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

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

ε -Samples [AB98] Hyp: domain S is a smooth curve or surface. S E 1

ε -Samples [AB98] Hyp: domain S is a smooth curve or surface. S E 1

ε -Samples [AB98] Hyp: domain S is a smooth curve or surface. Del | S ( E ) S E 1

ε -Samples [AB98] Hyp: domain S is a smooth curve or surface. E ⊂ S is an ε -sample of S if ∀ p ∈ S, d( p, E ) ≤ ε Def Del | S ( E ) Theorem If E is an ε -sample of S S , with ε 0 . 16 rch( S ) , then < Del | S ( E ) ≈ S and lies at Hausdorff distance O ( ε 2 ) of S . E p q 1

Curve/Surface Meshing and ε -samples Given S and E , • for which values of ε is E an ε -sample of S ? • which points should be added to E , to make it an ε -sample? 2

Loose ε -Samples [BO04] Hyp: S is C 1 , 1 , E ⊂ S Let V be the set of the edges of Vor( E ) E ⊂ S is a loose ε -sample of S if Def ∀ p ∈ V ∩ S, d( p, E ) ≤ ε candidate 3

Loose ε -Samples [BO04] Hyp: S is C 1 , 1 , E ⊂ S Let V be the set of the edges of Vor( E ) E ⊂ S is a loose ε -sample of S if Def ∀ p ∈ V ∩ S, d( p, E ) ≤ ε candidate ε -samples are loose ε -samples Theorem 1 3

Loose ε -Samples [BO04] Hyp: S is C 1 , 1 , E ⊂ S Let V be the set of the edges of Vor( E ) E ⊂ S is a loose ε -sample of S if Def ∀ p ∈ V ∩ S, d( p, E ) ≤ ε candidate ε -samples are loose ε -samples Theorem 1 ! The contraposal is false... ε 3

Loose ε -Samples [BO04] Hyp: S is C 1 , 1 , E ⊂ S Let V be the set of the edges of Vor( E ) E ⊂ S is a loose ε -sample of S if Def ∀ p ∈ V ∩ S, d( p, E ) ≤ ε candidate ε -samples are loose ε -samples Theorem 1 ! The contraposal is false... ... but true asymptotically Theorem 2 For ε ≤ 0 . 16 rch( S ) , � � ε loose ε -samples are ε 1 + -samples, rch( S ) loose ̺ rch( S ) -samples are ̺ (1 + ̺ ) rch( S ) - samples ( ̺ = ε / rch( S ) ) 3

Loose ε -Samples [BO04] What is the smallest value ε 0 of ε such that E is an ε -sample? • d Let d = max { d( p, E ) | p ∈ V ∩ S } → ∀ ε < d , E is not an ε -sample (Thm 1) � � d → ∀ ε ≥ d 1 + , E is an ε -sample (Thm 2) rch( S ) � � d ⇒ d ≤ ε 0 ≤ d 1 + rch( S ) 4

Loose ε -Samples [BO04] How to enrich E so that it becomes a (sparse) ε -sample? • while there are far away candidates > ε 5

Loose ε -Samples [BO04] How to enrich E so that it becomes a (sparse) ε -sample? • while there are far away candidates insert one far away candidate in E ; 5

Loose ε -Samples [BO04] How to enrich E so that it becomes a (sparse) ε -sample? • while there are far away candidates insert one far away candidate in E ; update Vor( E ) and the list of candidates ; end while 5

Loose ε -Samples [BO04] How to enrich E so that it becomes a (sparse) ε -sample? • while there are far away candidates insert one far away candidate in E ; update Vor( E ) and the list of candidates ; end while � � Area( S ) The process terminates and inserts O points in E ε 2 Theorem Upon termination, E is a loose ε -sample ( ≈ ε -sample) of S 5

Loose ε -Samples [BO04] How to enrich E so that it becomes a (sparse) ε -sample? • while there are far away candidates insert one far away candidate in E ; update Vor( E ) and the list of candidates ; end while Space complexity: O ( n log n ) O ( n log 2 n ) Time complexity: � � Area( S ) The process terminates and inserts O points in E ε 2 Theorem Upon termination, E is a loose ε -sample ( ≈ ε -sample) of S 5

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Curve/Surface meshing: a brief survey • 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] 7

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 → 8

Implicit surfaces 9

Level-sets Implicit surfaces 9

Level-sets Implicit surfaces Point sets 9

Level-sets Implicit surfaces Molecules Point sets 9

What if S is unknown and unsampled? Curve and surface probing [BGO05] Discover the (rest of the) shape by means of a tool, Idea called a probing device input S - a convex compact set Ω ⊇ O ∂ Ω - a point o ∈ O O tool a probing device P , able to - move freely outside O P - cast rays and detect their first o intersection point with O goal - command the probing device - process the outcomes of Ω probes 10

What if S is unknown and unsampled? Curve and surface probing [BGO05] Use our sampling algorithm, with the probing device as oracle Strategy • probes are issued along Voronoi edges, from reachable Voronoi vertices • a subset ˆ S of Del | S ( E ) is main- tained Theorem Upon termination, ˆ S = Del | S ( E ) ⇒ E is a loose ε -sample of S 11

Surface Probing: a brief survey Exact probing • Convex polytopes: finger probes, hyperplane probes, X-ray probes [CY84, DEY86, Sk89] • Polyhedra with no coplanar facets: enhanced finger probes [BY92] Approximate probing • Planar convex smooth objects: hyperplane probes [LB94, Ri97, Ro92] • Non-convex smooth objects in 2D/3D [BGO05] 12