Subdivision Surfaces

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Subdivision Surfaces

• Introduction to Computer Graphics
• CSE 470/598
• Arizona State University

Dianne Hansford
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Overview

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

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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/jav
a/chaikin.htm
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Advantages

• Easy to make complex geometry
• Rendering very efficient
• Animation tools easily developed

Pixars A Bugs Life first feature film to
use subdivision surfaces. (Toy Story used NURBS.)
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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

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Chaikens Algorithm

• Chaiken published in 74An algorithm for high
  speed curve generation

a corner cutting method on each edge
ratios 13 and 31
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Chaikens Algorithm

• Riesenfeld (Utah) 75Realized Chaikens
  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

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Doo-Sabin
one-level of subdivision
Input polyhedral mesh
many levels of subdivision
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Doo-Sabin 78

• Generalization of Chaikens idea to 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
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Doo-Sabin

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

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Catmull-Clark
one-level of subdivision
Input polyhedral mesh
many-levels of subdivision
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Catmull-Clark 78
Generalization of Chaikens idea to bicubic
B-spline surfaces

• Input Polyhedral mesh
• Algorithm
• Form F-vertices centroid of faces 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

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CC - Extraordinary

• Valence not 4
• 1) Input mesh had valence not 42) Face with
  ngt4 sides
• Creates extraordinary vertex (in limit)
• (Remember smoothness less there)

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Lets compare
D-S
C-C
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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
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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!

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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
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References

• Ken Joys class noteshttp//graphics.cs.ucdavis.e
  du
• 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/subd
  ivision/
• CGW articlehttp//cgw.pennnet.com/Articles/Articl
  e_Display.cfm?SectionArticlesSubsectionDisplay
  ARTICLE_ID196304