Tamal K. Dey The Ohio State University

Delaunay Meshing of Surfaces

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Point Cloud Data Surface Reconstruction

`

Point Cloud

Surface Reconstruction

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Voronoi Based Algorithms1. Alpha-shapes (Edelsbrunner, Mück 94)

2. Crust (Amenta, Bern 98)

3. Natural Neighbors (Boissonnat, Cazals 00)

4. Cocone (Amenta, Choi, Dey, Leekha, 00)

5. Tight Cocone (Dey, Goswami, 02)

6. Power Crust (Amenta, Choi, Kolluri 01)

7. Distance function (Edelsbrunner 95, Giesen 02, Chazal,

Lieutier,Cohen-Steiner 06)

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Medial axis

f(x) is the

distance

to medial axis

f(x)

Each x has a sample

within f(x) distance

Local Feature Size and ε-sample [ABE98]

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Reconstruction Guarantees

• Given an ε-sample from a smooth, compact surface without boundary, the output piecewise linear surface has the exact topology (homeomorphic/isotopic) and approximate geometry (Hausdorff distance O(ε)f(x)) if ε <0.06.

• Curve and Surface Reconstruction : Algorithms with Mathematical Analysis, Cambridge University Press (2006?)

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Polyhedral Surface (conforming)

Input PLC Output Mesh

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Basics of Delaunay Refinement

Chew 89, Ruppert 95• Maintain a Delaunay triangulation of

the current set of vertices.• If some property is not satisfied by

the current triangulation, insert a new point which is locally farthest.

• Burden is on showing that the algorithm terminates (shown by packing argument).

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Delaunay Refinement for Quality

• R/l = 1/(2sinθ)≥1/√3

• Choose a constant ≥ 1if R/l is greater than this constant, insert the circumcenter.

R

l

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Delaunay Refinement for 2D Point Sets

R/l ≥ 1.0

30 degree

R

l

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Polyhedral Volumes and Surface

[Shewchuk 98]

Input PLC Final Mesh

• No input angle is less than 90 degree

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Delaunay Refinement for Input Conformity

• Diametric ball of a subsegment empty.

• If encroached by a point p, insert the midpoint.

• Subfacets: 2D Delaunay triangles of vertices on a facet.

• If diametric ball of a subfacet encroached by a point p, insert the center.

p

p

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Small Angle Problem

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SOS-split

[Cohen-Steiner et al. 02]

Sharp Vertex Protection

( ) / 4f u

u

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Subfacet Splitting

• Trick to stop indefinite splitting of subfacets in the presence of small angles is to split only the non-Delaunay subfacets.

• It can be shown that the circumradius of such a subfacet is large when it is split.

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Summary of Results

• A simpler algorithm and an implementation.

• Local feature size needed at only the sharp vertices.

• No spherical surfaces to mesh.• Quality guarantees

• Most triangles have bounded radius-edge ratio.• Any skinny triangle is at a distance from

some sharp vertex or some point on a sharp edge.

f xx x

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Results

Delaunay Meshing for Smooth Surfaces

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Implicit Surface

F: R3 => R, Σ = F-1(0)

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Two Work• Boissonnat-Oudot 03: General

implicit surfaces, Ensure TBP with local feature size

• Cheng-Dey-Ramos-Ray 04: General implicit surface, no feature size computation.

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Restricted DelaunayRestricted Delaunay

• Del Q|Σ :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects Σ.

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Topological Ball PropertyTopological Ball Property

• A -dimensional Voronoi face intersects in Σ a -dimensional ball.

• Theorem : [ES’97] The underlying space of

the complex Del Q|Σ is homeomorphic to Σ if Vor Q has the topological ball property.

k

( 1)k

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Building Sample P

1. If topological ball property is not satisfied insert a point p in P.

2. Argue each point p is inserted > k f(p) away from all other points where k = 0.06.

-- Termination is guaranteed by 2. -- Topology is guaranteed by 1 and

the termination.

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Topological Disk TestTopoDiskK ( )TopoDiskK ( ) If is not a

topological disk, return furthest point in edge-surface intersections.

qq

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Topological Disk Test

TopoDiskK ( )TopoDiskK ( ) If is not a

topological disk, return furthest point in .

q

q

qG V

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Topology Sampling

Topology(P): If VorEdge, TopoDisk, FacetCycle or Silhouette

in order inserts a new point in P.

Continue till no new point is inserted.

Return P.

• Topology Lemma: If P includes critical

points of Σ and Topology(P) terminates then topological ball property is satisfied.

• Distance Lemma I: Each inserted point p is > k f(p) away from all

other points.

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Geometry Sampling• Quality(P): If a triangle t has ρ(t) > (1+k)2 , insert where e = dual t.• Smoothing(P): If two adjacent triangles make sharp edge,

insert where e = dual t.• Distance Lemma II: Each point is > k f(p) away from all other

points.

e

e

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Results

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Polyhedral Surfaces (non-Polyhedral Surfaces (non-conforming)conforming)[Dey-Li-Ray 05][Dey-Li-Ray 05]

Input:Input: Polyhedral surface G approximating .

Output:Output: A vertex set Q where each vertex lies on G and triangulation T

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AssumptionsAssumptions

• G approximates a smooth .

• G is -flat w.r.t .• Many designed

surfaces, reconstructed surfaces are -flat.

• Relation to Lipschitz surface (Boissonnat-Oudot 06)

p

p( ){f p

pn

pn

( , )

( , )

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Sparse Sampling and Termination

• Theorem:Theorem: If and are sufficiently small, such that each intersection point is away from all other points.

and

k

p ( )kf p

54 10 , 0.1 0.02k

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Results

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Conclusions• Different algorithms for Delaunay meshing of

surfaces/volumes in different input forms• All of them have theoretical guarantees• The implementations can be downloaded from http://www.cse.ohio-state.edu/~tamaldey/ Cocone: cocone.html Polyhedra: qualmesh.html Polyhedra (nonconforming): surfremesh.html• Meshing a nonsmooth curved surface [BO06],

remeshing polygonal surface with small angles.• Anisotropic meshing [CDRW06]• CGAL acknowledgement: www.cgal.org