Building Good Triangulations 1. Nets and thick triangulations 2. - PowerPoint PPT Presentation

Building Good Triangulations 1. Nets and thick triangulations 2. Triangulation of manifolds Jean-Daniel Boissonnat INRIA Hamilton Mathematics Institute 17-19 June, 2018 1 / 58

Triangulations A central subject since the early days of Computational Geometry Triangulations as data structures Triangulation of polygonal/polyhedral domains Optimal triangulations Delaunay triangulation A central subject in Mesh Generation, Manifold Learning Quality of approximation Quality of elements Higher dimensions More general topological spaces 2 / 58

Delaunay-like complexes 1 Nets and Delaunay refinement 2 Thick triangulations 3 3 / 58

Voronoi diagrams A set of points P in ( R d , � . � ) Voronoi cell V ( p i ) = { x : � x − p i � ≤ � x − p j � , ∀ j } Voronoi diagram ( P ) = { set of cells V ( p i ) , p i ∈ P } 4 / 58

Delaunay Triangulations Sur la sphère vide (On the empty sphere) , Boris Delaunay (1934) The Delaunay complex Del ( P ) is the nerve of Vor ( P ) Theorem If P contains no subset of d + 2 points on a same hypersphere, then Del ( P ) is a triangulation of P 5 / 58

Delaunay Triangulations Sur la sphère vide (On the empty sphere) , Boris Delaunay (1934) The Delaunay complex Del ( P ) is the nerve of Vor ( P ) Theorem If P contains no subset of d + 2 points on a same hypersphere, then Del ( P ) is a triangulation of P 5 / 58

Delaunay Triangulations Sur la sphère vide (On the empty sphere) , Boris Delaunay (1934) The Delaunay complex Del ( P ) is the nerve of Vor ( P ) Theorem If P contains no subset of d + 2 points on a same hypersphere, then Del ( P ) is a triangulation of P 5 / 58

Correspondence between structures i ) = h ∗ h p i : x d + 1 = 2 p i · x − p 2 p i = ( p i , p 2 ˆ p i i duality V ( P ) = h + p 1 ∩ . . . ∩ h + D ( P ) = conv − ( { ˆ − → p 1 , . . . , ˆ p n } ) p n ↑ ↓ nerve Voronoi diagram of P − → Delaunay triang. of P The diagram commutes if P is in general position wrt spheres 6 / 58

Corollaries Combinatorial complexity The Voronoi diagram of n points of R d has the same combinatorial complexity as the intersection of n half-spaces of R d + 1 The Delaunay triangulation of n points of R d has the same combinatorial complexity as the convex hull of n points of R d + 1 The two complexities are the same by duality : Θ( n ⌈ d 2 ⌉ ) [Mc Mullen 1970] Worst-case : points on the moment curve Γ( t ) = { t , t 2 , ..., t d } ⊂ R d Quadratic in R 3 7 / 58

Corollaries Algorithmic complexity Construction of Del ( P ) , P = { p 1 , ..., p n } ⊂ R d 1 Lift the points of P onto the paraboloid x d + 1 = x 2 of R d + 1 : p i = ( p i , p 2 p i → ˆ i ) 2 Compute conv ( { ˆ p i } ) 3 Project the lower hull conv − ( { ˆ p i } ) onto R d Complexity : Θ( n log n + n ⌈ d 2 ⌉ ) [Clarkson & Shor 1989] [Chazelle 1993] 8 / 58

Laguerre (power, weighted) diagrams D ( x , b ) = ( x − p ) 2 − r 2 B = { b 1 , ..., b n } Voronoi cell : V ( b i ) = { x : D ( x , b i ) ≤ D ( x , b j ) ∀ j } Voronoi diagram of B : = { set of cells V ( b i ) , b i ∈ B} 9 / 58

Delaunay triangulations of balls Vor ( B ) Del ( B ) is the nerve of Vor ( B ) Theorem If the balls are in general position, then Del ( B ) is a triangulation of a subset P ′ ⊆ P of the points General position for balls : no point of R d has same power wrt d + 2 balls 10 / 58

Delaunay triangulations of balls Vor ( B ) Del ( B ) is the nerve of Vor ( B ) Theorem If the balls are in general position, then Del ( B ) is a triangulation of a subset P ′ ⊆ P of the points General position for balls : no point of R d has same power wrt d + 2 balls 10 / 58

Correspondence between structures ˆ i ) = h ∗ h b i : x d + 1 = 2 p i · x − p 2 i + r 2 b i = ( p i , p 2 i − r 2 i b i duality V ( B ) = h + b 1 ∩ . . . ∩ h + D ( B ) = conv − ( { ˆ b 1 , . . . , ˆ − → b n } ) b n ↑ ↓ nerve Voronoi diagram of B − → Delaunay triang. of B The diagram commutes if B is in general position 11 / 58

Affine diagrams Sites + distance functions s.t. the bisectors are hyperplanes Theorem [Aurenhammer] Any affine diagram of R d is the Laguerre diagram of a set of balls of R d Examples : 12 / 58

Delaunay triangulation restricted to a union of balls Alpha-complex [Edelsbrunner et al.] C ( b ) = b ∩ V ( b ) Vor | U ( B ) = { f ∈ Vor ( B ) , f ∩ U � = ∅} U = � b ∈B C ( b ) Del | U ( B ) 13 / 58

Discrete metric spaces : Witness Complex [de Silva] L a finite set of points (landmarks) vertices of the complex W a dense sample (witnesses) pseudo circumcenters σ w Let σ be a (abstract) simplex with vertices in L , and let w ∈ W . We say that w is a witness of σ if � w − p � ≤ � w − q � ∀ p ∈ σ and ∀ q ∈ L \ σ The witness complex Wit ( L , W ) is the complex consisting of all simplexes σ such that for any simplex τ ⊆ σ , τ has a witness in W 14 / 58

Discrete metric spaces : Witness Complex [de Silva] L a finite set of points (landmarks) vertices of the complex W a dense sample (witnesses) pseudo circumcenters σ w Let σ be a (abstract) simplex with vertices in L , and let w ∈ W . We say that w is a witness of σ if � w − p � ≤ � w − q � ∀ p ∈ σ and ∀ q ∈ L \ σ The witness complex Wit ( L , W ) is the complex consisting of all simplexes σ such that for any simplex τ ⊆ σ , τ has a witness in W 14 / 58

Discrete metric spaces : Witness Complex [de Silva] L a finite set of points (landmarks) vertices of the complex W a dense sample (witnesses) pseudo circumcenters σ w Let σ be a (abstract) simplex with vertices in L , and let w ∈ W . We say that w is a witness of σ if � w − p � ≤ � w − q � ∀ p ∈ σ and ∀ q ∈ L \ σ The witness complex Wit ( L , W ) is the complex consisting of all simplexes σ such that for any simplex τ ⊆ σ , τ has a witness in W 14 / 58

Easy consequences of the definition The witness complex can be defined for any metric space and, in particular, for discrete metric spaces If W ′ ⊆ W , then Wit ( L , W ′ ) ⊆ Wit ( L , W ) Del ( L ) ⊆ Wit ( L , R d ) 15 / 58

Easy consequences of the definition The witness complex can be defined for any metric space and, in particular, for discrete metric spaces If W ′ ⊆ W , then Wit ( L , W ′ ) ⊆ Wit ( L , W ) Del ( L ) ⊆ Wit ( L , R d ) 15 / 58

Easy consequences of the definition The witness complex can be defined for any metric space and, in particular, for discrete metric spaces If W ′ ⊆ W , then Wit ( L , W ′ ) ⊆ Wit ( L , W ) Del ( L ) ⊆ Wit ( L , R d ) 15 / 58

Identity of Del ( L ) and Wit ( L , R d ) [de Silva 2008] Wit ( L , W ) ⊆ Wit ( L , R d ) = Del ( L ) Theorem : Remarks ◮ Faces of all dimensions have to be witnessed ◮ Wit ( L , W ) is embedded in R d if L is in general position wrt spheres 16 / 58

Identity of Del ( L ) and Wit ( L , R d ) [de Silva 2008] Wit ( L , W ) ⊆ Wit ( L , R d ) = Del ( L ) Theorem : Remarks ◮ Faces of all dimensions have to be witnessed ◮ Wit ( L , W ) is embedded in R d if L is in general position wrt spheres 16 / 58

Identity of Del ( L ) and Wit ( L , R d ) [de Silva 2008] Wit ( L , W ) ⊆ Wit ( L , R d ) = Del ( L ) Theorem : Remarks ◮ Faces of all dimensions have to be witnessed ◮ Wit ( L , W ) is embedded in R d if L is in general position wrt spheres 16 / 58

Proof of de Silva’s theorem Attali, Edelsbrunner, Mileyko 2007] τ = [ p 0 , ..., p k ] is a k -simplex of Wit ( L ) witnessed by a ball B τ (i.e. B τ ∩ L = τ ) We prove that τ ∈ Del ( L ) by induction on k Clearly true for k = 0 k ′ ≤ k − 1 Hyp. : true for B := B τ σ := ∂ B ∩ τ , l := | σ | B σ // σ ∈ Del ( L ) by the hyp. B τ while l + 1 = dim σ < k do c B ← the ball centered on [ cw ] s.t. σ w τ - σ ⊂ ∂ B , - B witnesses τ - | ∂ B ∩ τ | = l + 1 ( B witnesses τ ) ∧ ( τ ⊂ ∂ B ) ⇒ τ ∈ Del ( L ) 17 / 58

Proof of de Silva’s theorem Attali, Edelsbrunner, Mileyko 2007] τ = [ p 0 , ..., p k ] is a k -simplex of Wit ( L ) witnessed by a ball B τ (i.e. B τ ∩ L = τ ) We prove that τ ∈ Del ( L ) by induction on k Clearly true for k = 0 k ′ ≤ k − 1 Hyp. : true for B := B τ σ := ∂ B ∩ τ , l := | σ | B σ // σ ∈ Del ( L ) by the hyp. B τ while l + 1 = dim σ < k do c B ← the ball centered on [ cw ] s.t. σ w τ - σ ⊂ ∂ B , - B witnesses τ - | ∂ B ∩ τ | = l + 1 ( B witnesses τ ) ∧ ( τ ⊂ ∂ B ) ⇒ τ ∈ Del ( L ) 17 / 58

Case of sampled domains : Wit ( L , W ) � = Del ( L ) W a finite set of points ⊂ T d = R d / Z d (the flat torus of dimension d ) Wit ( L , W ) � = Del ( L ) , even if W is a dense sample of T d b a Vor( a, b ) [ ab ] ∈ Wit ( L , W ) ⇔ ∃ p ∈ W , Vor 2 ( a , b ) ∩ W � = ∅ 18 / 58

Relaxed witness complex Alpha-witness Let σ be a simplex with vertices in L . We say that a point w ∈ W is an α -witness of σ if � w − p � ≤ � w − q � + α ∀ p ∈ σ and ∀ q ∈ L \ σ Alpha-relaxed witness complex The α -relaxed witness complex Wit α ( L , W ) is the maximal simplicial complex with vertex set L whose simplices have an α -witness in W Wit 0 ( L , W ) = Wit ( L , W ) Wit α ( L , W ) ⊆ Wit β ( L , W ) Filtration : α ≤ β ⇒ 19 / 58