C1 Interpolation with Cumulative Chord Cubics

• Autorzy: Kozera R. , Noakes L.
• Języki publikacji: EN

Abstrakty

EN

Cumulative chord C¹ piecewise-cubics, for unparameterized data from regular curves in Rⁿ, are constructed as follows. In the first step derivatives at given ordered interpolation points are estimated from ordinary (non-C¹) cumulative chord piecewise-cubics. Then Hermite interpolation is used to generate a C¹ piecewise-cubic interpolant. Theoretical estimates of orders of approximation are established, and their sharpness verified through numerical experiments. Good performance of the interpolant is also confirmed experimentally on sparse data.

Słowa kluczowe

EN

interpolation; unparameterized data; cumulative chords; curve and length estimation; orders of approximation

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2004
• Tom: Vol. 61, nr 3,4
• Strony: 285--301
• Opis fizyczny: Bibliogr. 23 poz., wykr.

Twórcy

autor

Kozera R.

• School of Computer Science and Software Engineering, The University of Western Australia, 35 Strling Highway, Crawley W.A. 6009, Perth, Australia

autor

Noakes L.

• School of Computer Science and Software Engineering, The University of Western Australia, 35 Strling Highway, Crawley W.A. 6009, Perth, Australia

Bibliografia

• [1] de Boor, C: A Practical Guide to Spline, Revised edition. Springer-Verlag, New York Heidelberg Berlin, 2001.
• [2] de Boor. C, Höllig, K., Sabin, M.: High accuracy geometric Hermite interpolation, Computer Aided Geom. Design A, 1987,269-278.
• [3] Epstein, M. P.: On the influence of parameterization in parametric interpolation, SIAM J. Numer. Anal., 13, 1976,261-268.
• [4] Farin. G.: Curves and Surfaces for Computer Aided Geometric Design. A Practical Guide. Academic Press. San Diego, 1993.
• [5] Kiciak, P.: Postawy modelowania krzywych i powierzchni, Wydawnictwo Naukowo-Techniczne, Warszawa, 2000, In Polish.
• [6] Klingenberg, W.: A Course in Differential Geometry, Springer-Verlag, Berlin Heidelberg, 1978.
• [7] Kozera, R.: Asymptotics for length and trajectory from cumulative chord piecewise-quartics, Fundamenta Informaticae, 61(3-4), 2004, 267-283.
• [8] Kozera, R.: Cumulative chord piecewise-quartics for length and trajectory estimation, 10th Int. Conf. on Computer Analysis of Images and Patterns, CAIP 2003, Groningen, The Netherlands, August 2003 (N. Petkov. M. A. Westenberg, Eds.), LNCS, vol. 2756 of LNCS, Springer-Verlag, Berlin Heidelberg, 2003, 697-705.
• [9] Kozera, R., Noakes, L., Klette, R.: External versus internal parameterization for lengths of curves with nonuniform samplings, in: Theoretical Foundations of Computer Vision. Geometry and Computational Imaging (C. R. T. Asano, R. Klette, Ed.), vol. 2616 of LNCS, Springer-Verlag, Berlin Heidelberg, 2003, 403-418.
• [10] Kvasov, B. I.: Methods of Shape-Preserving Spline Approximation, World Scientific, Singapore, 2000.
• [11] Lachance, M. A., Schwartz, A. J.: Four point parabolic interpolation, Computer Aided Geom. Design, 8, 1991.143-149.
• [12] Lee, E. T. Y.: Comers, cusps, and parameterization: variations on a theorem of Epstein. SIAM J. Numer. Anal,29, 1992,553-565.
• [13] IVl0rken, K., Scherer, K.: A general framework for high-accuracy parametric interpolation. Math. Computation, 66(217), 1997,237-260.
• [14] Noakes, L., Kozera, R.: Cumulative chords piece wise-quadratics and piecewise-cubics, in: Geometrical Properties from Incomplete Data (R. Klette, R. Kozera, L. Noakes, J. Weickert, Eds.), Kluver Academic Publishing, Submitted.
• [15] Noakes, L., Kozera, R.: Cumulative chords and piecewise-quadratics. Int. Conf on Computer Vision and Graphics, ICCVG 2002, Zakopane, Poland. May 2002 (K. Wojciechowski, Ed.), vol. II, Association for image Processing Poland, Silesian University of Technology Gliwice Poland, Institute of Theoretical and Applied Informatics PAS Gliwice Poland, 2002, 589-595.
• [16] Noakes, L., Kozera, R.: Interpolating sporadic data, 7th Euro. Conf. on Computer Vision. FCCV 2002, Copenhagen, Denmark, May 2002 (A. Heyden, G. Sparr, M. Nielsen, P. Jobansen, Eds.), LNCS, vol. 2351/II of LNCS, Springer-Verlag, Berlin Heidelberg, 2002. 613-625.
• [17] Noakes, L., Kozera, R.: More-or-less uniform sampling and lengths of curves, Quar. Appl Math., 61(3), 2003,475-484.
• [18] Noakes, L., Kozera, R., Klette, R.: Length estimation for curves with different samplings, in: Digital Image Geometry (G. Bertrand, A. Imiya, R. Klette, Eds.), vol. 2243 of LNCS, Springer-Verlag, Berlin Heidelberg, 2001,339-351.
• [19] Noakes, L., Kozera, R., Klette, R.: Length estimation for curves with £-uniform sampling, 9th Int. Conf. on Computer Analysis of Images and Patterns. CAIP200I. Warsaw, Poland, Septemher2001 (W. Skarbek, Ed.), LNCS. vol. 2124 of LNCS, Springer-Verlag, Berlin Heidelberg, 2001, 518-526.
• [20] Piegl, L., Tiller, W.: The NURBS Book, 2nd edition, Springer-Verlag, Berlin Heidelberg, 1997.
• [21] Rababah, A.: High order approximation methods for curves, Computer Aided Geom. Design, 12, 1995, 89-102.
• [22] Ralston, A.: A First Course in Numerical Analysis, McGraw-Hill, 1965.
• [23] Schaback, R.: Optimal geometric Hermite interpolation of curves, in: Mathematical Methods for Curves and Surfaces II (M. Dæhlen, T. Lyche, L. Schumaker, Eds.}. Vanderbilt University Press, 1998, 1-12.