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Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches

Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches. Zhangjin Huang, Guoping Wang School of EECS, Peking University, China October 17, 2007. Generalization of uniform bicubic B-spline surface

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Improved Error Estimate for Extraordinary Catmull-Clark Subdivision Surface Patches

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  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) Uniform bicubic B-spline surface initial mesh step 1 limit surface

  3. 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 CCSS patch: regular vs. extraordinary Blue: regular Red : extraordinary Control mesh Limit surface

  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.)

  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.

  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.

  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!

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