Title: Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches
1Improved Error Estimate for Extraordinary
Catmull-Clark Subdivision Surface Patches
- Zhangjin Huang, Guoping Wang
- School of EECS, Peking University, China
- October 17, 2007
2Catmull-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
3CCSS 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
4Piecewise 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
5Distance 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.
6Second 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
7Error 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
8Error estimation for extraordinary patches (cont.)
(1)
9Distance 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
10Convergence 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,
11Comparsion 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
12Comparison 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.
13Application 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
14Comparison 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.
15Application CCSS intersection
16Conclusion
- 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
17Thank you!