Anisotropic Fast- Anisotropic Fast-Marching methods Marching Jean-Marie With applications to curvature penalization Mirebeau What exactly can solve the Fast- Jean-Marie Mirebeau Marching Algorithm ? The semi- University Paris Sud, CNRS, University Paris-Saclay Lagrangian paradigm February 2, 2017 The Hamiltonian paradigm Mathematical Coffees, Huawei-FSMP In collaboration Remco Duits (Eindhoven, TU/e University), Laurent Cohen, Da Chen (Univ. Paris-Dauphine) Johann Dreo (Thales TRT) This work was partly funded by ANR JCJC NS-LBR
Anisotropic Fast- Marching Jean-Marie Mirebeau What exactly can solve the What exactly can solve the Fast-Marching Algorithm ? Fast- Marching Algorithm ? The semi- Lagrangian paradigm The semi-Lagrangian paradigm The Hamiltonian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Semi-Lagrangian approach Fast- Marching Jean-Marie Let X be a finite set, and U : X → R be the unknown. Mirebeau What exactly can solve the Fast- Marching Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Semi-Lagrangian approach Fast- Marching Jean-Marie Let X be a finite set, and U : X → R be the unknown. Mirebeau A fixed point problem Λ U ≡ U is FM-solvable. . . What exactly can solve the provided operator Λ : R X → R X obeys, ∀ U , V ∈ R X , ∀ λ ∈ R Fast- Marching Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Semi-Lagrangian approach Fast- Marching Jean-Marie Let X be a finite set, and U : X → R be the unknown. Mirebeau A fixed point problem Λ U ≡ U is FM-solvable. . . What exactly can solve the provided operator Λ : R X → R X obeys, ∀ U , V ∈ R X , ∀ λ ∈ R Fast- Marching ◮ (Monotony) U ≤ V ⇒ Λ U ≤ Λ V . Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Semi-Lagrangian approach Fast- Marching Jean-Marie Let X be a finite set, and U : X → R be the unknown. Mirebeau A fixed point problem Λ U ≡ U is FM-solvable. . . What exactly can solve the provided operator Λ : R X → R X obeys, ∀ U , V ∈ R X , ∀ λ ∈ R Fast- Marching ◮ (Monotony) U ≤ V ⇒ Λ U ≤ Λ V . Algorithm ? The semi- ◮ (Causality) U <λ = V <λ ⇒ (Λ U ) ≤ λ = (Λ V ) ≤ λ . Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Semi-Lagrangian approach Fast- Marching Jean-Marie Let X be a finite set, and U : X → R be the unknown. Mirebeau A fixed point problem Λ U ≡ U is FM-solvable. . . What exactly can solve the provided operator Λ : R X → R X obeys, ∀ U , V ∈ R X , ∀ λ ∈ R Fast- Marching ◮ (Monotony) U ≤ V ⇒ Λ U ≤ Λ V . Algorithm ? The semi- ◮ (Causality) U <λ = V <λ ⇒ (Λ U ) ≤ λ = (Λ V ) ≤ λ . Lagrangian paradigm The Hamiltonian paradigm Example : Dijkstra’s algorithm For each p ∈ X let Neigh ( p ) ⊆ X be a collection of neighbors, and δ ( p , q ) the corresponding edge lengths. Λ U ( p ) := q ∈ Neigh ( p ) U ( q ) + δ ( q , p ) . min
Anisotropic Fast-Marching: the Hamiltonian approach Fast- Marching Let X be a finite set, and s : X → R + be a speed function. Jean-Marie Mirebeau What exactly can solve the Fast- Marching Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Hamiltonian approach Fast- Marching Let X be a finite set, and s : X → R + be a speed function. Jean-Marie And inverse problem HU ≡ s 2 is FM-solvable. . . Mirebeau provided operator H has the following form What exactly can solve the Fast- Marching HU ( p ) := H ( p , U ( p ) , ( U ( p ) − U ( q )) q ∈ X ) , Algorithm ? The semi- and satisfies Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Fast-Marching: the Hamiltonian approach Fast- Marching Let X be a finite set, and s : X → R + be a speed function. Jean-Marie And inverse problem HU ≡ s 2 is FM-solvable. . . Mirebeau provided operator H has the following form What exactly can solve the Fast- Marching HU ( p ) := H ( p , U ( p ) , ( U ( p ) − U ( q )) q ∈ X ) , Algorithm ? The semi- and satisfies Lagrangian paradigm ◮ (Monotony) H is non-decreasing w.r.t. 2nd and 3rd var. The Hamiltonian paradigm
Anisotropic Fast-Marching: the Hamiltonian approach Fast- Marching Let X be a finite set, and s : X → R + be a speed function. Jean-Marie And inverse problem HU ≡ s 2 is FM-solvable. . . Mirebeau provided operator H has the following form What exactly can solve the Fast- Marching HU ( p ) := H ( p , U ( p ) , ( U ( p ) − U ( q )) q ∈ X ) , Algorithm ? The semi- and satisfies Lagrangian paradigm ◮ (Monotony) H is non-decreasing w.r.t. 2nd and 3rd var. The Hamiltonian ◮ (Causality) H only depends on the positive part of the paradigm third variable(s).
