Xavier Pennec Asclepios team, INRIA Sophia-Antipolis – Mediterranée, France with V. Arsigny, P. Fillard, M. Lorenzi, etc . Geometric Structures for Statistics on Shapes and Deformations in Computational Anatomy Geometrical Models in Vision Workshop SubRiemannian Geometry Semester October 23, 2014, IHP, Paris, FR
Computational Anatomy 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 X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 2
Geometric features in Computational Anatomy Noisy geometric features Tensors, covariance matrices Curves, fiber tracts Surfaces Transformations Rigid, affine, locally affine, diffeomorphisms Goal: statistical modeling at the population level Deal with noise consistently on these non-Euclidean manifolds A consistent computing framework for simple statistics X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 3
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 - IHP, Sub-Riemannian Geom, 23/10/2014 4
Longitudinal deformation analysis Deformation trajectories in different reference spaces time Patient A ? ? Template Patient B How to transport longitudinal deformation across subjects? Convenient mathematical settings for transformations? X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 5
Outline Statistical computing on Riemannian manifolds Simple statistics on Riemannian manifolds Extension to manifold-values images Computing on Lie groups Lie groups as affine connection spaces The SVF framework for diffeomorphisms Towards more complex geometries X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 6
Bases of Algorithms in Riemannian Manifolds Riemannian metric : Dot product on tangent space Speed, length of a curve Shortest path: Riemannian Distance Geodesics characterized by 2 nd order diff eqs: locally unique for initial point and speed Exponential map (Normal coord. syst.) : Geodesic shooting: Exp x (v) = g (x,v) (1) Log: find vector to shoot right (geodesic completeness!) Reformulate algorithms with exp x and log x Vector -> Bipoint (no more equivalent class) Operator Euclidean space Riemannian manifold Subtraction ( y ) xy Log xy y x x Addition ( xy ) y Exp y x xy x Distance dist ( , ) dist ( , ) x y y x x y xy x ( t ) Gradient descent ( ( )) x x C x x Exp C x t x t t t t X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 7
Random variable in a Riemannian Manifold Intrinsic pdf of x For every set H 𝑄 𝐲 ∈ 𝐼 = 𝑞 𝑧 𝑒𝑁(𝑧) 𝐼 Lebesgue’s measure Uniform Riemannian Mesure 𝑒𝑁 𝑧 = det 𝐻 𝑧 𝑒𝑧 Expectation of an observable in M 𝑭 𝐲 𝜚 = 𝜚 𝑧 𝑞 𝑧 𝑒𝑁 𝑧 𝑁 𝜚 = 𝑒𝑗𝑡𝑢 2 (variance) : 𝑭 𝐲 𝑒𝑗𝑡𝑢 . , 𝑧 2 = 𝑒𝑗𝑡𝑢 𝑧, 𝑨 2 𝑞 𝑨 𝑒𝑁(𝑨) 𝑁 𝜚 = 𝑦 (mean) : 𝑭 𝐲 𝐲 = 𝑧 𝑞 𝑧 𝑒𝑁 𝑧 𝑁 X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 8
First Statistical Tools: Moments Frechet / Karcher mean minimize the variance Ε 2 E E x x . ( ). ( ) 0 ( ) 0 argmin dist( , ) M y p z d z P C x x x x x M y M Variational characterization: Exponential barycenters Existence and uniqueness (convexity radius) [Karcher / Kendall / Le / Afsari] Empirical mean: a.s. uniqueness [Arnaudon & Miclo 2013] Gauss-Newton Geodesic marching 1 n x exp ( ) with E y Log (x ) v v x 1 x x t i t n t 1 i [Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ] X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 9
First Statistical Tools: Moments Covariance (PCA) [higher moments] T T E x . x x . x . ( ). ( ) M z z p z d z x x xx x M Principal component analysis Tangent-PCA: principal modes of the covariance Principal Geodesic Analysis (PGA) [Fletcher 2004] [Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ] X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 10
Statistical Analysis of the Scoliotic Spine [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ] Database 307 Scoliotic patients from the Montreal’s Sainte-Justine Hospital. 3D Geometry from multi-planar X-rays Mean Main translation variability is axial (growth?) Main rot. var. around anterior-posterior axis X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 11
Statistical Analysis of the Scoliotic Spine [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ] AMDO’06 best paper award, Best French-Quebec joint PhD 2009 PCA of the Covariance: • Mode 1: King’s class I or III • Mode 3: King’s class IV + V 4 first variation modes • Mode 4: King’s class V (+II) • Mode 2: King’s class I, II, III have clinical meaning X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 12
Outline Statistical computing on Riemannian manifolds Simple statistics on Riemannian manifolds Extension to manifold-values images Computing on Lie groups Lie groups as affine connection spaces The SVF framework for diffeomorphisms Towards more complex geometries X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 13
Diffusion Tensor Imaging Covariance of the Brownian motion of water Filtering, regularization Interpolation / extrapolation Architecture of axonal fibers Symmetric positive definite matrices Cone in Euclidean space (not complete) Convex operations are stable mean, interpolation More complex operations are not PDEs, gradient descent… All invariant metrics under GLn T | Tr Tr( ). Tr( ) ( -1/n) W W W W W W 1 2 1 2 1 2 Id Exponential map 1 / 2 1 / 2 1 / 2 1 / 2 ( ) exp( . . ) Exp 1 / 2 1 / 2 1 / 2 1 / 2 Log map ( ) log( . . ) Log 2 Distance 2 1 / 2 1 / 2 ( , ) | log( . . ) dist Id X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 14
Manifold-valued image algorithms Integral or sum in M: weighted Fréchet mean Interpolation Linear between 2 elements: interpolation geodesic Bi- or tri-linear or spline in images: weighted means Gaussian filtering: convolution = weighted mean 2 ( ) min ( ) dist ( , ) x G x x i i i PDEs for regularization and extrapolation: the exponential map (partially) accounts for curvature Gradient of Harmonic energy = Laplace-Beltrami 1 2 ( ) ( ) ( ) x x x u O u S Anisotropic regularization using robust functions 2 Reg ( ) ( ) x dx ( x ) Simple intrinsic numerical schemes thanks the exponential maps! [ Pennec, Fillard, Arsigny, IJCV 66(1), 2005, ISBI 2006] X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 15
Filtering and anisotropic regularization of DTI Raw Euclidean Gaussian smoothing Riemann Gaussian smoothing Riemann anisotropic smoothing X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 16
Outline Statistical computing on Riemannian manifolds Simple statistics on Riemannian manifolds Extension to manifold-values images Computing on Lie groups Lie groups as affine connection spaces The SVF framework for diffeomorphisms Towards more complex geometries X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 17
Limits of the Riemannian Framework Lie group: Smooth manifold with group structure Composition g o h and inversion g -1 are smooth Left and Right translation L g (f) = g o f R g (f) = f o g 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 No bi-invariant metric in general Incompatibility of the Fréchet mean with the group structure Left of right metric: different Fréchet means The inverse of the mean is not the mean of the inverse Examples with simple 2D rigid transformations Can we design a mean compatible with the group operations? Is there a more convenient structure for statistics on Lie groups? X. Pennec - STIA - Sep. 18 2014 18
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