Xavier Pennec. Asclepios team, INRIA Sophia- Antipolis Mediterranée, France

Transcription

1 Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterranée, France With contributions from Vincent Arsigny, Marco Lorenzi, Christof Seiler, etc Statistics on Deformations in Computational Anatomy: Geometric Structures and Topological Questions Topological Structures in Computational Biology, IMA, Dec. 2013

2 Anatomy 2007 Science that studies the structure and the relationship in space of different organs and tissues in living systems [Hachette Dictionary] Visible Human Project, NLM, Voxel-Man, U. Hambourg, er cerebral atlas, Vesale, 1543 Antiquity Animal models Philosophical physiology Paré, 1585 Renaissance: Dissection, surgery Descriptive anatomy Talairach & Tournoux, e century: Anatomo-physiology Microscopy, histology : Explosion of imaging Computer atlases Brain decade Galien ( ) Vésale ( ) Paré ( ) Sylvius ( ) Willis ( ) Gall ( ) : Phrenology Talairach ( ) Revolution of observation means ( ) : From dissection to in-vivo in-situ imaging From representative individual to population From descriptive atlases to interactive and generative models (simulation) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

3 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 - Topol. Struct. in Comp. Bio., IMA, Dec

4 Statistical Analysis of Geometric Features? Geometric features belong to manifolds Curves, tracts Surfaces Tensors, covariance matrices Transformations / deformations Algorithms for statistics on geometric manifolds? Definition of mean / covariance / PCA / distributions of geometric features? Efficient and consistent algorithms to? X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

5 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 - Topol. Struct. in Comp. Bio., IMA, Dec

6 Geometric setting for cross-sectional and longitudinal deformation analysis? Patient A time Template?? Patient B Deformation representation? Parallel transport longitudinal deformation across subjects? Geometrical/statistical setting consistant with diffeo Lie group? X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

7 How to statistically decompose deformations into meaningful spatial components? Local rotations / vorticity Regional volume changes (growth / atrophy) Hierarchy of large to small scale deformations Topological analysis of the spatial deformation map? Decomposition into regions and their interactions? X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

8 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

9 Riemannian geometry is a powerful structure to build consistent statistical computing algorithms Shape spaces & directional statistics [Kendall StatSci 89, Small 96, Dryden & Mardia 98] Numerical integration, dynamical systems & optimization [Helmke & Moore 1994, Hairer et al 2002] Matrix Lie groups [Owren BIT 2000, Mahony JGO 2002] Optimization on Matrix Manifolds [Absil, Mahony, Sepulchre, 2008] Information geometry (statistical manifolds) [Amari 1990 & 2000, Kass & Vos 1997] [Oller & Corcuera Ann. Stat. 1995, Battacharya & Patrangenaru, Ann. Stat & 2005] Statistics for image analysis Rigid body transformations [Pennec PhD96] General Riemannian manifolds [Pennec JMIV98, NSIP99, JMIV06] PGA for M-Reps [Fletcher IPMI03, TMI04] Planar curves [Klassen & Srivastava PAMI 2003] Geometric computing Subdivision scheme [Rahman, Donoho, Schroder SIAM MMS 2005] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

10 The geometric framework: Riemannian Manifolds 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 map (Normal coord. syst.) : Geodesic shooting: Exp x (v) = g (x,v) (1) Unfolding (Log x ), folding (Exp x ) Log: find vector to shoot right (geodesic completeness!) Vector -> Bipoint (no more equivalent class) Operator Euclidean space Riemannian manifold Subtraction Addition Distance xy y x y x xy dist( x, y) y x Gradient descent x x C x ) t t ( t xy Log x (y) y Exp (xy) dist( x, y) xy x Exp C( x )) t t x x ( t x X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

11 Statistical tools on Riemannian manifolds Metric -> Volume form (reference measure) dm(x) Probability density functions X, P( x X ) px( y). d M( y) X Expectation of a function from M into R : Definition : E (x) M (y). p x ( y). dm ( y) Variance : ( y) E dist( Information (neg. entropy): 2 y, x). p ( z). dm ( ) 2 2 x dist( y, z) x z M I x Elog( p ( x)) x X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

12 First statistical tools: moments Fréchet / Karcher mean: Minimize the variance Optimum: exponential barycenter Gauss-Newton Geodesic marching Existence and uniqueness : Karcher / Kendall / Le / Afsari Covariance (tpca) and higher orders moments xx E T xx. xx xz. xz M T. dp( z) Ε E 1 x argmin Edist( y, x) ym xx xx. dp( z) 0 P( C) 0 x t M 1 exp x ( v) with v E yx t [Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec NSIP 99 & JMIV06] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

