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 1: Simple Statistics on Riemannian Manifolds Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis
Science that studies the structure and the relationship in Anatomy space of different organs and tissues in living systems [Hachette Dictionary] 2007 Visible Human Project, NLM, 1996-2000 Voxel-Man, U. Hambourg, 2001 1st cerebral atlas, Vesale, 1543 Talairach & Tournoux, 1988 Paré, 1585 1990-2000: 17-20e century: Antiquity • Explosion of imaging Renaissance: • Animal models • Anatomo-physiology • Dissection, surgery • Computer atlases • Microscopy, histology • Philosophical physiology • Descriptive anatomy • Brain decade Vésale (1514-1564) Sylvius (1614-1672) Gall (1758-1828) : Phrenology Galien (131-201) Paré (1509-1590) Willis (1621-1675) Talairach (1911-2007) Revolution of observation means (1988-2007) : From dissection to in-vivo in-situ imaging From representative individual to population From descriptive atlases to interactive and generative models (simulation) X. Pennec - ESI - Shapes, Feb 10-13 2015 2
Computational Anatomy Design Mathematical Methods and Algorithms to Model and Analyze the Anatomy Statistics of organ shapes across species, populations, diseases… Model organ development across time (heart- beat, growth, ageing, ages…) To understand and to model how life is functioning Classify structural deviations ( taxonomy ), Relate anatomy and function To detect, understand and correct dysfunctions From generic (atlas-based) to patients-specific models Very active topic in medical image analysis Keyword of Medic Image Computing and Computer Assisted Intervention (MICCAI) X. Pennec - ESI - Shapes, Feb 10-13 2015 3
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 - 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 Reference template = Mean (atlas) Shape variability encoded by the deformations Statistics on groups of transformations (Lie groups, diffeomorphism)? Consistency with group operations (non commutative)? X. Pennec - ESI - Shapes, Feb 10-13 2015 6
Methods of computational anatomy Remodeling of the right ventricle of the heart in tetralogy of Fallot Mean shape Shape variability Correlation with clinical variables Predicting remodeling effect Shape of RV in 18 patients X. Pennec - ESI - Shapes, Feb 10-13 2015 7
Longitudinal deformation analysis Dynamic obervations time Patient A ? ? Template Patient B How to transport longitudinal deformation across subjects? What are the convenient mathematical settings? X. Pennec - ESI - Shapes, Feb 10-13 2015 8
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 9
Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Introduction to computational anatomy The Riemannian manifold computational structure Simple statistics on Riemannian manifolds Applications to the spine shape and registration accuracy Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb 10-13 2015 10
Basic probabilities and statistics Measure: random vector x of pdf ( z ) p x Σ ( x ) Approximation: ~ , x xx • Mean: x E( ) . ( ). z p z dz x x • Covariance: T E ( x ).( x ) x x xx Propagation: T h h Σ ( ) x , . . h ~ h y x xx x x Noise model: additive, Gaussian... Principal component analysis 2 Statistical distance: Mahalanobis and X. Pennec - ESI - Shapes, Feb 10-13 2015 11
Differentiable manifolds Définition: Locally Euclidean Topological space which can be globally curved Same dimension + differential regularity Simple Examples Sphere Saddle (hyperbolic space) Surface in 3D space And less simple ones Projective spaces Rotations of R 3 : SO 3 ~ P 3 Rigid, affine Transformation Diffeomorphisms X. Pennec - ESI - Shapes, Feb 10-13 2015 12
Differentiable manifolds Computing in a a manifold Extrinsic Embedding in ℝ 𝑜 Intrinsic Coordinates : charts Atlas = consistent set of charts Measuring? Volumes (surfaces) Lengths Straight lines X. Pennec - ESI - Shapes, Feb 10-13 2015 13
Measuring extrinsic distances Basic tool: the scalar product , t v w v w • Norm of a vector , v v v • Angle between vectors , cos( ) v w v w g (t) • Length of a curve v w g g ( ) || ( ) || L t dt p X. Pennec - ESI - Shapes, Feb 10-13 2015 14
Measuring extrinsic distances Basic tool: the scalar product , t t , ( ) v v w w v v w G p w p • Norm of a vector , v v v Bernhard Riemann p 1826-1866 p • Angle between vectors Bernhard Riemann 1826-1866 , cos( ) v w v w p p p • Length of a curve g g ( ) || ( ) || t L t dt g ( ) X. Pennec - ESI - Shapes, Feb 10-13 2015 15
Riemannian manifolds Basic tool: the scalar product t , ( ) v w v G p w p Bernhard Riemann 1826-1866 • Geodesic between 2 points Bernhard Riemann 1826-1866 • Shortest path • Calculus of variations (E.L.) : • Length of a curve 2 nd order differential equation g g (specifies acceleration) ( ) || ( ) || t L t dt g ( ) • Free parameters: initial speed and starting point X. Pennec - ESI - Shapes, Feb 10-13 2015 16
Bases of Algorithms in Riemannian Manifolds Exponential map (Normal coordinate system): Exp x = geodesic shooting parameterized by the initial tangent Log x = unfolding the manifold in the tangent space along geodesics Geodesics = straight lines with Euclidean distance Local global domain: star-shaped, limited by the cut-locus Covers all the manifold if geodesically complete Reformulate algorithms with exp x and log x Vector -> Bi-point (no more equivalence classes) Operation Euclidean space Riemannian Subtraction ( y ) xy y x xy Log x ( xy ) y Exp Addition y x xy x dist ( , ) dist ( , ) x y y x Distance x y xy x ( ) ( ( )) x x C x x Exp C x Gradient descent t x t t t t t X. Pennec - ESI - Shapes, Feb 10-13 2015 17
Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Introduction to computational anatomy The Riemannian manifold computational structure Simple statistics on Riemannian manifolds Applications to the spine shape and registration accuracy Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb 10-13 2015 21
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 = log 𝑞 (information) : 𝑭 𝐲 log 𝑞 = 𝑞 𝑧 log (𝑞 𝑧 )𝑒𝑁 𝑧 𝑁 𝜚 = 𝑦 (mean) : 𝑭 𝐲 𝐲 = 𝑧 𝑞 𝑧 𝑒𝑁 𝑧 𝑁 X. Pennec - ESI - Shapes, Feb 10-13 2015 22
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