1. Computing Medial Axis and Curve Skeleton from Voronoi Diagrams Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Joint work with Wulue Zhao, Jian Sun http://web.cse.ohio-state.edu/~tamaldey/medialaxis.htm

2. 2 Department of Computer and Information Science Medial Axis for a CAD model http://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm CAD model Point Sampling Medial Axis

3. 3 Department of Computer and Information Science Medial axis approximation for smooth models

4. 4 Department of Computer and Information Science Amenta-Bern 98: Pole and Pole Vector Tangent Polygon Umbrella U p Voronoi Based Medial Axis

5. 5 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.

6. 6 Department of Computer and Information Science Angle Condition Angle Condition [θ ]: 

7. 7 Department of Computer and Information Science ‘Only Angle Condition’ Results  = 18 degrees  = 3 degrees  = 32 degrees

8. 8 Department of Computer and Information Science ‘Only Angle Condition’ Results  = 15 degrees  = 20 degrees  = 30 degrees

9. 9 Department of Computer and Information Science Ratio Condition Ratio Condition [  ]:

10. 10 Department of Computer and Information Science ‘Only Ratio Condition’ Results  = 2  = 4  = 8

11. 11 Department of Computer and Information Science ‘Only Ratio Condition’ Results  = 2  = 4  = 6

12. 12 Department of Computer and Information Science Medial axis approximation for smooth models

13. 13 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 

14. 14 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:

15. 15 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.

16. 16 Department of Computer and Information Science Experimental Results

17. 17 Department of Computer and Information Science Experimental Results

18. 18 Department of Computer and Information Science Experimental Results

19. 19 Department of Computer and Information Science Medial Axis Medial Axis from a CAD model CAD model Point Sampling

20. 20 Department of Computer and Information Science Medial Axis Medial Axis from a CAD model http://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm CAD model Point Sampling

21. 21 Department of Computer and Information Science Further work Only Ratio condition provides theoretical convergence: Noisy sample [Chazal-Lieutier] Topology guarantee.

22. 22 Department of Computer and Information Science Stable parts  = 8, θ = 22.5 degrees  = 12, θ = 50 degrees

23. Curve-skeletons with Medial Geodesic Function Joint work with J. Sun 2006

24. 24 Department of Computer and Information Science Curve Skeleton

25. 25 Department of Computer and Information Science 1D representation of 3D shapes, called curve-skeleton, useful in some applications Geometric modeling, computer vision, data analysis, etc Reduce dimensionality Build simpler algorithms Desirable properties [Cornea et al. 05] centered, preserving topology, stable, etc Issues No formal definition enjoying most of the desirable properties Existing algorithms often application specific Motivation (D.-Sun 2006)

26. 26 Department of Computer and Information Science Medial axis: set of centers of maximal inscribed balls The stratified structure [Giblin-Kimia04]: g enerically, the medial axis of a surface consists of five types of points based on the number of tangential contacts. M 2 : inscribed ball with two contacts, form sheets M 3 : inscribed ball with three contacts, form curves Others: Medial axis

27. 27 Department of Computer and Information Science Medial geodesic function (MGF)

28. 28 Department of Computer and Information Science Properties of MGF Property 1 (proved): f is continuous everywhere and smooth almost everywhere. The singularity of f has measure zero in M 2. Property 2 (observed): There is no local minimum of f in M 2. Property 3 (observed): At each singular point x of f there are more than one shortest geodesic paths between a x and b x.

29. 29 Department of Computer and Information Science Defining curve-skeletons Sk 2 =Sk Å M 2 : the set of singular points of MGF or points with negative divergence w.r.t. r f Sk 3 =Sk Å M 3 : A point of other three types is on the curve-skeleton if it is the limit point of Sk 2 [ Sk 3

30. 30 Department of Computer and Information Science Defining curve-skeletons Sk 2 =SkM 2 : set of singular points of MGF on M 2 (negative divergence of Grad f. Sk 3 =SkM 3 : extending the view of divergence A point of other three types is on the curve-skeleton if it is the limit point of Sk 2 U Sk 3 Sk=Cl(Sk 2 U Sk 3 )

31. 31 Department of Computer and Information Science Examples

32. 32 Department of Computer and Information Science Properties of curve-skeletons Thin (1D curve) Centered Homotopy equivalent Junction detective Stable Prop1: set of singular points of MGF is of measure zero in M 2 Medial axis is in the middle of a shape Prop3: more than one shortest geodesic paths between its contact points Medial axis homotopy equivalent to the original shape Curve-skeleton homotopy equivalent to the medial axis

33. 33 Department of Computer and Information Science Shape eccentricity and computing tubular regions Eccentricity: e(E)=g(E) / c(E)

34. 34 Department of Computer and Information Science Conclusions Voronoi based approximation algorithms Scale and density independent Fine tuning is limited Provable guarantees Software Medial: www.cse.ohio-state.edu/~tamaldey/cocone.html Cskel: www.cse.ohio-state.edu/~tamaldey/cskel.html

35. Thank you!