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

2/52 Department of Computer and Information Science Surface Reconstruction ` Point Cloud Surface Reconstruction

3/52 Department of Computer and Information Science Algorithms 1.Alpha-shapes (Edelsbrunner, Mücke 94) 2.Crust (Amenta, Bern 98) 3.Natural Neighbors (Boissonnat, Cazals 00) 4.Cocone (Amenta, Choi, Dey, Leekha, 00) 5.Tight Cocone (Dey, Goswami, 02) 6.Power Crust (Amenta, Choi, Kolluri 01)

4/52 Department of Computer and Information Science Basic Topology d-ball B d {x in R d | ||x|| ≤ 1} d-sphere S d {x in R d+1 | ||x||=1} Homeomorphism h: T 1 → T 2 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(B d )=S (d-1)

5/52 Department of Computer and Information Science Basic Topology Smooth Manifolds Triangulation k-simplex Simplicial complex K: (i) t in K if t is a face of t' in K (ii) t 1, t 2 in K => t 1 ∩ t 2 face of both K is a triangulation of a topological space T if T ≈ |K|

6/52 Department of Computer and Information Science Sampling

7/52 Department of Computer and Information Science Medial Axis

8/52 Department of Computer and Information Science f(x) is the distance to medial axis Local Feature Size Amenta-Bern-Eppstein 98 f(x) f(x)

9/52 Department of Computer and Information Science Each x has a sample within  f(x) distance  -sampling Amenta-Bern-Eppstein 98 x

10/52 Department of Computer and Information Science  -sample ε-sample is also ε'-sample for ε' > ε

11/52 Department of Computer and Information Science 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||

12/52 Department of Computer and Information Science FTL: Feature translation lemma Lemma (Feature Translation): If ||x-y|| ≤ ε f(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)

13/52 Department of Computer and Information Science 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

14/52 Department of Computer and Information Science Voronoi/Delaunay Diagrams Voronoi diagram V P : collection of Voronoi cells {V p } V p ={x in R 3 | ||x-p|| ≤ ||x-q|| for all q in P} Voronoi facet, Voronoi edge, Voronoi vertex Delaunay triangulation D P : Dual of V P, a simplicial complex Delaunay edge, triangle, tetrahedra

15/52 Department of Computer and Information Science Delaunay properties Emptiness : A simplex t is in D P 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 D P, 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 D P

16/52 Department of Computer and Information Science Restricted Voronoi/Delaunay Restricted Voronoi: V P,Σ = {V p Σ =V p ∩ Σ | p in P} Restricted Delaunay: D p, Σ ={A k-simplex is Conv R where ∩ V p, Σ ≠ Ø for p in R}

17/52 Department of Computer and Information Science Curve samples and Voronoi

18/52 Department of Computer and Information Science Crust Algorithm (2D) Amenta-Bern-Eppstein 98 Compute V P Add Voronoi vertices Compute Delaunay Retain edges between samples only

19/52 Department of Computer and Information Science Nearest Neighbor Algorithm Dey-Kumar 99 Compute D P For each p, compute nearest neighbor For each p, compute its half-neighbor.

20/52 Department of Computer and Information Science Difficulties in 3D Voronoi vertices can come close to the surface… slivers are nasty. There is no unique `correct’ surface for reference or …… Voronoi vertex

21/52 Department of Computer and Information Science Normals and Voronoi Cells(3D) Amenta-Bern 98

22/52 Department of Computer and Information Science Long Voronoi cells Lemma (Medial): Let m1 and m2 be the centers of two medial balls at p. V p contains m 1, m 2. Exercise 4: Prove it

23/52 Department of Computer and Information Science NL : Normal Lemma Lemma (Normal) : Let v be a point in V p with ||v-p||>μf(p). Then, angle((v-p),n p )≤ arcsin ε/μ(1- ε) + arcsin ε /(1- ε). Exercise 5: Prove NL

24/52 Department of Computer and Information Science 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(n x,n y ) ≤ r/(1-3r).

25/52 Department of Computer and Information Science ENL: Edge Normal Lemma Lemma (Edge Normal): angle((p-q),n p ) >  /2 – arcsin ||p-q||/2f(p). Proof: sin θ = ||p-q||/2||m-p|| ≤ ||p-q||/2f(p)

26/52 Department of Computer and Information Science TNL: Triangle Normal Lemma Lemma (Triangle Normal) : angle(n pqr,n p ) ≤ α + arcsin((2/√3) sin 2α) where α ≤ arcsin d/f(p) and d, the circumradius, is sufficiently small.