13 Distributions for parametric tests 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 N( y) T xx. Γ. xx / k.exp 2 k Γ Σ ( 1) 1 3 r Ric O / n/ 2 1/ det( Σ).1 O / r Mahalanobis D2 distance / test: Any distribution: Gaussian: t 2 ( 1) x xy. xx (y) ( ) n E 2 x x.xy 2 3 ( x) O( ) x 2 n / r [ Pennec, NSIP 99, JMIV 2006 ] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

14 Natural Riemannian Metrics on Lie Groups 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 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 (-1) g.x, DLg (-1).y >Id Practical computations using left (or right) translations X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

15 Statistical Analysis of the Scoliotic Spine [ J. Boisvert et al. ISBI 06, AMDO 06 and IEEE TMI 27(4), 2008 ] Database Mean 307 Scoliotic patients from the Montreal s Sainte-Justine Hospital. 3D Geometry from multi-planar X-rays Main translation variability is axial (growth?) Main rot. var. around anterior-posterior axis X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

16 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: 4 first variation modes have clinical meaning Mode 1: King s class I or III Mode 2: King s class I, II, III Mode 3: King s class IV + V Mode 4: King s class V (+II) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

17 General Non-Compact and Non-Commutative case Which metric? Left- and right- invariant metrics generally differ! 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 - Topol. Struct. in Comp. Bio., IMA, Dec

18 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

19 Basics of Lie groups Flow of a left invariant vector field X = DL. x starting from e 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 a neighborhood of 0 in g to a neighborhood of e in G (not true in general for inf. dim) 3 curves at each point parameterized by the same tangent vector Left / Right-invariant geodesics, one-parameter subgroups Question: Can one-parameter subgroups be geodesics? X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

20 Affine connection Affine connection spaces Infinitesimal parallel transport : GG x GG GG Smooth and linear in first variable Satisfies Leibnitz rule in second variable Geodesics = straight lines Covariant derivative of tangent vector is null γ γ = 0 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) Two connections have the same geodesics iff they differ by torsion T(x,y)= X Y - Y X [X,Y] Generally focus on torsion-free connections X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

21 Canonical connections on Lie groups Left invariant connections DL.X DL. Y = DL. X Y Characterized by the bilinear form a x, y = X Y e g Symmetric part specifies geodesics Skew symmetric part specifies torsion along them Cartan Schouten connections (def. of Postnikov) L-inv connections for which one-parameter subgroups are geodesics Uniquely determined by a x, x = 0 (skew symmetry) Bi-invariant connections Ad is an isometry: a Ad g. x, Ad g. y = Ad g. a x, y Compatibility of a and ad: a [z, x], y + a x, [z, y] = z, a x, y X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

22 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 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) Curvature R x, y z = 1 4 x, y, z Parallel transport along geodesics: Π exp (y) x = DL y exp ( ). DL exp ( y ).x 2 2 Cartan Connections are generally not metric X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

23 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

24 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 M x X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

25 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) Fine existence: in a sufficiently small normal convex neighborhood of some point, there exists a unique bi-invariant mean. Iterative algorithm with linear convergence: m t+1 = m t Exp 1 n Log m t 1. g i Left, right and inverse consistency 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) ( 1) is a mean of g i [Pennec & Arsigny, Ch.7 p , Matrix Information Geometry, Springer, 2012] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

26 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 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 - Topol. Struct. in Comp. Bio., IMA, Dec

27 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 - Topol. Struct. in Comp. Bio., IMA, Dec

28 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 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 - Topol. Struct. in Comp. Bio., IMA, Dec

29 Sym. Cartan connection vs left-invariant metrics What is similar Geodesics uniquely defined in a local geodesics convex neighbourhood Curved spaces with no torsion Riemannian Exp and Log maps: normal coordinate system in finite dimension What we gain Geodesics are left (and right) translations of one-parameter subgroups Invariance by left and right translations + inversion Efficiency (stationarity) What we loose No compatible metric in general for non compact & non abelian groups Geodesic completeness but no Hopf-Rinow theorem There is not always a smooth geodesic joining two points (e.g. SL 2, no pb for GL n ) Infinite dimensions: exponential might not be locally diffeomorphic Known examples on Diff(S 1 ) but with k H k X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

30 Cartan connection give a nice affine connection space setting Open questions Global existence and uniqueness of bi-invariant mean? Adapt other algorithms and statistics to affine geometry? Generalizing to infinite dimensions? What are the diffeomorphisms that we cannot reach? Left/Right invariant metrics vs symmetric Cartan connection Differences between geodesics is quantified by ad*(x) Geodesics are identical for normal elements (screw motions for SE(3)) Can we bound the difference for other elements? Evaluate the practical impact on statistics (power, localization) Preliminary experiments indicate a better stability for Cartan connection in multiple registration X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

