Shape Analysis and Computational Anatomy: A Geometrical Perspective on the Statistical Analysis of Population of Manifolds Alain Trouvé CMLA, Ecole Normale Supérieure de Cachan Grenoble, Statlearn’11 2011-03-17 Alain Trouvé () Statistics on Shape Spaces 1 / 42
Building a differentiable and riemannian setting on shape spaces Outline Building a differentiable and riemannian setting on shape spaces Normal coordinates for statistical analysis Means and Atlases Currents and Manifold Representation Statistics and statistical models Challenges Alain Trouvé () Statistics on Shape Spaces 2 / 42
Building a differentiable and riemannian setting on shape spaces Anatomical shapes Few anatomical structures segmented in MRI Sulcal Lines Internal Structures Fiber Bundles Various shape spaces: points, surfaces, pieces of submanifolds, grey-level images, tensor fields, etc. Alain Trouvé () Statistics on Shape Spaces 3 / 42
Building a differentiable and riemannian setting on shape spaces Anatomical shapes Few anatomical structures segmented in MRI Sulcal Lines Internal Structures Fiber Bundles Various shape spaces: points, surfaces, pieces of submanifolds, grey-level images, tensor fields, etc. Alain Trouvé () Statistics on Shape Spaces 3 / 42
Building a differentiable and riemannian setting on shape spaces Trivial metric don’t work ◮ Differentiable structure should be compatible with “smooth” transformation of a shape (e.g. geometrical transformation) ◮ Not true for the L 2 metric on image and “smooth” transformations f ( x ) . τ → τ ˙ = f ( x + τ ) ◮ τ → t · f is not smooth (but τ → τ is !). Alain Trouvé () Statistics on Shape Spaces 4 / 42
Building a differentiable and riemannian setting on shape spaces A global geometrical setting through Riemannian submersion from a group of transformation onto a homogeneous space ◮ Consider a transitive group action i.e. G × M → M ( g , m ) → g . m where M = G ˙ m 0 with m 0 ∈ M (“template”). ◮ If G is equipped with a G 0 (isotropy group) equivariant metric then d M ( m , m ′ ) = inf { d G ( g , g ′ ) | g . m 0 = m , g ′ . m 0 = m ′ } is a distance on M (coming from the projected riemannian distance on M ) G / G 0 ≃ M ◮ Simple framework: Right invariant metric on G (standard construction on finite dimensional Lie group). (Video) Example on the sphere (landmarks matching) (J. Glaunes) Alain Trouvé () Statistics on Shape Spaces 5 / 42
Building a differentiable and riemannian setting on shape spaces Thm (T.) 0 ( R d , R d ) there exists G V ⊂ Diff 1 ( R d ) with a right invariant → C 1 For any V ֒ distance d G for which ◮ G V is complete ◮ There exists a minimizing geodesic between any two elements in G V � 1 d G ( Id , φ ) 2 = inf { | v t | 2 dt | ˙ φ = v ◦ φ, φ 1 = φ } 0 Alain Trouvé () Statistics on Shape Spaces 6 / 42
Normal coordinates for statistical analysis Outline Building a differentiable and riemannian setting on shape spaces Normal coordinates for statistical analysis Means and Atlases Currents and Manifold Representation Statistics and statistical models Challenges Alain Trouvé () Statistics on Shape Spaces 7 / 42
Normal coordinates for statistical analysis Common representation framework ◮ Exponential mapping exp m 0 : T m 0 M → M δ m 0 → exp m 0 ( δ m 0 ) such that t → exp m 0 ( t δ m 0 ) is the solution of the geodesic equation m ˙ ∇ ˙ m = 0 starting from m 0 with initial velocity δ m 0 . ◮ Local diffeomorphism (when dim M < ∞ ) Alain Trouvé () Statistics on Shape Spaces 8 / 42
Normal coordinates for statistical analysis Feasibility of the normal coordinate computation ◮ Optimal control problem, π ( φ ) = φ. m 0 : � � min 1 | v t | 2 V dt + g ( m 1 ) g ( m ) = r ( m , m obs ) � 2 � � � m = d π ( m ) . v = v . m ˙ � ◮ Associated hamiltonian H ( m , p , v ) = ( p | v . m ) − 1 2 | v | 2 V ◮ Metric operator: 1 2 | v | 2 V = ( Lv | v ) with L : V → V ∗ Alain Trouvé () Statistics on Shape Spaces 9 / 42
Normal coordinates for statistical analysis Reduction via momentum map ◮ v = Kj ( m , p ) (Horizontal lift) ◮ Maximum Pontryagin Principle H r ( m , p ) = max H ( m , p , v ) v ◮ Hamiltonian evolution: � ∂ H r ˙ � m = ∂ p � � − ∂ H r ˙ p = � ∂ q Alain Trouvé () Statistics on Shape Spaces 10 / 42
