Download presentation

Presentation is loading. Please wait.

Published byJesse Lucas Modified over 3 years ago

1
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
2/16 Department 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

3
3/16 Department of Computer Science and Engineering 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

4
4/16 Department of Computer Science and Engineering Roadmap

5
5/16 Department of Computer Science and Engineering Medial axis: set of centers of maximal inscribed balls The stratified structure [Giblin-Kimia04]: g enerically, the medial axis of a surface consists of five types of points based on the number of tangential contacts. M 2 : inscribed ball with two contacts, form sheets M 3 : inscribed ball with three contacts, form curves Others: Medial axis

6
6/16 Department of Computer Science and Engineering Medial geodesic function (MGF)

7
7/16 Department of Computer Science and Engineering Properties of MGF Property 1 (proved): f is continuous everywhere and smooth almost everywhere. The singularity of f has measure zero in M 2. Property 2 (observed): There is no local minimum of f in M 2. Property 3 (observed): At each singular point x of f there are more than one shortest geodesic paths between a x and b x.

8
8/16 Department of Computer Science and Engineering Defining curve-skeletons Sk 2 =Sk Å M 2 : the set of singular points of MGF or points with negative divergence w.r.t. r f Sk 3 =Sk Å M 3 : A point of other three types is on the curve-skeleton if it is the limit point of Sk 2 [ Sk 3

9
9/16 Department of Computer Science and Engineering Defining curve-skeletons Sk 2 =Sk Å M 2 : set of singular points of MGF or points with negative divergence w.r.t. r f Sk 3 =Sk Å M 3 : extending the view of divergence A point of other three types is on the curve-skeleton if it is the limit point of Sk 2 [ Sk 3 Sk=Cl(Sk 2 [ Sk 3 )

10
10/16 Department of Computer Science and Engineering Computing curve-skeletons MA approximation [Dey-Zhao03] : subset of Voronoi facets MGF approximation: f(F) and (F) Marking: E is marked if (F) ² n < for all incident Voronoi facets Erosion: proceed in collapsing manner and guided by MGF

11
11/16 Department of Computer Science and Engineering Examples

12
12/16 Department of Computer Science and Engineering 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 M 2 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

13
13/16 Department of Computer Science and Engineering Effect of

14
14/16 Department of Computer Science and Engineering 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

15
15/16 Department of Computer Science and Engineering Shape eccentricity and computing tubular regions Eccentricity: e(E)=g(E) / c(E)

16
16/16 Department of Computer Science and Engineering Timing

17
Department of Computer Science and Engineering Thank you!

Similar presentations

OK

A Minimum Cost Path Search Algorithm Through Tile Obstacles Zhaoyun Xing and Russell Kao Sun Microsystems Laboratories.

A Minimum Cost Path Search Algorithm Through Tile Obstacles Zhaoyun Xing and Russell Kao Sun Microsystems Laboratories.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian automobile industries Detail ppt on filariasis ppt Ppt on national parks of india Ppt on water activity definition Ppt on hydrostatic forces on submerged surfaces in static fluids Ppt on organisational structure Ppt on extinct plants and animals Ppt on new technology free download Ppt on metro rail in hyderabad india Ppt on computer languages of the future