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

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Statistical Computing

n Manifolds for

Computational Anatomy 3: Metric and Affine Geometric Settings for Lie Groups

Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis

Xavier Pennec

Asclepios team, INRIA Sophia- Antipolis – Mediterranée, France

With contributions from Vincent Arsigny, Pierre Fillard, Marco Lorenzi, Christof Seiler, Jonathan Boisvert, Nicholas Ayache, etc

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups

 Riemannian frameworks on Lie groups  Lie groups as affine connection spaces  Bi-invariant statistics with Canonical Cartan connection  The SVF framework for diffeomorphisms

Analysis of Longitudinal Deformations

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Analysis of the Scoliotic Spine

Data



307 Scoliotic patients from the Montreal’s St-Justine Hosp



3D Geometry from multi-planar X-rays



Articulated model:17 relative pose of successive vertebras

Statistics



Main translation variability is axial (growth?)



Main rot. var. around anterior-posterior axis



4 first variation modes related to King’s classes [ J. Boisvert et al. ISBI’06, AMDO’06 and IEEE TMI 27(4), 2008 ]

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Bronze Standard Rigid Registration Validation

Best explanation of the observations (ML) :

 LSQ criterion  Robust Fréchet mean  Robust initialization and Newton gradient descent

Result Derive tests on transformations for accuracy / consistency

X. Pennec - ESI - Shapes, Feb 10-13 2015

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 

2 2 1 2 2 1 2

), , ( min ) , (   T T T T d 

trans rot j i

T   , ,

,





ij ij ij T

T d C ) ˆ , (

2

[ T. Glatard & al, MICCAI 2006,

Int. Journal of HPC Apps, 2006 ]

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Morphometry through Deformations

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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

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Natural Riemannian Metrics on Transformations

Transformation are Lie groups: Smooth manifold G compatible with group structure

 Composition g o h and inversion g-1 are smooth  Left and Right translation Lg(f) = g ○ f Rg (f) = f ○ g  Conjugation Conjg(f) = g ○ f ○ g-1

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

Implementation

 Practical computations using left (or right) translations

X. Pennec - ESI - Shapes, Feb 10-13 2015

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 

g) (f .Log DL (g) Log fg x) . DL ( Exp f x Exp

1) ( Id f f f f

1) (

 



  



Id

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Example on 3D rotations

Space of rotations SO(3):

 Manifold: RT.R=Id and det(R)=+1  Lie group ( R1 o R2 = R1.R2 & Inversion: R(-1) = RT )

Metrics on SO(3): compact space, there exists a bi-invariant metric

 Left / right invariant / induced by ambient space <X, Y> = Tr(XT Y)

Group exponential

 One parameter subgroups = bi-invariant Geodesic starting at Id

 Matrix exponential and Rodrigue’s formula: R=exp(X) and X = log(R)

 Geodesic everywhere by left (or right) translation

LogR(U) = R log(RT U) ExpR(X) = R exp(RT X) Bi-invariant Riemannian distance

 d(R,U) = ||log(RT U)|| = q( RT U )

X. Pennec - ESI - Shapes, Feb 10-13 2015

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General Non-Compact and Non-Commutative case

No Bi-invariant Mean for 2D Rigid Body Transformations

 Metric at Identity: 𝑒𝑗𝑡𝑢(𝐽𝑒, 𝜄; 𝑢1; 𝑢2 )2 = 𝜄2 + 𝑢1

2+ 𝑢2 2

 𝑈

1 = 𝜌 4 ; − 2 2 ; 2 2

𝑈

2 = 0; 2; 0 𝑈 3 = − 𝜌 4 ; − 2 2 ; − 2 2

 Left-invariant Fréchet mean: 0; 0; 0  Right-invariant Fréchet mean: 0;

2 3 ; 0 ≃ (0; 0.4714; 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 - ESI - Shapes, Feb 10-13 2015

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups

 Riemannian frameworks on Lie groups  Lie groups as affine connection spaces  Bi-invariant statistics with Canonical Cartan connection  The SVF framework for diffeomorphisms

Analysis of Longitudinal Deformations

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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 - ESI - Shapes, Feb 10-13 2015

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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 (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)

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Adapted from Lê Nguyên Hoang, science4all.org

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Canonical connections on Lie groups

