1 / 18

Subdivision Surfaces

Subdivision Surfaces. Introduction to Computer Graphics CSE 470/598 Arizona State University. Dianne Hansford. Overview. What are subdivision surfaces in a nutshell ? Advantages Chaiken’s algorithm The curves that started it all Classic methods Doo-Sabin and Catmull-Clark

sanam
Download Presentation

Subdivision Surfaces

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. Subdivision Surfaces Introduction to Computer Graphics CSE 470/598 Arizona State University Dianne Hansford

  2. Overview • What are subdivision surfaces in a nutshell ? • Advantages • Chaiken’s algorithm The curves that started it all • Classic methods Doo-Sabin and Catmull-Clark • Extensions on the concept

  3. What is subdivision? http://www.multires.caltech.edu/teaching/demos/java/chaikin.htm

  4. Advantages • Easy to make complex geometry • Rendering very efficient • Animation tools “easily” developed Pixar’s A Bug’s Life first feature film to use subdivision surfaces. (Toy Story used NURBS.)

  5. Disadvantages • Precision difficult to specify in general • Analysis of smoothness very difficult to determine for a new method • No underlying parametrization Evaluation at a particular point difficult

  6. Chaiken’s Algorithm Chaiken published in ’74An algorithm for high speed curve generation a corner cutting method on each edge: ratios 1:3 and 3:1

  7. Chaiken’s Algorithm Riesenfeld (Utah) ’75Realized Chaiken’s algorithm an evaluation method for quadratic B-spline curves (parametric curves) Theoretical foundation sparked more interest in idea. Subdivision surface schemes Doo-Sabin Catmull-Clark

  8. one-level of subdivision Doo-Sabin Input: polyhedral mesh many levels of subdivision

  9. Doo-Sabin ‘78 Generalization of Chaiken’s ideato biquadratic B-spline surfaces Input: Polyhedral mesh Algorithm: 1) Form points within each face 2) Connect points to form new faces: F-faces, E-faces, V-faces Repeat ... Output: polyhedral mesh; mostly 4-sided faces except some F- & V-faces; valence = 4 everywhere

  10. Doo-Sabin Repeatedly subdivide ... Math analysis will say that a subdivision scheme’s smoothness tends to be the same everywhere but at isolated points. extraordinary points Doo-Sabin: non-four-sided patches become extraordinary points

  11. Catmull-Clark one-level of subdivision Input: polyhedral mesh many-levels of subdivision

  12. Catmull-Clark ‘78 Generalization of Chaiken’s idea to bicubic B-spline surfaces Input: Polyhedral mesh Algorithm: • Form F-vertices: centroid of face’s vertices • Form E-points: combo of edge vertices and F-points • Form V-points: average of edge midpoints • Form new faces (F-E-V-E) Repeat.... Output: mesh with all 4-sided faces but valence not = 4

  13. CC - Extraordinary Valence not = 4 1) Input mesh had valence not = 42) Face with n>4 sides Creates extraordinary vertex (in limit) (Remember: smoothness less there)

  14. Let’s compare D-S C-C

  15. Convex Combos Note: D-S & C-C use convex combinations !(Weighting of each point in [0,1]) Guarantees the following properties: • new points in convex hull of old • local control • affinely invariant (All schemes use barycentric combinations) See references at end for exact equations

  16. Data Structures • Each scheme demands a slightly different structure to be most efficient • Basic structure for mesh must exist plus more info • Schemes tend to have bias – faces, vertices, edges .... as foundation of method • Lots of room for creativity!

  17. Extensions Many schemes have been developed since.... interpolation(butterfly scheme) more control (notice sharp edges) See NYU reference for variety of schemes Pixar: tailored for animation

  18. References • Ken Joy’s class noteshttp://graphics.cs.ucdavis.edu • Gerald Farin & DCHThe Essentials of CAGD, AK Petershttp://eros.cagd.eas.asu.edu/~farin/essbook/essbook.html • Joe Warren & Heinrik Weimer www.subdivision.org • NYU Media Labhttp://www.mrl.nyu.edu/projects/subdivision/ • CGW articlehttp://cgw.pennnet.com/Articles/Article_Display.cfm?Section=Articles&Subsection=Display&ARTICLE_ID=196304

More Related