Defining and Computing Curve-skeletons with Medial Geodesic Function

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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
• 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
• 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
• 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
• 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
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 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