Left invariant connections 𝛂𝑬𝑴.𝒀𝑬𝑴. 𝒁 = 𝑬𝑴. 𝛂𝒀𝒁

 Characteric bilinear form on the Lie algebra 𝑏 𝑦, 𝑧 = 𝛼𝑌

𝑍

|𝑓 ∈ g

 Symmetric part

1 2 (𝑏 𝑦, 𝑧 + 𝑏 𝑧, 𝑦 ) specifies geodesics

 Skew symmetric part

1 2 (𝑏 𝑦, 𝑧 − 𝑏 𝑧, 𝑦 ) specifies torsion along them

Bi-invariant connections

 𝑏 𝐵𝑒 𝑕 . 𝑦, 𝐵𝑒 𝑕 . 𝑧 = 𝐵𝑒 𝑕 . 𝑏 𝑦, 𝑧  𝑏 [𝑨, 𝑦], 𝑧 + 𝑏 𝑦, [𝑨, 𝑧] = 𝑨, 𝑏 𝑦, 𝑧 , x, y, z in g

Cartan Schouten connections (def. of Postnikov)

 LInv connections for which one-parameter subgroups are geodesics

 Matrices : M(t) = exp(t.V)  Diffeos : translations of Stationary Velocity Fields (SVFs)

 Uniquely determined by 𝑏 𝑦, 𝑦 = 0 (skew symmetry)

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Canonical Cartan connections on Lie groups

Bi-invariant Cartan Schouten connections

 Family 𝑏 𝑦, 𝑧 = 𝜇 𝑦, 𝑧 (-,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 (𝑏 𝑦, 𝑧 + 𝑏 𝑧, 𝑦 = 0):

ne-parameter subgroups and their left and right translations

 Curvature: R 𝑦, 𝑧 = λ 𝜇 − 1 [ 𝑦, 𝑧 , 𝑨]  Torsion: 𝑈 𝑦, 𝑧 = 2𝑏 𝑦, 𝑧 − 𝑦, 𝑧

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)

 𝑏 𝑦, 𝑧 = 1

2 𝑦, 𝑧

 Curvature 𝑆 𝑦, 𝑧 𝑨 = − 1

4

𝑦, 𝑧 , 𝑨

 Parallel transport along geodesics: Πexp

(𝑧)𝑦 = 𝐸𝑀exp (𝑧

2). 𝐸𝑀exp

(𝑧

2).x

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Cartan Connections are generally not metric

Levi-Civita Connection of a left-invariant (pseudo) metric is left-invariant

 Metric dual of the bracket < 𝑏𝑒∗ 𝑦, 𝑧 , 𝑨 > = < 𝑦, 𝑨 , 𝑧 >  𝑏 𝑦, 𝑧 =

1 2 𝑦, 𝑧 − 1 2 𝑏𝑒∗ 𝑦, 𝑧 + 𝑏𝑒∗ 𝑧, 𝑦

Bi-invariant (pseudo) metric => Symmetric Cartan connection

 A left-invariant (pseudo) metric is right-invariant if it is Ad-invariant

< 𝑦, 𝑧 > = < 𝐵𝑒𝑕 𝑦 , 𝐵𝑒𝑕 𝑧 >

 Infinitesimaly: < 𝑦, 𝑨 , 𝑧 > + < 𝑦, 𝑧, 𝑨 > = 0 or 𝑏𝑒∗ 𝑦, 𝑧 + 𝑏𝑒∗ 𝑧, 𝑦 = 0

Existence of bi-invariant (pseudo) metrics

 A Lie group admits a bi-invariant metric iff Ad(G) is relatively compact

𝐵𝑒 𝐻 ⊂ 𝑃 g ⊂ 𝐻𝑀(g)

 No bi-invariant metrics for rigid-body transformations

 Bi-invariant pseudo metric (Quadratic Lie groups):

Medina decomposition [Miolane MaxEnt 2014]

 Bi-inv. pseudo metric for SE(n) for n=1 or 3 only

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Group Geodesics

Group geodesic convexity

 Take an NCN V at Id  There exists a NCN 𝑊

𝑕 = 𝑕 ∘ 𝑊 ∩ 𝑊 ∘ 𝑕 at each g s.t.:

