Applications of right-invariant Riemannian metrics on

LDDMM and beyond
Fran¸cois-Xavier
Vialard
Applications of right-invariant Riemannian
metrics on diffeomorphism groups to
biomedical imaging
Fran¸cois-Xavier Vialard
Joint work with Marc Niethammer, Laurent Risser and Tanya Schmah and Alain Trouv´
e.
University Paris-Dauphine
30 Janvier 2014
Outline
1
Introduction to diffeomorphisms group and Riemannian tools
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Motivation
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
• Developing geometrical and statistical tools to analyse
biomedical shapes distributions/evolutions,
• Developing the associated numerical algorithms.
Soft available here:
http://sourceforge.net/projects/utilzreg/
Example of problems of interest
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Given two shapes, find a diffeomorphism of R3 that maps one
shape onto the other
LDDMM and beyond
Example of problems of interest
Fran¸cois-Xavier
Vialard
3
Given two shapes, find a diffeomorphism of R that maps one
shape onto the other
Different data types and different way of representing them.
Figure: Two slices of 3D brain images of the same subject at different
ages
Introduction to
diffeomorphisms group
and Riemannian tools
Example of problems of interest
Given two shapes, find a diffeomorphism of R3 that maps one
shape onto the other
Deformation by a diffeomorphism
Figure: Diffeomorphic deformation of the image
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Variety of shapes
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Figure: Different anatomical structures extracted from MRI data
Variety of shapes
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Figure: Different anatomical structures extracted from MRI data
About Computational Anatomy
Old problems:
1
to find a framework for registration of biological shapes,
2
to develop statistical analysis in this framework.
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
About Computational Anatomy
Old problems:
1
to find a framework for registration of biological shapes,
2
to develop statistical analysis in this framework.
Action of a transformation group on shapes or images
Idea pioneered by Grenander and al. (80’s), then developed by
M.Miller, A.Trouv´e, L.Younes.
Figure: deforming the shape of a fish by D’Arcy Thompson, author of
On Growth and Forms (1917)
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
About Computational Anatomy
Old problems:
1
to find a framework for registration of biological shapes,
2
to develop statistical analysis in this framework.
Action of a transformation group on shapes or images
Idea pioneered by Grenander and al. (80’s), then developed by
M.Miller, A.Trouv´e, L.Younes.
Figure: deforming the shape of a fish by D’Arcy Thompson, author of
On Growth and Forms (1917)
New problems like study of Spatiotemporal evolution of
shapes within a diffeomorphic approach
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
A Riemannian approach to diffeomorphic
registration
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Several diffeomorphic registration methods are available:
• Free-form deformations B-spline-based diffeomorphisms by D.
Rueckert
• Log-demons (X.Pennec et al.)
• Large Deformations by Diffeomorphisms (M. Miller,A.
Trouv´e, L. Younes)
Only the last one provides a Riemannian framework.
LDDMM and beyond
Set of deformations
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
• Let V Hilbert space of C 1 vector fields (V ,→ C 1 ).
• vt ∈ V a time dependent vector field on Rn .
• φt ∈ Diff , the flow defined by
∂t φt = vt (φt ) .
(1)
Definition
GV = {φ(1) | v ∈ L2 ([0, 1], V )} is the optimization set.
Remarks
• Stable under composition and inverse.
• GV is a group but not a Lie group (unless finite dimensional).
LDDMM and beyond
Variational formulation
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Find the best deformation, minimize
J (φ) = inf
φ∈GV
d(φ.A, B)2
| {z }
similarity measure
(2)
LDDMM and beyond
Variational formulation
Fran¸cois-Xavier
Vialard
Find the best deformation, minimize
Introduction to
diffeomorphisms group
and Riemannian tools
2
J (φ) = inf d(φ.A,
B)
φ∈G
{z
}
|
V similarity measure
(2)
Tychonov regularization:
J (φ) =
R(φ)
| {z }
+
Regularization
1
d(φ.A, B)2 .
