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Locally optimal knots and tension parameters for exponential splines. (English) Zbl 1098.65010
Summary: Tension can be applied to cubic splines in order to avoid undesired spurious oscillations. This leads to the well-known (exponential) spline in tension. It is crucial but unfortunately difficult to find suitable tension parameters of interpolating splines in tension. Instead of heuristics, we propose a simultaneous knot placing and tension setting algorithm for least-squares splines in tension which includes interpolating splines in tension as a special case. Moreover, the splines presented here are the foundation of exponential surface splines on fairly arbitrary meshes [cf. K. O. Riedel, ZAMM Z. Angew. Math. Mech. 85, No. 3, 176–188 (2005; Zbl 1067.65017)].
Reviewer: Reviewer (Berlin)

MSC:
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
65D05 Numerical interpolation
41A15 Spline approximation
Software:
FITPACK; pchip; TSPACK
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References:
[1] Barsky, B.A., Exponential and polynomial methods for applying tension to an interpolating spline curve, Computer vision, graphics, image process., 27, 1-18, (1984) · Zbl 0601.65007
[2] de Boor, C., A practical guide to splines, (1978), Springer Berlin, Heidelberg, New York · Zbl 0406.41003
[3] Costantini, P.; Pelosi, F., Shape-preserving approximation by space curves, Numer. algorithms, 27, 237-264, (2001) · Zbl 1079.65018
[4] Costantini, P.; Pelosi, F., Shape-preserving approximation of spatial data, Adv. comput. math., 20, 25-51, (2004) · Zbl 1045.65010
[5] Delbourgo, R.; Gregory, J.A., \(C^2\) rational quadratic spline interpolation to monotonic data, IMA J. numer. anal., 3, 141-152, (1983) · Zbl 0523.65005
[6] Delbourgo, R.; Gregory, J.A., The determination of the derivative parameters for a monotonic rational quadratic interpolant, IMA J. numer. anal., 3, 141-152, (1983) · Zbl 0523.65005
[7] Dierckx, P., Curve and surface Fitting with splines, (1993), Oxford Science Publications Clarendon Press, Oxford · Zbl 0782.41016
[8] Fritsch, F.N.; Carlson, R.E., Monotone piecewise cubic interpolation, SIAM, J. numer. anal., 17, 238-246, (1980) · Zbl 0423.65011
[9] Gregory, J.A.; Delbourgo, R., Piecewise rational quadratic interpolation to monotonic data, IMA J. numer. anal., 2, 123-130, (1982) · Zbl 0481.65004
[10] Gregory, A.; Sarfraz, M., A rational cubic spline with tension, Comput. aided geom. design, 7, 1-13, (1990) · Zbl 0717.65003
[11] Hayes, J.G., Curve Fitting by polynomials in one variable, (), 43-64
[12] Heidemann, U., Linearer ausgleich mit exponentialsplines bei automatischer bestimmung der intervallteilungspunkte, Computing, 36, 217-227, (1986) · Zbl 0582.65008
[13] Heß, W.; Schmidt, J.W., Convexity preserving interpolation with exponential splines, Computing, 36, 335-342, (1986) · Zbl 0581.41013
[14] D.J. Higham, Monotonic piecewise cubic interpolation, with applications to ODE plotting, TR 229/90, Department of Computer Science, University of Toronto, 1990. · Zbl 0768.65002
[15] Jüttler, B., Shape preserving least-squares approximation by polynomial parametric spline curves, Comput. aided geom. design, 14, 731-747, (1997) · Zbl 0893.68156
[16] P.E. Koch, T. Lyche, Exponential B-splines in tension, in: C.K. Chui, L.L. Schumaker, J.D. Ward (Eds.), Approximation Theory VI, vol. 2, ISBN 0-12-174587-2, Academic Press, London, 1989, pp. 361-364. · Zbl 0754.41006
[17] Koch, P.E.; Lyche, T., Construction of exponential tension B-splines of arbitrary order, (), 255-258 · Zbl 0736.41013
[18] Koch, P.E.; Lyche, T., Interpolation with exponential B-splines in tension, computing supplementum 8, (), 173-190 · Zbl 0855.41004
[19] R.W. Lynch, A method for choosing a tension factor for spline under tension interpolation, M.S. Thesis, University of Texas at Austin, 1982.
[20] Nielson, G.M., Computation of \(\nu\)-splines, TR 044-433-11, department of mathematics, (1974), Arizona State University
[21] G.M. Nielson, Some piecewise polynomial alternatives to splines under tension, in: R.E. Barnhill, R.F. Riesenfeld (Eds.), Computer Aided Geometric Design, 1974, pp. 209-235.
[22] Pruess, S., Properties of splines in tension, J. approx. theory, 17, 86-96, (1976) · Zbl 0327.41009
[23] Renka, R.J., Interpolatory tension splines with automatic selection of tension factors, SIAM J. sci. statist. comput., 8, 3, 393-415, (1987) · Zbl 0629.65009
[24] Renka, R.J., Tspack: tension spline curve-Fitting package, ACM trans. math. software, 19, 81-94, (1993) · Zbl 0889.65007
[25] Rentrop, P., An algorithm for the computation of the exponential spline, Numer. math., 35, 81-93, (1980) · Zbl 0461.65009
[26] K.O. Riedel, Aspects of image processing—splines, anisotropic diffusion, and biological models, Ph.D. Thesis, ISBN 3-8322-1236-1, Shaker Verlag, Aachen, 2003. · Zbl 1101.68931
[27] Riedel, K.O., Two-dimensional splines on fairly arbitrary meshes, ZAMM—Z. angew. math. mech., 85, 3, 176-188, (2005) · Zbl 1067.65017
[28] Sapidis, N.S.; Kaklis, P.D.; Loukakis, T.A., A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation, Numer. math., 54, 179-192, (1988) · Zbl 0636.65010
[29] Sarfraz, M., A geometrical rational spline with tension controls: an alternative to the weighted nu-spline, Punjab univ. J. math., 26, 27-40, (1993) · Zbl 0816.41006
[30] Sarfraz, M., Freeform rational bicubic spline surfaces with tension control, Facta univ. ser. math. inform., 9, 83-93, (1994) · Zbl 0832.65009
[31] Schweikert, D.G., An interpolating curve using a spline in tension, J. math. phys., 45, 312-317, (1966) · Zbl 0146.14102
[32] Späth, H., Exponential spline interpolation, Computing, 4, 225-233, (1968) · Zbl 0184.19803
[33] Stoer, J., Numerische Mathematik 1, (1999), Springer, Berlin Heidelberg, New York
[34] Stoer, J.; Bulirsch, R., Introduction to numerical analysis, (1980), Springer, Berlin Heidelberg, New York · Zbl 0423.65002
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