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Defining and Computing Curve-skeletons with Medial Geodesic Function PowerPoint PPT Presentation

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

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Defining and computing curve skeletons with medial geodesic function l.jpg

Defining and Computing Curve-skeletons with Medial Geodesic Function

Tamal K. Dey and Jian Sun

The Ohio State University


Motivation l.jpg

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


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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


Roadmap l.jpg

Roadmap


Medial axis l.jpg

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:


Medial geodesic function mgf l.jpg

Medial geodesic function (MGF)


Properties of mgf l.jpg

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 l.jpg

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)


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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 l.jpg

Examples


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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


Effect of l.jpg

Effect of 


Shape eccentricity and computing tubular regions l.jpg

Shape eccentricity and computing tubular regions

  • Eccentricity: e(E)=g(E) / c(E)


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Thank you!


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