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

Han, Bin; Jia, Rong-Qing

doi:10.1090/S0025-5718-06-01821-7

Optimal $C^2$ two-dimensional interpolatory ternary subdivision schemes with two-ring stencils
HTML articles powered by AMS MathViewer

by Bin Han and Rong-Qing Jia PDF

Math. Comp. 75 (2006), 1287-1308 Request permission

Abstract:

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed $\log _3 11 (\approx 2.18266)$, where the critical Hölder smoothness exponent of a function $f : \mathbb {R}^2\mapsto \mathbb {R}$ is defined to be \[ \nu _\infty (f) \coloneq \sup \{ \nu : f\in \operatorname {Lip} \nu \}. \] 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 Hölder 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 $\ell _p$-norm joint spectral radius.

References

Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
Di-Rong Chen, Rong-Qing Jia, and S. D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002), no. 1, 128–149. MR 1874918, DOI 10.1006/acha.2001.0363
Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), no. 5, 1388–1410. MR 1112515, DOI 10.1137/0522089
N. A. Dodgson, M. A. Sabin, L. Barthe, M. F. Hassan, Towards a ternary interpolatory subdivision scheme for the triangular mesh. preprint.
Serge Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986), no. 1, 185–204. MR 829123, DOI 10.1016/0022-247X(86)90077-6
Nira Dyn and David Levin, Subdivision schemes in geometric modelling, Acta Numer. 11 (2002), 73–144. MR 2008967, DOI 10.1017/S0962492902000028
N. Dyn, D. Levin and J. A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9 (1990), 160–169.
T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR 1244611
Bin Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM J. Math. Anal. 31 (1999/00), no. 2, 274–304. MR 1740939, DOI 10.1137/S0036141098336418
Bin Han, Construction of multivariate biorthogonal wavelets by CBC algorithm, Wavelet analysis and multiresolution methods (Urbana-Champaign, IL, 1999) Lecture Notes in Pure and Appl. Math., vol. 212, Dekker, New York, 2000, pp. 105–143. MR 1777991
Bin Han, Projectable multivariate refinable functions and biorthogonal wavelets, Appl. Comput. Harmon. Anal. 13 (2002), no. 1, 89–102. MR 1930178, DOI 10.1016/S1063-5203(02)00007-6
Bin Han, Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra Appl. 353 (2002), 207–225. MR 1919638, DOI 10.1016/S0024-3795(02)00307-5
Bin Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24 (2003), no. 3, 693–714. MR 1972675, DOI 10.1137/S0895479801390868
Bin Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003), no. 1, 44–88. MR 2010780, DOI 10.1016/S0021-9045(03)00120-5
Bin Han, Classification and construction of bivariate subdivision schemes, Curve and surface fitting (Saint-Malo, 2002) Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003, pp. 187–197. MR 2042446
Bin Han and Rong-Qing Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), no. 5, 1177–1199. MR 1618691, DOI 10.1137/S0036141097294032
Bin Han and Rong-Qing Jia, Optimal interpolatory subdivision schemes in multidimensional spaces, SIAM J. Numer. Anal. 36 (1999), no. 1, 105–124. MR 1654587, DOI 10.1137/S0036142997325611
Bin Han and Rong-Qing Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Math. Comp. 71 (2002), no. 237, 165–196. MR 1862994, DOI 10.1090/S0025-5718-00-01300-4
M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson, and M. A. Sabin, An interpolating $4$-point $C^2$ ternary stationary subdivision scheme, Comput. Aided Geom. Design 19 (2002), no. 1, 1–18. MR 1879678, DOI 10.1016/S0167-8396(01)00084-X
Rong Qing Jia, Subdivision schemes in $L_p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309–341. MR 1339166, DOI 10.1007/BF03028366
Rong-Qing Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998), no. 222, 647–665. MR 1451324, DOI 10.1090/S0025-5718-98-00925-9
Rong-Qing Jia, Sherman D. Riemenschneider, and Ding-Xuan Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 1–28. MR 1709723, DOI 10.1137/S089547989732383X
Qingtang T. Jiang, Peter Oswald, and Sherman D. Riemenschneider, $\sqrt {3}$-subdivision schemes: maximal sum rule orders, Constr. Approx. 19 (2003), no. 3, 437–463. MR 1979060, DOI 10.1007/s00365-002-0521-2
Charles Loop, Smooth ternary subdivision of triangle meshes, Curve and surface fitting (Saint-Malo, 2002) Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003, pp. 295–302. MR 2042456
C. A. Micchelli and Thomas Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997), no. 4, 455–545. MR 1470295, DOI 10.1023/A:1018971524949
Sherman D. Riemenschneider and Zuowei Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Anal. 34 (1997), no. 6, 2357–2381. MR 1480385, DOI 10.1137/S0036142995294310
Luiz Velho and Denis Zorin, 4-8 subdivision, Comput. Aided Geom. Design 18 (2001), no. 5, 397–427. Subdivision algorithms (Schloss Dagstuhl, 2000). MR 1841458, DOI 10.1016/S0167-8396(01)00039-5
Ding-Xuan Zhou, The $p$-norm joint spectral radius for even integers, Methods Appl. Anal. 5 (1998), no. 1, 39–54. MR 1631335, DOI 10.4310/MAA.1998.v5.n1.a2