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

2/52 Department of Computer and Information Science Problems Surface reconstruction (Cocone) Medial axis (Medial) Shape segmentation and matching (SegMatch)

3/52 Department of Computer and Information Science Surface Reconstruction ` Point Cloud Surface Reconstruction

4/52 Department of Computer and Information Science Voronoi based algorithms 1.Alpha-shapes (Edelsbrunner, Mucke 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)

5/52 Department of Computer and Information Science Medial Axis

6/52 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)

7/52 Department of Computer and Information Science Each x has a sample within  f(x) distance  -Sampling [ABE98] x

8/52 Department of Computer and Information Science Voronoi/Delaunay

9/52 Department of Computer and Information Science Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]

10/52 Department of Computer and Information Science Poles P+P+ P-P-

11/52 Department of Computer and Information Science Normal Lemma The angle between the pole vector v p and the normal n p is O(  ). P+P+ P-P- npnp vpvp

12/52 Department of Computer and Information Science Cocone Algorithm [Amenta-Choi-Dey-Leekha SoCG00] Simplified/improved the Crust Only single Voronoi computation Analysis is simpler No normal filtering step Proof of homeomorphism

13/52 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

14/52 Department of Computer and Information Science Cocone Algorithm

15/52 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 output surface has a point x   within O(  )f(x) distance. Theorem: The output surface computed by Cocone from an  -sample is homeomorphic to the sampled surface for sufficiently small .

16/52 Department of Computer and Information Science Undersampling [Dey-Giesen SoCG01] Boundaries Small features Non-smoothness

17/52 Department of Computer and Information Science Boundaries

18/52 Department of Computer and Information Science Small Features High curvature regions are often undersampled

19/52 Department of Computer and Information Science Data Set Engine

20/52 Department of Computer and Information Science Nonsmoothness

21/52 Department of Computer and Information Science Watertight Surfaces

22/52 Department of Computer and Information Science Tight Cocone [Dey-Goswami SM03]

23/52 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.

24/52 Department of Computer and Information Science Result Sharp corners and edges of AutoPart can be reconstructed.

25/52 Department of Computer and Information Science Timings PIII, 933Mhz, 512MB

26/52 Department of Computer and Information Science Noisy Data – Ram Head Front view Rear view

27/52 Department of Computer and Information Science Example movie file Mannequin

28/52 Department of Computer and Information Science Bunny data Bunny Point data Tight Cocone Robust Cocone

29/52 Department of Computer and Information Science Medial axis from point sample Dey-Zhao SM02 [Hoffman-Dutta 90],[Culver-Keyser-Manocha 99],[Giblin-Kimia 00], [Foskey-Lin-Manocha 03] Voronoi based [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]

30/52 Department of Computer and Information Science Medial Axis Medial Ball Medial Axis  -Sampling

31/52 Department of Computer and Information Science Geometric Definitions Delaunay Triangulation Voronoi Diagram Pole and Pole Vector Tangent Polygon Umbrella U p

32/52 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.

33/52 Department of Computer and Information Science Angle Condition Angle Condition [θ ]: 

34/52 Department of Computer and Information Science Ratio Condition Ratio Condition [  ]:

35/52 Department of Computer and Information Science Algorithm

36/52 Department of Computer and Information Science Theorem Let F be the subcomplex computed by M EDIAL. As  approaches zero: Each point in F converges to a medial axis point. Each point in the medial axis is converged upon by a point in F.

37/52 Department of Computer and Information Science Experimental Results

38/52 Department of Computer and Information Science Experimental Results

39/52 Department of Computer and Information Science Experimental Results

40/52 Department of Computer and Information Science Computation Time Pentium PC 933 MHz CPU 512 MB memory CGAL 2.3 C++ O1 optimization

41/52 Department of Computer and Information Science Medial Axis from a CAD model CAD model Point Sampling Medial Axis

42/52 Department of Computer and Information Science Medial Axis Medial Axis from a CAD model CAD model Point Sampling

43/52 Department of Computer and Information Science Example movie file Anchor Medial

44/52 Department of Computer and Information Science Segmentation and matching Siddiqui-Shokoufandeh-Dickinson-Zucker 99 (Shock graphs) Hilaga-Shinagawa-Kohmura-Kunni 01 (Reeb graph) Osada-Funkhouser-Chazelle-Dobkin 01 (Shape distribution) Bespalov-Shokoufandeh-Regli-Sun 03(spectral decomposition) Dey-Giesen-Goswami 03 (Morse theory)

45/52 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

46/52 Department of Computer and Information Science Feature definition Flow Continuous Discrete flow Discretization

47/52 Department of Computer and Information Science Anchor set Anchor set: Height fuinction:

48/52 Department of Computer and Information Science Driver and critical points Driver : d(x) is the closest point on the anchor hull Critical points Anchor Hull : H(x) is convex hull of A(x)

49/52 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

50/52 Department of Computer and Information Science Features F(x) = closure( S(x) ) for a maximum x

51/52 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 

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

53/52 Department of Computer and Information Science Stable manifolds Feature F(x) = closure( S(x) ) for a maximum x

54/52 Department of Computer and Information Science Stable manifolds in 3D Stable manifolds are not subcomplexes of Delaunay We approximate the stable manifolds with Delaunay simplices

55/52 Department of Computer and Information Science Algorithm for

56/52 Department of Computer and Information Science Merging Small perturbations create insignificant features Sampling artifacts introduce more segmentations Merge stable manifolds

57/52 Department of Computer and Information Science Results (2D)

58/52 Department of Computer and Information Science Results (3D)

59/52 Department of Computer and Information Science Results (3D)

60/52 Department of Computer and Information Science Matching CAD models

61/52 Department of Computer and Information Science Conclusions Noisy samples: Reconstruction and segmentation Improving segmentation and matching for CAD models (requires understanding of non- smoothness) Software available from Acknowledgement: NSF, DARPA, ARO, CGAL