Abstract
Smooth cumulative chord piecewise-cubics, for unparameterised 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 regular (geometrically smooth) piecewise-cubic interpolant. Sharpness of theoretical estimates of orders of approximation for length and trajectory is verified by numerical experiments. Good performance of the interpolant is also confirmed experimentally on sparse data. This may be applicable in computer graphics and vision, image segmentation, medical image processing, and in computer aided geometrical design.
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REFERENCES
Blake, A. and Isard, M. (1998). Active Contours. Springer-Verlag, Berlin Heidelberg New York.
de Boor, C. (2001). A Practical Guide to Splines. Springer-Verlag, New York Berlin Heidelberg.
de Boor, C., Höllig, K., and Sabin, M. (1987). High accuracy geometric Hermite interpolation. Computer Aided Geom. Design, 4:269–278.
Desbleds-Mansard, C., Anwander, A., Chaabane, L., Orkisz, M., Neyran, B., Douek, P. C., and Magnin, I. E. (2001). Dynamic active contour model for size independent blood vessel lumen segmentation and quantification in high-resolution magnetic resonance images. In Skarbek, W., editor, Proc. 9th Int. Conf. Computer Anal. of Images and Patterns, Warsaw Poland, volume 2124 of Lect. Notes Comp. Sc., pages 264–273, Berlin Heidelberg. Springer-Verlag.
Epstein, M. P. (1976). On the influence of parameterization in parametric interpolation. SIAM J. Numer. Anal., 13:261–268.
Kass, M., Witkin, A., and Terzopoulos, D. (1988). Active contour models. Int. J. Comp. Vision, 1:321–331.
Kozera, R. (2003). Cumulative chord piecewise-quartics for length and trajectory estimation. In Petkov, N. and Westenberg, M. A., editors, Proc. 10th Int. Conf. Computer Anal. of Images and Patterns, Groningen The Netherlands, volume 2756 of Lect. Notes Comp. Sc., pages 697–705, Berlin Heidelberg. Springer-Verlag.
Kozera, R. (2004). Asymptotics for length and trajectory from cumulative chord piecewisequartics. Fundamenta Informaticae, 61(3-4):267–283.
Kozera, R. and Noakes, L. (2004). C1 interpolation with cumulative chord cubics. Fundamenta Informaticae, 61(3-4):285–301.
Kozera, R., Noakes, L., and Klette, R. (2003). External versus internal parameterization for lengths of curves with nonuniform samplings. In Asano, T., Klette, R., and Ronse, C., editors, Theoret. Found. Comp. Vision, Geometry Computat. Imaging, volume 2616 of Lect. Notes Comp. Sc., pages 403–418, Berlin Heidelberg. Springer-Verlag.
Kvasov, B. I. (2000). Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore.
Lachance, M. A. and Schwartz, A. J. (1991). Four point parabolic interpolation. Computer Aided Geom. Design, 8:143–149.
Lee, E. T. Y. (1992). Corners, cusps and parameterization: variations on a theorem of Epstein. SIAM J. of Numer. Anal., 29:553–565.
Mørken, K. and Scherer, K. (1997). A general framework for high-accuracy parametric interpolation. Math. Computat., 66(217):237–260.
Noakes, L. and Kozera, R. (2002). Cumulative chords and piecewise-quadratics. In Wojciechowski, K., editor, Proc. Int. Conf. Computer Vision and Graphics, Zakopane Poland, volume II, pages 589–595. Association for Image Processing Poland, Silesian University of Technology Gliwice Poland, Institute of Theoretical and Applied Informatics PAS Gliwice Poland.
Noakes, L. and Kozera, R. (2003). More-or-less uniform sampling and lengths of curves. Quar. Appl. Math., 61(3):475–484.
Noakes, L. and Kozera, R. (2004). Cumulative chord piecewise-quadratics and piecewise-cubics. In Klette, R., Kozera, R., Noakes, L., and J., Weickert, editors, Geometric Properties from Incomplete Data. Kluwer Academic Publishers. In press.
Noakes, L., Kozera, R., and Klette, R. (2001a). Length estimation for curves with different samplings. In Bertrand, G., Imiya, A., and Klette, R., editors, Digit. Image Geometry, volume 2243 of Lect. Notes Comp. Sc., pages 339–351, Berlin Heidelberg. Springer-Verlag.
Noakes, L., Kozera, R., and Klette, R. (2001b). Length estimation for curves with ε-uniform samplings. In Skarbek, W., editor, Proc. 9th Int. Conf. Computer Anal. of Images and Patterns, Warsaw Poland, volume 2124 of Lect. Notes Comp. Sc., pages 339–351, Berlin Heidelberg. Springer-Verlag.
Piegl, L. and Tiller, W. (1997). The NURBS Book. Springer-Verlag, Berlin Heidelberg.
Rababah, A. (1995). High order approximation methods for curves. Computer Aided Geom. Design, 12:89–102.
Schaback, R. (1989). Interpolation in ℝ2 by piecewise quadratic visually C2 Bézier polynomials. Computer Aided Geom. Design, 6:219–233.
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Kozera, R., Noakes, L. (2006). SMOOTH INTERPOLATION WITH CUMULATIVE CHORD CUBICS. In: Wojciechowski, K., Smolka, B., Palus, H., Kozera, R., Skarbek, W., Noakes, L. (eds) Computer Vision and Graphics. Computational Imaging and Vision, vol 32. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4179-9_14
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DOI: https://doi.org/10.1007/1-4020-4179-9_14
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