Defining and Computing Curveskeletons 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

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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

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Roadmap
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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

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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

• Sk2SkÅM2 the set of singular points of MGF or
  points with negative divergence w.r.t. rf
• Sk3SkÅM3
• A point of other three types is on the
  curve-skeleton if it is the limit point of Sk2
  Sk3

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Defining curve-skeletons

• Sk2SkÅM2 set of singular points of MGF or
  points with negative divergence w.r.t. rf
• Sk3SkÅM3 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
• SkCl(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 lt ? 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 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
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Effect of ?
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Shape eccentricity and computing tubular regions

• Eccentricity e(E)g(E) / c(E)
• Compute tubular regions
• classify skeleton edges and mesh faces based on a
  given threshold
• depth first search

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Shape eccentricity and computing tubular regions

• Eccentricity e(E)g(E) / c(E)

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Timing
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Thank you!