2
2σ
|
{z
}
(3)
similarity measure
Riemannian metric on GV :
R(φ) =
1
2
Z
1
|vt |2V dt
0
is a right invariant metric on GV .
(4)
LDDMM and beyond
Optimization problem
Fran¸cois-Xavier
Vialard
Minimizing
J (v ) =
1
2
Introduction to
diffeomorphisms group
and Riemannian tools
1
Z
|vt |2V dt +
0
1
d(φ0,1 .A, B)2 .
2σ 2
In the case of landmarks:
J (φ) =
1
2
Z
1
|vt |2V dt +
0
k
1 X
kφ(xi ) − yi k2 ,
2σ 2
i=1
In the case of images:
d(φ0,1 .I0 , Itarget )2 =
Z
U
|I0 ◦ φ1,0 − Itarget |2 dx .
LDDMM and beyond
Optimization problem
Fran¸cois-Xavier
Vialard
Minimizing
J (v ) =
1
2
Introduction to
diffeomorphisms group
and Riemannian tools
1
Z
|vt |2V dt +
0
1
d(φ0,1 .A, B)2 .
2σ 2
In the case of landmarks:
J (φ) =
1
2
Z
1
|vt |2V dt +
0
k
1 X
kφ(xi ) − yi k2 ,
2σ 2
i=1
In the case of images:
d(φ0,1 .I0 , Itarget )2 =
Z
|I0 ◦ φ1,0 − Itarget |2 dx .
U
Main issues for practical applications:
• choice of the metric (prior),
• choice of the similarity measure.
A Riemannian framework on the orbit
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Proposition
Right-invariant metric + left action =⇒ Riemannian metrics on
the orbits. The map Πq0 : G 3 g 7→ g · q0 ∈ Q is a Riemannian
submersion.
Proposition
The inexact matching functional
Z
J (v ) =
0
1
|vt |2V dt +
1
d(φ0,1 .A, B)2
σ2
leads to geodesics on the orbit of A for the induced Riemannian
metric.
Why does the Riemannian framework matter?
Generalizations of statistical tools in Euclidean space:
• Distance often given by a Riemannian metric.
• Straight lines → geodesic defined by
Z 1
Variational definition: arg min
kck
˙ 2c(t) dt = 0 ,
c(t)
0
Equivalent (local) definition: ∇c˙ c˙ = c¨ + Γ(c)(c,
˙ c)
˙ = 0.
• Average → Fr´echet/K¨archer mean.
Variational definition:
Critical point definition:
arg min{x → E [d 2 (x, y )]dµ(y )}
E [∇x d 2 (x, y )]dµ(y )] = 0 .
• PCA → Tangent PCA or PGA.
• Geodesic regression, cubic regression...(variational or
algebraic)
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Why does the Riemannian framework matter?
Generalizations of statistical tools in Euclidean space:
• Distance often given by a Riemannian metric.
• Straight lines → geodesic defined by
Z 1
Variational definition: arg min
kck
˙ 2c(t) dt = 0 ,
c(t)
0
Equivalent (local) definition: ∇c˙ c˙ = c¨ + Γ(c)(c,
˙ c)
˙ = 0.
• Average → Fr´echet/K¨archer mean.
Variational definition:
Critical point definition:
arg min{x → E [d 2 (x, y )]dµ(y )}
E [∇x d 2 (x, y )]dµ(y )] = 0 .
• PCA → Tangent PCA or PGA.
• Geodesic regression, cubic regression...(variational or
algebraic)
Riemannian metric needed, or at least a connection.
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Why does the Riemannian framework matter?
Generalizations of statistical tools in Euclidean space:
• Distance often given by a Riemannian metric.
• Straight lines → geodesic defined by
Z 1
Variational definition: arg min
kck
˙ 2c(t) dt = 0 ,
c(t)
0
Equivalent (local) definition: ∇c˙ c˙ = c¨ + Γ(c)(c,
˙ c)
˙ = 0.
• Average → Fr´echet/K¨archer mean.
Variational definition:
Critical point definition:
arg min{x → E [d 2 (x, y )]dµ(y )}
E [∇x d 2 (x, y )]dµ(y )] = 0 .