27/52 Department of Computer and Information Science Topology Closed Ball property (Edelsbrunner, Shah 94): If restricted Voronoi cell is a closed ball in each dimension, then D P, 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)

28/52 Department of Computer and Information Science SDL: Short Distance Lemma Lemma Short Distance : x, y two points in V p,Σ. (i) ||x-p||< ε/(1- ε)f(p) (ii) ||x-y|| < 2ε/(1- ε)f(x). Exercise 6: Prove it.

29/52 Department of Computer and Information Science LDL: Long Distance Lemma Lemma (Long Distance) : Suppose L intersects S in two points x, y and makes angle less than ξ with n x. Then ||x-y||>2f(x)cos ξ.

30/52 Department of Computer and Information Science VEL: Voronoi Edge Lemma Lemma (Voronoi Edge) : A Voronoi edge intersects Σ in a single point. Proof: x ≤ angle(n pqr, n p ) + angle(n p, n x ) ≤ 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 ε is sufficiently small Exercise 7: Prove it

31/52 Department of Computer and Information Science VFL: Voronoi Facet Lemma Lemma (Voronoi Facet): A Voronoi facet intersects Σ in a 1-ball. Proof: angle(Ln x )≤ angle(Ln p ) + angle(n p, n x ) ≤ 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 ε is sufficiently small

32/52 Department of Computer and Information Science VCL: Voronoi Cell Lemma Lemma (Voronoi Cell) : A Voronoi cell intersects Σ in a 2-ball. Proof: show that handles and connected components of Σ cannot be in the cell. Then show that if the cell intersects Σ in multiple disks, we reach a contradiction with SDL and LDL.

33/52 Department of Computer and Information Science Poles P+P+ P-P-

34/52 Department of Computer and Information Science PVL: Pole Vector Lemma P+P+ P-P- npnp vpvp Lemma (Pole Vector) : angle((p + -p),n p )=2arcsin ε /(1- ε). Proof: ||p + -p||> f(p) since V p contains a medial axis point (Medial Lemma). Plug this in Normal Lemma.

35/52 Department of Computer and Information Science Crust in 3D Amenta-Bern 98 Introduce poles Filter crust triangles from Delaunay Filter by normals Extract manifold

36/52 Department of Computer and Information Science Manifold Extraction: Prunning Remove Sharp edges with their triangles

37/52 Department of Computer and Information Science 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

38/52 Department of Computer and Information Science Manifold Extraction: Walk Walk inside or outside the possibly thickened surface

39/52 Department of Computer and Information Science Cocone Algorithm Amenta-Choi-Dey-Leekha 00 Simplified/improved the Crust Only single Voronoi computation Analysis is simpler No normal filtering step Proof of homeomorphism

40/52 Department of Computer and Information Science Cocone v p = p + - p is the pole vector Space spanned by vectors within the Voronoi cell making angle > 3  /8 with v p or -v p

41/52 Department of Computer and Information Science Cocone Algorithm

42/52 Department of Computer and Information Science Candidate triangles computation e=(a,b); a=a-p; b= b-p

43/52 Department of Computer and Information Science Candidate Triangle Properties Candidate triangles include the restricted Delaunay triangles Their circumradii are small O(  )f(p) Their normals make only O(  ) angle with the surface normals at the vertices

44/52 Department of Computer and Information Science Restricted Delaunay property Claim: Let y in V p ∩ Σ. Then, angle(n p,(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),v p )>angle((y-p),n p )-angle(n p,v p ) >  /2- ε -angle(n p,v p ) > 3  /8 by PVL for ε < 0.1.

45/52 Department of Computer and Information Science No sharp edge Lemma Sharp: No restricted Delaunay triangle has a sharp edge for ε < 0.06

46/52 Department of Computer and Information Science 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 < Proof: There is y in dual edge so that angle(v p,(y-p))>3  /8. By PVL angle(n p,(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(n pqr,n p )=O(ε) Proof: Follows from STL and TNL.

47/52 Department of Computer and Information Science Homeomorphism Let M be the triangulated surface obtained after the manifold extraction. Define h: R 3 -> 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.

48/52 Department of Computer and Information Science Cocone Guarantees Theorem: Any point x   is within O(  f(x) distance from a point in the output. Conversely, any point of the output surface has a point x   within O(  )f(x) distance for ε < Theorem: The output surface computed by Cocone from an  -sample is homeomorphic to the sampled surface for ε < 0.06.

