Right-invariant metrics on diffeomorphisms groups with applications - PowerPoint PPT Presentation

LDDMM and beyond Fran¸ cois-Xavier Vialard Right-invariant metrics on diffeomorphisms groups with applications to diffeomorphic image registration. Fran¸ cois-Xavier Vialard Joint work with Colin Cotter, Marc Niethammer, Laurent Risser, Alain Trouv´ e. University Paris-Dauphine October 4th 2013

LDDMM and beyond Outline Fran¸ cois-Xavier Vialard 1 Introduction to Large Deformation by Diffeomorphisms Metric Mapping (LDDMM) 2 Higher-order models 3 Statistics on initial momenta 4 Another right-invariant metrics

LDDMM and beyond Motivation Fran¸ cois-Xavier Vialard Introduction to Large Deformation by Diffeomorphisms Metric Mapping (LDDMM) Higher-order models Statistics on initial momenta Another • Developing geometrical and statistical tools to analyse right-invariant metrics biomedical shapes distributions/evolutions, • Developing the associated numerical algorithms.

LDDMM and beyond Example of problems of interest Fran¸ cois-Xavier Given two shapes, find a diffeomorphism of R 3 that maps one Vialard shape onto the other Introduction to Large Deformation by Diffeomorphisms Different data types and different way of representing them. Metric Mapping (LDDMM) Higher-order models Statistics on initial momenta Another right-invariant metrics Figure: Two slices of 3D brain images of the same subject at different ages

LDDMM and beyond About Computational Anatomy Fran¸ cois-Xavier Vialard Old problems: Introduction to Large Deformation by to find a framework for registration of biological shapes, 1 Diffeomorphisms Metric Mapping (LDDMM) to develop statistical analysis in this framework. 2 Higher-order models Action of a transformation group on shapes or images Statistics on initial momenta Idea pioneered by Grenander and al. (80’s), then developed by Another right-invariant metrics 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 A Riemannian approach to diffeomorphic Fran¸ cois-Xavier Vialard registration Introduction to Large Several diffeomorphic registration methods are available: Deformation by Diffeomorphisms • Free-form deformations B-spline-based diffeomorphisms by D. Metric Mapping (LDDMM) Rueckert Higher-order models • Log-demons (X.Pennec et al.) Statistics on initial momenta • Large Deformations by Diffeomorphisms (M. Miller,A. Another Trouv´ e, L. Younes) right-invariant metrics Only the last one provides a Riemannian framework. • v t ∈ V a time dependent vector field on R n . • φ t ∈ Diff , the flow defined by ∂ t φ t = v t ( φ t ) . (1) Action of the group of diffeomorphism G 0 (flow at time 1): Π : G 0 × C → C , Π( φ, X ) . = φ. X � 1 Right-invariant metric on G 0 : d ( φ 0 , 1 , Id) 2 = 1 0 | v t | 2 V dt . 2 − → Strong metric needed on V (Mumford and Michor: Vanishing Geodesic Distance on... )

LDDMM and beyond Left action and right-invariant metric Fran¸ cois-Xavier Vialard Definition (Left action) Introduction to Large Deformation by Diffeomorphisms A left action for the group G is a map G × M → M satisfying Metric Mapping (LDDMM) Id · q = q for q ∈ M. 1 Higher-order models g 2 · ( g 1 · M ) = ( g 2 g 1 ) · q. Statistics on initial 2 momenta Another Example: The group on itself, GL n ( R ) acting by multiplication on right-invariant metrics R n . Definition (Right-invariant length) Let G be a Lie group and � · � a scalar product on its Lie algebra g := T Id G. Let g ( t ) be a C 1 path on the group. The length of the path ℓ ( g ( t )) can be defined by: � 1 � ∂ t g ( t ) g ( t ) − 1 � 2 dt . ℓ ( g ( t )) = (2) 0 Note that v ( t ) = ∂ t g ( t ) g ( t ) − 1 ∈ g . This is called right-trivialized velocity.

LDDMM and beyond Left action and right-invariant metric Fran¸ cois-Xavier Vialard Introduction to Large Deformation by Definition (Right-invariant metric) Diffeomorphisms Metric Mapping (LDDMM) Let g 1 , g 2 ∈ G be two group elements, the distance between g 1 Higher-order models and g 2 can be defined by: Statistics on initial momenta �� 1 � � ∂ t g ( t ) g ( t ) − 1 � 2 dt | g (0) = g 1 and g (1) = g 1 Another d 2 ( g 1 , g 2 ) = inf right-invariant metrics g ( t ) 0 Minimizers are called geodesics. Right-invariance simply means: d 2 ( g 1 g , g 2 g ) = d ( g 1 , g 2 ) . It comes from: ∂ t ( g ( t ) g 0 )( g ( t ) g 0 ) − 1 = ∂ t g ( t ) g 0 g − 1 0 g ( t ) − 1 = ∂ t g ( t ) g ( t ) − 1 .

