defining and computing curve skeletons with medial geodesic function
Download
Skip this Video
Download Presentation
Defining and Computing Curve-skeletons with Medial Geodesic Function

Loading in 2 Seconds...

play fullscreen
1 / 14

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


  • 400 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Defining and Computing Curve-skeletons with Medial Geodesic Function' - bernad


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
defining and computing curve skeletons with medial geodesic function

Defining and Computing Curve-skeletons with Medial Geodesic Function

Tamal K. Dey and Jian Sun

The Ohio State University

motivation
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
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
  • 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
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
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
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
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

ad