) Cardiac motion tracking for each subject [McLeod, Miccai 2013] Log demons projected but with 204 parameters instead of a few millions expp AHA regions Stationary velocity fields Diffeomorphism X. Pennec - ESI - Shapes, Feb

33 A powerful framework for statistics Parametric diffeomorphisms [Arsigny et al., MICCAI 06, JMIV 09] One affine transformation per region (polyaffines transformations) Cardiac motion tracking for each subject [McLeod, Miccai 2013] Log demons projected but with 204 parameters instead of a few millions Group analysis using tensor reduction : reduced model 8 temporal modes x 3 spatial modes = 24 parameters (instead of 204) X. Pennec - ESI - Shapes, Feb

34 Population level: Hierarchical Deformation model Spatial structure of the anatomy common to all subjects w 0 w 1 w 2 w 3 w 4 w 5 w 6 Subject level: Varying deformation atoms for each subject M 0 1 M 0 K M 1 M 2 M 1 M 2 M 3 M 4 M 5 M 6 M 3 M 4 M 5 M 6 Aff(3) valued trees X. Pennec - ESI - Shapes, Feb

35 Hierarchical Estimation of the Variability Orient ed bounding boxes Weight s Structure First m ode of variat ion Level 0 Global Global scaling scaling Level 1 Thickness Level 2 47 subjects Angle and and ram us ramus [Seiler, Pennec, Reyes, Medical Image Analysis 16(7): , 2012] Level X. Pennec 3 - ESI - Shapes, Feb

36 Level 2 Hierarchical Estimation of the Variability Angle and ram us Level 3 Two Two sides sides Level 4 Teeth Teeth Level 5 47 subjects [Seiler, Pennec, Reyes, Medical Image Analysis 16(7): , 2012] X. Pennec - ESI - Shapes, Feb

37 Physics Which space for anatomical shapes? Homogeneous space-time structure at large scale (universality of physics laws) [Einstein, Weil, Cartan ] Heterogeneous structure at finer scales: embedded submanifolds (filaments ) The universe of anatomical shapes? Affine, Riemannian of fiber bundle structure? Learn locally the topology and metric Very High Dimensional Low Sample size setup Geometric prior might be the key! Modélisation de la structure de l'univers. NASA X. Pennec - ESI - Shapes, Feb

38 Advertisement Geometric Sciences of Information - GSI 2013 Paris, August , Computational Information Geometry Hessian/Symplectic Information Geometry Optimization on Matrix Manifolds Probability on Manifolds Optimal Transport Geometry Shape Spaces: Geometry and Statistic Geometry of Shape Variability. Organizers: S. Bonnabel, J. Angulo, A. Cont, F. Nielsen, F. Barbaresco Scientific committee: F. Nielsen, M. Boyom P. Byande, F. Barbaresco, S. Bonnabel, R. Sepulchre, M. Arnaudon, G. Peyré, B. Maury, M. Broniatowski, M. Basseville, M. Aupetit, F. Chazal, R. Nock, J. Angulo, N. Le Bihan, J. Manton, A. Cont, A.Dessein, A.M. Djafari, H. Snoussi, A. Trouvé, S. Durrleman, X. Pennec, J.F. Marcotorchino, M. Petitjean, M. Deza X. Pennec - ESI - Shapes, Feb

39 Advertisement Mathematical Foundations of Computational Anatomy Workshop at MICCAI 2013 (MFCA 2013) Nagoya, Japan, September 22 or 26, 2013 Organizers: S. Bonnabel, J. Angulo, A. Cont, F. Nielsen, F. Barbaresco Proceedings of previous editions: Scientific committee: F. Nielsen, M. Boyom P. Byande, F. Barbaresco, S. Bonnabel, R. Sepulchre, M. Arnaudon, G. Peyré, B. Maury, M. Broniatowski, M. Basseville, M. Aupetit, F. Chazal, R. Nock, J. Angulo, N. Le Bihan, J. Manton, A. Cont, A.Dessein, A.M. Djafari, H. Snoussi, A. Trouvé, S. Durrleman, X. Pennec, J.F. Marcotorchino, M. Petitjean, M. Deza X. Pennec - ESI - Shapes, Feb

40 X. Pennec - ESI - Shapes, Feb