Modified subdivision surfaces with continuous curvature

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Modified subdivision surfaces with continuous
curvature
Adi Levin
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Overview

• Catmull-Clark subdivision
• The C2 problem
• Fixing smoothness by blending
• Implementation issues
• Optimal blending

NEW
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Catmull-Clark subdivision (quads only)
Initial mesh

• Recursive quadrilateral subdivision
• Every new vertex position is defined by a linear
  combination of vertex positions from the previous
  step
• The limit surface is C2 continuous
• Except for extraordinary points

limit surface
step 1
step 2
step 3
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Subdivision surfaces are not C2

• The overall shape is nice

Catmull-Clark surface
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Subdivision surfaces are not C2

• But a close examination shows shape problems near
  the extraordinary points

Shape problem near a point of valence 6
Isophotes
Mean curvature
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Subdivision surfaces can be modified to become C2
Original Catmull-Clark
Phong shading
mean curvature
isophotes
C2 Modified Catmull-Clark
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Fixing smoothness by blending

• Given f(x), which is smooth except at x0.
• We want to modify it in the interval -1,1
• p(x) a polynomial approximation to f(x) in
  -1,1
• w(x)

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Guidelines for C2 blending

• f(x) is C2 except at x0 (and C1 everywhere)
• w(x) should be C2
• w(x) should decay to zero fast enough at x0

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Guidelines for C2 blending

• p should approximate f (but not too closely)
• w and w should be small

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Guidelines for C2 blending
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Blending for surfaces
(The blending region)
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Blending for surfaces

• For subdivision surfaces, we need
• Evaluation of points on the limit surface
• A local parameterization around each
  extraordinary point

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Evaluation of Catmull-Clark surfaces

• Every vertex is mapped to a limit point
• Can be evaluated by local a linear mask

Initial mesh
step 1
step 2
step 3
limit surface
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A smooth local parameterizationThe
characteristic map

• Subdominant eigenvectors of the subdivision
  operator

2D initial data
After a single subdivision step
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A smooth local parameterizationThe
characteristic map

• Subdominant eigenvectors of the subdivision
  operator
• Every vertex is mapped to a pair of (u,v)
  parameters

v
u
Initial mesh
step 1
step 2
step 3
limit
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A smooth local parameterizationThe
characteristic map
Example After 2 steps of subdivision
We ignore neighboring extraordinary vertices
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The blending region
the subdominant eigenvalue of the subdivision
operator
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The blending region

• Blending regions of neighboring extraordinary
  vertices do not overlap

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An algorithm for C2 modification of a
Catmull-Clark surface

• Preprocessing
• At each extraordinary vertex, calculate a local
  polynomial fit p(u,v) with respect to the
  characteristic map
• Evaluation (after k steps of subdivision)
• For a given vertex of the k-times refined mesh,
  evaluate (u,v) from the characteristic map and
  apply the blending formula

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Polynomial fit at an extraordinary vertex

• Subdivide the mesh 3 times and evaluate limit
  positions
• Around each extraordinary vertex,
• Fit a polynomial p to the points where
• Use cubic polynomials

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Polynomial fit at an extraordinary vertex

• Subdivide the mesh 3 times and evaluate limit
  positions
• Around each extraordinary vertex,
• Fit a polynomial p to the points where
• Use cubic polynomials
• Exception for valence 3 vertices use quadratic
  polynomials

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The modified surfaces are C2 and they look better
Original Catmull-Clark
Phong shading
mean curvature
isophotes
C2 Modified Catmull-Clark
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Initial mesh
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(top view)
3
6
Original Catmull-Clark
(Mean curvature)
C2 Modified Catmull-Clark
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3
5
Initial mesh
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Original Catmull-Clark
(Gaussian curvature)
C2 Modified Catmull-Clark
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Initial mesh
(Isophotes)
Original Catmull-Clark
C2 Modified Catmull-Clark
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z1
Initial mesh
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z-1
Original Catmull-Clark
(Isophotes)
(Mean curvature)
C2 Modified Catmull-Clark
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A more efficient way to compute the polynomial fit

• Subdivide the mesh once
• The polynomial at an extraordinary point depends
  only on the (16n) vertices in the 2-ring
• Precompute the 10 (16n) matrix for every
  valence n

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Optimal blend (not in the paper)

• Problem
• p should approximate s, but not too closely
• A better approach
• Calculate p such that the blend between p and s
  is optimal
• Assumes a quadratic measure of the fairness
• Does not require more computations!

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Optimal blend example

• The polynomial at an extraordinary point depends
  only on the (16n) vertices in the 2-ring
• Precompute the (d1)(d2)/2 (16n) matrix for
  every valence n

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Optimal blend more generally

• If
• ltf,ggt is linear in f and in g
• L and R are linear multi-functionals
• ltf,fgt depends on values inside the blending
  region
• ltf,fgt can be evaluated
• Then p depends linearly on the (16n) 2-ring
  vertices after one subdivision step

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Conclusion further work

• After a simple modification, subdivision surfaces
  become C2 continuous.
• Same asymptotic time complexity
• The modified surface looks almost similar to the
  original.
• The method can be applied to any subdivision
  scheme.
• It remains to investigate how to get the best
  surface quality
• Degree of polynomial
• Optimal blending, with a fairness functional
• Different weight functions