-Samples [AB98] Hyp: domain S is a smooth curve or surface. S 1 - - PowerPoint PPT Presentation

ε-Samples

Hyp: domain S is a smooth curve or surface.

[AB98]

S

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ε-Samples

Hyp: domain S is a smooth curve or surface.

[AB98]

S

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E

ε-Samples

Hyp: domain S is a smooth curve or surface.

[AB98]

S

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ε-Samples

Hyp: domain S is a smooth curve or surface.

[AB98]

S

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Del|S(E) E

ε-Samples

Hyp: domain S is a smooth curve or surface. Def E ⊂ S is an ε-sample of S if ∀p ∈ S, d(p, E) ≤ ε

[AB98]

p q

S

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Del|S(E) E Theorem If E is an ε-sample of S, with ε < 0.16 rch(S), then Del|S(E) ≈ S and lies at Hausdorff distance O(ε2) of S.

Given S and E,

• which points should be added

to E, to make it an ε-sample?

• for which values of ε is E an

ε-sample of S?

Curve/Surface Meshing and ε-samples

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Let V be the set of the edges of Vor(E) Def E ⊂ S is a loose ε-sample of S if ∀p ∈ V ∩ S, d(p, E) ≤ ε

candidate [BO04]

Loose ε-Samples

Hyp: S is C1,1, E ⊂ S 3

Let V be the set of the edges of Vor(E) Def E ⊂ S is a loose ε-sample of S if ∀p ∈ V ∩ S, d(p, E) ≤ ε

candidate

Theorem 1 ε-samples are loose ε-samples

[BO04]

Loose ε-Samples

Hyp: S is C1,1, E ⊂ S 3

Let V be the set of the edges of Vor(E) Def E ⊂ S is a loose ε-sample of S if ∀p ∈ V ∩ S, d(p, E) ≤ ε

candidate

Theorem 1 ε-samples are loose ε-samples ! The contraposal is false... ε

[BO04]

Loose ε-Samples

Hyp: S is C1,1, E ⊂ S 3

Let V be the set of the edges of Vor(E) Def E ⊂ S is a loose ε-sample of S if ∀p ∈ V ∩ S, d(p, E) ≤ ε

candidate

Theorem 1 ε-samples are loose ε-samples ! The contraposal is false... Theorem 2 For ε ≤ 0.16 rch(S), loose ε-samples are ε

• 1 +

ε rch(S)

• samples,

loose ̺rch(S)-samples are ̺ (1 + ̺) rch(S)- samples (̺ = ε/rch(S))

[BO04]

Loose ε-Samples

... but true asymptotically

Hyp: S is C1,1, E ⊂ S 3

Let d = max{d(p, E) | p ∈ V ∩ S} → ∀ε < d, E is not an ε-sample (Thm 1) → ∀ε ≥ d

• 1 +

d rch(S)

• , E is an ε-sample (Thm 2)

⇒ d ≤ ε0 ≤ d

• 1 +

d rch(S)

• d

[BO04]

Loose ε-Samples

• What is the smallest value ε0 of ε such that E is an ε-sample?

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while there are far away candidates > ε

[BO04]

Loose ε-Samples

• How to enrich E so that it becomes a (sparse) ε-sample?

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while there are far away candidates insert one far away candidate in E ;

[BO04]

Loose ε-Samples

• How to enrich E so that it becomes a (sparse) ε-sample?

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while there are far away candidates insert one far away candidate in E ; update Vor(E) and the list of candidates ; end while

[BO04]

Loose ε-Samples

• How to enrich E so that it becomes a (sparse) ε-sample?

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Theorem The process terminates and inserts O

• Area(S)

ε2

• points in E

Upon termination, E is a loose ε-sample (≈ ε-sample) of S while there are far away candidates insert one far away candidate in E ; update Vor(E) and the list of candidates ; end while

[BO04]

Loose ε-Samples

• How to enrich E so that it becomes a (sparse) ε-sample?

5

Theorem The process terminates and inserts O

• Area(S)

ε2

• points in E

Upon termination, E is a loose ε-sample (≈ ε-sample) of S while there are far away candidates insert one far away candidate in E ; update Vor(E) and the list of candidates ; end while

[BO04]

Loose ε-Samples

• How to enrich E so that it becomes a (sparse) ε-sample?

5 Space complexity: O(n log n) Time complexity: O(n log2 n)

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Curve/Surface meshing: a brief survey

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• Marching Cubes [LC87, Ch95]
• Surface mesher based on Delaunay refinement [Ch93]
• Implicit surface mesher based on subdivision [Bl94]
• Implicit surface mesher based on critical points theory [HS97]
• Our variant of Chew’s algorithm [BO03]
• Implicit surface mesher based on critical point theory [BCV04]
• Surface mesher based on the Closed Ball Property [CDRR04]
• Variant of Marching Cubes with adaptive grid [PV04]

Geometric predicates

• 1. Is the sphere passing through 4 given points of E empty?

(Delaunay/Voronoi)

• 2. Does a given segment intersect S?

(restricted Delau- nay/candidates) → little prior knowledge of S is required

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Implicit surfaces 9

Implicit surfaces Level-sets 9

Implicit surfaces Level-sets Point sets 9

Implicit surfaces Level-sets Point sets Molecules 9

What if S is unknown and unsampled?

Curve and surface probing [BGO05]

Idea Discover the (rest of the) shape by means of a tool, called a probing device P Ω ∂Ω S O

• input
• a convex compact set Ω ⊇ O
• a point o ∈ O

tool a probing device P, able to

• move freely outside O
• cast rays and detect their first

intersection point with O goal

• command the probing device
• process the outcomes of

probes

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Strategy Use our sampling algorithm, with the probing device as oracle Theorem Upon termination, ˆ S = Del|S(E) ⇒ E is a loose ε-sample of S

• probes are issued along Voronoi

edges, from reachable Voronoi vertices

• a subset ˆ

S of Del|S(E) is main- tained

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What if S is unknown and unsampled?

Curve and surface probing [BGO05]

• Convex polytopes: finger probes, hyperplane probes, X-ray probes

[CY84, DEY86, Sk89]

• Polyhedra with no coplanar facets: enhanced finger probes [BY92]

Exact probing Approximate probing

• Planar convex smooth objects: hyperplane probes

[LB94, Ri97, Ro92]

• Non-convex smooth objects in 2D/3D [BGO05]

Surface Probing: a brief survey

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