• PCA → Tangent PCA or PGA.
• Geodesic regression, cubic regression...(variational or
algebraic)
Riemannian metric needed, or at least a connection.
Pitfalls:
• Loose uniqueness of geodesic or average (positive curvature).
• Equivalent definitions diverge (generalisation of PCA).
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
LDDMM and beyond
Geodesics and exponential map
Fran¸cois-Xavier
Vialard
On a given Riemannian manifold M, geodesics are given by
Exp : Tp M 7→ M
Figure: The exponential map encodes geodesics
1
Always defined locally.
2
Not always defined globally (geodesic completeness)
3
Not always surjective.
(5)
Introduction to
diffeomorphisms group
and Riemannian tools
Interpolation, Extrapolation
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Figure: Interpolation of happiness
Interpolation, Extrapolation
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Figure: Extrapolation of happiness
LDDMM and beyond
K¨archer mean on 3D images
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Init. guesses
1 iteration
2 iterations
3 iterations
A1i
A2i
A3i
A4i
Figure: Average image estimates Am
i , m ∈ {1, · · · , 4} after i =0, 1, 2
and 3 iterations.
Longitudinal data
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Figure: Slices of 3D volumic images: 33 / 36 / 43 weeks of gestational
age of the same subject.
LDDMM and beyond
Longitudinal data
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Movie
Figure: Representation of the surface - Back of the brain
LDDMM and beyond
Riemannian cubic splines
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Acceleration on a Riemannian manifold M: let c : I → M be a C 2
curve. The notion of acceleration is:
X
D
c(t)
˙
= ∇c˙ c(=
˙
c¨k +
c˙ i Γki,j c˙ j )
dt
i,j
with ∇ the Levi-Civita connection.
(6)
LDDMM and beyond
Riemannian cubic splines
Fran¸cois-Xavier
Vialard
Acceleration on a Riemannian manifold M: let c : I → M be a C 2
curve. The notion of acceleration is:
X
D
c(t)
˙
= ∇c˙ c(=
˙
c¨k +
c˙ i Γki,j c˙ j )
dt
(6)
i,j
with ∇ the Levi-Civita connection.
Riemannian splines: Crouch, Silva-Leite (90’s)
Z
On SO(3) inf
c
0
1
1
|∇c˙t c˙t |2M dt .
2
subject to c(i) = ci and c(i)
˙
= vi for i = 0, 1.
(7)
Introduction to
diffeomorphisms group
and Riemannian tools
Numerical examples on points
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
• First Column: Geodesic Regression
• Second column: Linear Interpolation
• Third Column: Spline Interpolation
Robustness to noise
Due to the spatial regularisation of the kernel:
Figure: Gaussian noise added to the position of 50 landmarks
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Robustness to noise
Due to the spatial regularisation of the kernel:
Figure: Gaussian noise added to the position of 50 landmarks
• Left: no noise.
• Center: standard deviation of 0.02.
• Right: standard deviation of 0.09.
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
A generative model of shape evolutions
A stochastic model:
Proposition
If k is C 1 , the solutions of the stochastic differential equation
defined by
(
dpt = −∂x H0 (pt , xt )dt + ut (xt )dt + ε(pt , xt )dBt
(8)
dxt = ∂p H0 (pt , xt )dt.
are non exploding with few assumptions on ut and ε.
Figure: The first figure represents a calibrated spline interpolation and the
three others √
are white noise perturbations ot the spline interpolation with
respectively n set to 0.25, 0.5 and 0.75.
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Connecting with mathematics
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Calculation of Euler-Lagrange equation:
1
Henri Poincar´e, Sur une forme nouvelle des ´equations de la
M´ecanique. [1901]
2
Vladimir Arnold, Sur la g´eom´etrie diff´erentielle des groupes
de Lie de dimension innie et ses applications `a
l’hydrodynamique des fluides parfaits. [1966]
3
Ebin, Marsden: Groups of diffeomorphisms and the motion of
an incompressible fluid. [1970]
Right-invariant metric
Definition (Right-invariant metric)
Let g1 , g2 ∈ G be two group elements, the distance between g1
and g2 can be defined by:
Z 1
d 2 (g1 , g2 ) = inf
k∂t g (t)g (t)−1 k2 dt |g (0) = g0 and g (1) = g1
g (t)
0
Minimizers are called geodesics.