31 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

32 Riemannian metrics on diffeomorphisms Space of deformations Transformation y=(x) Curves in transformation spaces: (x,t) Tangent vector = speed vector field Right invariant metric Eulerian scheme Sobolev Norm H k or H (RKHS) in LDDMM diffeomorphisms [Miller, Trouve, Younes, Dupuis ] Geodesics determined by optimization of a time-varying vector field Distance d 2 v t v t Geodesics characterized by initial momentum (, ) arg min( 0 1 d( x, t) v t ( x) dt Point supported objects (Currents, e.g. curves, surface): finite dimensional parameterization with Dirac currents t Images: more difficult implementation [Beg IJCV 2005, Niethammer 09] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec v t t v t 2 Id t. dt) [ Glaunes PhD 06 ]

33 The SVF framework for Diffeomorphisms Framework of [Arsigny et al., MICCAI 06] Use one-parameter subgroups Exponential of a smooth vector field is a diffeomorphism u is a smooth stationary velocity field Exponential: solution at time 1 of ODE x(t) / t = u( x(t) ) exp Stationary velocity field Diffeomorphism X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

34 Efficient numerical methods The SVF framework for Diffeomorphisms Take advantage of algebraic properties of exp and log. exp(t.v) is a one-parameter subgroup. Direct generalization of numerical matrix algorithms. Efficient parametric diffeomorphisms Computing the deformation: Scaling and squaring recursive use of exp(v)=exp(v/2) o exp(v/2) [Arsigny MICCAI 2006] 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 - Topol. Struct. in Comp. Bio., IMA, Dec

35 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 time-consistent log-demons for AD is publicly available 3D myocardium strain / incompressible deformations (Mansi MICCAI 10) Hierarchichal multiscale polyaffine log-demons (Seiler, Media 2012) [MICCAI 2011 Young Scientist award] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

36 Longitudinal structural damage in Alzheimer s Disease baseline 2 years follow-up Widespread Hippocampal Ventricle s cortical expansion atrophy thinning X. Pennec - FUN13, April

37 Measuring Temporal Evolution with deformations: Deformation-based morphometry Fast registration with deformation parameterized by SVF φ t x = exp(t. v x ) [ Lorenzi, Ayache, Frisoni, Pennec, Neuroimage 81, 1 (2013) ] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

38 Longitudinal deformation analysis in AD From patient specific deformations to population trend (parallel transport of deformation trajectories) Patient A Template?? Patient B PhD Marco Lorenzi - Collaboration With G. Frisoni (IRCCS FateBenefratelli, Brescia) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

39 Discrete Parallel transport along arbitrary curves Infinitesimal parallel transport = connection g X : TMTM A numerical scheme to integrate for symmetric connections: Schild s Ladder [Elhers et al, 1972] Build geodesic parallelogrammoid Iterate along the curve A P 1 P 1 l P N PA) P N C P 2 l A P P 0 0 A P 0 P 0 [Lorenzi, Ayache, Pennec: Schild's Ladder for the parallel transport of deformations in time series of images, IPMI 2011 ] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

40 Analysis of longitudinal datasets Multilevel framework Single-subject, two time points Log-Demons (LCC criteria) Single-subject, multiple time points 4D registration of time series within the Log-Demons registration. Multiple subjects, multiple time points Pole or Schild s Ladder [Lorenzi et al, in Proc. of MICCAI 2011] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

41 Longitudinal changes in Alzheimer s disease (141 subjects ADNI data) AD vs Ctrl Student s t statistic on log- Jacobian Expansion Contraction X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

42 Longitudinal model for AD Modeled changes from 70 AD subjects (ADNI data) Deformation amplification (linear extrapolation on SVF) year Extrapolated Observed Extrapolated X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

43 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

44 Morphological analysis of stationary velocity fields Volume changes Atrophy!! Helmholtz decomposition Structural readjustments (Det Jac=1) MICCAI

45 Analysis of Stationary Velocity Field Discovery Virtual Pressure p Defines sources and sinks of the atrophy process Quantification Divergence p Defines flux across expanding/contracting regions Divergence Theorem p n ds p dv V V M Lorenzi, N Ayache, X Pennec. Regional flux analysis of longitudinal atrophy in Alzheimer's disease. MICCAI 201 X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

46 Regional analysis of Pressure maps E C C E Nice Step1. local maxima (sources) and minima (sinks) of pressure field Step2. Expansion and Contraction: areas of maximal outwards/inwards flux M Lorenzi, N Ayache, X Pennec. Regional flux analysis of longitudinal atrophy in Alzheimer's disease. MICCAI 201 X. Pennec - Topol. - 53