LDDMM and beyond Euler-Poincar´ e equation Fran¸ cois-Xavier Vialard Compute the Euler-Lagrange equation of the distance functional: Introduction to Large Deformation by Diffeomorphisms ∂ g − d ∂ L ∂ L Metric Mapping g = 0 (LDDMM) dt ∂ ˙ Higher-order models With a change of variable, let’s do ”reduction” on the Lie algebra: Statistics on initial momenta � 1 � 1 Special case of 0 L ( g , ˙ g ) dt = 0 ℓ ( v ( t ) , Id ) dt . Another right-invariant metrics v ) ∂ℓ ( ∂ t + ad ∗ ∂ v = 0 . Proof. Compute variations of v ( t ) in terms of u ( t ) = δ g ( t ) g ( t ) − 1 . Find that admissible variations on g can be written as: δ v ( t ) = ˙ u − ad v u for any u vanishing at 0 and 1. Recall that ad v u = [ u , v ].

LDDMM and beyond EPDiff equation Fran¸ cois-Xavier Vialard Let’s formally apply this to the group of diffeomorphisms of R d Introduction to Large Deformation by with a metric � u , v � = � u , Lv � L 2 . Denoting m = Lu , Diffeomorphisms Metric Mapping (LDDMM) ∂ t m + Dm . u + Du T . m + div ( u ) m = 0 . (3) Higher-order models Statistics on initial For example, the L 2 metric gives: momenta Another right-invariant metrics ∂ t m + Du . u + Du T . u + div ( u ) m = 0 . (4) On the group of volume preserving diffeomorphisms of ( M , µ ) with the L 2 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 ∇� u � L 2 ) Other equations: Camassa-Holm equation, Hunter-Saxton equation...

LDDMM and beyond Left action and momentum map Fran¸ cois-Xavier Vialard Introduction to Large Suppose that the action is transitive and a submersion at identity, Deformation by Diffeomorphisms if g (0) = Id Metric Mapping (LDDMM) v ∈ g → v . q := d dt | 0 g ( t ) · q Higher-order models Statistics on initial momenta surjective. Define a Riemannian metric on TM by: Another right-invariant metrics � δ q � 2 = inf v ∈ g {� v � 2 | v · q = δ q } . Definition Let p ∈ T ∗ q M be a co-tangent vector at q then the momentum map is J : T ∗ M → g ∗ ( q , p ) → � J ( q , p ) , v � g = ( p , v · q )

LDDMM and beyond A Riemannian framework on the orbit Fran¸ cois-Xavier Vialard Introduction to Large Deformation by Diffeomorphisms Metric Mapping Proposition (LDDMM) Higher-order models Right-invariant metric + left action = ⇒ Riemannian metrics on Statistics on initial the orbits. The map Π q 0 : G ∋ g �→ g · q 0 ∈ Q is a Riemannian momenta submersion. Another right-invariant metrics Proposition The inexact matching functional � 1 V dt + 1 | v t | 2 σ 2 d ( φ 1 . q 0 , q target ) 2 J ( v ) = 0 leads to geodesics on the orbit of A for the induced Riemannian metric.

LDDMM and beyond Optimal control viewpoint Fran¸ cois-Xavier Vialard Introduction to Large Deformation by � 1 Move q 0 in order to minimize 1 0 � v � 2 dt + 1 2 � q (1) − q target � 2 Diffeomorphisms Metric Mapping 2 (LDDMM) under the constraint ˙ q = v · q . Higher-order models Pontryagin principle implies extremals are solutions of: Statistics on initial momenta Another right-invariant metrics q = v · q ˙ (5) p = − v ∗ · p ˙ (6) Lv = J ( q , p ) . (7) and p (1) + ∂ q [ 1 2 � q − q target � 2 ]( q (1)) = 0. Proposition J ( q , p ) satisfies the EPDiff equation.

LDDMM and beyond Matching problems in a diffeomorphic framework Fran¸ cois-Xavier Vialard Introduction to Large Deformation by U a domain in R n 1 Diffeomorphisms Metric Mapping V a Hilbert space of C 1 vector fields such that: (LDDMM) 2 Higher-order models Statistics on initial � v � 1 , ∞ ≤ C | v | V . momenta Another right-invariant metrics V is a Reproducing kernel Hilbert Space (RKHS): (pointwise evaluation continuous) = ⇒ Existence of a matrix function k V (kernel) defined on U × U such that: � v ( x ) , a � = � k V ( ., x ) a , v � V . Right invariant distance on G 0 � 1 d (Id , φ ) 2 = | v t | 2 inf V dt , v ∈ L 2 ([0 , 1] , V ) 0 − → geodesics on G 0 .

LDDMM and beyond Matching problems in a diffeomorphic framework Fran¸ cois-Xavier Vialard Action of G 0 on group of points (Landmarks): Introduction to Large Deformation by Diffeomorphisms φ. ( x 1 , . . . , x k ) = ( φ ( x 1 ) , . . . , φ ( x k )) , Metric Mapping (LDDMM) Higher-order models Momentum map: � k i =1 δ p i x i . Statistics on initial momenta Action of G 0 on images ( I ∈ L ∞ ( U )): Another right-invariant metrics φ. I = I ◦ φ − 1 . Momentum map: J ( I , P ) = − P ∇ I . Action of G 0 on surfaces: φ. S = φ ( S ) , Action on measures: ( φ.µ, f ) . = ( µ, f ◦ φ ) Generalized to currents (linear form on Ω k c ( R d )) and varifolds.