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 Jean-Daniel Boissonnat Laboratoire LIS PRISME-INRIA

SM’2002 2 Introduction • Applications : – mesh generation – medial axis approximation – surface reconstruction Question : Complexity of the Delaunay triangulation of points scattered over a surface ?

SM’2002 3 Complexity of the Delaunay triangulation • Spheres circumscribing tetrahedra are empty Data points Convex hull

SM’2002 4 Complexity of the Delaunay triangulation • Complexity = | Edges | > | Tetrahedra | > | Triangles | / 4 Delaunay neighbours Convex hull

SM’2002 5 Complexity of the Delaunay triangulation • For n points, in the worst-case: – in R 3 , Ω( n 2 ) Goal : exhibit practical geometric constraints for subquadratic / linear bounds.

SM’2002 6 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 log 4 n ) [Golin & Na 2002]

SM’2002 7 Deterministic results • Wrt spread : O ( spread 3 ) [Erickson 2002] largest interpoint distance Spread = smallest interpoint distance

SM’2002 8 Deterministic results • Wrt spread : O ( Spread 3 ) [Erickson 2002] – surfaces sampled with spread O ( √ n ) : O ( n √ n ) largest interpoint distance Spread = smallest interpoint distance = O ( √ n ) Ω( 1 √ n )

SM’2002 9 Deterministic results • Wrt spread : O ( Spread 3 ) [Erickson 2002] – surfaces sampled with spread O ( √ n ) : O ( n √ n ) – Well-sampled cylinder : Ω( n √ n )

SM’2002 10 Our main result For points distributed on a polyedral surface in R 3 : the Delaunay triangulation is linear • Deterministic result – polyedral surface – sampling condition – proof

SM’2002 11 Polyedral surface • Polyedral surface = Finite collection of facets that form a pur piece-wise linear complex • Facet = bounded polygon

SM’2002 12 Sampling condition • ( ε, κ ) -sample E : 1. 2.

SM’2002 13 Sampling condition • ( ε, κ ) -sample E : 1. ∀ x ∈ F , B ( x, ε ) encloses at least one point of E ∩ F 2. ≥ 1 x F

SM’2002 14 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, 2 ε ) encloses at most κ points of E ∩ F ≥ 1 ≤ κ x F

SM’2002 15 Sampling condition � 1 � • n = Θ ε 2 • n (Γ ⊕ ε ) = O ( length (Γ) × √ n ) F Γ

SM’2002 16 Delaunay triangulation • Assumptions : ( ε, κ ) -sample of a polyedral surface • Proof : Count Delaunay edges Empty sphere Delaunay edge

SM’2002 17 Proof • Count Delaunay edges

SM’2002 18 Counting Delaunay edges • 2 zones on the surface ε -regular zone

SM’2002 19 Counting Delaunay edges • 2 zones on the surface ε -regular zone ε -singular zone

SM’2002 20 Counting Delaunay edges • 3 types of edges ① regular – regular

SM’2002 21 Counting Delaunay edges • 3 types of edges ① regular – regular ② singular – singular

SM’2002 22 Counting Delaunay edges • 3 types of edges ① regular – regular ② singular – singular ③ singular – regular

SM’2002 23 Regular - Regular • A sample point has at most κ neighbours in its own facet F ε m

SM’2002 24 Regular - Regular • A sample point has at most κ neighbours in its own facet F m

SM’2002 25 Regular - Regular • A sample point has at most κ neighbours in its own facet F 2 ε m

SM’2002 26 Regular - Regular • A sample point has at most κ neighbours in any facet F ′ 2 ε m ′ F m

SM’2002 27 Regular - Regular • A sample point has at most κ neighbours in any facet F ′ 2 ε m ′ F m

SM’2002 28 Regular - Regular • A sample point has at most κ neighbours in any facet F ′ 2 ε m ′ F m

SM’2002 29 Regular - Regular • Number of Delaunay edges in the regular zone : O ( n ) F ′ F m

SM’2002 30 Singular - Singular • Brutal force : O ( √ n ) × O ( √ n ) = O ( n )

SM’2002 31 Singular - Regular • Locate the neighbours of x in F x F ? Neighbours of x

SM’2002 32 Singular - Regular • Locate the neighbours of x in F Empty sphere x F Neighbours of x

SM’2002 33 Singular - Regular • Locate the neighbours of x in F Empty sphere Tangent sphere x F Neighbours of x

SM’2002 34 Singular - Regular • Neighbours of x : V ( x ) enlarged by 2 ε Tangent sphere x F Neighbours of x V ( x )

SM’2002 35 Singular - Regular Singular points : E s Tangent empty sphere x V ( x ) F

SM’2002 36 Singular - Regular Singular points : E s x V ( x ) F

SM’2002 37 Diagram associated to F and points E s P x x F p V ( x ) Tangent empty sphere

SM’2002 38 Diagram associated to F and points E s • Bissector of two points : a circle or a line

SM’2002 39 Diagram associated to F and points E s

SM’2002 40 Delaunay edges between F and E s V ( x ) = ( ∩ disks ) \ ( ∪ disks )

SM’2002 41 Delaunay edges between F and E s V ( x ) Neighbours of x

SM’2002 42 Delaunay edges between F and E s length ( ∂V ( x )) × √ n n ( V ( x )) +

SM’2002 43 Singular - Regular • Length of edges ≤ n ( E s ) × ∂F = O ( √ n ) V ( x ) V ( y ) F

SM’2002 44 Singular - Regular • Length of edges ≤ n ( E s ) × ∂F = O ( √ n ) V ( x ) V ( y ) F

SM’2002 45 Singular - Regular • Length of edges ≤ n ( E s ) × ∂F = O ( √ n ) V ( x ) V ( y ) F

SM’2002 46 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 : + 612 π κ 2 L 2 � 1 + C κ � n 2 A C : number of facets A : area L : � length( ∂ facet)

SM’2002 47 Conclusion and perspective • Linear bound for polyhedral surfaces • Extend this result to generic surfaces