47 Discovery Most important in- and out- flow regions From group-wise ADNI dataset ( C2 - insula C3 inf front gyrus C5 - hippocampi C6 - temporal poles 20 Alzheimer s patients 1 year follow-up two time points M Lorenzi, G B. Frisoni, N Ayache, and X Pennec. Probabilistic Flux Analysis of Cerebral Longitudinal Atrophy. In MICCAI workshop NIBAD 2012 X. Pennec - Topol. - 56

48 Quantification: Flux through personalized regions to subject specific MICCAI 2012 grand Challenge Effect size on Atrophy Measure Ranked 1 st & 2 nd on Hippocampus Probabilistic masks in the subject space Among competitors: Freesurfer (Harvard, USA) Montreal Neurological Institute, Canada Mayo Clinic, USA University College of London, UK University of Pennsylvania, USA M Lorenzi, G B. Frisoni, N Ayache, and X Pennec. Probabilistic Flux Analysis of Cerebral Longitudinal Atrophy. In MICCAI workshop NIBAD 2012 X. Pennec - Topol. - 57

49 Roadmap Statistics on Riemannian manifolds An alternative affine connection space structure on Lie groups Lie groups as affine connection spaces Bi-invariant statistics with Canonical Cartan connection Statistical analysis of deformations: from geometry to topology Mean longitudinal deformations with SVF deformations Statistical analysis of growth/atrophy: spatial pattern in SVFs Polyaffine deformation trees: hierarchical description X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

50 Statistical analysis of multiscale deformations Registration from template to subject = population of deformations How to extract the most meaningful transformation parameters? How to decompose the inter-subject variability across scales? 47 mandibles (isosurfaces extracted form CT images) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

51 Local discrete deformation description = deformation atom Example deformation: two local rotations Atom: translation Type: vector field Parameters = (number of pixels) x (image dimension) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

52 Local discrete deformation description = deformation atom Example deformation: two local rotations [Arsigny et al., JMIV, 2009] Atom: translation Type: vector field Parameters = (number of pixels) x (image dimension) Atom: affine Type: polyaffine transformation Parameters = 12 x number of regions X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

53 Adaptive hierarchical space decomposition Hierarchical space decomposition using Oriented Bounding Boxes (OBB) Each region is the support of an affine transformation (deformation atoms) All locally affine regions are fused into a global poly-affine diffeomorphism Level 0 Level 1 Level 2 [ Seiler, Pennec, Reyes. Capturing the Multiscale Anatomical Shape Variability with Polyaffine Transformation Trees. Medical Image Analysis (MedIA), ] X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

54 Spatial hierarchy is an anatomical ontology X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

55 Hierarchical Model of Anatomy and Deformations Population level: 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 [Seiler, X. Pennec Pennec, - Topol. Reyes, Struct. MedIA in Comp. 16(7): , Bio., IMA, Dec ] [MICCAI Young Scientist Award 2001] 64

56 Data Driven Mandible Trees Registration of 47 CT Images to a Fixed Reference Orient ed bounding boxes Weight s Structure First m ode of variat ion Level 0 Global Global scaling scaling Level 1 Thickness Level 2 Angle and and ram us ramus X. Pennec - Topol. [Seiler, Struct. in Pennec, Comp. Bio., Reyes, IMA, Dec. Medical 2013 Image Analysis 16(7): , 2012] 66

57 Data Driven Mandible Trees Level 2 Registration of 47 CT Images to a Fixed Reference Angle and ram us Level 3 Two Two sides sides Level 4 Teeth Teeth Level 5 X. Pennec - Topol. [Seiler, Struct. in Pennec, Comp. Bio., Reyes, IMA, Dec. Medical 2013 Image Analysis 16(7): , 2012] 67

58 Statistical topology challenges in computational anatomy From Geometry to multiscale topology Statistics of spatial properties of Potential functions (pressure maps) Morse-Smale complex Divergence free velocity fields (vorticity maps): topology of vorticity? Complete vector fields (momenta, SVFs)? Statistics on hierarchal space decompositions Minimal spanning tree description of anatomical regions? Statistics on manifold-valued trees? (deformation atoms: polyaffine, higher order kernels) X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec

59 Advertisement Mathematical Foundations of Computational Anatomy Workshop at MICCAI Conference every 2 year Proceedings of previous editions: 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 - Topol. Struct. in Comp. Bio., IMA, Dec

60 Thank You! Publications: Software: X. Pennec - Topol. Struct. in Comp. Bio., IMA, Dec