Optimal C2 two-dimensional interpolatory ternary subdivision schemes with two-ring stencils

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Holder smoothness exponent of its basis function cannot exceed log 3 11(≈ 2.18266), where the critical Holder smoothness exponent of a function f: R 2 R is defined to be v ∞ (f):= sup{v: f ∈ Lipv}. On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Holder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound log 3 11. Consequently, we obtain optimal smoothest C 2 interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the l p -norm joint spectral radius.

[1]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[2]  R. Jia,et al.  Optimal Interpolatory Subdivision Schemes in Multidimensional Spaces , 1998 .

[3]  Bin Han,et al.  Symmetry Property and Construction of Wavelets With a General Dilation Matrix , 2001 .

[4]  S. Dubuc.  Interpolation through an iterative scheme , 1986 .

[5]  D. Levin,et al.  Subdivision schemes in geometric modelling , 2002, Acta Numerica.

[6]  Charles A. Micchelli,et al.  Regularity of multiwavelets , 1997, Adv. Comput. Math..

[7]  B. Han.  Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm , 1998 .

[8]  R. Jia,et al.  Multivariate refinement equations and convergence of subdivision schemes , 1998 .

[9]  Bin Han,et al.  Vector cascade algorithms and refinable function vectors in Sobolev spaces , 2003, J. Approx. Theory.

[10]  Neil A. Dodgson,et al.  An interpolating 4-point C2 ternary stationary subdivision scheme , 2002, Comput. Aided Geom. Des..

[11]  I. Daubechies,et al.  Two-scale difference equations I: existence and global regularity of solutions , 1991 .

[12]  R. Jia.  Subdivision Schemes in L p Spaces , 1995 .

[13]  S. Riemenschneider,et al.  Convergence of Vector Subdivision Schemes in Sobolev Spaces , 2002 .

[14]  Rong-Qing Jia,et al.  Subdivision schemes inLp spaces , 1995, Adv. Comput. Math..

[15]  Luiz Velho,et al.  4-8 Subdivision , 2001, Comput. Aided Geom. Des..

[16]  Jiang,et al.  Square root 3 -Subdivision Schemes: Maximal Sum Rule Orders , 2003 .

[17]  C. Micchelli,et al.  Stationary Subdivision , 1991 .

[18]  Zuowei Shen,et al.  Multidimensional Interpolatory Subdivision Schemes , 1997 .

[19]  Bin Han,et al.  Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function , 2002, SIAM J. Matrix Anal. Appl..

[20]  Bin Han,et al.  Quincunx fundamental refinable functions and quincunx biorthogonal wavelets , 2002, Math. Comput..

[21]  M. F. Hassan,et al.  Towards a ternary interpolating subdivision scheme for the triangular mesh , 2002 .

[22]  Charles T. Loop.  Smooth Ternary Subdivision of Triangle Meshes , 2002 .

[23]  Rong-Qing Jia,et al.  Smoothness of Multiple Refinable Functions and Multiple Wavelets , 1999, SIAM J. Matrix Anal. Appl..

[24]  Ding-Xuan Zhou.  The $p$-norm joint spectral radius for even integers , 1998 .

[25]  Bin Han,et al.  Analysis and Construction of Optimal Multivariate Biorthogonal Wavelets with Compact Support , 1999, SIAM J. Math. Anal..

[26]  Rong-Qing Jia,et al.  Approximation properties of multivariate wavelets , 1998, Math. Comput..

[27]  B. Han.  Projectable multivariate refinable functions and biorthogonal wavelets , 2002 .