 𝑕 ∘ 𝐹𝑦𝑞 𝑦 = 𝐹𝑦𝑞(𝐵𝑒 𝑕 . 𝑦) ∘ 𝑕  𝑀𝑝𝑕 𝑕 ∘ ℎ ∘ 𝑕 −1

= 𝐵𝑒 𝑕 . 𝑀𝑝𝑕 ℎ

Group geodesics in Vg

 𝐹𝑦𝑞𝑕 𝑤 = 𝑕 ∘ 𝐹𝑦𝑞 𝐸𝑀𝑕 −1 . 𝑤 = 𝐹𝑦𝑞 𝐸𝑆𝑕 −1 . 𝑤 ∘ 𝑕  𝑀𝑝𝑕𝑕 ℎ = 𝐸𝑀𝑕. 𝑀𝑝𝑕 𝑕 −1 ∘ ℎ = 𝐸𝑆𝑕. 𝑀𝑝𝑕 ℎ ∘ 𝑕 −1

X. Pennec - ESI - Shapes, Feb 10-13 2015

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups

 Riemannian frameworks on Lie groups  Lie groups as affine connection spaces  Bi-invariant statistics with Canonical Cartan connection  The SVF framework for diffeomorphisms

Analysis of Longitudinal Deformations

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Mean value on an affine connection space

Fréchet / Karcher means not usable (no distance) but: Exponential barycenters

 [Emery & Mokobodzki 91, Corcuera & Kendall 99]

𝑀𝑝𝑕𝑦 𝑧 𝜈(𝑒𝑧) = 0 or 𝑀𝑝𝑕𝑦 𝑧𝑗

𝑗

= 0

 Existence? Uniqueness?  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

 Algorithm to compute the mean: fixed point iteration (stability?)

X. Pennec - ESI - Shapes, Feb 10-13 2015

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 

   

 

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

2

    





C P z d z p y

y M M

M

x

x x x x Ε

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Bi-invariant Mean on Lie Groups

Exponential barycenter of the symmetric Cartan connection

 Locus of points where 𝑀𝑝𝑕 𝑛−1. 𝑕𝑗 = 0 (whenever defined)  Iterative algorithm: 𝑛𝑢+1 = 𝑛𝑢 ∘ 𝐹𝑦𝑞

1 𝑜 𝑀𝑝𝑕 𝑛𝑢 −1. 𝑕𝑗

 First step corresponds to the Log-Euclidean mean  Corresponds to the first definition of bi-invariant mean of [V. Arsigny, X. Pennec,

and N. Ayache. Research Report RR-5885, INRIA, April 2006.]

Mean is 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)

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[Pennec & Arsigny, Ch.7 p.123-166 , Matrix Information Geometry, Springer, 2012]

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Bi-invariant Mean on Lie Groups

Fine existence

 If the data points belong to a sufficiently small normal convex

neighborhood of some point, then there exists a unique solution in this NCN.

 Moreover, the iterated point strategy converges at least at a

linear rate towards this unique solution, provided the initialization is close enough.

 Proof: using an auxiliary metric, the iteration is a contraction.

Closed-form for 2 points

 𝑛(𝑢) = 𝑦 ∘ 𝐹𝑦𝑞 (𝑢. 𝑀𝑝𝑕 𝑦 −1 ∘ 𝑧 )

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Special Matrix Groups

Scaling and translations ST(n)

 No bi-invariant metric  Group geodesics defined globally, all points are reachable  Existence and uniqueness of bi-invariant mean (closed form)

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Group / left-invariant / right-invariant geodesics

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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 - ESI - Shapes, Feb 10-13 2015

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Example mean of 2D rigid-body transformation

𝑈

1 =

𝜌 4 ; − 2 2 ; 2 2 𝑈2 = 0; 2; 0 𝑈3 = − 𝜌 4 ; − 2 2 ; − 2 2

 Metric at Identity: 𝑒𝑗𝑡𝑢(𝐽𝑒, 𝜄; 𝑢1; 𝑢2 )2 = 𝜄2 + 𝑢1

2+ 𝑢2 2

 Left-invariant Fréchet mean: 0; 0; 0  Log-Euclidean mean: 0;

2−𝜌/4 3

; 0 ≃ (0; 0.2096; 0)

 Bi-invariant mean: 0;

2−𝜌/4 1+𝜌/4( 2+1) ; 0 ≃ (0; 0.2171; 0)

 Right-invariant Fréchet mean: 0;

2 3 ; 0 ≃ (0; 0.4714; 0)

X. Pennec - ESI - Shapes, Feb 10-13 2015

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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

𝜈 𝑛,Σ (𝑕) = 𝑀𝑝𝑕𝑛 𝑕

𝑗Σ𝑗𝑘 (−1) 𝑀𝑝𝑕𝑛 𝑕 𝑘𝜈(𝑒𝑧)

 Tangent Principal Component Analysis (t-PCA)  Principal Geodesic Analysis (PGA), provided a data likelihood  Independent Component Analysis (ICA)

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Generalizing the statistical setting

Covariance matrix & higher order moments

 Can be defined as a 2-covariant tensor

Σ = 𝑀𝑝𝑕𝑦 𝑧 ⊗ 𝑀𝑝𝑕𝑦 𝑧 𝜈(𝑒𝑧)

 Cannot be defined as Sij = E( <x|ei><x|ej>) (no dot product)  Diagonalization depends on the basis: change PCA to ICA?

