Computing Medial Axis and Curve Skeleton from Voronoi

Diagrams

Tamal K. Dey

Department of Computer Science and EngineeringThe Ohio State University

Joint work with Wulue Zhao, Jian Sun

http://web.cse.ohio-state.edu/~tamaldey/medialaxis.htm

2Department of Computer and Information Science

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

CAD model

Point Sampling Medial Axis

3Department of Computer and Information Science

Medial axis approximation for smooth models

4Department of Computer and Information Science

• Amenta-Bern 98: Pole and Pole Vector

• Tangent Polygon

• Umbrella Up

Voronoi Based Medial Axis

5Department of Computer and Information Science

Filtering conditions

• Medial axis point m• Medial angle θ• Angle and Ratio

Conditions

Our goal: : approximate the medial axis as a approximate the medial axis as a subset of Voronoi facets.subset of Voronoi facets.

6Department of Computer and Information Science

Angle Condition

• Angle Condition [θ ]:

pqσ,tnpUσ

max

2

7Department of Computer and Information Science

‘Only Angle Condition’ Results

= 18 degrees

= 3 degrees = 32 degrees

8Department of Computer and Information Science

‘Only Angle Condition’ Results

= 15 degrees

= 20 degrees

= 30 degrees

9Department of Computer and Information Science

Ratio Condition

• Ratio Condition []:

R

qpmin

pU

||||

10Department of Computer and Information Science

‘Only Ratio Condition’ Results

= 2

= 4

= 8

11Department of Computer and Information Science

‘Only Ratio Condition’ Results

= 2

= 4

= 6

12Department of Computer and Information Science

Medial axis approximation for smooth models

15Department of Computer and Information Science

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.

18Department of Computer and Information Science

Experimental Results

19Department of Computer and Information Science

Medial AxisMedial Axis

Medial Axis from a CAD model

CAD model

Point Sampling

20Department of Computer and Information Science

Medial AxisMedial Axis

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

CAD model

Point Sampling

21Department of Computer and Information Science

Further work

• Only Ratio condition provides theoretical convergence:• Noisy sample

• [Chazal-Lieutier] Topology guarantee.

Curve-skeletons with Medial Geodesic Function

Joint work with J. Sun 2006

24Department of Computer and Information Science

Curve Skeleton

25Department 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)

26Department of Computer and Information Science

• 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:

Medial axis

27Department of Computer and Information Science

Medial geodesic function (MGF)

28Department 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 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.

30Department of Computer and Information Science

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)

31Department of Computer and Information Science

Examples

33Department of Computer and Information Science

Shape eccentricity and computing tubular regions

• Eccentricity: e(E)=g(E) / c(E)

34Department 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

Thank you!