Xavier Pennec Asclepios team, INRIA Sophia-Antipolis Mediterrane, - - PowerPoint PPT Presentation

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 ) ( t

t t

x C x x   







) (y Log xy

x

 xy x y   x y y x   ) , ( dist

x

xy y x  ) , ( dist

) (xy Exp y

x



)) ( (

t x t

x C Exp x

t

  







x y xy  

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

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

 𝑭𝐲 𝜚 = 𝜚 𝑧 𝑞 𝑧 𝑒𝑁 𝑧 𝑁

 𝜚 = 𝑒𝑗𝑡𝑢2 (variance) : 𝑭𝐲 𝑒𝑗𝑡𝑢 . , 𝑧 2 = 𝑒𝑗𝑡𝑢 𝑧, 𝑨 2𝑞 𝑨 𝑒𝑁(𝑨)

𝑁

 𝜚 = 𝑦 (mean) : 𝑭𝐲 𝐲 = 𝑧 𝑞 𝑧 𝑒𝑁 𝑧

𝑁

• X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014

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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 i t

t t

n v v

1 x x 1

) (x Log 1 y E with ) ( exp x x

 

 

 

 

 

) ( ) ( ). ( . x x E ) , dist( E argmin

2

    





C P z d z p y

y M M

M

x

x x x x Ε

[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 . x x . x E

T T

z d z p z z

x xx

x x

• 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

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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

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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 / 1 2 / 1 2 / 1 2 / 1

) . . exp( ) (       

  

Exp

2 / 1 2 / 1 2 / 1 2 / 1

) . . log( ) (         

  

Log

2 2 / 1 2 / 1 2

) . . log( | ) , (

Id

dist

  

        

 

• 1/n)

( ) Tr( ). Tr( Tr |

2 1 2 1 2 1

     W W W W W W

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 i i

x x G x ) , ( dist ) ( min ) (

2 

 

2 1

) ( ) ( ) (  



O u x x x

S u

     

 

 





     dx x

x 2 ) (

) ( ) ( Reg

[ Pennec, Fillard, Arsigny, IJCV 66(1), 2005, ISBI 2006]

• X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014

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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 𝑌 = 𝐸𝑀. 𝑦 from identity

 𝛿𝑦 𝑢 exists for all time  One parameter subgroup: 𝛿𝑦 𝑡 + 𝑢 = 𝛿𝑦 𝑡 . 𝛿𝑦 𝑢

Lie group exponential

 Definition: 𝑦 ∈ 𝔥  𝐹𝑦𝑞 𝑦 = 𝛿𝑦 1 𝜗 𝐻  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: 𝛼𝛿

𝛿 = 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

 𝑀𝑝𝑕𝑦 𝑧𝑗

𝑗

= 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 𝑀𝑝𝑕 𝑦−1. 𝑧𝑗 = 0  Algorithm: fixed point iteration (local convergence)

𝑦𝑢+1 = 𝑦𝑢 ∘ 𝐹𝑦𝑞 1 𝑜 𝑀𝑝𝑕 𝑦𝑢

−1. 𝑧𝑗

 Mean stable by left / right composition and inversion

 If 𝑛 is a mean of 𝑕𝑗 and ℎ is any group element, then

ℎ ∘ 𝑛 is a mean of ℎ ∘ 𝑕𝑗 , 𝑛 ∘ ℎ is a mean of the points 𝑕𝑗 ∘ ℎ and 𝑛(−1) is a mean of 𝑕𝑗

(−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 𝜄𝑗 < 𝜌

 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

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Generalization of the Statistical Framework

Covariance matrix & higher order moments

 Defined as tensors in tangent space

Σ = 𝑀𝑝𝑕𝑦 𝑧 ⊗ 𝑀𝑝𝑕𝑦 𝑧 𝜈(𝑒𝑧)

 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

𝝌𝒖 𝒚 = 𝒇𝒚𝒒(𝒖. 𝒘 𝒚 )

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 : TMTM 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 PA)

[Lorenzi, Pennec: Efficient Parallel Transport of Deformations in Time Series

• f 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

• f Images: from Schild's to pole Ladder, JMIV 50(1-2):5-17, 2013 ]

P0 P’0 P1 A P’1 PA)     C P0 P’0 P1 A PA)     P’1 P0 P’0 T0 A T’0 PA)

• 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

* )) ( ~ ( ) ( T t v Exp t T 

Multivariate group-wise comparison

• f 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

• n (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

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• X. Pennec - IHP, Sub-Riemannian Geom, 23/10/2014

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Publications: https://team.inria.fr/asclepios/publications/ Software: https://team.inria.fr/asclepios/software/

Thank You!