computing shapes and their features from point samples n.
Skip this Video
Loading SlideShow in 5 Seconds..
Computing Shapes and Their Features from Point Samples PowerPoint Presentation
Download Presentation
Computing Shapes and Their Features from Point Samples

Computing Shapes and Their Features from Point Samples

133 Vues Download Presentation
Télécharger la présentation

Computing Shapes and Their Features from Point Samples

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  1. Computing Shapes and Their Features from Point Samples Tamal K. Dey The Ohio State University

  2. ` Surface Reconstruction Point Cloud Surface Reconstruction

  3. Algorithms • Alpha-shapes (Edelsbrunner, Mücke 94) • Crust (Amenta, Bern 98) • Natural Neighbors (Boissonnat, Cazals 00) • Cocone (Amenta, Choi, Dey, Leekha, 00) • Tight Cocone (Dey, Goswami, 02) • Power Crust (Amenta, Choi, Kolluri 01)

  4. Basic Topology • d-ball Bd {x in Rd | ||x|| ≤ 1} • d-sphere Sd {x in Rd+1 | ||x||=1} • Homeomorphism h: T1 → T2 where h is continuous, bijective and has continuous inverse • K-Manifold : neighborhoods homeomorphic to open k-ball 2-sphere, torus, double torus are 2-manifolds • K-manifold with boundary: interior points, boundary points Bd is a d-manifold with boundary where bd(Bd)=S(d-1)

  5. Basic Topology • Smooth Manifolds • Triangulation • k-simplex • Simplicial complex K: (i) t in K if t is a face of t' in K (ii) t1, t2 in K => t1∩ t2 face of both • K is a triangulation of a topological space T if T ≈ |K|

  6. Sampling

  7. Medial Axis

  8. f(x) • f(x) is the distance to medial axis Local Feature SizeAmenta-Bern-Eppstein 98

  9. x -samplingAmenta-Bern-Eppstein 98 • Each x has a sample within f(x) distance

  10. -sample ε-sample is also ε'-sample for ε'> ε

  11. Lipschitz Property of f() Lemma (Lipschitz Continuity): f(x) ≤ f(y) + ||x-y|| Proof: Let m be a point on M with f(y)=||y-m|| By triangular inequality ||x-m|| ≤||y-m|| + ||x-y|| f(x) ≤ ||x-m|| ≤ f(y)+||x-y||

  12. FTL: Feature translation lemma Lemma (Feature Translation): If ||x-y|| ≤ εf(x) then 1/(1+ ε)f(y) ≤ f(x) ≤ 1/(1- ε)f(y) Exercise 1: Prove it. Also prove ||x-y|| ≤ε/(1- ε)f(y)

  13. FBL: Feature Ball Lemma Lemma (Feature Ball): If a d-ball B intersects a k-manifold Σ at more than two points with either (i) B∩Σ is not a k-ball, or (ii) bd(B)∩Σ is not a (k-1)-sphere, then B contains a medial axis point. Exercise 2: Prove

  14. Voronoi/Delaunay Diagrams Voronoi diagram VP: collection of Voronoi cells {Vp} Vp={x in R3 | ||x-p|| ≤ ||x-q|| for all q in P} Voronoi facet, Voronoi edge, Voronoi vertex Delaunay triangulation DP: Dual of VP, a simplicial complex Delaunay edge, triangle, tetrahedra

  15. Delaunay properties Emptiness : A simplex t is in DP if and only if there is a circumscribing ball of t that does not contain any point of P inside. Proof: If a k-simplex t, 0 ≤k ≤3, is in a DP, its dual (3-k)-dimensional Voronoi element has a point x that is equidistant from the (k+1) vertices of t. Also these vertices are closest to x among all points of P. This only means the ball centered at x with the vertices of t on the bounding sphere is empty. Exercise 3: Show that if t has an empty circumball, t is in DP

  16. Restricted Voronoi/Delaunay Restricted Voronoi: VP,Σ= {Vp Σ=Vp∩ Σ | p in P} Restricted Delaunay: Dp, Σ ={A k-simplex is Conv R where ∩Vp, Σ ≠ Ø for p in R}

  17. Curve samples and Voronoi

  18. Crust Algorithm (2D)Amenta-Bern-Eppstein 98 • Compute VP • Add Voronoi vertices • Compute Delaunay • Retain edges between samples only

  19. Nearest Neighbor AlgorithmDey-Kumar99 • Compute DP • For each p, compute nearest neighbor • For each p, compute its half-neighbor.

  20. Voronoi vertex Difficulties in 3D • There is no unique `correct’ surface for reference • Voronoi vertices can come close to the surface… slivers are nasty. or or ……

  21. Normals and Voronoi Cells(3D) Amenta-Bern 98

  22. Long Voronoi cells Lemma (Medial): Let m1 and m2 be the centers of two medial balls at p. Vp contains m1, m2. Exercise 4: Prove it

  23. NL : Normal Lemma Lemma (Normal) : Let v be a point in Vp with ||v-p||>μf(p). Then, angle((v-p),np)≤ arcsin ε/μ(1- ε) + arcsin ε/(1- ε). Exercise 5: Prove NL

  24. NVL: Normal Variation lemma Lemma (Normal Variation) : Let x and y be two points with ||x-y||≤r f(x) for r< 1/3. Then, angle(nx,ny) ≤ r/(1-3r).

  25. ENL: Edge Normal Lemma Lemma (Edge Normal): angle((p-q),np) > /2 – arcsin ||p-q||/2f(p). Proof: sin θ = ||p-q||/2||m-p|| ≤ ||p-q||/2f(p)

  26. TNL: Triangle Normal Lemma Lemma (Triangle Normal) : angle(npqr,np) ≤ α + arcsin((2/√3) sin 2α) where α ≤ arcsin d/f(p) and d, the circumradius, is sufficiently small.

