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BIT Numerical Mathematics

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01.09.2015

Shape preserving \(HC^2\) interpolatory subdivision

verfasst von: Davide Lettieri, Carla Manni, Francesca Pelosi, Hendrik Speleers

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2015

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Abstract

A subdivision procedure is developed to solve a \(C^2\) Hermite interpolation problem with the further request of preserving the shape of the initial data. We consider a specific non-stationary and non-uniform variant of the Merrien \(HC^2\) subdivision family, and we provide a data dependent strategy to select the related parameters which ensures convergence and shape preservation for any set of initial monotone and/or convex data. Each step of the proposed subdivision procedure can be regarded as the midpoint evaluation of an interpolating function—and of its first and second derivatives—in a suitable space of \(C^2\) functions of dimension \(6\) which has tension properties. The limit function of the subdivision procedure is a \(C^2\) piecewise quintic polynomial interpolant.

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The monotone decreasing case can be addressed in a similar way.

The concave case can be addressed in a similar way.

The Maple worksheets with all the computations can be found at http://www.mat.uniroma2.it/~manni/Maple_HC2.html.

Akima, H.: A new method of interpolation and smooth curve fitting based on local procedures. J. ACM 17, 589–602 (1970)MATHCrossRef

Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Am. Math. Soc. 93(453), (1991)

Costantini, P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des. 17, 419–446 (2000)MATHMathSciNetCrossRef

Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)MATHMathSciNetCrossRef

Costantini, P., Kaklis, P., Manni, C.: Polynomial cubic splines with tension properties. Comput. Aided Geom. Des. 27, 592–610 (2010)MATHMathSciNetCrossRef

Costantini, P., Manni, C.: On constrained nonlinear Hermite subdivision. Constr. Approx. 28, 291–331 (2008)MATHMathSciNetCrossRef

Cravero, I., Manni, C.: Shape-preserving interpolants with high smoothness. J. Comput. Appl. Math. 157, 383–405 (2003)MATHMathSciNetCrossRef

Daubechies, I., Guskov, I., Sweldens, W.: Regularity of irregular subdivision. Constr. Approx. 15, 381–426 (1999)MATHMathSciNetCrossRef

Delbourgo, R., Gregory, J.A.: Shape preserving piecewise rational interpolation. SIAM J. Sci. Stat. Comput. 6, 967–976 (1985)MATHMathSciNetCrossRef

10.

Dubuc, S.: Scalar and Hermite subdivision schemes. Appl. Comput. Harm. Anal. 21, 376–394 (2006)MATHMathSciNetCrossRef

11.

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

12.

Goodman, T.: Shape preserving interpolation by curves. In: Levesley, J., Anderson, I.J., Mason, J.C. (eds.) Algorithms for Approximation IV, pp. 24–35. University of Huddersfield Proceedings, Huddersfield (2002)

13.

Goodman, T., Mazure, M.L.: Blossoming beyond extended Chebyshev spaces. J. Approx. Theory 109, 48–81 (2001)MATHMathSciNetCrossRef

14.

Goodman, T.N.T.: Total positivity and the shape of curves. In: Total positivity and its applications (Jaca, 1994), Math. Appl., vol. 359, pp. 157–186. Kluwer Acad. Publ., Dordrecht (1996)

15.

Guglielmi, N., Manni, C., Vitale, D.: Convergence analysis of \(C^2\) Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas. Linear Algebra Appl. 434, 884–902 (2011)MATHMathSciNetCrossRef

16.

Koch, P.E., Lyche, T.: Interpolation with exponential B-splines in tension. In: Geometric Modelling, Comput. Suppl., vol. 8, pp. 173–190. Springer, Vienna (1993)

17.

Kvasov, B.I.: Algorithms for shape preserving local approximation with automatic selection of tension parameters. Comput. Aided Geom. Des. 17, 17–37 (2000)MathSciNetCrossRef

18.

Lettieri, D., Manni, C., Speleers, H.: Piecewise rational quintic shape-preserving interpolation with high smoothness. Jaen J. Approx. 6, 233–260 (2014)

19.

Lyche, T., Merrien, J.L.: Hermite subdivision with shape constraints on a rectangular mesh. BIT Numer. Math. 46, 831–859 (2006)MATHMathSciNetCrossRef

20.

Lyche, T., Merrien, J.L.: \(C^1\) interpolatory subdivision with shape constraints for curves. SIAM J. Numer. Anal. 44, 1095–1121 (2006)MATHMathSciNetCrossRef

21.

Manni, C.: On shape preserving \(C^2\) Hermite interpolation. BIT 41, 127–148 (2001)MATHMathSciNetCrossRef

22.

Manni, C., Mazure, M.L.: Shape constraints and optimal bases for \(C^1\) Hermite interpolatory subdivision schemes. SIAM J. Numer. Anal. 48, 1254–1280 (2010)MATHMathSciNetCrossRef

23.

Mazure, M.L.: Chebyshev–Bernstein bases. Comput. Aided Geom. Des. 16, 649–669 (1999)MATHMathSciNetCrossRef

24.

Merrien, J.L.: A family of Hermite interpolants by bisection algorithms. Numer. Algor. 2, 187–200 (1992)MATHMathSciNetCrossRef

25.

Merrien, J.L.: Interpolants d’Hermite \(C^2\) obtenus par subdivision. M2AN. Math. Model. Numer. Anal. 33, 55–65 (1999)MATHMathSciNetCrossRef

26.

Merrien, J.L., Sablonnière, P.: Monotone and convex \(C^1\) Hermite interpolants generated by a subdivision scheme. Constr. Approx. 19, 279–298 (2003)MATHMathSciNetCrossRef

27.

Merrien, J.L., Sablonnière, P.: Rational splines for Hermite interpolation with shape constraints. Comput. Aided Geom. Des. 30, 296–309 (2013)MATHCrossRef

28.

Pelosi, F., Sablonnière, P.: Shape-preserving \(C^1\) Hermite interpolants generated by a Gori–Pitolli subdivision scheme. J. Comput. Appl. Math. 220, 686–711 (2008)MATHMathSciNetCrossRef

29.

Peña, J.: Shape Preserving Representations in Computer Aided Geometric Design. Nova Science, Massachusetts (1999)

30.

Sablonnière, P.: Bernstein-type bases and corner cutting algorithms for \(C^1\) Merrien’s curves. Adv. Comput. Math. 20, 229–246 (2004)MATHMathSciNetCrossRef

31.

Schweikert, D.G.: An interpolation curve using a spline in tension. J. Math. Phys. 45, 312–317 (1966)MATHMathSciNet

Titel: Shape preserving interpolatory subdivision
verfasst von: Davide Lettieri
Carla Manni
Francesca Pelosi
Hendrik Speleers
Publikationsdatum: 01.09.2015
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 3/2015
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-014-0530-0

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