Voronoi-Based Medial Axis Approximation for Smooth Models

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. CAD model Point Sampling Medial Axis for a CAD modelhttp://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm Medial Axis

3. Medial axis approximation for smooth models

4. Voronoi Based Medial Axis • Amenta-Bern 98: Pole and Pole Vector • Tangent Polygon • Umbrella Up

5. Filtering conditions Our goal: approximate the medial axis as a subset of Voronoi facets. • Medial axis point m • Medial angle θ • Angle and Ratio Conditions

6. Angle Condition • Angle Condition [θ ]: 

7. ‘Only Angle Condition’ Results  = 32 degrees  = 18 degrees  = 3 degrees

8. ‘Only Angle Condition’ Results  = 30 degrees  = 20 degrees  = 15 degrees

9. Ratio Condition • Ratio Condition []:

10. ‘Only Ratio Condition’ Results  = 4  = 8  = 2

11. ‘Only Ratio Condition’ Results  = 2  = 4  = 6

12. Medial axis approximation for smooth models

13. Theorem • Let F be the subcomplex computed by MEDIAL. 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.

14. Experimental Results

15. CAD model Point Sampling Medial Axis from a CAD model Medial Axis

16. Medial Axis from a CAD modelhttp://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm CAD model Medial Axis Point Sampling

17. Further work • Only Ratio condition provides theoretical convergence: • Noisy sample • [Chazal-Lieutier] Topology guarantee.

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

19. Curve Skeleton

20. Motivation (D.-Sun 2006) • 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

21. Medial axis • Medial axis: set of centers of maximal inscribed balls • The stratified structure [Giblin-Kimia04]: generically, the medial axis of a surface consists of five types of points based on the number of tangential contacts. • M2: inscribed ball with two contacts, form sheets • M3: inscribed ball with three contacts, form curves • Others:

22. Medial geodesic function (MGF)

23. Properties of MGF • Property 1 (proved): f is continuous everywhere and smooth almost everywhere. The singularity of f has measure zero in M2. • Property 2 (observed): There is no local minimum of f in M2. • Property 3 (observed): At each singular point x of f there are more than one shortest geodesic paths between ax and bx.

24. Defining curve-skeletons • Sk2=SkM2: set of singular points of MGF on M2 (negative divergence of Grad f. • Sk3=SkM3: extending the view of divergence • A point of other three types is on the curve-skeleton if it is the limit point of Sk2 U Sk3 • Sk=Cl(Sk2 U Sk3)

25. Examples

26. Shape eccentricity and computing tubular regions • Eccentricity: e(E)=g(E) / c(E)

27. 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

28. Thank you!