Department of Computer Science and Engineering Defining and Computing Curve-skeletons with Medial Geodesic Function Tamal K. Dey and Jian Sun The Ohio State University

2/16 Department of Computer Science and Engineering 1D representation of 3D shapes, called curve-skeleton, useful in many application 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

3/16 Department of Computer Science and Engineering Give a mathematical definition of curve-skeletons for 3D objects bounded by connected compact surfaces Enjoy most of the desirable properties Give an approximation algorithm to extract such curve- skeletons Practically plausible Contributions

4/16 Department of Computer Science and Engineering Roadmap

5/16 Department of Computer Science and Engineering 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

6/16 Department of Computer Science and Engineering Medial geodesic function (MGF)

7/16 Department of Computer Science and Engineering 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.

8/16 Department of Computer Science and Engineering 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

9/16 Department of Computer Science and Engineering Defining curve-skeletons Sk 2 =Sk Å M 2 : set of singular points of MGF or points with negative divergence w.r.t. r f Sk 3 =Sk Å M 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 [ Sk 3 Sk=Cl(Sk 2 [ Sk 3 )

10/16 Department of Computer Science and Engineering Computing curve-skeletons MA approximation [Dey-Zhao03] : subset of Voronoi facets MGF approximation: f(F) and (F) Marking: E is marked if (F) ² n < for all incident Voronoi facets Erosion: proceed in collapsing manner and guided by MGF

11/16 Department of Computer Science and Engineering Examples

12/16 Department of Computer Science and Engineering 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

13/16 Department of Computer Science and Engineering Effect of

14/16 Department of Computer Science and Engineering Shape eccentricity and computing tubular regions Eccentricity: e(E)=g(E) / c(E) Compute tubular regions classify skeleton edges and mesh faces based on a given threshold depth first search

15/16 Department of Computer Science and Engineering Shape eccentricity and computing tubular regions Eccentricity: e(E)=g(E) / c(E)

16/16 Department of Computer Science and Engineering Timing

Department of Computer Science and Engineering Thank you!