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Defining and Computing Curve-skeletons with Medial Geodesic Function Tamal K. Dey and Jian Sun The Ohio State University Motivation 1D representation of 3D shapes, called curve-skeleton, useful in many application Geometric modeling, computer vision, data analysis, etc

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### Defining and Computing Curve-skeletons with Medial Geodesic Function

Tamal K. Dey and Jian Sun

The Ohio State University

Motivation Function

• 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

Contributions Function

• 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

Medial axis Function

• 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:

Properties of MGF Function

• 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 Function

• 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 Function

• 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 Function

Properties of curve-skeletons Function

• Thin (1D curve)

• Centered

• Homotopy equivalent

• Junction detective

• Stable

Prop1: set of singular points of MGF is of measure zero in M2

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

Effect of Function

• Eccentricity: e(E)=g(E) / c(E)