1. Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches Zhangjin Huang, Guoping Wang School of EECS, Peking University, China October 17, 2007

2.  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 meshstep 1limit surface Uniform bicubic B-spline surface

3. CCSS patch: regular vs. extraordinary  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 meshLimit surface Blue: regular Red : extraordinary

4. 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

5. 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.

6. 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]:  : the second order norm of the level-k control points of  Recurrence formula: : the k-step convergence rate of second order norm

7. 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

8. Error estimation for extraordinary patches (cont.)  For,  We have  (1)

9. 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

10. 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,

11. 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 356789101216 0.666670.720000.750000.801020.750000.830250.830000.805560.81250 Old 0.666670.720000.888890.801021.007810.830251.055001.229171.33398 0.291670.401630.468750.512120.484380.551570.559750.549190.56146 Old 0.291670.401630.509840.512120.569090.551570.621380.687650.73257

12. 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 356789101216 1.000000.714290.666670.717950.500000.736360.735290.642860.66667 Old 1.000000.714290.705880.717950.695650.736360.757580.765960.73563 0.784310.574890.549020.527360.424240.510180.519590.500640.51663 Old 0.784310.574890.642260.527360.582440.510180.678440.892081.09095

13. 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

14. 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. 356789101216 0.01 91113141316222836 Old91116141816171617 0.001 121619221924 25 Old121622 2624324050

15. Application: CCSS intersection

16. 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

17. Thank you!