Department of Computer Science and Engineering Defining and Computing Curve-skeletons with Medial...
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Transcript of Department of Computer Science and Engineering Defining and Computing Curve-skeletons with Medial...
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
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• 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
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Roadmap
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• 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
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Medial geodesic function (MGF)
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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.
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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
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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 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
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Effect of
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Shape eccentricity and computing tubular regions
• Eccentricity: e(E)=g(E) / c(E)
Department of Computer Science and Engineering
Thank you!