49/52 Department of Computer and Information Science Undersampling Dey-Giesen 01 Boundaries Small features Non-smoothness

50/52 Department of Computer and Information Science Boundaries

51/52 Department of Computer and Information Science Small Features High curvature regions are often undersampled

52/52 Department of Computer and Information Science Well Sampled Patch and Boundary Vertices     is well sampled if ε-sampling holds for   Restricted Voronoi on  defines boundary vertices p is interior if restricted cell has no boundary point otherwise p is boundary vertex

53/52 Department of Computer and Information Science Radius and Height Radius r(p): radius of cocone Height h(p): distance to the negative pole p - cocone neighbors Np

54/52 Department of Computer and Information Science Flatness Condition Vertex p is flat if 1. Ratio condition: r(p)   h(p) 2. Normal condition:  v p,v q    q with p  N q

55/52 Department of Computer and Information Science Boundary Detection (1st phase) IsFlat( p, ,  ) check ratio and normal condition for V p ; if both are satisfied return true else return false end

56/52 Department of Computer and Information Science Boundary Detection (2nd phase) Boundary (P, ,  ) Compute the set R of flat vertices; while  p  R and p  N q with q  R and r(p)  h(p) and  v p,v q  R:=R  p; endwhile return P\R end

57/52 Department of Computer and Information Science Reconstruction Cocone (P, ,  ) Compute V P ; for each p  P if p  B compute T of triangles with duals intersecting C p ; endif enfor; Extract manifold; end B:= Boundary( P, ,  )

58/52 Department of Computer and Information Science Data Set Sat

59/52 Department of Computer and Information Science Data Set Engine

60/52 Department of Computer and Information Science Nonsmoothness

61/52 Department of Computer and Information Science Watertight Surfaces

62/52 Department of Computer and Information Science Tight Cocone Dey-Goswami 02

63/52 Department of Computer and Information Science Tight COCONE Principle Compute the Delaunay triangulation of the input point set. Use COCONE along with detection of undersampling to get an initial surface with undersampled regions identified. Stitch the holes from the existing Delaunay triangles without inserting any new point. Effectively, the output surface bounds one or more solids.

64/52 Department of Computer and Information Science Result Sharp corners and edges of AutoPart can be reconstructed.

65/52 Department of Computer and Information Science Dinosaur

66/52 Department of Computer and Information Science Large Data Octree subdivision

67/52 Department of Computer and Information Science Cracks Cracks appear in surface computed from octree boxes

68/52 Department of Computer and Information Science Padding Include a fraction from the neighbors to form the extended box

69/52 Department of Computer and Information Science Surface Matching

70/52 Department of Computer and Information Science Experimental Data Pentium III,733Mhz,512Mb Time Memory

71/52 Department of Computer and Information Science Lucy million points, 198 mints

72/52 Department of Computer and Information Science David’s Head 2 mil points, 93 minutes

73/52 Department of Computer and Information Science Noisy Data - Bunny Front view Rear view

74/52 Department of Computer and Information Science Noisy Data – Ram Head Front view Rear view

75/52 Department of Computer and Information Science Example movie file Mannequin

76/52 Department of Computer and Information Science Female Point dataTight CoconeRobust Cocone Examples

77/52 Department of Computer and Information Science Mannequin Point dataTight CoconeRobust Cocone Examples

78/52 Department of Computer and Information Science Cocone Software Cocone: Reconstructs surfaces with boundaries. Tight Cocone: Reconstructs watertight surfaces. Available from Acknowledgement: CGAL

79/52 Department of Computer and Information Science Medial axis from point sample Earlier work did not have guarantees [Attali-Montanvert-Lachaud 01] Power shape : guarantees topology, uses power diagram [Amenta-Choi-Kolluri 01] Medial : Approximates the medial axis as a Voronoi subcomplex and has converegence guarantee. [Dey-Zhao 02]

80/52 Department of Computer and Information Science Medial Axis Medial Ball Medial Axis  -Sampling

81/52 Department of Computer and Information Science Geometric Definitions Pole and Pole Vector Tangent Polygon Umbrella U p

82/52 Department of Computer and Information Science Filtering conditions Medial axis point m Medial angle θ Angle and Ratio Conditions : approximate the medial axis as a subset of Voronoi facets. Our goal: approximate the medial axis as a subset of Voronoi facets.

