X. Pennec With contributions from V. Arsigny, N. Ayache, J. - PowerPoint PPT Presentation

X. Pennec With contributions from V. Arsigny, N. Ayache, J. Boisvert, P. Fillard, et al. Statistical Computing on Riemannian manifolds From Riemannian Geometry to Computational Anatomy September 18, 2006 Mathematics and Image Analysis 2006 1

Standard Medical Image Analysis Methodological / algorithmically axes � Registration � Segmentation � Image Analysis/Quantification Measures are geometric and noisy � Feature extracted from images � Registration = determine a transformations � Diffusion tensor imaging We need: � Statistiques � A stable computing framework Mathematics and Image Analysis 2006 September 18, 2006 2

Historical examples of geometrical features Geometric features • Lines, oriented points… • Extremal points: semi-oriented frames Transformations • Rigid, Affine, locally affine, families of deformations How to deal with noise consistently on these features? Mathematics and Image Analysis 2006 September 18, 2006 3

Per-operative registration of MR/US images MR Image Registered US Initial US Performance Evaluation: statistics on transformations Mathematics and Image Analysis 2006 September 18, 2006 4

Interpolation, filtering of tensor images Raw Anisotropic smoothing Computing on Manifold-valued images Mathematics and Image Analysis 2006 September 18, 2006 5

Computational Anatomy Computational Anatomy, an emerging discipline , P. Thompson, M. Miller, NeuroImage special issue 2004 Mathematical Foundations of Computational Anatomy, X. Pennec and S. Joshi, MICCAI workshop, 2006 Modeling and Analysis of the Human Anatomy � Estimate representative / average organ anatomies � Model organ development across time � Establish normal variability � To detect and classify of pathologies from structural deviations � To adapt generic (atlas-based) to patients-specific models � Statistical analysis on (and of) manifolds Mathematics and Image Analysis 2006 September 18, 2006 6

Overview The geometric computational framework � (Geodesically complete) Riemannian manifolds Statistical tools on pointwise features � Mean, Covariance, Parametric distributions / tests � Application examples on rigid body transformations Manifold-valued images: Tensor Computing � Interpolation, filtering, diffusion PDEs � Diffusion tensor imaging Metric choices for Computational Neuroanatomy � Morphometry of sulcal lines on the brain � Statistics of deformations for non-linear registration Mathematics and Image Analysis 2006 September 18, 2006 7

Riemannian Manifolds: geometrical tools Riemannian metric : � Dot product on tangent space � Speed, length of a curve � Distance and geodesics � Closed form for simple metrics/manifolds � Optimization for more complex Exponential chart (Normal coord. syst.) : � Development in tangent space along geodesics � Geodesics = straight lines � Distance = Euclidean � Star shape domain limited by the cut-locus � Covers all the manifold if geodesically complete Mathematics and Image Analysis 2006 September 18, 2006 8

Computing on Riemannian manifolds Operation Euclidean space Riemannian manifold = − = log ( ) Subtraction xy y x xy y x = + = Addition exp ( ) y xy y x xy x = = − Distance ( , ) ( , ) dist x y xy dist x y y x x Σ + = Σ − ε ∇ C Σ Σ = − ε ∇ Σ Gradient descent ( t ) exp ( ( )) C ε + ε Σ t t t t t Mathematics and Image Analysis 2006 September 18, 2006 9

Overview � The geometric computational framework � (Geodesically complete) Riemannian manifolds Statistical tools on pointwise features � Mean, Covariance, Parametric distributions / tests � Application examples on rigid body transformations Manifold-valued images: Tensor Computing � Interpolation, filtering, diffusion PDEs � Diffusion tensor imaging Metric choices for Computational Neuroanatomy � Morphometry of sulcal lines on the brain � Statistics of deformations for non-linear registration Mathematics and Image Analysis 2006 September 18, 2006 11

Statistical tools on Riemannian manifolds Metric -> Volume form (measure) M ( x ) d ∫ ∀ ∈ = , ( ) ( ). M( ) Probability density functions X P x X p y d y x X Expectation of a function φ from M into R : ] ∫ [ φ = φ � Definition : E . ( ). M ( ) (x) (y) p y d y x M [ ] ∫ = = � Variance : 2 2 σ 2 ( ) E . ( ). M ( ) dist( , x ) dist( , ) y y y z p z d z x x M [ ] [ ] = � Information (neg. entropy): x x I E log( ( )) p x Mathematics and Image Analysis 2006 September 18, 2006 12

Statistical tools: Moments Frechet / Karcher mean minimize the variance [ ] [ ] ( ) [ ] [ ] ∫ Ε = ⇒ = = = 2 x x x x argmin E dist( , ) E x x . ( ). M ( ) 0 ( ) 0 p z d z P C y x ∈ y M M Geodesic marching [ ] = = x x exp ( ) with E y v v + 1 x t t Covariance et higher moments [ ] ( )( ) ( )( ) T ∫ T Σ = = x x E x . x x . x . ( ). M ( ) z z p z d z xx x M [ Pennec, JMIV06, RR-5093, NSIP’99 ] Mathematics and Image Analysis 2006 September 18, 2006 13

Distributions for parametric tests Uniform density: = ( ) Ind ( ) / Vol( ) � maximal entropy knowing X p z z X x X 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 ( ) ( ) − ( ) ( ) = − + σ + ε σ Γ Σ ( 1 ) 1 Ric / O r ⎛ ⎞ T 3 = ⎜ x Γ x ⎟ ( ( ) ) ( ) . exp x . . x / 2 N y k ( ) ( ) − − = π + σ + ε σ ⎝ ⎠ n / 2 Σ 1 / 2 3 2 . det( ) . 1 / k O r t μ = Σ − 2 ( 1 ) ( y ) x y . . x y Mahalanobis D2 distance / test: x xx [ ] 2 x μ = E ( ) � Any distribution: n x ( ) μ ∝ χ + σ + ε σ 2 x 2 3 ( ) ( ) / O r � Gaussian: x n [ Pennec, JMIV06, NSIP’99 ] Mathematics and Image Analysis 2006 September 18, 2006 15

Gaussian on the circle = θ ∈ − π π Exponential chart: ] . ; . [ x r r r Gaussian: truncated standard Gaussian → ∞ : standard Gaussian r (Ricci curvature → 0) γ → uniform pdf with 0 : σ = π 2 2 ( . ) / 3 r (compact manifolds) γ → ∞ : Dirac Mathematics and Image Analysis 2006 September 18, 2006 16

Overview � The geometric computational framework Statistical tools on pointwise features � Mean, Covariance, Parametric distributions / tests � Application examples on rigid body transformations Manifold-valued images: Tensor Computing � Interpolation, filtering, diffusion PDEs � Diffusion tensor imaging Metric choices for Computational Neuroanatomy � Morphometry of sulcal lines on the brain � Statistics of deformations for non-linear registration Mathematics and Image Analysis 2006 September 18, 2006 17

Validation of the error prediction Comparing two transformations and their Covariance matrix : μ ≈ χ 2 2 ( , ) T T 1 2 6 Mean: 6, Var: 12 KS test 2 ≈ μ Intra-echo: , KS test OK 6 2 > μ Inter-echo: , KS test failed, Bias ! 50 Bias estimation: (chemical shift, susceptibility effects) σ = 0 . 06 deg (not significantly different from the identity) � rot σ = (significantly different from the identity) 0 . 2 mm � trans 2 ≈ μ Inter-echo with bias corrected: , KS test OK 6 [ X. Pennec et al., Int. J. Comp. Vis. 25(3) 1997, MICCAI 1998 ] Mathematics and Image Analysis 2006 September 18, 2006 20

Liver puncture guidance using augmented reality S. Nicolau, IRCAD / INRIA S. Nicolau, IRCAD / INRIA 3D (CT) / 2D (Video) registration � 2D-3D EM-ICP on fiducial markers � Certified accuracy in real time Validation � Bronze standard (no gold-standard) � Phantom in the operating room (2 mm) � 10 Patient (passive mode): < 5mm (apnea) PhD S. Nicolau, MICCAI05, ECCV04, ISMAR04, IS4TM03, Comp. Anim. & Virtual World 2005, IEEE TMI (soumis) Mathematics and Image Analysis 2006 September 18, 2006 23

Statistical Analysis of the Scoliotic Spine [ J. Boisvert, X. Pennec, N. Ayache, H. Labelle, F. Cheriet,, ISBI’06 ] 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 rotation var. around anterior-posterior axis � PCA of the Covariance 4 first variation modes have clinical meaning � Mathematics and Image Analysis 2006 September 18, 2006 24

Statistical Analysis of the Scoliotic Spine • Mode 1: King’s class I or III • Mode 3: King’s class IV + V • Mode 2: King’s class I, II, III • Mode 4: King’s class V (+II) Mathematics and Image Analysis 2006 September 18, 2006 25

Overview � The geometric computational framework � Statistical tools on pointwise features Manifold-valued images: Tensor Computing � Interpolation, filtering, diffusion PDEs � Diffusion tensor imaging Metric choices for Computational Neuroanatomy � Morphometry of sulcal lines on the brain � Statistics of deformations for non-linear registration Mathematics and Image Analysis 2006 September 18, 2006 26