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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
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 2
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
Computational Anatomy
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
4 X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
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]
Patient 3
Atlas
Patient 1
Patient 2
Patient 4
Patient 5
1
2
3
4
5
Longitudinal deformation analysis
5
time
Deformation trajectories in different reference spaces
How to transport longitudinal deformation across subjects? Convenient mathematical settings for transformations?
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Patient A
Patient B
? ? Template
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 6
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 7
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 2nd order diff eqs:
locally unique for initial point and speed
Operator Euclidean space Riemannian manifold
Subtraction Addition Distance
Gradient descent )( ttt xCxx
)(yLogxy x
xyxy
xyyx ),(distx
xyyx ),(dist
)(xyExpy x
))(( txt xCExpxt
xyxy
Reformulate algorithms with expx and logx Vector -> Bipoint (no more equivalent class)
Exponential map (Normal coord. syst.) : Geodesic shooting: Expx(v) = g(x,v)(1) Log: find vector to shoot right (geodesic completeness!)
Random variable in a Riemannian Manifold
Intrinsic pdf of x For every set H
𝑃 𝐱 ∈ 𝐻 = 𝑝 𝑦 𝑑𝑀(𝑦)𝐻
Lebesgue’s measure
Uniform Riemannian Mesure 𝑑𝑀 𝑦 = det 𝐺 𝑦 𝑑𝑦
Expectation of an observable in M 𝑬𝐱 𝜙 = 𝜙 𝑦 𝑝 𝑦 𝑑𝑀 𝑦
𝑀
𝜙 = 𝑑𝑖𝑠𝑡2 (variance) : 𝑬𝐱 𝑑𝑖𝑠𝑡 . , 𝑦2 = 𝑑𝑖𝑠𝑡 𝑦, 𝑧 2𝑝 𝑧 𝑑𝑀(𝑧)
𝑀
𝜙 = 𝑥 (mean) : 𝑬𝐱 𝐱 = 𝑦 𝑝 𝑦 𝑑𝑀 𝑦𝑀
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 8
First Statistical Tools: Moments Frechet / Karcher mean minimize the variance
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
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 9
n
i
it tt nvv
1
xx1 )(xLog1
yE with )(expx x
0)( 0)().(.xxE ),dist(E argmin 2
CPzdzpyy MM
MxxxxxΕ
[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]
10
First Statistical Tools: Moments
Covariance (PCA) [higher moments]
Principal component analysis Tangent-PCA: principal modes of the covariance
Principal Geodesic Analysis (PGA) [Fletcher 2004]
M
M )().(.x.xx.xE TT
zdzpzz xxx xx
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
[Oller & Corcuera 95, Battacharya & Patrangenaru 2002, Pennec, JMIV06, NSIP’99 ]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 11
Statistical Analysis of the Scoliotic Spine
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
[ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 12
Statistical Analysis of the Scoliotic Spine
• 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)
PCA of the Covariance: 4 first variation modes have clinical meaning
[ 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
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 13
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
14
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 Exponential map
Log map
Distance
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
2/12/12/12/1 )..exp()(
Exp2/12/12/12/1 )..log()(
Log
22/12/12 )..log(|),(
Iddist
-1/n)( )Tr().Tr( Tr| 212121 WWWWWW T
Id
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 15
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
PDEs for regularization and extrapolation: the exponential map (partially) accounts for curvature
Gradient of Harmonic energy = Laplace-Beltrami
Anisotropic regularization using robust functions
Simple intrinsic numerical schemes thanks the exponential maps!
i iixxGx ),(dist )(min)( 2
21)()()(
Ouxxx
Su
dxx
x
2
)()()(Reg
[ Pennec, Fillard, Arsigny, IJCV 66(1), 2005, ISBI 2006]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 16
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 17
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
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 Lg(f) = g o f Rg (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 = < DLg(-1) .x , DLg(-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
Basics of Lie groups
Flow of a left invariant vector field 𝑋 = 𝐷𝐿. 𝑥 from identity 𝛾𝑥 𝑡 exists for all time One parameter subgroup: 𝛾𝑥 𝑠 + 𝑡 = 𝛾𝑥 𝑠 . 𝛾𝑥 𝑡
Lie group exponential Definition: 𝑥 ∈ 𝔤 𝐸𝑥𝑝 𝑥 = 𝛾𝑥 1 𝜖 𝐺 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 - IHP, Sub-Riemannian Geom, 23/10/2014 19
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: 𝛻𝛾 𝛾 = 0 2nd order differential equation:
Normal coordinate system Local exp and log maps
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 20
Adapted from Lê Nguyên Hoang, science4all.org
Cartan-Schouten Connection on Lie Groups
A unique connection Symmetric (no torsion) and bi-invariant For which geodesics through Id are one-parameter
subgroups (group exponential) Matrices : M(t) = A.exp(t.V) Diffeos : translations of Stationary Velocity Fields (SVFs)
Levi-Civita connection of a bi-invariant metric (if it exists) Continues to exists in the absence of such a metric
(e.g. for rigid or affine transformations)
Two flat connections (left and right) Absolute parallelism: no curvature but torsion (Cartan / Einstein)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 21
Statistics on an affine connection space
Fréchet mean: exponential barycenters 𝐿𝑜𝑔𝑥 𝑦𝑖𝑖 = 0 [Emery, Mokobodzki 91, Corcuera, Kendall 99]
Existence local uniqueness if local convexity [Arnaudon & Li, 2005]
For Cartan-Schouten connections [Pennec & Arsigny, 2012] Locus of points x such that 𝐿𝑜𝑔 𝑥−1. 𝑦𝑖 = 0 Algorithm: fixed point iteration (local convergence)
𝑥𝑡+1 = 𝑥𝑡 ∘ 𝐸𝑥𝑝1
𝑛 𝐿𝑜𝑔 𝑥𝑡
−1. 𝑦𝑖
Mean stable by left / right composition and inversion If 𝑚 is a mean of 𝑔𝑖 and ℎ is any group element, then
ℎ ∘ 𝑚 is a mean of ℎ ∘ 𝑔𝑖 , 𝑚 ∘ ℎ is a mean of the points 𝑔𝑖 ∘ ℎ
and 𝑚(−1) is a mean of 𝑔𝑖(−1)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 22
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 𝜃𝑖 < 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 - IHP, Sub-Riemannian Geom, 23/10/2014 23
Generalization of the Statistical Framework
Covariance matrix & higher order moments Defined as tensors in tangent space
Σ = 𝐿𝑜𝑔𝑥 𝑦 ⊗ 𝐿𝑜𝑔𝑥 𝑦 𝜇(𝑑𝑦)
Matrix expression changes according to the basis
Other statistical tools Mahalanobis distance well defined and bi-invariant Tangent Principal Component Analysis (t-PCA) Principal Geodesic Analysis (PGA), provided a data likelihood Independent Component Analysis (ICA)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 24
25
Cartan Connections vs Riemannian
What is similar Standard differentiable geometric structure [curved space without torsion] Normal coordinate system with Expx et Logx [finite dimension]
Limitations of the affine framework No metric (but no choice of metric to justify) The exponential does always not cover the full group
Pathological examples close to identity in finite dimension In practice, similar limitations for the discrete Riemannian framework
Global existence and uniqueness of bi-invariant mean? Use a bi-invariant pseudo-Riemannian metric? [Miolane poster]
What we gain A globally invariant structure invariant by composition & inversion Simple geodesics, efficient computations (stationarity, group exponential) The simplest linearization of transformations for statistics?
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 26
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
28
Idea: [Arsigny MICCAI 2006, Bossa MICCAI 2007, Ashburner Neuroimage 2007] Exponential of a smooth vector field is a diffeomorphism Parameterize deformation by time-varying Stationary Velocity Fields
Direct generalization of numerical matrix algorithms Computing the deformation: Scaling and squaring [Arsigny MICCAI 2006]
recursive use of exp(v)=exp(v/2) o exp(v/2)
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)
The SVF framework for Diffeomorphisms
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
•exp
Stationary velocity field Diffeomorphism
Optimize LCC with deformation parameterized by SVF
- 29 X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Measuring Temporal Evolution with deformations
𝝋𝒕 𝒙 = 𝒆𝒙𝒑(𝒕. 𝒗 𝒙 )
https://team.inria.fr/asclepios/software/lcclogdemons/
[ Lorenzi, Ayache, Frisoni, Pennec, Neuroimage 81, 1 (2013) 470-483 ]
Longitudinal deformation analysis in AD From patient specific evolution to population trend
(parallel transport of SVS parameterizing deformation trajectories) Inter-subject and longitudinal deformations are of different nature
and might require different deformation spaces/metrics Consistency of the numerical scheme with geodesics?
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 30
Patient A
Patient B
? ? Template
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
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 31
P0 P’0
P1
A
P2
P’1 A’
C
P0 P’0
PN
A
P’N PA)
[Lorenzi, Pennec: Efficient Parallel Transport of Deformations in Time Series of Images: from Schild's to pole Ladder, JMIV 50(1-2):5-17, 2013 ]
Parallel transport along geodesics Along geodesics: Pole Ladder [Lorenzi and Pennec, JMIV 2013]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 32
[Lorenzi, Pennec: Efficient Parallel Transport of Deformations in Time Series of Images: from Schild's to pole Ladder, JMIV 50(1-2):5-17, 2013 ]
P0 P’0
P1
A
P’1 PA)
C
P0 P’0
P1
A
PA)
P’1
P0 P’0
T0
A
T’0 PA)
-A’ A’ C geodesic
Analysis of longitudinal datasets Multilevel framework
34
Single-subject, two time points
Single-subject, multiple time points
Multiple subjects, multiple time points
Log-Demons (LCC criteria)
4D registration of time series within the Log-Demons registration.