83/52 Department of Computer and Information Science Angle Condition Angle Condition [θ ]: Max{σ in U p angle(n σ,(q-p)) }<  /2- θ

84/52 Department of Computer and Information Science ‘Only Angle Condition’ Results  = 18 degrees  = 3 degrees  = 32 degrees

85/52 Department of Computer and Information Science ‘Only Angle Condition’ Results  = 15 degrees  = 20 degrees  = 30 degrees

86/52 Department of Computer and Information Science Ratio Condition Ratio Condition [  ]: Min{σ in U p ||p-q||/R σ > 

87/52 Department of Computer and Information Science ‘Only Ratio Condition’ Results  = 2  = 4  = 8

88/52 Department of Computer and Information Science ‘Only Ratio Condition’ Results  = 2  = 4  = 6

89/52 Department of Computer and Information Science Algorithm Each of Angle and Ratio conditions individually is not sufficient. Combination of both conditions First Angle, then Ratio The Angle condition captures the Delaunay edges which lie away from the surface. The Ratio condition captures the the Delaunay edges which make small angles with the umbrella triangles but are comparatively larger than their circumradii. Allows f ixed values of θ and 

90/52 Department of Computer and Information Science Algorithm

91/52 Department of Computer and Information Science Analysis Lemma 1: If w  lies in the segment mm’,  p = O (  ) S m1m1 m’ 

92/52 Department of Computer and Information Science Analysis (continued) Lemma 2: Let F = Dual pq be a Voronoi facet where pq satisfies the angle condition [  ] with   2  p +  p. Any point w in F with is at a distance from m where m and  are the center and radius of the medial ball at p with.  p,  p = O (  )  >(  -  p )

93/52 Department of Computer and Information Science Analysis (continued) Lemma 3: Let w be any point in the Voronoi facet F=Dual pq with where  is the radius of the medial ball at p with center m so that. If pq satisfies the Ratio condition then either or Proof If, Lemma 2 gives If, ……

94/52 Department of Computer and Information Science Analysis (continued) Lemma 4: Let w  V p be a point such that, where  is the radius of a medial ball at p with the center m and, and. Then, for sufficiently small  >0, if the medial angle of p is larger than  1/3.

95/52 Department of Computer and Information Science Theorems Theorem: if by Angle: (i), apply Lemma 4 Otherwise Lemma 2 if by Ratio: (i), apply Lemma 4 Otherwise Lemma 3 Theorem:

96/52 Department of Computer and Information Science Experimental Results

97/52 Department of Computer and Information Science Experimental Results

98/52 Department of Computer and Information Science Experimental Results

99/52 Department of Computer and Information Science Experimental Results

100/52 Department of Computer and Information Science Medial Axis from a CAD model CAD model Point Sampling Medial Axis

101/52 Department of Computer and Information Science Medial Axis Medial Axis from a CAD model CAD model Point Sampling

102/52 Department of Computer and Information Science Medial Axis Medial Axis from a CAD model CAD model Point Sampling

103/52 Department of Computer and Information Science Example movie file Anchor Medial

104/52 Department of Computer and Information Science Segmentation and matching Dey-Giesen-Goswami 03 Segment a shape into `features’ Match two shapes based on the segmentation

105/52 Department of Computer and Information Science Feature definition Flow Continuous Discrete flow Discretization

106/52 Department of Computer and Information Science Flow Anchor set: Height fuinction:

107/52 Department of Computer and Information Science Flow Vector field v : if x is regular and 0 otherwise Flow  induced by v Fix points of  are the critical points of h

108/52 Department of Computer and Information Science Features F(x) = closure( S(x) ) for a maximum x

109/52 Department of Computer and Information Science Flow by discrete set Driver d(x) : closest point on  dual to the Voronoi object containing x Vector field: This also induces a flow 

110/52 Department of Computer and Information Science Stable manifolds Gabriel edges are stable manifolds of saddles Stable manifolds of maxima are shaded

111/52 Department of Computer and Information Science Stable manifolds Feature F(x) = closure( S(x) ) for a maximum x

112/52 Department of Computer and Information Science Flow relation t < t’ if the circumcenters of t and t’ lie on the same side of the edge shared by them. Collect all triangles related by the transitive closure of <

113/52 Department of Computer and Information Science Flow relation in 3D In 2D there is at most one t’ so that t< t’ Exercise 9: Show an example in 3D where a tetrahedron t < t1 and t < t2. Show that there cannot be any third tetrahedron t3 so that t< t3.

114/52 Department of Computer and Information Science Algorithm for

115/52 Department of Computer and Information Science Merging Small perturbations create insignificant features Sampling artifacts introduce more segmentations Merge stable manifolds

116/52 Department of Computer and Information Science Results (2D)

117/52 Department of Computer and Information Science Results (3D)

118/52 Department of Computer and Information Science Results (2D)

119/52 Department of Computer and Information Science Results (3D)

120/52 Department of Computer and Information Science Open problems Design an algorithm to reconstruct non-smooth surface Design an algorithm for medial axis approximaion with topological guarantee Prove an approximation result for feature segmentation Software: Cocone, Medial : Segmatch: