A linear bound on the Complexity of the Delaunay Triangulation of - - PowerPoint PPT Presentation

A linear bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Dominique Attali Laboratoire LIS Jean-Daniel Boissonnat PRISME-INRIA

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Introduction

• Applications :

– mesh generation – medial axis approximation – surface reconstruction Question : Complexity of the Delaunay triangulation of points scattered over a surface ?

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Complexity of the Delaunay triangulation

• Spheres circumscribing tetrahedra are empty

Data points Convex hull

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Complexity of the Delaunay triangulation

• Complexity = | Edges | > | Tetrahedra | > |Triangles|/4

Delaunay neighbours Convex hull

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Complexity of the Delaunay triangulation

• For n points, in the worst-case:

– in R3, Ω(n2) Goal : exhibit practical geometric constraints for subquadratic / linear bounds.

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Probabilistic results

• Expected complexity for n random points on

– a ball : Θ(n) [Dwyer 1993] – a convex polytope : Θ(n) [Golin & Na 2000] – a polytope : O(n log4 n) [Golin & Na 2002]

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Deterministic results

• Wrt spread : O(spread3)

[Erickson 2002]

Spread =

largest interpoint distance smallest interpoint distance

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Deterministic results

• Wrt spread : O(Spread3)

[Erickson 2002] – surfaces sampled with spread O(√n) : O(n√n)

largest interpoint distance smallest interpoint distance

Spread =

= O(√n) Ω( 1

√n)

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Deterministic results

• Wrt spread : O(Spread3)

[Erickson 2002] – surfaces sampled with spread O(√n) : O(n√n) – Well-sampled cylinder : Ω(n√n)

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Our main result

For points distributed on a polyedral surface in R3 : the Delaunay triangulation is linear

• Deterministic result

– polyedral surface – sampling condition – proof

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Polyedral surface

• Polyedral surface = Finite collection of facets that form a pur

piece-wise linear complex

• Facet = bounded polygon

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Sampling condition

• (ε, κ)-sample E :

1. 2.

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Sampling condition

• (ε, κ)-sample E :
• 1. ∀x ∈ F , B(x, ε) encloses at least one point of E ∩ F

2. ≥ 1 F x

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Sampling condition

• (ε, κ)-sample E :
• 1. ∀x ∈ F , B(x, ε) encloses at least one point of E ∩ F
• 2. ∀x ∈ F , B(x, 2ε) encloses at most κ points of E ∩ F

x ≤ κ ≥ 1 F

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Sampling condition

• n = Θ

1

ε2

• n(Γ ⊕ ε) = O( length(Γ) × √n )

Γ F

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Delaunay triangulation

• Assumptions : (ε, κ)-sample of a polyedral surface
• Proof : Count Delaunay edges

Empty sphere Delaunay edge

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Proof

• Count Delaunay edges

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Counting Delaunay edges

• 2 zones on the surface

ε-regular zone

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Counting Delaunay edges

• 2 zones on the surface

ε-regular zone ε-singular zone

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Counting Delaunay edges

• 3 types of edges

① regular – regular

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Counting Delaunay edges

• 3 types of edges

① regular – regular ② singular – singular

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Counting Delaunay edges

• 3 types of edges

① regular – regular ② singular – singular ③ singular – regular

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Regular - Regular

• A sample point has at most κ neighbours in its own facet

F ε m

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Regular - Regular

• A sample point has at most κ neighbours in its own facet

F m

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Regular - Regular

• A sample point has at most κ neighbours in its own facet

F 2ε m

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Regular - Regular

• A sample point has at most κ neighbours in any facet

F F ′ 2ε m m′

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Regular - Regular

• A sample point has at most κ neighbours in any facet

F F ′ 2ε m′ m

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Regular - Regular

• A sample point has at most κ neighbours in any facet

F F ′ 2ε m′ m

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Regular - Regular

• Number of Delaunay edges in the regular zone : O(n)

F F ′ m

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Singular - Singular

• Brutal force : O(√n) × O(√n) = O(n)

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Singular - Regular

• Locate the neighbours of x in F

F

Neighbours of x ?

x

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Singular - Regular

• Locate the neighbours of x in F

Empty sphere

F

Neighbours of x

x

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Singular - Regular

• Locate the neighbours of x in F

Empty sphere

F

Neighbours of x Tangent sphere

x

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Singular - Regular

• Neighbours of x : V (x) enlarged by 2ε

Tangent sphere

F

Neighbours of x

V (x) x

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Singular - Regular

F V (x) Singular points : Es

Tangent empty sphere

x

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Singular - Regular

Singular points : Es F V (x) x

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Diagram associated to F and points Es

Tangent empty sphere Px F

V (x)

x p

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Diagram associated to F and points Es

• Bissector of two points : a circle or a line

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Diagram associated to F and points Es

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Delaunay edges between F and Es V (x) = (∩disks) \ (∪disks)

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Delaunay edges between F and Es

Neighbours of x

V (x)

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Delaunay edges between F and Es n(V (x)) +

length(∂V (x)) × √n

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Singular - Regular

• Length of edges ≤ n(Es) × ∂F = O(√n)

V (x) V (y) F

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Singular - Regular

• Length of edges ≤ n(Es) × ∂F = O(√n)

V (x) V (y) F

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Singular - Regular

• Length of edges ≤ n(Es) × ∂F = O(√n)

V (x) V (y) F

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Main result

Let S be a polyhedral surface and E a (ε, κ)-sample of S of size

|E| = n. The number of edges in the Delaunay triangulation of E is at

most :

• 1 + C κ

2 + 612 π κ2 L2 A

• n

C : number of facets A : area L :

length(∂facet)

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Conclusion and perspective

• Linear bound for polyhedral surfaces
• Extend this result to generic surfaces