Slides - Workshop on Geometrical Models in Vision

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
2
Patient 2
3
4
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 - 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 2nd order diff eqs:
locally unique for initial point and speed
Exponential map (Normal coord. syst.) :
 Geodesic shooting: Expx(v) = g
(x,v)(1)

Log: find vector to shoot right (geodesic completeness!)
Reformulate algorithms with expx and logx

Vector -> Bipoint (no more equivalent class)
Operator
Subtraction
Euclidean space
Distance
xy  y  x
y  x  xy
dist ( x, y)  y  x
Gradient descent
xt   xt  C( xt )
Addition
Riemannian manifold
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
xy  Log x ( y )
y  Exp x (xy )
dist ( x, y)  xy
x
xt   Exp xt (C ( xt ))
7
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 =
𝑀
𝜙 𝑦 𝑝 𝑦 𝑑𝑀 𝑦

𝜙 = 𝑑𝑖𝑠𝑡 2 (variance) : 𝑬𝐱 𝑑𝑖𝑠𝑡 . , 𝑦

𝜙 = 𝑥 (mean) : 𝑬𝐱 𝐱 =
𝑀
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
2
=
𝑀
𝑑𝑖𝑠𝑡 𝑦, 𝑧 2 𝑝 𝑧 𝑑𝑀(𝑧)
𝑦 𝑝 𝑦 𝑑𝑀 𝑦
8
First Statistical Tools: Moments
Frechet / Karcher mean minimize the variance

Εx  argmin E dist( y, x)2
yM





 
E xx   xx. px ( z ).dM( z )  0 P(C )  0
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 t 1  exp x t (v) with v  E yx   Log x t (x i )
n i 1
[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]
     xz . xz  . p (z).dM(z)
 xx  E xx . xx
T
T
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:
4 first variation modes
have clinical meaning
• 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)
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
W1 | W2
Id


 Tr W1T W2   Tr(W1 ).Tr(W2 ) (   -1/n)

Exponential map

Log map
  Log  ( )  1/ 2 log( 1/ 2 .. 1/ 2 )1/ 2

Distance
dist (,  ) 2   | 
Exp ( )  1/ 2 exp( 1/ 2 .. 1/ 2 )1/ 2
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014

 log( 1/ 2 .. 1/ 2 )
2
Id
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
( x)  min i G ( x  xi ) dist 2 (, i )
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!
 
( x)  1 uS ( x)( x  u)  O  2


Reg ()    ( x) ( x ) dx
2
[ 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
Riemann Gaussian smoothing
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Euclidean Gaussian smoothing
Riemann anisotropic smoothing
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 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 .x , DLg .y >Id
(-1)
(-1)
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
Adapted from Lê Nguyên Hoang, science4all.org
20
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


𝑖 𝐿𝑜𝑔𝑥
𝑦𝑖 = 0
[Emery, Mokobodzki 91, Corcuera, Kendall 99]
Existence local uniqueness if local convexity [Arnaudon & Li, 2005]
For Cartan-Schouten connections
[Pennec & Arsigny, 2012]
𝐿𝑜𝑔 𝑥 −1 . 𝑦𝑖 = 0

Locus of points x such that

Algorithm: fixed point iteration (local convergence)
𝑥𝑡+1

1
= 𝑥𝑡 ∘ 𝐸𝑥𝑝
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
(−1)
𝑖
𝑔
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
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
25
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
26
The SVF framework for Diffeomorphisms
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
•exp
Stationary velocity field
Diffeomorphism
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)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
28
Measuring Temporal Evolution with deformations
Optimize LCC with deformation parameterized by SVF
𝝋𝒕 𝒙 = 𝒆𝒙𝒑(𝒕. 𝒗 𝒙 )
https://team.inria.fr/asclepios/software/lcclogdemons/
[ Lorenzi, Ayache, Frisoni, Pennec, Neuroimage 81, 1 (2013) 470-483 ]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
- 29
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?
Patient A
Template
?
?
Patient B
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
30
Parallel transport along arbitrary curves
Infinitesimal parallel transport = connection
g’X : TMTM
A numerical scheme to integrate for symmetric connections:
Schild’s Ladder [Elhers et al, 1972]
 Build geodesic parallelogrammoid
 Iterate along the curve
PN
PA)
A’
P’1
P
P’N
1

C
P0

P2


A
P’0
A
P0
P’0
[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 ]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
31
Parallel transport along geodesics
Along geodesics: Pole Ladder [Lorenzi and Pennec, JMIV 2013]
T0
PA)
P’1P1


C
P0

P0
P1
PA)
T’0
P’1

C geodesic
A’
-A’
 
AA
PA)
P’0
P’0
P0
A
P’0
[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 ]
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
32
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 - IHP, Sub-Riemannian Geom, 23/10/2014
34
Longitudinal model for AD
Estimated from 1 year changes – Extrapolation to 15 years
70 AD subjects (ADNI data)
year
Extrapolated
Observed
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
Extrapolated
35
Mean deformation / atrophy per group
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)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
36
Study of prodromal Alzheimer’s disease
Linear regression of the SVF over time: interpolation + prediction
Multivariate group-wise comparison
of the transported SVFs shows
statistically significant differences
(nothing significant on log(det) )
T (t )  Exp (v~(t )) *T0
[Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011]
X. Pennec - NZMRI, Jan 13-17 2013
37
Group-wise flux analysis in Alzheimer’s
disease: Quantification
…to subject specific
From group-wise…
Effect size on left hippocampus
NIBAD’12 Challenge:
Top-ranked on Hippocampal atrophy measures
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
38
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
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
40
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?
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
41
Towards more complex geometries?
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)


Original Shape (1476 delta currents)
Compressed Shape (281 delta currents)
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
42
Towards more complex geometries?
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
Standard contact structure of the Heisenberg group
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
43
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…)
The universe of anatomical shapes?


Modélisation de la structure de l'Univers. NASA
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!
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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 manifoldvalued 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
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Thank You!
Publications: https://team.inria.fr/asclepios/publications/
Software: https://team.inria.fr/asclepios/software/
X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014
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