Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface PatchesZhangjin Huang, Guoping Wang

School of EECS, Peking University, ChinaOctober 17, 2007

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

Catmull-Clark subdivision surface (CCSS)

initial mesh step 1 limit surface

2C

Uniform bicubic B-spline surface

CCSS patch: regular vs. extraordinary

F

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, 2n+8 control points

Control mesh Limit surface

Blue: regularRed : extraordinary

1F

2F1S

2S

SF

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

S F

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 : Stam’s 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.

( , )u vS

S

( , )L u v

S

F

F

( , )max ( , ) ( , )u v

u v L u v

S

[0,1] [0,1]

( , )max ( , ) ( , ) ( )u v

u v L u v C n M

S

SM( )C n 1(4)

3C

( )C n

Second order norm of an extraordinary CCSS patch Second order norm : the maximum norm of 2n+10 second

order differences of the 2n+8 level-0 control points of an extraordinary CCSS patch [Cheng et al. 2006]: S

: the second order norm of the level-k control points of Recurrence formula:

: the k-step convergence rate of second order norm

0M

0( )kM r n Mk( )r nk

SkM

Error estimation for extraordinary patches

Stam’s parameterization: Partition an extraordinary patch into an infinite sequence of uniform bicubic B-spline patches Partition the unit square into tiles

{ }, 1, 1,2,3km k m S

S

{ }, 1, 1,2,3km k m

( , ) ( , )km

kmu v u v

S S

12S

22S

11S

21S

23S

13S

Error estimation for extraordinary patches (cont.)

For ,

We have

( , ) kmu v

( , ) ( , ) ( , ) ( , )

1( , ) ( , ) ( , ) ( , )0

2 1( , ) ( , )0 00

kS u v L u v S u v L u vm

k k k kS u v L u v L u v L u vm m mk i iL u v L u vi

1( , ) ( , )3

11 1( , ) ( , )40

11( , ) ( , )min{ ,8}

k k kS u v L u v Mm mk k kL u v L u v Mmi i iL u v L u v M

n

(1)

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

21 1 1 11 0 0( , ) ( , ) ( ) ( ) , k3 4 min{ ,8} min{ ,8}0 0

kk kS u v L u v M M r n M r n Mn ni ii i

S

S

M

max ( , ) ( , ) ( )( , )

S u v L u v C n Mu v

1( ) ( )

min{ ,8} 0C n r nin i

( ), i>1r ni

max ( , ) ( , ) ( ) , 1( , )

S u v L u v C n Mu v

11( ) ( )min{ ,8}(1 ( )) 0

iC n r nn r n i

( ), i=0,...,ir n

F

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, 1 1(3) 2 / 3, (5) 18/ 25r r

2

21

2

23 , 2 12

3 2 16( ) , 4 4

12 , 4 2

n kn

r n n knnn k

n

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 12 160.66667 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

1( )r n

2 ( )r n

1( )r n

Comparison of bound constants

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.

n 3 5 6 7 8 9 10 12 161.00000 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

( ), 1,2C n

1( )C n

2 ( )C n

( )C n

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

0

1/ ( )

( ) ( )log , 0 1, 1j

j r n

r n C n Ml j

min0 1

k l jjj

SS

Comparison of subdivision depths

The second order norm is assumed to be 2 Our approach has a 20% improvement over the

matrix based method if n is even.

3 5 6 7 8 9 10 12 160.01 9 11 13 14 13 16 22 28 36Old 9 11 16 14 18 16 17 16 170.001 12 16 19 22 19 24 24 24 25Old 12 16 22 22 26 24 32 40 50

Application: CCSS intersection

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 ( )C n

Thank you!