Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterrane, - PowerPoint PPT Presentation

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

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 10-13 2015 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  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 10-13 2015 3

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  X. Pennec - ESI - Shapes, Feb 10-13 2015 4 [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ]

Bronze Standard Rigid Registration Validation  ˆ Best explanation of the observations (ML) :  2 ( , ) C d T ij T ij ij  LSQ criterion   2   2  2 ( , ) min ( , ),  Robust Fréchet mean d T T T T 1 2 1 2  Robust initialization and Newton gradient descent [ T. Glatard & al, MICCAI 2006,   Result , , T , i j rot trans Int. Journal of HPC Apps, 2006 ] Derive tests on transformations for accuracy / consistency X. Pennec - ESI - Shapes, Feb 10-13 2015 5

Morphometry through Deformations Atlas  1  5 Patient 1 Patient 5  4  3  2 Patient 4 Patient 3 Patient 2 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 10-13 2015 6

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 g (-1) .x , DL g (-1) .y > Id Implementation  Practical computations using left (or right) translations       ( 1) Exp x f Exp ( DL . x) fg Log (g) DL .Log (f g)    f ( 1) f f Id Id f X. Pennec - ESI - Shapes, Feb 10-13 2015 7

Example on 3D rotations Space of rotations SO(3):  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 10-13 2015 8

General Non-Compact and Non-Commutative case No Bi-invariant Mean for 2D Rigid Body Transformations  Metric at Identity: 𝑒𝑗𝑡𝑢(𝐽𝑒, 𝜄; 𝑢 1 ; 𝑢 2 ) 2 = 𝜄 2 + 𝑢 1 2 + 𝑢 2 2 𝜌 2 2 𝜌 2 2  𝑈 1 = 4 ; − 2 ; 𝑈 2 = 0; 2; 0 𝑈 3 = − 4 ; − 2 ; − 2 2  Left-invariant Fréchet mean: 0; 0; 0 2  Right-invariant Fréchet mean: 0; 3 ; 0 ≃ (0; 0.4714; 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 10-13 2015 9

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 10-13 2015 10

Basics of Lie groups = 𝐸𝑀. 𝑦 from identity Flow of a left invariant vector field 𝑌  𝛿 𝑦 𝑢 exists for all time  One parameter subgroup: 𝛿 𝑦 𝑡 + 𝑢 = 𝛿 𝑦 𝑡 . 𝛿 𝑦 𝑢 Lie group exponential  Definition: 𝑦 ∈ 𝔥  𝐹𝑦𝑞 𝑦 = 𝛿 𝑦 1 𝜗 𝐻  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 10-13 2015 11

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: 𝛼 𝛿 𝛿 = 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 10-13 2015 14

Canonical connections on Lie groups Left invariant connections 𝛂 𝑬𝑴.𝒀 𝑬𝑴. 𝒁 = 𝑬𝑴. 𝛂 𝒀 𝒁 | 𝑓 ∈ g  Characteric bilinear form on the Lie algebra 𝑏 𝑦, 𝑧 = 𝛼 𝑌 𝑍 1 2 (𝑏 𝑦, 𝑧 + 𝑏 𝑧, 𝑦 ) specifies geodesics  Symmetric part 1 2 (𝑏 𝑦, 𝑧 − 𝑏 𝑧, 𝑦 ) specifies torsion along them  Skew symmetric part Bi-invariant connections  𝑏 𝐵𝑒 𝑕 . 𝑦, 𝐵𝑒 𝑕 . 𝑧 = 𝐵𝑒 𝑕 . 𝑏 𝑦, 𝑧  𝑏 [𝑨, 𝑦], 𝑧 + 𝑏 𝑦, [𝑨, 𝑧] = 𝑨, 𝑏 𝑦, 𝑧 , 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 𝑏 𝑦, 𝑦 = 0 (skew symmetry) X. Pennec - ESI - Shapes, Feb 10-13 2015 15

Canonical Cartan connections on Lie groups Bi-invariant Cartan Schouten connections  Family 𝑏 𝑦, 𝑧 = 𝜇 𝑦, 𝑧 (-,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 ( 𝑏 𝑦, 𝑧 + 𝑏 𝑧, 𝑦 = 0) : one-parameter subgroups and their left and right translations  Curvature: R 𝑦, 𝑧 = λ 𝜇 − 1 [ 𝑦, 𝑧 , 𝑨]  Torsion: 𝑈 𝑦, 𝑧 = 2𝑏 𝑦, 𝑧 − 𝑦, 𝑧 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)  𝑏 𝑦, 𝑧 = 1 2 𝑦, 𝑧  Curvature 𝑆 𝑦, 𝑧 𝑨 = − 1 𝑦, 𝑧 , 𝑨 4  Parallel transport along geodesics: Π exp (𝑧) 𝑦 = 𝐸𝑀 exp 2 ) . 𝐸𝑀 exp 2 ) .x ( 𝑧 ( 𝑧 X. Pennec - ESI - Shapes, Feb 10-13 2015 16

Cartan Connections are generally not metric Levi-Civita Connection of a left-invariant (pseudo) metric is left-invariant  Metric dual of the bracket < 𝑏𝑒 ∗ 𝑦, 𝑧 , 𝑨 > = < 𝑦, 𝑨 , 𝑧 > 2 𝑏𝑒 ∗ 𝑦, 𝑧 + 𝑏𝑒 ∗ 𝑧, 𝑦 1 1  𝑏 𝑦, 𝑧 = 2 𝑦, 𝑧 − Bi-invariant (pseudo) metric => Symmetric Cartan connection  A left-invariant (pseudo) metric is right-invariant if it is Ad-invariant < 𝑦, 𝑧 > = < 𝐵𝑒 𝑕 𝑦 , 𝐵𝑒 𝑕 𝑧 >  Infinitesimaly: < 𝑦, 𝑨 , 𝑧 > + < 𝑦, 𝑧, 𝑨 > = 0 or 𝑏𝑒 ∗ 𝑦, 𝑧 + 𝑏𝑒 ∗ 𝑧, 𝑦 = 0 Existence of bi-invariant (pseudo) metrics  A Lie group admits a bi-invariant metric iff Ad(G) is relatively compact 𝐵𝑒 𝐻 ⊂ 𝑃 g ⊂ 𝐻𝑀( 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 10-13 2015 17