Anisotropic Fast-Marching: the Hamiltonian approach Fast- Marching Let X be a finite set, and s : X → R + be a speed function. Jean-Marie And inverse problem HU ≡ s 2 is FM-solvable. . . Mirebeau provided operator H has the following form What exactly can solve the Fast- Marching HU ( p ) := H ( p , U ( p ) , ( U ( p ) − U ( q )) q ∈ X ) , Algorithm ? The semi- and satisfies Lagrangian paradigm ◮ (Monotony) H is non-decreasing w.r.t. 2nd and 3rd var. The Hamiltonian ◮ (Causality) H only depends on the positive part of the paradigm third variable(s). Example : upwind discretization of �∇ u � 2 = s 2 Assume that X ⊆ h Z d is a cartesian grid, and let ( e i ) be the canonical basis. Define for U ∈ R X , p ∈ X HU ( p ) := h − 2 � max { 0 , U ( p ) − U ( p + he i ) , U ( p ) − U ( p − he i ) } 2 . 1 ≤ i ≤ d
Anisotropic What we want to solve Fast- Marching Jean-Marie Mirebeau What exactly can solve the Fast- Marching Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic What we want to solve Fast- Marching Jean-Marie Setting: Finsler geometry Mirebeau Consider a domain, a metric, and a speed function What exactly can solve the F : Ω × R d → [ 0 , + ∞ ] , Fast- Ω ⊆ R d , s : Ω → ] 0 , ∞ [ . Marching Algorithm ? The semi- Define for each smooth path γ : [ 0 , 1 ] → Ω Lagrangian paradigm � 1 The d t Hamiltonian length F ( γ ) := F γ ( t ) (˙ γ ( t )) s ( γ ( t )) . paradigm 0
Anisotropic What we want to solve Fast- Marching Jean-Marie Setting: Finsler geometry Mirebeau Consider a domain, a metric, and a speed function What exactly can solve the F : Ω × R d → [ 0 , + ∞ ] , Fast- Ω ⊆ R d , s : Ω → ] 0 , ∞ [ . Marching Algorithm ? The semi- Define for each smooth path γ : [ 0 , 1 ] → Ω Lagrangian paradigm � 1 The d t Hamiltonian length F ( γ ) := F γ ( t ) (˙ γ ( t )) s ( γ ( t )) . paradigm 0 Objective: compute a front arrival time Given a set of seeds S ⊆ Ω compute u : Ω → R defined by u ( p ) := inf { length F ( γ ); γ ( 0 ) ∈ S , γ ( 1 ) = p } , and extract the corresponding minimal paths.
Anisotropic Fast- Marching Jean-Marie Mirebeau What exactly can solve the What exactly can solve the Fast-Marching Algorithm ? Fast- Marching Algorithm ? The semi- Lagrangian paradigm The semi-Lagrangian paradigm The Hamiltonian paradigm The Hamiltonian paradigm
Anisotropic Using notations Ω (domain), S (seeds), u (front arrival time), Fast- Marching F (metric), s (speed function). Jean-Marie Mirebeau What exactly can solve the Fast- Marching Algorithm ? The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Using notations Ω (domain), S (seeds), u (front arrival time), Fast- Marching F (metric), s (speed function). Jean-Marie Bellman’s optimality principle Mirebeau What exactly can solve the q ∈ V ⊆ Ω \ S ⇒ u ( q ) = inf p ∈ ∂ V u ( p ) + d F ( p , q ) . Fast- Marching Algorithm ? where d F ( q , p ) is the length of the shortest path from p to q . The semi- Lagrangian paradigm The Hamiltonian paradigm
Anisotropic Using notations Ω (domain), S (seeds), u (front arrival time), Fast- Marching F (metric), s (speed function). Jean-Marie Bellman’s optimality principle Mirebeau What exactly can solve the q ∈ V ⊆ Ω \ S ⇒ u ( q ) = inf p ∈ ∂ V u ( p ) + d F ( p , q ) . Fast- Marching Algorithm ? where d F ( q , p ) is the length of the shortest path from p to q . The semi- Lagrangian paradigm Discretization The Let X ⊆ Ω and ∂ X ⊆ R d \ Ω be finite sets. Let V ( p ) be a Hamiltonian paradigm polytope enclosing each p ∈ X , with vertices in X ∪ ∂ X . Define Λ U ( x ) = q ∈ ∂ V ( p ) F p ( q − p ) + I V ( p ) U ( q ) , min where I V denotes piecewise linear interpolation.
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