Subdivision Surfaces

Introduction to Computer Graphics

CSE 470/598Arizona 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• Extensions on the concept

What is subdivision?

Input: polygon or polyhedral mesh

Process: repeatedly refine (subdivide) geometry

Output: “smooth” curve or surface

http://www.multires.caltech.edu/teaching/demos/java/chaikin.htm

Advantages

• Easy to make complex geometry

• Rendering very efficient

• Animation tools “easily” developed

Pixar’s A Bug’s Life first feature film to usesubdivision surfaces.(Toy Story used NURBS.)

Disadvantages

• Precision difficult to specify in general

• Analysis of smoothness very difficult to determine for a new method

• No underlying parametrizationEvaluation at a particular point difficult

Chaiken’s AlgorithmChaiken published in ’74

An algorithm for high speed curve generation

a corner cutting method on each edge: ratios 1:3 and 3:1

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 schemesDoo-SabinCatmull-Clark

Doo-Sabin

Input: polyhedral mesh

one-levelofsubdivision

many levelsofsubdivision

Doo-Sabin ‘78Generalization of Chaiken’s idea to biquadratic B-spline surfaces

Input: Polyhedral meshAlgorithm: 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

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

Catmull-Clark

Input: polyhedral mesh

one-levelofsubdivision

many-levelsof subdivision

Catmull-Clark ‘78

Input: Polyhedral meshAlgorithm:1) Form F-vertices: centroid

of face’s vertices2) Form E-points: combo of

edge vertices and F-points

3) Form V-points: average of edge midpoints

4) Form new faces (F-E-V-E)Repeat....Output: mesh with all 4-sided

faces but valence not = 4

Generalization of Chaiken’s idea to bicubic B-spline surfaces

CC - Extraordinary

Valence not = 41) Input mesh had valence not = 42) Face with n>4 sides

Creates extraordinary vertex (in limit)(Remember: smoothness less there)

Let’s compare D-S

C-C

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

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!

ExtensionsMany schemes have been developed since....

more control (notice sharp edges)

See NYU reference for variety of schemes

interpolation(butterfly scheme)

Pixar: tailored for animation

References• Ken Joy’s class notes

http://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