1 / 14

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 1D representation of 3D shapes, called curve-skeleton, useful in many application Geometric modeling, computer vision, data analysis, etc

bernad
Download Presentation

Defining and Computing Curve-skeletons with Medial Geodesic Function

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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!

More Related