Pole or Schild’s Ladder
[Lorenzi et al, in Proc. of MICCAI 2011] X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Longitudinal model for AD
35
Estimated from 1 year changes – Extrapolation to 15 years
70 AD subjects (ADNI data)
Observed Extrapolated Extrapolated year
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Mean deformation / atrophy per group
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 36
M Lorenzi, N Ayache, X Pennec G B. Frisoni, for ADNI. Disentangling the normal aging from the pathological Alzheimer's disease progression on structural MR images. 5th Clinical Trials in Alzheimer's Disease (CTAD'12), Monte Carlo, October 2012. (see also MICCAI 2012)
Study of prodromal Alzheimer’s disease Linear regression of the SVF over time: interpolation + prediction
X. Pennec - NZMRI, Jan 13-17 2013 37
0*))(~()( TtvExptT
Multivariate group-wise comparison of the transported SVFs shows statistically significant differences (nothing significant on log(det) )
[Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011]
Group-wise flux analysis in Alzheimer’s disease: Quantification
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 38
From group-wise… …to subject specific
NIBAD’12 Challenge: Top-ranked on Hippocampal atrophy measures
Effect size on left hippocampus
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.
A zoo of log-demons registration algorithms: Log-demons: Open-source ITK implementation (Vercauteren MICCAI 2008)
http://hdl.handle.net/10380/3060 [MICCAI Young Scientist Impact award 2013]
Tensor (DTI) Log-demons (Sweet WBIR 2010): https://gforge.inria.fr/projects/ttk
LCC log-demons for AD (Lorenzi, Neuroimage. 2013) https://team.inria.fr/asclepios/software/lcclogdemons/
Hierarchichal multiscale polyaffine log-demons (Seiler, Media 2012) http://www.stanford.edu/~cseiler/software.html [MICCAI 2011 Young Scientist award]
3D myocardium strain / incompressible deformations (Mansi MICCAI’10)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 39
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 40
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 41
Expx / Logx is the basis of algorithms to compute on (fields of) Riemannian / affine manifolds
Simple statistics Mean through an exponential barycenter iteration Covariance matrices and higher order moments
Interpolation / filtering / convolution weighted means
Diffusion, extrapolation: standard discrete Laplacian = Laplace-Beltrami
Discrete parallel transport using Schild / Pole ladder
The Fréchet mean/exponential barycenter is the key Existance & uniqueness?
Statistics on surfaces seen as currents Characterize curves or surfaces by the flux (along or through them) of
all smooth vector fields (in a RKHS)
Extrinsinc statistical analysis in space of currents (mean, PCA) [Durrleman et al, MFCA 2008] (mean current is not a surface)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 42
Towards more complex geometries?
Original Shape (1476 delta currents) Compressed Shape (281 delta currents)
Fibre bundles and non integrable geometries Locally affine atoms of transformation:
Polyaffine deformations [Arsigny et al., MICCAI 06, JMIV 09] Jetlets diffeomorphisms [Sommer SIIMS 2013, Jacobs / Cotter 2014]
Multiscale LDDMM [Sommer et al, JMIV 2013] Fibers and sheets in the myocardium
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 43
Standard contact structure of the Heisenberg group
Towards more complex geometries?
Which space for anatomical shapes?
Physics Homogeneous space-time structure at large
scale (universality of physics laws) [Einstein, Weil, Cartan…]
Heterogeneous structure at finer scales: embedded submanifolds (filaments…)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 48
The universe of anatomical shapes? Affine, Riemannian of fiber bundle structure? Learn locally the topology and metric
Very High Dimensional Low Sample size setup Geometric prior might be the key!
Modélisation de la structure de l'Univers. NASA
Some references Statistics on Riemannnian manifolds
Xavier Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision, 25(1):127-154, July 2006. http://www.inria.fr/sophia/asclepios/Publications/Xavier.Pennec/Pennec.JMIV06.pdf
Invariant metric on SPD matrices and of Frechet mean to define manifold-valued image processing algorithms Xavier Pennec, Pierre Fillard, and Nicholas Ayache. A Riemannian Framework for
Tensor Computing. International Journal of Computer Vision, 66(1):41-66, Jan. 2006. http://www.inria.fr/sophia/asclepios/Publications/Xavier.Pennec/Pennec.IJCV05.pdf
Bi-invariant means with Cartan connections on Lie groups Xavier Pennec and Vincent Arsigny. Exponential Barycenters of the Canonical Cartan
Connection and Invariant Means on Lie Groups. In Frederic Barbaresco, Amit Mishra, and Frank Nielsen, editors, Matrix Information Geometry, pages 123-166. Springer, May 2012. http://hal.inria.fr/hal-00699361/PDF/Bi-Invar-Means.pdf
Cartan connexion for diffeomorphisms: Marco Lorenzi and Xavier Pennec. Geodesics, Parallel Transport & One-parameter
Subgroups for Diffeomorphic Image Registration. International Journal of Computer Vision, 105(2), November 2013 https://hal.inria.fr/hal-00813835/document
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 49
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014 50
Publications: https://team.inria.fr/asclepios/publications/
Software: https://team.inria.fr/asclepios/software/
Thank You!
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