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Defining and Computing Curve-Skeletons with Medial Geodesic Function

This study presents a mathematical definition of curve-skeletons for 3D objects enclosed by connected compact surfaces. Curve-skeletons provide a 1D representation of 3D shapes, useful in geometric modeling, computer vision, and data analysis. The proposed method preserves desirable properties such as topology and stability. We introduce an approximation algorithm to effectively extract these curve-skeletons and delve into the properties of the medial geodesic function (MGF), demonstrating its continuity and smoothness across the structure, alongside its significance in extracting geometrical insights.

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Defining and Computing Curve-Skeletons with Medial Geodesic Function

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  1. Defining and Computing Curve-skeletons with Medial Geodesic Function Tamal K. Dey and Jian Sun The Ohio State University

  2. 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

  3. 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

  4. Roadmap

  5. 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:

  6. Medial geodesic function (MGF)

  7. 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.

  8. 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)

  9. 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

  10. Examples

  11. 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

  12. Effect of 

  13. Shape eccentricity and computing tubular regions • Eccentricity: e(E)=g(E) / c(E)

  14. Thank you!

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