Geodesics and Fast Marching Methods Gabriel Peyré Laurent Cohen Jean-Marie Mirebeau É C O L E N O R M A L E S U P É R I E U R E
Geodesic computation on a graph Graph: ( V, E ), V = { 1 , . . . , n } , E ⊂ V 2 (symmetric). j y w i,j i Cost: ( w i,j ) ( i,j ) ∈ E , w i,j > 0. γ Path: γ = ( γ 1 , . . . , γ K ) , ( γ k , γ k +1 ) ∈ E . x 2
Geodesic computation on a graph Graph: ( V, E ), V = { 1 , . . . , n } , E ⊂ V 2 (symmetric). j y w i,j i Cost: ( w i,j ) ( i,j ) ∈ E , w i,j > 0. γ Path: γ = ( γ 1 , . . . , γ K ) , ( γ k , γ k +1 ) ∈ E . x = P K − 1 def. Length: L ( γ ) k =1 w γ k , γ k +1 . Geodesic distance: d ( x, y ) = γ 1 = x, γ K = y L ( γ ). min 2
Geodesic computation on a graph Graph: ( V, E ), V = { 1 , . . . , n } , E ⊂ V 2 (symmetric). j y w i,j i Cost: ( w i,j ) ( i,j ) ∈ E , w i,j > 0. γ Path: γ = ( γ 1 , . . . , γ K ) , ( γ k , γ k +1 ) ∈ E . x = P K − 1 def. Length: L ( γ ) k =1 w γ k , γ k +1 . Geodesic distance: d ( x, y ) = γ 1 = x, γ K = y L ( γ ). min Di ffi culty: metrication error. 2
Connections with Maxflow Problems Flow on edge: f j,i = − f i,j . i f i,j 0 < 0 0 > f j i , def. def. = � div > div( f ) i = P j ⇠ i f i,j , r j 0 j 3
Connections with Maxflow Problems Flow on edge: f j,i = − f i,j . i f i,j 0 < 0 0 > f j i , def. def. = � div > div( f ) i = P j ⇠ i f i,j , r j 0 j nP o d ( x, y ) = min ( i,j ) ∈ E w i,j | f i,j | ; div( f ) = δ x − δ y f ∈ R E → recast as max-flow. y x f i,j 6 = 0 3
Connections with Maxflow Problems Flow on edge: f j,i = − f i,j . i f i,j 0 < 0 0 > f j i , def. def. = � div > div( f ) i = P j ⇠ i f i,j , r j 0 j nP o d ( x, y ) = min ( i,j ) ∈ E w i,j | f i,j | ; div( f ) = δ x − δ y f ∈ R E → recast as max-flow. = max u ∈ R N { u y ; | ( r u ) i,j | 6 w i,j , u x = 0 } → recast as min-cut. u i y x d ( x, y ) 0 ( r u ) i,j /w i,j 0 1 f i,j 6 = 0 3
Parametric Surfaces Parameterized surface: u ⇥ R 2 ⇤� ϕ ( u ) ⇥ M . ∂ϕ u 1 ∂ u 1 u 2 ϕ ∂ϕ ∂ u 2 4
Parametric Surfaces Parameterized surface: u ⇥ R 2 ⇤� ϕ ( u ) ⇥ M . ∂ϕ u 1 ∂ u 1 u 2 ϕ γ γ ∂ϕ ∂ u 2 Curve in parameter domain: t ⇥ [0 , 1] ⇤� γ ( t ) ⇥ D . 4
Parametric Surfaces Parameterized surface: u ⇥ R 2 ⇤� ϕ ( u ) ⇥ M . ∂ϕ u 1 ∂ u 1 u 2 ϕ ¯ γ γ γ ∂ϕ ¯ γ ∂ u 2 Curve in parameter domain: t ⇥ [0 , 1] ⇤� γ ( t ) ⇥ D . def. Geometric realization: ¯ γ ( t ) = ϕ ( γ ( t )) ∈ M . 4
Parametric Surfaces Parameterized surface: u ⇥ R 2 ⇤� ϕ ( u ) ⇥ M . ∂ϕ u 1 ∂ u 1 u 2 ϕ ¯ γ γ γ ∂ϕ ¯ γ ∂ u 2 Curve in parameter domain: t ⇥ [0 , 1] ⇤� γ ( t ) ⇥ D . def. Geometric realization: ¯ γ ( t ) = ϕ ( γ ( t )) ∈ M . For an embedded manifold M ⊂ R n : � ⇥ � ∂ϕ , ∂ϕ ⇥ First fundamental form: I ϕ = . ∂ u i ∂ u j i,j =1 , 2 Length of a curve � 1 � 1 ⇥ def. γ � ( t ) | L ( γ ) = | | ¯ | d t = γ � ( t ) I γ ( t ) γ � ( t )d t. 4 0 0
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . W ( x ) 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . 2-D shape: M ⊂ R 2 , H ( x ) = Id 2 . W ( x ) 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . 2-D shape: M ⊂ R 2 , H ( x ) = Id 2 . Isotropic metric: H ( x ) = W ( x ) 2 Id n . W ( x ) 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . 2-D shape: M ⊂ R 2 , H ( x ) = Id 2 . Isotropic metric: H ( x ) = W ( x ) 2 Id n . Image processing: image I , W ( x ) 2 = ( ε + | | ) − 1 . | � I ( x ) | W ( x ) 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . 2-D shape: M ⊂ R 2 , H ( x ) = Id 2 . Isotropic metric: H ( x ) = W ( x ) 2 Id n . Image processing: image I , W ( x ) 2 = ( ε + | | ) − 1 . | � I ( x ) | Parametric surface: H ( x ) = I x (1 st fundamental form). W ( x ) 5
Riemannian Manifold Riemannian manifold: M ⊂ R n (locally) Riemannian metric: H ( x ) ∈ R n × n , symmetric, positive definite. � 1 ⇥ γ � ( t ) T H ( γ ( t )) γ � ( t )d t. def. Length of a curve γ ( t ) ∈ M : L ( γ ) = 0 Euclidean space: M = R n , H ( x ) = Id n . 2-D shape: M ⊂ R 2 , H ( x ) = Id 2 . Isotropic metric: H ( x ) = W ( x ) 2 Id n . Image processing: image I , W ( x ) 2 = ( ε + | | ) − 1 . | � I ( x ) | Parametric surface: H ( x ) = I x (1 st fundamental form). DTI imaging: M = [0 , 1] 3 , H ( x )=di ff usion tensor. W ( x ) 5
Geodesic Distances Geodesic distance metric over M ⊂ R n d M ( x, y ) = γ (0)= x, γ (1)= y L ( γ ) min Geodesic curve: γ ( t ) such that L ( γ ) = d M ( x, y ). def. Distance map to a starting point x 0 ∈ M : U x 0 ( x ) = d M ( x 0 , x ). metric Shape Anisotropic Isotropic Euclidean Surface geodesics
What’s Next? Laurent Cohen: Dijkstra and Fast Marching algorithms. Jean-Marie Mirebeau: anisotropy and adaptive stencils. Last v v u u v u First v u u u u v v v First Last
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