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