provably good sampling and meshing of surfaces

Provably Good Sampling and Meshing of Surfaces

Jean-Daniel Boissonnat, Steve Oudot


Not a smooth surface


Smooth surface




Well distributed sample points

Smooth surface



Good triangula9on



Smooth surface


25 All angles are greater than 25 degrees


Good triangula9on:


Smooth surface


25

All triangles are equilateral


Good triangula9on:


Smooth surface

All angles are greater than 25 degrees


25

The best approximates




Smooth surface

Good triangula9on:



25

The best approximates



Smooth surface

Jean-‐Daniel Boissonnat, Steve Oudot

Good triangula9on:


Presented by Anisimov Dmitry

1. Take a smooth surface Compact, orientable, at least C2 – con9nuous closed surface.

Completely suitable Not completely suitable

2. Sample this surface •  Medial axis of the surface

2d medial axis 3d medial axis

M S


2. Sample this surface •  Distance to the medial axis that is

2d medial axis 3d medial axis

dM

dM

dMM


2. Sample this surface •  Minimum distance to the medial axis that is

dMinf = inf dM (x), x ∈ S{ }

dMinfM


2. Sample this surface •  Some user-‐defined func9on σ : S→ R

Ø  Posi9ve that is

Ø  1-‐Lipschitz that is σ (x)−σ (y) ≤ x − y

σ > 0


2. Sample this surface •  Ball of center and radius that is B(c, r)cB r

B(c, r)


2. Sample this surface •  Construc9on of the ini9al point sample E

Pick up at least one point x on each connected component of S and insert it in !E



Consider a ball Bx centered at x of radius less

min 16dist(x, !E \ {x}),dM (x),

16σ (x)

"#$

%&'



Repeatedly shoot rays inside Bx and pick up three points (ux, vx, wx) of S Bx

Insert (ux, vx, wx) in E



Connec9ng these points we get a persistent facet

All persistent facets are Delaunay facets restricted to S



All persistent facets remain restricted Delaunay facets throughout the course of algorithm


3. Triangulate this surface •  Compute the 3-‐dimensional Delaunay triangula9on of E

Del(E)


3. Triangulate this surface •  Compute the set of all edges of the Voronoi diagram of E

V(E)


3. Triangulate this surface •  Compute Delaunay triangula9on of E restricted to S

DelS(E)


3. Triangulate this surface •  Compute Delaunay triangula9on of E restricted to S

DelS(E)

Not constrained


3. Triangulate this surface •  Surface Delaunay ball BD of restricted Delaunay facet f

DelS(E)


3. Triangulate this surface •  Surface Delaunay ball BD of restricted Delaunay facet f

DelS(E)

Any ball centered at some point of where f* is Voronoi edge dual to f

S f *


3. Triangulate this surface •  Bad surface Delaunay ball BD which is stored in L

DelS(E)

c

It is ball B(c, r) such that r > σ(c)

r


3. Triangulate this surface •  Surface Delaunay patch

DelS(E)

The intersec9on of a surface Delaunay ball with S


3. Triangulate this surface •  Loose ε-sample

dM c


3. Triangulate this surface •  E is a loose ε-‐sample of S if:

dM c

1. ∀c ∈ SV (E),E B(c,εdM (c)) ≠∅2. DelS(E) has ver9ces on all the connected components of S


3. Triangulate this surface •  DelS(E) has ver9ces on all the connected components of S

V(E)


3. Triangulate this surface •  Algorithm While L is not empty

•  Take an element B(c,r) from L •  Insert c into E and update Del(E) •  Update DelS(E) by tes9ng all the Voronoi edges that have changed or appeared:

Ø  Delete from DelS(E) the Delaunay facets whose dual Voronoi edges no longer intersect S Ø  Add to DelS(E) the new Delaunay facets whose dual Voronoi edges intersect S

•  Update L by Ø  Dele9ng all the elements of L which are no longer bad surface Delaunay balls Ø  Adding all the new surface Delaunay balls that are bad


3. Triangulate this surface •  Termina9on and output of the Algorithm

Ø  The Algorithm terminates


3. Triangulate this surface •  Termina9on and output of the Algorithm

Ø  The Algorithm outputs E and DelS(E)

E is a loose ε-‐sample of S

DelS(E) is homeomorphic to the input surface S and approximates it in terms of its Hausdorff distance, normals, curvature, and area.


3. Triangulate this surface •  Output of the Algorithm


Magic Epsilon •  To find ε you must solve this simple inequality:

2ε1−8ε

+ arcsin ε1−ε

≥π4

•  Or just take this:

ε = 0.091


Applica9ons

Smooth surface


Applica9ons

Not smooth surface


Applica9ons

Bad triangula9on Good triangula9on


References J.-‐D. Boissonnat and S. Oudot. “Provably Good Sampling and Meshing of Surfaces.” Graphical Models 67 (2005), 405-‐51.


References M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, and B. Levy. “Polygon Mesh Processing.” Chapter 6, Sec9on 6.5.1 (2010), 92-‐96.


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provably good sampling and meshing of surfaces

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