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On a family of non-oscillatory subdivision schemes having regularity Cr with r > 1

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Abstract

In this paper, the properties of a new family of nonlinear dyadic subdivision schemes are presented and studied depending on the conditions imposed to the mean used to rewrite the linear scheme upon which the new scheme is based. The convergence, stability, and order of approximation of the schemes of the family are analyzed in general. Also, the elimination of the Gibbs oscillations close to discontinuities is proved in particular cases. It is proved that these schemes converge towards limit functions that are Hölder continuous with exponent larger than 1. The results are illustrated with several examples.

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Notes

  1. \(L^{\infty }\)stability of the limit function: Let S be a linear uniformly convergent subdivision scheme and let ϕ be its limit function defined by \(\phi =S^{\infty } \delta \)with \(\delta _{n}=0 \quad \forall n \in \mathbb {N}\backslash \left \{0\right \}\)and δ0 = 1. The limit function ϕ is said to be \(L^{\infty }\)stable if:

    $$ \exists A,B>0 \text{ s.t. } \forall f \in l^{\infty}(\mathbb{Z}), A||f||_{\infty} \leq ||{\sum}_{n \in \mathbb{Z}} f_{n}\phi(.-n)||_{L^{\infty}} \leq B ||f||_{\infty}, $$

    where \( ||f||_{\infty }=sup_{n\in \mathbb {Z}}\left \{ |f_{n}|\right \}\).

References

  1. Amat, S., Dadourian, K., Liandrat, J.: Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. Adv. Comput. Math. 34(3), 253–277 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Amat, S., Dadourian, K., Liandrat, J.: On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards Cs functions with s > 1. Math. Comp. 80, 959–971 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Amat, S., Donat, R., Liandrat, J., Trillo, J.C.: Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing, Found. Comput. Math. 6(2), 193–225 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Amat, S., Liandrat, J.: On the stability of the PPH nonlinear multiresolution. Appl. Comp. Harm. Anal. 18(2), 198–206 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Amat, S., Liandrat, J., Ruiz, J., Trillo, J.C.: On a nonlinear mean and its application to image compression using multiresolution schemes. Numer. Algor. 71, 729–752 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Amat, S., Ruiz, J., Trillo, J.C. , Yáñez, D.F.: On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon. J. Comp. App. Math. 354, 310–325 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Chaikin, G.: An algorithm for high speed curve generation. Comput. Gr. Image Process. 3, 346–349 (1974)

    Google Scholar 

  8. Catmull, E., Clark, J.H.: Recursively generated B-spline surfaces on topological meshes. Comput. Aided Design 19(453), 350–355 (1978)

    Google Scholar 

  9. Cohen, A., Dyn, N., Matei, B.: Quasilinear subdivision schemes with applications to ENO interpolation. Appl. Comp. Harm. Anal. 15, 89–116 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Amer. Math. Soc. 93(453), 89–116 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Daubechies, I., Runborg, O., Sweldens, W.: Normal multiresolution approximation of curves. Const. Approx. 20(3), 399–363 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Donat, R., López-Ureña, S., Santágueda, M.: A family of non-oscillatory 6-point interpolatory subdivision schemes. Adv. Comput. Math. 43(4), 849–883 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Donoho, D., Yu, T.P.-Y.: Nonlinear pyramid transforms based on median interpolation. SIAM J. Math. Anal. 31(5), 1030–1061 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Dyn, N.: Subdivision schemes in computer aided geometric design, Advances in Numerical Analysis II., Subdivision algorithms and radial functions. In: Light, W.A. (ed.) , pp 36–104. Prentice-Hall, Oxford University Press (1992)

  16. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numerica 11, 73–144 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Dyn, N., Hormann, K., Sabin, M.A., Shen, Z.: Polynomial reproduction by symmetric subdivision schemes. J. Appr. Theory 155, 28–42 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Dyn, N., Floater, M.S., Hormann, K.: A C2 four-point subdivision scheme with fourth order accuracy and its extensions. Methods for Curves and Surfaces: Tromsø 2004, 145-156, Nashboro Press, Editors M. Dæhlen and K. Mørken and L. L. Schumaker, Series: Modern Methods in Mathematics (2005)

  19. Dyn, N., Gregory, J., Levin, D.: A four-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design 4, 257–268 (1987)

    MathSciNet  MATH  Google Scholar 

  20. Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114, 185–204 (1986)

    MathSciNet  MATH  Google Scholar 

  21. Floater, M.S., Michelli, C.A.: Nonlinear stationary subdivision, Approximation theory: in memory of A.K. Varna, edt: Govil N.K, Mohapatra N., Nashed Z., Sharma A., Szabados J., 209–224 (1998)

  22. Guessab, A., Moncayo, M., Gerhard, S.: A class of nonlinear four-point subdivision schemes. Adv. Comput. Math. 37(2), 151–190 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Harizanov, S., Oswald, P.: Stability of nonlinear subdivision and multiscale transforms. Constr. Approx. 31(3), 359–393 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Hormann, K., Sabin, M.A.: A family of subdivision schemes with cubic precision. Comput. Aided Geom. Design 25(1), 41–52 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Kuijt, F.: Convexity preserving interpolation: Nonlinear subdivision and splines. University of Twente, PhD thesis (1998)

    MATH  Google Scholar 

  27. Marquina, A., Serna, S.: Power ENO methods: A fifth-order accurate Weighted Power ENO method. J. Comput. Phys. 194(2), 632–658 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Oswald, P.: Smoothness of nonlinear median-interpolation subdivision. Adv. Comput. Math. 20(4), 401–423 (2004)

    MathSciNet  MATH  Google Scholar 

  29. Pitolli, F.: Ternary shape-preserving subdivision schemes. Math. Comput. Simulat.. 106, 185–194 (2014)

    MathSciNet  Google Scholar 

  30. Schaefer, S., Vouga, E., Goldman, R.: Nonlinear subdivision through nonlinear averaging. Comput. Aided Geom. Design 25(3), 162–180 (2008)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank the referees for their useful suggestions and comments that, with no doubt, have helped to improve the quality of this paper.

Funding

Sergio Amat, Juan Ruiz and Juan C. Trillo have been supported through the Programa de apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and through the national research project MTM2015-64382-P (MINECO/FEDER). Dionisio F. Yáñez has been supported through the Programa de apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714, through the national research project 431 MTM2017-83942-P (MINECO/FEDER).

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Sergio Amat, Juan Ruiz & Juan C. Trillo

  2. Deptartamento de Matemáticas, Universidad de Valencia (EG), Valencia, Spain

    Dionisio F. Yáñez

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  1. Sergio Amat
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  2. Juan Ruiz
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  4. Dionisio F. Yáñez
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Correspondence to Juan Ruiz.

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Amat, S., Ruiz, J., Trillo, J.C. et al. On a family of non-oscillatory subdivision schemes having regularity Cr with r > 1. Numer Algor 85, 543–569 (2020). https://doi.org/10.1007/s11075-019-00826-3

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