Normal coordinates for statistical analysis Reduction and statistical power ◮ A geodesic on M comes from a geodesic on G but its initial velocities v 0 or its momentum Lv 0 belongs to a subspace V ∗ 0 = j ( m 0 , T ∗ m 0 M ) ⊂ V ∗ So, geodesic optimization ⇒ dimensionality reduction ⇒ better statistical power. Alain Trouvé () Statistics on Shape Spaces 11 / 42
Normal coordinates for statistical analysis ◮ Curve example: γ 0 : [ 0 , 1 ] → R d a continuous curve. Then p 0 is a vectorial measure M f ([ 0 , 1 ] , R d ) = C ([ 0 , 1 ] , R d ) ∗ and � 1 v 0 ( x ) = K ( x , γ 0 ( s )) dp 0 ( s ) K ( x , y ) ∈ M d ( R ) kernel 0 ◮ Moreover, if the geodesic comes from a smooth inexact matching problem e.g. � | γ obs − γ 1 ( s ) | 2 ds g ( γ 1 ) = then p 1 + ∂ g ∂γ ( γ 1 ) = 0 and p 1 ∈ C ([ 0 , 1 ] , R d ) . Same is true for p 0 ◮ For images, if the template is smooth and the data attachment term is � | I obs − I 1 | 2 ( x ) dx then smooth e.g. g ( I 1 ) = � v 0 ( x ) = K ( x , y ) p 0 ( y ) ∇ I 0 ( y ) dy Lv 0 = p 0 ∇ I 0 distribution of vector fields normal to the level set of I 0 . Alain Trouvé () Statistics on Shape Spaces 12 / 42
Means and Atlases Outline Building a differentiable and riemannian setting on shape spaces Normal coordinates for statistical analysis Means and Atlases Currents and Manifold Representation Statistics and statistical models Challenges Alain Trouvé () Statistics on Shape Spaces 13 / 42
Means and Atlases Karcher means For a data set { m i } 1 ≤ i ≤ n the Karcher mean is the point m 0 minimizing n V ( m 0 ) . � d M ( m 0 , m 1 ) 2 = i = 1 ◮ Existence and uniqueness for finite dimensional M and { m i } sufficiently closed or under negative curvature condition. Situation unclear for dim M = ∞ . Alain Trouvé () Statistics on Shape Spaces 14 / 42
Means and Atlases ◮ Usually observations are y i belongs to an observation space Y different from M . Need the introduction of a data attachment term. n n � � d M ( m 0 , m 1 ) 2 + λ minimize r ( m i , y i ) m 0 , m 1 , ··· , m n i = 1 i = 1 or equivalently (lift on the group G ) n � � d G ( Id , φ i ) 2 + λ minimize r ( φ i . m i , y i ) m 0 ,φ 1 , ··· ,φ n i = 1 i = 1 Alain Trouvé () Statistics on Shape Spaces 15 / 42
Means and Atlases Hypertemplate setting n � � d G ( Id , φ i ) 2 + λ minimize r ( φ i . m i , y i ) m 0 ,φ 1 , ··· ,φ n i = 1 i = 1 ◮ Variational problem: if r is smooth enough we have existence of a solution ˆ φ i , · · · , ˆ φ n for m 0 fixed: n pairwise matching problems. ◮ If m 0 is let free, existence issues if dim ( M ) = + ∞ . ◮ Introduce an hypertemplate m h and look for m 0 = ψ. m h solution of n � � ψ,φ 1 , ··· ,φ n d G ( Id , ψ ) 2 + d G ( Id , φ i ) 2 + λ minimize r ( φ i ◦ ψ. m h , y i ) i = 1 i = 1 m 0 = ˆ m 1 = ˆ ˆ ψ. m h , ˆ φ i . ˆ m 0 Actually used to build atlases in medical imaging Alain Trouvé () Statistics on Shape Spaces 16 / 42
Means and Atlases Atlas learning through hypertemplate Figure: 3D hippocampuses data.- Ma, Miller, T., Younes Neuroimage’08 Alain Trouvé () Statistics on Shape Spaces 17 / 42
Currents and Manifold Representation Outline Building a differentiable and riemannian setting on shape spaces Normal coordinates for statistical analysis Means and Atlases Currents and Manifold Representation Statistics and statistical models Challenges Alain Trouvé () Statistics on Shape Spaces 18 / 42
Currents and Manifold Representation Why using currents ? ◮ Challenging problem for submanifold data : ◮ if X is a submanifold, it does not depend as a manifold on any particular parametrization (up to smooth chart changes) ◮ Noisy observation of manifolds may not be a smooth manifold or a manifold at all ! Alain Trouvé () Statistics on Shape Spaces 19 / 42
Currents and Manifold Representation What is a current ? ◮ Goes back to De Rham. Introduced in this setting by Glaunes and Vaillant. ◮ Currents integrate differential forms � ω ∈ Ω p 0 ( R d ) → C X ( ω ) = ω ∈ R � �� � X cont. p -form On a chart γ : U → X ⊂ R d � � ω γ ( s ) ( ∂γ ∧ · · · ∧ ∂γ ω . = ) ds ∂ s 1 ∂ s p γ ( U ) U but the expression in independent of a smooth positively oriented reparametrization of the coordinate space ψ : U → U ∂γ ◦ ψ ∧ · · · ∧ ∂γ ◦ ψ = Jac ( ψ )( ∂γ ∧ · · · ∧ ∂γ ) ◦ ψ ∂ s 1 ∂ s p ∂ s 1 ∂ s p ◮ X can be seen as an element of (Ω p 0 ( R d )) ∗ . Depends on the orientation of X ( X is an orientable manifold). Alain Trouvé () Statistics on Shape Spaces 20 / 42
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