Varifolds and Surface Approximation Blanche Buet joint work with - PowerPoint PPT Presentation

Varifolds and Surface Approximation Blanche Buet joint work with Gian Paolo Leonardi (Modena), Simon Masnou (Lyon) and Martin Rumpf (Bonn) Laboratoire de math´ ematiques d’Orsay, Paris Sud 7 february 2019 The Mathematics of imaging, Paris

Why varifolds ? ◮ flexible : you can endow both discrete and continuous objects with a varifold structure. ◮ encode order 1 information (tangent bundle): unoriented objects . ◮ provide weak notion of curvatures . ◮ natural distances to compare varifolds.

Plan A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arc length parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) .

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arclength parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) . Let ϕ : R 2 → R be C 1 : � L d dtϕ ( γ ( t )) γ ′ ( t ) dt 0 � L � � L ϕ ( γ ( t )) γ ′ ( t ) ϕ ( γ ( t )) γ ′′ ( t ) = − dt ���� t =0 � �� � � �� � 0 by parts κ ( γ ( t )) =0

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arclength parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) . Let ϕ : R 2 → R be C 1 : � L � d dtϕ ( γ ( t )) γ ′ ( t ) dt = − ϕ ( x ) κ ( x ) 0 Γ

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arclength parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) . Let ϕ : R 2 → R be C 1 : � L � d dtϕ ( γ ( t )) γ ′ ( t ) dt = − ϕ ( x ) κ ( x ) 0 Γ � L � L � � d dtϕ ( γ ( t )) γ ′ ( t ) dt = ∇ ϕ ( γ ( t )) · γ ′ ( t ) γ ′ ( t ) dt 0 0 � � = ( ∇ ϕ ( x ) · τ ( x )) τ = Π θ ( x ) ( ∇ ϕ ( x )) Γ Γ

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arclength parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) . Let ϕ : R 2 → R be C 1 : ✎ ☞ � � ϕ ( x ) κ ( x ) = − Π θ ( x ) ( ∇ ϕ ( x )) ✍ ✌ Γ Γ

We start with ◮ Γ ⊂ R 2 a C 2 closed curve, ◮ γ : [0 , L ] → R 2 an injective ( 0 ∼ L ) arclength parametrization of Γ . ◮ unit tangent vector τ : for x = γ ( t ) ∈ Γ , τ ( x ) = γ ′ ( t ) and θ ( x ) the angle between τ ( x ) and the horizontal. ◮ curvature vector κ : for x = γ ( t ) ∈ Γ , κ ( x ) = γ ′′ ( t ) . Let ϕ : R 2 → R be C 1 : ✎ ☞ � � ϕ ( x ) κ ( x ) = − Π θ ( x ) ( ∇ ϕ ( x )) ✍ ✌ Γ Γ ◮ weak formulation of curvature, ◮ relies only on the knowledge of �� � � ψ : R 2 × R → R continuous � � ψ ( x, θ ( x )) . � ∀ x ∈ R 2 , ω �→ ψ ( x, ω ) is π –periodic Γ

Our first varifold � � ψ : � R 2 × R � → R ψ continuous and π –periodic � C = . � ( x, ω ) �→ ψ ( x, ω ) w.r.t. ω The continuous linear form   : C → V Γ R � �→ ψ ( x, θ ( x ))  ψ Γ is the 1 –varifold naturally associated with Γ .

Our first varifold � � ψ : � R 2 × R � → R ψ continuous and π –periodic � C = . � ( x, ω ) �→ ψ ( x, ω ) w.r.t. ω The continuous linear form on C   : C → V Γ R � �→ ψ ( x, θ ( x ))  ψ Γ ✎ ☞ is the 1 –varifold naturally associated with Γ . � With ψ ( x, ω ) = Π ω ∇ ϕ ( x ) , ϕ ( x ) κ ( x ) = − V Γ ( ψ ) and ✍ ✌ Γ ◮ Knowing V Γ is enough to recover the curvature κ . ◮ Conversely, it is possible to define a notion of generalized curvature for any continuous linear form on C , that is for ANY 1 –varifold in R 2 .

Plan A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Varifolds ? Generalized surface : couples a weighted spatial information and a non oriented direction information. Introduced by Almgren in the 60’ : weak notion of surface allowing good compactness properties. A d –varifold is a Radon measure in R n × G d,n .

The Grassmannian Grassmannian of d –planes : G d,n = { d –vector sub-spaces of R n } . � non-oriented d –planes. We identify P ∈ G d,n with the orthogonal projector Π P onto P , so that G d,n can be seen as a compact subset of M n ( R ) :  �  A 2 = A �   � A T = A � G d,n ≃  A ∈ M n ( R ) �  � Trace( A ) = d Distance on G d,n : d ( P, Q ) = � Π P − Π Q � .

