Abstract
This paper presents new univariate linear non-uniform interpolatory subdivision constructions that yield high smoothness, C 3 and C 4, and are based on least-degree spline interpolants. This approach is motivated by evidence, partly presented here, that constructions based on high-degree local interpolants fail to yield satisfactory shape, especially for sparse, non-uniform samples. While this improves on earlier schemes, a broad consideration of alternatives yields two technically simpler constructions that result in comparable shape and smoothness: careful pre-processing of sparse, non-uniform samples and interlaced fitting with splines of increasing smoothness. We briefly compare these solutions to recent non-linear interpolatory subdivision schemes.
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References
Ursula, H., Augsdörfer, N.A.: Dodgson, and Malcolm A. Sabin. Variations on the four-point subdivision scheme. Computer Aided Geometric Design 27(1), 78–95 (2010)
Beccari, C., Casciola, G., Romani, L.: Polynomial-based non-uniform interpolatory subdivision with features control. Journal of Computational and Applied Mathematics 16(235), 4754–4769 (2011)
Beccari, C.V., Casciola, G., Romani, L.: Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines. BIT Numerical Mathematics 51(4), 781–808 (2011)
Beccari, C.V., Casciola, G., Romani, L.: Construction and characterization of non-uniform local interpolating polynomial splines. J. Computational Applied Mathematics 240 (2013)
Dyn, N., Floater, M.S., Hormann, K.: A C 2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Daehlen, M., Morken, K., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces, Tromsoe, pp. 145–156 (2004)
Daubechies, I., Guskov, I., Sweldens, W.: Regularity of irregular subdivision. Constr. Approx. 15(3), 381–426 (1999)
Dyn, N., Hormann, K.: Geometric conditions for tangent continuity of interpolatory planar subdivision curves. Computer Aided Geometric Design 29(6), 332–347 (2012)
Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numerica 11, 73–144 (2002)
Dyn, N., Levin, D., Gregory, J.: A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4(4), 257–268 (1988)
Dyn, N.: Subdivision schemes in computer-aided geometric design. In: Light, W. (ed.) Advances in Numerical Analysis II, pp. 36–104. Oxford University Press (1992)
Dyn, N.: Interpolatory subdivision schemes. In: Iske, A., Quak, E., Floater, M.S. (eds.) Tutorials on Multiresolurion in Geometric Modelling, pp. 25–50. Springer, Heidelberg (2000)
Floater, M., Beccari, C.V., Cashman, T., Romani, L.: A smoothness criterion for monotonicity-preserving subdivision. Advances in Computational Mathematics 240 (2013)
Hormann, K.: Private communication (October 2012)
Ko, K.P., Lee, B.-G., Yoon, G.J.: A study on the mask of interpolatory symmetric subdivision schemes. Applied Mathematics and Computation 187(2), 609–621 (2007)
Karčiauskas, K., Peters, J.: Curvature-sensitive splines. Presentation at: 8th Itl. Conference on Mathematical Methods for Curves and Surfaces, Oslo, Norway (2012)
Karčiauskas, K., Peters, J.: Curvature-sensitive splines and design with basic curves. Computer-Aided Design (45), 415–423 (2013)
Karčiauskas, K., Peters, J.: Non-uniform interpolatory subdivision via splines. Journal of Computational and Applied Mathematics, MATA 2012 Issue 240, 31–41 (2013)
Lang, S.: Complex Analysis, 2nd edn. Springer, New York (1985)
Lee, E.: Choosing nodes in parametric curve interpolation. Computer Aided Design 21(6) (1989); Presented at the SIAM Applied Geometry Meeting, Albany, N.Y. (1987)
Sabin, M.: Analysis and Design of Univariate Subdivision Schemes. Geometry and Computing, vol. 6. Springer, New York (2010)
Sabin, M.A., Dodgson, N.A.: A circle-preserving variant of the four-point subdivision scheme. In: Mathematical Methods for Curves and Surfaces: Tromsø 2004, Modern Methods in Mathematics (2005)
Warren, J.: Binary subdivision schemes for functions over irregular knot sequences. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Proceedings of the First Conference on Mathematical Methods for Curves and Surfaces (MMCS 1994), Nashville, USA, June 16-21, pp. 543–562. Vanderbilt University Press (1995)
Weissman, A.: A 6-point interpolatory subdivision scheme for curve design. PhD thesis, Tel Aviv University (1990)
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Karčiauskas, K., Peters, J. (2014). Non-uniform Interpolatory Subdivision Based on Local Interpolants of Minimal Degree. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_15
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DOI: https://doi.org/10.1007/978-3-642-54382-1_15
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