Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches

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Improved Error Estimate for Extraordinary
Catmull-Clark Subdivision Surface Patches

• Zhangjin Huang, Guoping Wang
• School of EECS, Peking University, China
• October 17, 2007

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Catmull-Clark subdivision surface (CCSS)

• Generalization of uniform bicubic B-spline
  surface
• continuous except at extraordinary points,
  whose valences are not 4
• The limit of a sequence of recursively refined
  control meshes

Uniform bicubic B-spline surface
initial mesh
step 1
limit surface
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CCSS patch regular vs. extraordinary
Blue regular Red extraordinary
Control mesh
Limit surface

• Assume each mesh face in the control mesh
• a quadrilateral
• at most one extraordinary point
• An interior mesh face ? a surface patch
• Regular patch bicubic B-spline patch, 16 control
  points
• Extraordinary patch not B-spline patch, 2n8
  control points

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Piecewise linear approximation and error
estimation

• Control mesh is often used as a piecewise linear
  approximation to a CCSS

• How to estimate the error (distance) between a
  CCSS and its control mesh?
• Wang et al. measured the maximal distance between
  the control points and their limit positions
• Cheng et al. devised a more rigorous way to
  measure the distance between a CCSS patch and
  its mesh face
• We improve Cheng et al.s estimate for
  extraordinary CCSS patches

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Distance between a CCSS patch and its control mesh

• Distance between a CCSS patch and its mesh
  face (or control mesh) is defined as
• a unit square
• Stams parameterization of over
• bilinear parameterization of over
• Cheng et al. bounded the distance by
• the second order norm of
• a constant that depends on valence n,
• We derive a more precise if n is even.

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Second order norm of an extraordinary CCSS patch

• Second order norm the maximum norm of
  2n10 second order differences of the 2n8
  level-0 control points of an extraordinary CCSS
  patch Cheng et al. 2006

• the second order norm of the level-k
  control points of
• Recurrence formula
• the k-step convergence rate of second
  order norm

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Error estimation for extraordinary patches

• Stams parameterization
• Partition an extraordinary patch into an
  infinite sequence of uniform bicubic B-spline
  patches
• Partition the unit square into tiles

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Error estimation for extraordinary patches (cont.)

• For ,
• We have

(1)
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Distance bounds for extraordinary CCSS patches

• It follows that
• Theorem 1. The distance between an extraordinary
  CCSS patch and the corresponding mesh face
  is bounded by
• , is the
  second order norm of
• There are no explicit expression for
  , we have the following practical bound for error
  estimation
• ,
  are the convergence rates of
  second order norm

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Convergence rates of second order norm

• By solving constrained minimization problems, we
  can get the optimal estimates for the convergence
  rates of second order norm.
• One-step convergence rate,

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Comparsion of the convergence rates

• If n is odd, our estimates equal the results of
  the matrix based method derived by Cheng et al.
• If n is even, our technique gives better
  estimates
• Cheng et al.s method gives wrong estimates if n
  is even and greater than 6. ( should be
  less than 1.)

n 3 5 6 7 8 9 10 12 16
0.66667 0.72000 0.75000 0.80102 0.75000 0.83025 0.83000 0.80556 0.81250
Old 0.66667 0.72000 0.88889 0.80102 1.00781 0.83025 1.05500 1.22917 1.33398
0.29167 0.40163 0.46875 0.51212 0.48438 0.55157 0.55975 0.54919 0.56146
Old 0.29167 0.40163 0.50984 0.51212 0.56909 0.55157 0.62138 0.68765 0.73257
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Comparison of bound constants
n 3 5 6 7 8 9 10 12 16
1.00000 0.71429 0.66667 0.71795 0.50000 0.73636 0.73529 0.64286 0.66667
Old 1.00000 0.71429 0.70588 0.71795 0.69565 0.73636 0.75758 0.76596 0.73563
0.78431 0.57489 0.54902 0.52736 0.42424 0.51018 0.51959 0.50064 0.51663
Old 0.78431 0.57489 0.64226 0.52736 0.58244 0.51018 0.67844 0.89208 1.09095

• If n is even, our bound is sharper than the
  result derived by the matrix based method.
• should decreases as increases. If n is
  quite large such as 12 and 16, the matrix based
  method may give improper estimates.

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Application subdivision depth estimation

• Theorem 2. Given an error tolerance , after
• steps of subdivision on the control mesh of a
  patch , the distance between and its
  level-k control mesh is smaller than . Here

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Comparison of subdivision depths
3 5 6 7 8 9 10 12 16
0.01 9 11 13 14 13 16 22 28 36
Old 9 11 16 14 18 16 17 16 17
0.001 12 16 19 22 19 24 24 24 25
Old 12 16 22 22 26 24 32 40 50

• The second order norm is assumed to be 2
• Our approach has a 20 improvement over the
  matrix based method if n is even.

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Application CCSS intersection
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Conclusion

• By solving constrained minimization problems, the
  optimal convergence rates of second order norm
  are derived.
• An improved error estimate for an extraordinary
  CCSS patch is obtained if the valence is even.
• More precise subdivision depths can be obtained.
• Open problems
• Whether is there an explicit expression for the
  multi-step convergence rate
• Whether can we determine the value of

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Thank you!