Metric learning for diffeomorphic image registration. Fran - PowerPoint PPT Presentation

Metric learning for diffeomorphic image registration. Fran¸ cois-Xavier Vialard Metric learning for diffeomorphic image registration. Fran¸ cois-Xavier Vialard Universit´ e Paris-Est Marne-la-Vall´ ee joint work with M. Niethammer and R. Kwitt. IHP, March 2019.

Metric learning for Outline diffeomorphic image registration. Fran¸ cois-Xavier Vialard 1 Introduction to diffeomorphisms group and Riemannian tools 2 Choice of the metric 3 Spatially dependent metrics 4 Metric learning 5 SVF metric learning

Metric learning for Example of problems of interest diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent metrics Metric learning Given two shapes, find a diffeomorphism of R 3 that maps one SVF metric learning shape onto the other

Metric learning for Example of problems of interest diffeomorphic image registration. Given two shapes, find a diffeomorphism of R 3 that maps one Fran¸ cois-Xavier Vialard shape onto the other Introduction to diffeomorphisms group and Riemannian tools Different data types and different way of representing them. Choice of the metric Spatially dependent metrics Metric learning SVF metric learning Figure – Two slices of 3D brain images of the same subject at different ages

Metric learning for Example of problems of interest diffeomorphic image registration. Given two shapes, find a diffeomorphism of R 3 that maps one Fran¸ cois-Xavier Vialard shape onto the other Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent metrics Metric learning SVF metric learning Deformation by a diffeomorphism Figure – Diffeomorphic deformation of the image

Metric learning for Variety of shapes diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent metrics Metric learning SVF metric learning Figure – Different anatomical structures extracted from MRI data

Metric learning for Variety of shapes diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent metrics Metric learning SVF metric learning Figure – Different anatomical structures extracted from MRI data

Metric learning for A Riemannian approach to diffeomorphic diffeomorphic image registration. registration Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent Several diffeomorphic registration methods are available: metrics Metric learning • Free-form deformations B-spline-based diffeomorphisms by D. SVF metric learning Rueckert • Log-demons (X.Pennec et al.) • Large Deformations by Diffeomorphisms (M. Miller,A. Trouv´ e, L. Younes) • ANTS Only the two last ones provide a Riemannian framework.

Metric learning for A Riemannian approach to diffeomorphic diffeomorphic image registration. registration Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools • v t ∈ V a time dependent vector field on R n . Choice of the metric Spatially dependent • ϕ t ∈ Diff , the flow defined by metrics Metric learning ∂ t ϕ t = v t ( ϕ t ) . (1) SVF metric learning 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... )

Metric learning for Matching problems in a diffeomorphic framework diffeomorphic image registration. Fran¸ cois-Xavier Vialard U a domain in R n Introduction to 1 diffeomorphisms group V a Hilbert space of C 1 vector fields such that: and Riemannian tools 2 Choice of the metric Spatially dependent � v � 1 , ∞ � C | v | V . metrics Metric learning SVF metric learning

Metric learning for Matching problems in a diffeomorphic framework diffeomorphic image registration. Fran¸ cois-Xavier Vialard U a domain in R n Introduction to 1 diffeomorphisms group V a Hilbert space of C 1 vector fields such that: and Riemannian tools 2 Choice of the metric Spatially dependent � v � 1 , ∞ � C | v | V . metrics Metric learning V is a Reproducing kernel Hilbert Space (RKHS): (pointwise SVF metric learning 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 .

Metric learning for Matching problems in a diffeomorphic framework diffeomorphic image registration. Fran¸ cois-Xavier Vialard U a domain in R n Introduction to 1 diffeomorphisms group V a Hilbert space of C 1 vector fields such that: and Riemannian tools 2 Choice of the metric Spatially dependent � v � 1 , ∞ � C | v | V . metrics Metric learning V is a Reproducing kernel Hilbert Space (RKHS): (pointwise SVF metric learning 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 .