Radon measure X = R n , X = R n × G d,n . A Radon measure in X will equivalently be (thanks to Riesz theorem) : ◮ a Borel mesure on X that takes finite values on compact sets. ◮ a positive linear form on C c (X) . Weak star convergence : � � ∗ µ i − ⇀ µ ⇔ ∀ ϕ ∈ C c (X) , ϕ d µ i → ϕ d µ . X X locally metrized for instance by the flat distance : � �� � � � ϕ is 1 –Lipschitz � ∆( µ, ν ) = sup ϕ dµ − ϕ dν � sup X | ϕ | ≤ 1 X X

About ∆ • Condition sup | ϕ | ≤ 1 : for ε > 0 and µ = (1 + ε ) δ 0 , ν = δ 0 , � � � � � � � � ϕ dµ − ϕ dν � = ε | ϕ (0) | − ϕ (0) → + ∞ + ∞ . − − − − − → � • Condition ϕ 1 –Lipschitz: for ε > 0 and µ = δ ε , ν = δ 0 , � � � � � � � � ϕ dµ − ϕ dν � = | ϕ ( ε ) − ϕ (0) | = 2 . � with ϕ ( ε ) = 1 and ϕ (0) = − 1 . ✗ ✔ • Localized version : B ⊂ R n  �  � � � ϕ is 1 –Lipschitz   � � ∆ B ( µ, ν ) = sup ϕ dµ − sup X | ϕ | ≤ 1 ϕ dν �   ✖ ✕ � X X spt ϕ ⊂ B

First examples 1 –Varifold associated with ◮ a segment S ⊂ R n whose direction is P ∈ G 1 ,n : V = H 1 | S ⊗ δ P , ◮ a union of segments M = ∪ 8 i =1 S i , S i of direction P i ∈ G 1 ,n : � 8 H 1 V = | S i ⊗ δ P i . i =1 � 2 –Varifold associated with a triangulated surface M = T , where T has direction P T ∈ G 2 ,n : T ∈T � L 2 V = | T ⊗ δ P T . T ∈T

Point cloud varifolds d –Varifold associated with a point cloud in R n , that is ◮ a finite set of points { x i } N i =1 ⊂ R n , i =1 ⊂ R ∗ ◮ weighted by masses { m i } N + , ◮ provided with directions { P i } N i =1 ⊂ G d,n . � N V = m i δ x i ⊗ δ P i . i =1

Regular varifolds When M ⊂ R n is a d –sub-manifold (or a d –rectifiable set) : 1. µ measure in R n supported in M : µ = H d | M . 2. a family ( ν x ) x ∈ M of probabilities in G d,n : ν x = δ T x M . Then define V = µ ⊗ ν x Radon measure in R n × G d,n in the sense: for ψ ∈ x C c ( R n × G d,n ) , � � � ψ ( x, P ) dν x ( P ) dµ ( x ) V ( ψ ) = ψ dV = x ∈ R n P ∈ G d,n � ψ ( x, T x M ) d H d ( x ) = M

Regular varifolds When M ⊂ R n is a d –sub-manifold (or a d –rectifiable set) : 1. µ measure in R n supported in M : µ = H d | M . 2. a family ( ν x ) x ∈ M of probabilities in G d,n : ν x = δ T x M . Then define V = µ ⊗ ν x Radon measure in R n × G d,n in the sense: for ψ ∈ C c ( R n × G d , n ) , � � � ψ ( x, P ) dν x ( P ) dµ ( x ) V ( ψ ) = ψ dV = x ∈ R n P ∈ G d,n � ψ ( x, T x M ) d H d ( x ) = M � Remember, for Γ : V Γ ( ψ ) = ψ ( x, θ ( x )) . Γ

Disintegration Mass of a varifold V : it’s the Radon measure � V � in R n defined as � V � ( A ) = V ( A × G d,n ) . Disintegration : a d –varifold V can be decomposed as V = µ ⊗ ν x with µ = � V � where for � V � –a.e. x , ν x is a probability measure in G d,n .

More varifolds ... Point cloud Volumic approx Rectifiable � � m K L n θ ( x ) H d m j δ x j ⊗ δ P j | K ⊗ δ P K | M ⊗ δ T x M j K ∈K � � m K L n � V � = θ ( x ) H d � V � = m j δ x j � V � = | K | M j K ∈K

Plan A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form