Defining and Computing Curve-skeletons with Medial Geodesic Function

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

2/16Department 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


• 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


Roadmap


• 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


Medial geodesic function (MGF)


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.


Defining curve-skeletons• Sk2=SkÅM2: set of singular

points of MGF or points with negative divergence w.r.t. rf

• Sk3=SkÅM3: extending the view of divergence

• A point of other three types is on the curve-skeleton if it is the limit point of Sk2[ Sk3

• Sk=Cl(Sk2[ Sk3)


Computing curve-skeletons

• MA approximation [Dey-Zhao03]: subset of Voronoi facets • MGF approximation: f(F) and (F)• Marking: E is marked if (F)²n < for all incident Voronoi

facets• Erosion: proceed in collapsing manner and guided by

MGF


Examples


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 M2Medial 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


Effect of


Shape eccentricity and computing tubular regions

• Eccentricity: e(E)=g(E) / c(E)

Department of Computer Science and Engineering

Thank you!

Defining and Computing Curve-skeletons with Medial Geodesic Function

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