Mahalanobis distance is well defined and bi-invariant

𝜈 𝑛,Σ (𝑕) = 𝑀𝑝𝑕𝑛 𝑕

𝑗Σ𝑗𝑘 (−1) 𝑀𝑝𝑕𝑛 𝑕 𝑘𝜈(𝑒𝑧)

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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 MaxEnt 2014]

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 - ESI - Shapes, Feb 10-13 2015

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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Statistical Computing on Manifolds for Computational Anatomy

Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups

 Riemannian frameworks on Lie groups  Lie groups as affine connection spaces  Bi-invariant statistics with Canonical Cartan connection  The SVF framework for diffeomorphisms

Analysis of Longitudinal Deformations

SLIDE 28

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 Hk or H∞ (RKHS) in LDDMM  diffeomorphisms [Miller,

Trouve, Younes, Holm, Dupuis, Beg… 1998 – 2009] Geodesics determined by optimization of a time-varying vector field

 Distance  Geodesics characterized by initial velocity / momentum  Optimization for images is quite tricky (and lenghty)

dt t x d x vt ) , ( ) (  

Id t t t

v v

t

1 

 



 ) ( min arg ) , (

1 2 1 2

dt v d

t t

t v 





 

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31

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)

 Computing the Jacobian :Dexp(v) = Dexp(v/2) o exp(v/2) . Dexp(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 - ESI - Shapes, Feb 10-13 2015
exp

Stationary velocity field Diffeomorphism

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Optimize LCC with deformation parameterized by SVF

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X. Pennec - ESI - Shapes, Feb 10-13 2015

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 ]

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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)

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/

 3D myocardium strain / incompressible deformations (Mansi MICCAI’10)  Hierarchichal multiscale polyaffine log-demons (Seiler, Media 2012)

http://www.stanford.edu/~cseiler/software.html [MICCAI 2011 Young Scientist award]

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A powerful framework for statistics

Parametric diffeomorphisms [Arsigny et al., MICCAI 06, JMIV 09]

 One affine transformation per region (polyaffines transformations)  Cardiac motion tracking for each subject [McLeod, Miccai 2013]

Log demons projected but with 204 parameters instead of a few millions

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expp

Stationary velocity fields Diffeomorphism AHA regions

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A powerful framework for statistics

Parametric diffeomorphisms [Arsigny et al., MICCAI 06, JMIV 09]

 One affine transformation per region (polyaffines transformations)  Cardiac motion tracking for each subject [McLeod, Miccai 2013]

Log demons projected but with 204 parameters instead of a few millions

 Group analysis using tensor reduction : reduced model

8 temporal modes x 3 spatial modes = 24 parameters (instead of 204)

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Hierarchical Deformation model

Varying deformation atoms for each subject

M3 M4 M5 M6 M1 M2 M0

K

M3 M4 M5 M6 M1 M2 M0

1

… Subject level:

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Spatial structure of the anatomy common to all subjects

w0 w1 w2 w3 w4 w5 w6

Population level: Aff(3) valued trees

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Level 0 Level 1 Level 3 Level 4 Level 2 Level 5 T wo sides T eeth Angle and ramus Thickness Global scaling Oriented bounding boxes Weights First mode of variation Structure

47 subjects Global scaling Thickness Angle and ramus

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[Seiler, Pennec, Reyes, Medical Image Analysis 16(7):1371-1384, 2012]

Hierarchical Estimation of the Variability

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Level 0 Level 1 Level 3 Level 4 Level 2 Level 5 T wo sides T eeth Angle and ramus Thickness Global scaling Oriented bounding boxes Weights First mode of variation Structure

Two sides Teeth [Seiler, Pennec, Reyes, Medical Image Analysis 16(7):1371-1384, 2012]

X. Pennec - ESI - Shapes, Feb 10-13 2015

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Hierarchical Estimation of the Variability

47 subjects

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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 - ESI - Shapes, Feb 10-13 2015

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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

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