Right-invariance simply means:
d 2 (g1 g , g2 g ) = d(g1 , g2 ) .
It comes from:
∂t (g (t)g0 )(g (t)g0 )−1 = ∂t g (t)g0 g0−1 g (t)−1 = ∂t g (t)g (t)−1 .
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Euler-Poincar´e equation
Compute the Euler-Lagrange equation of the distance functional:
d ∂L
∂L
−
=0
∂g
dt ∂ g˙
With a change of variable, let’s do ”reduction” on the Lie algebra:
R1
R1
Special case of 0 L(g , g˙ )dt = 0 `(v (t), Id)dt.
(
∂`
(∂t + ad∗v ) ∂v
= 0,
∂t g = v · g
Proof.
Compute variations of v (t) in terms of u(t) = δg (t)g (t)−1 . Find
that admissible variations on g can be written as:
δv (t) = u˙ − adv u for any u vanishing at 0 and 1. Recall that
adv u = [u, v ].
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
LDDMM and beyond
Euler-Poincar´e/Euler-Arnold equation
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Let’s formally apply this to the group of diffeomorphisms of Rd
with a metric hu, v i = hu, Lv iL2 . Denoting m = Lu,
∂t m + Dm.u + Du T .m + div(u)m = 0 .
(9)
For example, the L2 metric gives:
∂t u + Du.u + Du T .u + div(u)u = 0 .
(10)
On the group of volume preserving diffeomorphisms of (M, µ) with
the L2 metric:
Euler’s equation for ideal fluid where div(u) = 0
∂t u + ∇u u = −∇p ,
(use div(u) = 0 and write the term Du T .u as a gradient as
∇ 21 kukL2 )
LDDMM and beyond
Other equations...
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
Burgers equation L = Id on Diff (S1 ):
ut + 3uux = 0
(11)
Camassa-Holm equation L = Id − ∆ on Diff (S1 ):
ut − utxx + 3uux − 2ux uxx − uuxxx = 0
(12)
Korteweg de Vries equation on S1 for the Bott-Virasoro group
(central extension of Diff (S1 )):
ut − utxx + 3uux − 2ux uxx − uuxxx = 0
Hunter-Saxton equation...
(13)
LDDMM and beyond
About strong metrics on Diff s (Rd )
Fran¸cois-Xavier
Vialard
Definition
s
d
s
d
For s > d/2 + 1, Diff (R ) := {Id + f | f ∈ H (R )} ∩
1
CDiff
(Rd ) .
T Dn ×Dn T Dn → H n (Rd , Rd )×H n (Rd , Rd ) → R
(ϕ, X , Y ) 7→ (X ◦ ϕ−1 , Y ◦ ϕ−1 ) 7→ hX ◦ ϕ−1 , Y ◦ ϕ−1 iH n
|
{z
}
only continuous
|
{z
smooth!
}
1
Fredholm properties of Riemannian exponential maps on
diffeomorphism groups (Misiolek and Preston), Inventiones
Math.
→ Improve surjectivity of the exponential map.
2
A variational approach on groups of diffeomorphisms.
(Bruveris and FXV). [In preparation]
→ Complete surjectivity + higher order models.
Introduction to
diffeomorphisms group
and Riemannian tools
Beyond right-invariant metrics
LDDMM and beyond
Fran¸cois-Xavier
Vialard
Introduction to
diffeomorphisms group
and Riemannian tools
What metric to choose?
Right-invariant metric setting is constrained!
Ongoing work
1
Using left-invariant metrics,
2
Designing Riemannian metrics on shapes for statistics,
3
Supervised learning of left invariant metrics,
4
Supervised learning of connections.

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