  27. Topology Closed Ball property (Edelsbrunner, Shah 94): If restricted Voronoi cell is a closed ball in each dimension, then DP, Σ is homeomorphic to Σ. Assume P is an e-sample of Σ where e is sufficiently small. It can be shown that (P, Σ) satisfies the closed ball property. (proof from Cheng-Dey-Edelsbrunner-Sullivan 02)

  28. SDL: Short Distance Lemma Lemma Short Distance : x, y two points in Vp,Σ. (i) ||x-p||< ε/(1- ε)f(p) (ii) ||x-y|| < 2ε/(1- ε)f(x). Exercise 6: Prove it.

  29. LDL: Long Distance Lemma Lemma (Long Distance) : Suppose L intersects S in two points x, y and makes angle less than ξ with nx. Then ||x-y||>2f(x)cos ξ.

  30. VEL: Voronoi Edge Lemma Lemma (Voronoi Edge) : A Voronoi edge intersects Σ in a single point. Proof: x ≤ angle(npqr,np) + angle(np,nx) ≤ O(ε) + O(ε) by TNL and NVL. 2f(x)cos O(ε) ≤ ||x-y|| ≤ O(ε) f(x) by SDL and LDL Contradiction when εis sufficiently small Exercise 7: Prove it

  31. VFL: Voronoi Facet Lemma Lemma (Voronoi Facet): A Voronoi facet intersects Σin a 1-ball. Proof: angle(Lnx)≤ angle(Lnp) + angle(np,nx) ≤ O(ε) + O(ε) by ENL and NVL. 2f(x)cos O(ε) ≤ ||x-y|| ≤ O(ε) f(x) by SDL and LDL Contradiction when εis sufficiently small

  32. VCL: Voronoi Cell Lemma Lemma (Voronoi Cell) : A Voronoi cell intersects Σ in a 2-ball. Proof: show that handles and connected components of Σcannot be in the cell. Then show that if the cell intersects Σ in multiple disks, we reach a contradiction with SDL and LDL.

  33. P+ P- Poles

  34. P+ P- PVL: Pole Vector Lemma Lemma (Pole Vector) : angle((p+-p),np)=2arcsin ε/(1- ε). Proof: ||p+-p||> f(p) since Vp contains a medial axis point (Medial Lemma). Plug this in Normal Lemma. vp np

  35. Crust in 3DAmenta-Bern 98 • Introduce poles • Filter crust triangles from Delaunay • Filter by normals • Extract manifold

  36. Manifold Extraction: Prunning Remove Sharp edges with their triangles

  37. Why Prunning Works? • Crust triangles include restricted Delaunay triangles • The underlying space of the restricted Delaunay triangles is homeomorphic to the sampled surface • No edge of the restricted triangles is sharp • After prunning, at least the surface made by the restricted Delaunay triangles remains

  38. Manifold Extraction: Walk Walk inside or outside the possibly thickened surface

  39. Cocone AlgorithmAmenta-Choi-Dey-Leekha 00 • Simplified/improved the Crust • Only single Voronoi computation • Analysis is simpler • No normal filtering step • Proof of homeomorphism

  40. Cocone • vp= p+ - pis the pole vector • Space spanned by vectors within the Voronoi cell making angle > 3/8 with vp or -vp

  41. Cocone Algorithm

  42. Candidate triangles computation e=(a,b); a=a-p; b= b-p

  43. Candidate Triangle Properties • Candidate triangles include the restricted Delaunay triangles • Their circumradii are small O()f(p) • Their normals make only O(e) angle with the surface normals at the vertices

  44. Restricted Delaunay property Claim: Let y in Vp∩Σ . Then, angle(np,(y-p)) > /2- ε Exercise 8: Prove it. Lemma (Restricted Delaunay): All restricted triangles are in T for ε<0.1. Proof: Let y in e∩Σ where e is the dual edge for a triangle. angle((y-p),vp)>angle((y-p),np)-angle(np,vp) > /2- ε-angle(np,vp) > 3/8 by PVL for ε < 0.1.

  45. No sharp edge Lemma Sharp: No restricted Delaunay triangle has a sharp edge for ε < 0.06

  46. Small radius and flatness Lemma (Small Triangle): The circumradius r of any candidate triangle is O(ε)f(p) where p is any of its vertex and ε < 0.06. Proof: There is y in dual edge so that angle(vp,(y-p))>3/8. By PVL angle(np,(y-p)) > 3/8-2arcsin ε/(1- ε). Use contrapositive of NL to conclude ||y-p||=O(ε)f(p) for ε<0.06. Lemma (Flat Triangle): For each candidate triangle pqr angle(npqr,np)=O(ε) Proof: Follows from STL and TNL.

  47. Homeomorphism Let M be the triangulated surface obtained after the manifold extraction. Define h: R3 -> Σ where h(q) is the closest point on Σ. h is well defined except at the medial axis points. Lemma Homeomorphism: The restriction of h to M, h: M -> Σ, is a homeomorphism. Proof: Use STL, FTL for the proof, see ACDL00.

  48. Cocone Guarantees Theorem: Any point x S is within O(e)f(x) distance from a point in the output. Conversely, any point of the output surface has a point x S within O(e)f(x) distance for ε<0.06. Theorem: The output surface computed by Cocone from an e -sample is homeomorphic to the sampled surface for ε < 0.06.

  49. Undersampling Dey-Giesen 01 • Boundaries • Small features • Non-smoothness

  50. Boundaries