Metric learning for Variational formulation diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to Find the best deformation, minimize diffeomorphisms group and Riemannian tools ϕ ∈ G V d ( ϕ. A , B ) 2 Choice of the metric J ( ϕ ) = inf (2) Spatially dependent � �� � metrics similarity measure Metric learning SVF metric learning

Metric learning for Variational formulation diffeomorphic image registration. Fran¸ cois-Xavier Vialard Find the best deformation, minimize Introduction to diffeomorphisms group ϕ ∈ G V d ( ϕ. A , B ) 2 ✘ ✘✘✘✘✘✘✘✘✘✘✘ J ( ϕ ) = inf (2) and Riemannian tools � �� � Choice of the metric similarity measure Spatially dependent metrics Tychonov regularization: Metric learning SVF metric learning 1 2 σ 2 d ( ϕ. A , B ) 2 J ( ϕ ) = R ( ϕ ) + . (3) � �� � � �� � Regularization similarity measure Riemannian metric on G V : � 1 R ( ϕ ) = 1 | v t | 2 V dt (4) 2 0 is a right-invariant metric on G V .

Metric learning for Optimization problem diffeomorphic image registration. Fran¸ cois-Xavier Vialard Minimizing Introduction to � 1 diffeomorphisms group J ( v ) = 1 1 2 σ 2 d ( ϕ 0 , 1 . A , B ) 2 . and Riemannian tools | v t | 2 V dt + 2 Choice of the metric 0 Spatially dependent metrics In the case of landmarks: Metric learning � 1 k SVF metric learning J ( ϕ ) = 1 1 � � ϕ ( x i ) − y i � 2 , | v t | 2 V dt + 2 σ 2 2 0 i =1 In the case of images: � d ( ϕ 0 , 1 . I 0 , I target ) 2 = | I 0 ◦ ϕ 1 , 0 − I target | 2 dx . U

Metric learning for Optimization problem diffeomorphic image registration. Fran¸ cois-Xavier Vialard Minimizing Introduction to � 1 diffeomorphisms group J ( v ) = 1 1 2 σ 2 d ( ϕ 0 , 1 . A , B ) 2 . and Riemannian tools | v t | 2 V dt + 2 Choice of the metric 0 Spatially dependent metrics In the case of landmarks: Metric learning � 1 k SVF metric learning J ( ϕ ) = 1 1 � � ϕ ( x i ) − y i � 2 , | v t | 2 V dt + 2 σ 2 2 0 i =1 In the case of images: � d ( ϕ 0 , 1 . I 0 , I target ) 2 = | I 0 ◦ ϕ 1 , 0 − I target | 2 dx . U Main issues for practical applications: • choice of the metric (prior), • choice of the similarity measure.

Metric learning for Why does the Riemannian framework matter? diffeomorphic image registration. Fran¸ cois-Xavier Vialard Generalizations of statistical tools in Euclidean space: Introduction to • Distance often given by a Riemannian metric. diffeomorphisms group and Riemannian tools • Straight lines → geodesic defined by Choice of the metric Spatially dependent � 1 metrics c � 2 Variational definition: arg min � ˙ c ( t ) dt = 0 , Metric learning c ( t ) 0 SVF metric learning Equivalent (local) definition: ∇ ˙ c ˙ c = ¨ c + Γ( c )( ˙ c , ˙ c ) = 0 . • Average → Fr´ echet/Karcher mean. arg min { x → E [ d 2 ( x , y )] d µ ( y ) } Variational definition: E [ ∇ x d 2 ( x , y )] d µ ( y )] = 0 . Critical point definition: • PCA → Tangent PCA or PGA. • Geodesic regression, cubic regression...(variational or algebraic)

Metric learning for Karcher mean on 3D images diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Init. guesses Choice of the metric Spatially dependent metrics Metric learning SVF metric learning 1 iteration 2 iterations 3 iterations A 1 A 2 A 3 A 4 i i i i Figure – Average image estimates A m i , m ∈ { 1 , · · · , 4 } after i =0, 1, 2 and 3 iterations.

Metric learning for Interpolation, Extrapolation diffeomorphic image registration. Fran¸ cois-Xavier Vialard Introduction to diffeomorphisms group and Riemannian tools Choice of the metric Spatially dependent metrics Metric learning SVF metric learning Figure – Geodesic regression (MICCAI 2011)