Effective Computational Geometry for constructing Voronoi diagrams of curved by davebusters

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									Effective Computational Geometry for constructing Voronoi diagrams of curved objects

J-D Boissonnat
INRIA Sophia-Antipolis
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Overview 1 State of the art before ECG 2 ECG results Voronoi diagrams of circles, line segments and other convex objects in the plane ¨ Mobius diagrams and diagrams of spheres Approximation of Voronoi diagrams of general objects 3 Publications and codes 4 Conclusion

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State of the art before ECG Affine Voronoi diagrams

well understood because of their relation with polytopes exact combinatorial bounds, efficient algorithms, fast and robust code (CGAL)
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State of the art before ECG (2) Abstract Voronoi diagrams A general framework for planar Voronoi diagrams Apply to (non intersecting) line segments circles (not in full generality though)
[Klein 93]

¨ Does not apply to multiplicatively weighted (and Mobius) diagrams Voronoi diagrams of curved objects [ Alt & Schawrzkopf 95]

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State of the art before ECG (3) Algebraic and arithmetic issues Point-curve and curve-curve bisectors of planar parametric curves
[Farouki et al. 94-98]

exact but very slow implementation of Voronoi diagrams of line segments
[Burnikel et al. 94]

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State of the art before ECG (4) In higher dimensional spaces no good combinatorial bounds : lines in 3-space, spheres no efficient algorithm no implementation some preliminary (mainly theoretical) work relation between Voronoi diagrams of spheres and affine diagrams
[Aurenhammer, Will]

general results on lower envelopes of surfaces

[Sharir et al.]
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ECG main results • Planar Voronoi diagrams of circles, line segments and general convex objects ¨ • Mobius diagrams and Voronoi diagrams of spheres • PL approximation of Voronoi diagrams of general surfaces

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ECG main results • Planar Voronoi diagrams of circles, line segments and general convex objects ¨ • Mobius diagrams and Voronoi diagrams of spheres • PL approximation of Voronoi diagrams of general surfaces

• Voronoi diagrams and Delaunay triangulations restricted to manifolds, ε-samples, surface mesh generation, skin surfaces [G. Vegter’s talk] • Natural neighbor interpolation, flow complex and surface reconstruction [J. Giesen’s talk]

Voronoi diagrams of circles

[Emiris, Karavelas, Yvinec]

• First fully dynamic algorithm

• Internal representation = dual graph = TDS

• Exact evaluation of predicates (degree 16, Sturm sequences, multivariate resultants etc) • CGAL code : 100k circles ≈ 40 s (Pentium III, 1GHz)
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Voronoi diagrams of general planar convex objects • The algorithm for circles extends to convex pseudo-circles • and to general convex objects provided that their union has been computed

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A special case : Voronoi diagrams of line segments

• Line segments are allowed to intersect • Uses exact (filtered) arithmetic • CGAL package : 30 k segments ≈ 35s (Pentium 4, 2MHz)

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¨ Mobius diagrams

• Weighted points : Qi = (pi, λi, µi), pi ∈ Rd, λi ∈ R \ {0}, µi ∈ R • Distance function : δM (x, Qi) = λi x − pi 2 − µi Generalization of • Voronoi diagrams of points • Laguerre (power) diagrams • multiplicatively weighted Voronoi diagrams λi = 1 and µi = 0 λi = 1 µi = 0
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Main properties ¨ • A Mobius Diagram is the projection of a power diagram of Rd+1 restricted to the paraboloid xd+1 = x · x ¨ • Any diagram whose bisectors are spheres is a Mobius Diagram ¨ • Mobius diagrams can be defined on a sphere • Tight combinatorial bounds • A worst-case optimal algorithm implemented in CGAL

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Voronoi diagrams of (d − 1)-spheres of Rd

¨ • The radial projection of a Voronoi cell is a Mobius diagram • Tight bounds, worst-case optimal algorithm
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A CGAL based package

[C. Delage]

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PL approximation of the medial axis

[Attali & B.]

Hyp.

S are S two sets s.t. dH (S, S ) ≤ ε

Th. B = B(c, r) a S-empty maximal ball (i.e. intB ∩ S = ∅, ∂B ∩ S = {p, q}) √ If pq ≥ 4 ε, ∃ a S -empty maximal ball B = B(c , r ) √ |r − r |, c − c = O( 4 ε)

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PL approximation of the medial axis

Algorithm
1. Sample S (surface meshing algorithm) 2. Compute the 3D Delaunay triangulation of the sample points 3. Return the Voronoi facets whose nearest sample points are sufficiently far apart

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PL approximation of the medial axis

PL approximation of the medial axis

Combinatorial complexity of Delaunay triangulation of well-sampled surfaces Worst-case The worst-case combinatorial complexity of the DT of n points in Rd is Θ n
d 2

Probabilistic results

Expected complexity for n points randomly distributed
[Dwyer 93] [Golin & Na 00] [Golin & Na 02]

in a ball Θ(n) on the boundary of a convex polytope Θ(n) general O(n log4 n)

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Deterministic results • O(spread3)
[Erickson 02]

√ ⇒ Θ(n n) for a well-sampled surface • For a less restrictive sampling model On polyhedral surfaces On smooth generic surfaces Θ(n) O(n log n)
[Attali & B. 02] [Attali, B. & Lieutier 03]

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Dissemination of results ECG papers
1. Dynamic additively weighted Voronoi diagrams in 2D : Karavelas & Yvinec, ESA 2002 2. The Voronoi Diagram of Planar Convex Objects : Karavelas & Yvinec, ESA 2003 3. Root comparison techniques applied to computing the additively weighted Voronoi diagram : Karavelas & Emiris, SODA 2003 4. A practical and efficient algorithm for the segment Voronoi diagram : Karavelas, submitted 5. On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres : Boissonnat & Karavelas, SODA 2003 Complexity of the Delaunay triangulation of points on 6. polyhedral surfaces : Attali & Boissonnat, SM 2003, DCG 2004 7. smooth surfaces : Attali, Boissonnat & Lieutier, SoCG 2003 8. Approximation of the medial axis : Attali & Boissonnat, DCG 2003
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Dissemination of results (2) ECG codes 1. 2. 3. 4. Voronoi diagrams of circles Voronoi diagrams of line segments Planar Moebius diagrams Voronoi diagrams of spheres Karavelas Karavelas Delage Delage CGAL 3.0 CGAL (under review) prototype prototype

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Conclusion Main contributions • New structural results • New data structures and algorithms • A thorough study of the algebraic and arithmetic issues • Effective implementations • Close interactions with surface meshing and surface reconstruction A multidisciplinary research Exact versus PL approximations
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WP 1.1 1.2 2.2 3.1

Further research Anisotropic Voronoi diagrams Voronoi diagrams of ellipses and other algebraic objects PL approximation of the medial axis of manifolds with topological guarantees Applications of Voronoi diagrams of circles and spheres Shortest paths, Biology, Packing

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