Abstract
The problem of fitting sparse reduced data in arbitrary Euclidean space is discussed in this work. In our setting, the unknown interpolation knots are determined upon solving the corresponding optimization task. This paper outlines the non-linearity and non-convexity of the resulting optimization problem and illustrates the latter in examples. Symbolic computation within Mathematica software is used to generate the relevant optimization scheme for estimating the missing interpolation knots. Experiments confirm the theoretical input of this work and enable numerical comparisons (again with the aid of Mathematica) between various schemes used in the optimization step. Modelling and/or fitting reduced sparse data constitutes a common problem in natural sciences (e.g. biology) and engineering (e.g. computer graphics).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bézier, P.E.: Numerical Control: Mathematics and Applications. Wiley, New York (1972)
Boehm, E., Farin, G., Kahmann, J.: A survey of curve and surface methods in CAGD. Comput. Aided Geom. Des. 1(1), 1–60 (1988)
de Boor, C.: A Practical Guide to Spline. Springer, New York (1985)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Budzko, D.A., Prokopenya, A.N.: On the stability of equilibrium positions in the circular restricted four-body problem. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 88–100. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23568-9_8
Epstein, M.P.: On the influence of parameterization in parametric interpolation. SIAM J. Numer. Anal. 13, 261–268 (1976)
Farin, G.: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, San Diego (1993)
Farouki, R.T.: Optimal parameterizations. Comput. Aided Geom. Des. 14(2), 153–168 (1997)
Floater, M.S.: Chordal cubic spline interpolation is fourth order accurate. IMA J. Numer. Anal. 26, 25–33 (2006)
Hoschek, J.: Intrinsic parametrization for approximation. Comput. Aided Geom. Des. 5(1), 27–31 (1988)
Janik, M., Kozera, R., Kozioł, P.: Reduced data for curve modeling - applications in graphics, computer vision and physics. Adv. Sci. Tech. 7(18), 28–35 (2013)
Kocić, L.M., Simoncinelli, A.C., Della, V.B.: Blending parameterization of polynomial and spline interpolants. Facta Universitatis (NIŠ), Ser. Math. Inform. 5, 95–107 (1990)
Kozera, R.: Curve modelling via interpolation based on multidimensional reduced data. Stud. Informatica 25, 1–140 (2004). (4B(61))
Kozera, R., Noakes, L.: Piecewise-quadratics and exponential parameterizations for reduced data. Appl. Maths Comput. 221, 620–638 (2013)
Kozera, R., Noakes, L.: \(C^1\) Interpolation with cumulative chord cubics. Fundamenta Informaticae 61(3–4), 285–301 (2004)
Noakes, L., Kozera, R.: Interpolating sporadic data. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 613–625. Springer, Heidelberg (2002). doi:10.1007/3-540-47967-8_41
Kozera, R., Noakes, L.: Optimal knots selection for sparse reduced data. In: Huang, F., Sugimoto, A. (eds.) PSIVT 2015. LNCS, vol. 9555, pp. 3–14. Springer, Cham (2016). doi:10.1007/978-3-319-30285-0_1
Kozera, R., Noakes, L.: Modeling reduced sparse data. In: Romaniuk, R.S. (ed.) Photonics, Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2016. SPIE 2016, vol. 10031. Society of Photo-Optical Instrumentation Engineers, Bellingham (2016)
Kozera, R., Noakes, L.: Fitting Data via Optimal Interpolation Knots. (Submitted)
Kvasov, B.I.: Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)
Kuznetsov, E.B., Yakimovich, A.Y.: The best parameterization for parametric interpolation. J. Comp. Appl. Maths 191, 239–245 (2006)
Marin, S.P.: An approach to data parameterization in parametric cubic spline interpolation problems. J. Approx. Theory 41, 64–86 (1984)
Mørken, K., Scherer, K.: A general framework for high-accuracy parametric interpolation. Math. Comput. 66(217), 237–260 (1997)
Noakes, L.: A global algorithm for geodesics. J. Math. Austral. Soc. Ser. A 64, 37–50 (1999)
Noakes, L., Kozera, R.: Cumulative chords piecewise-quadratics and piecewise-cubics. In: Klette, R., Kozera, R., Noakes, L., Weickert, J. (eds.) Geometric Properties from Incomplete Data. Computational Imaging and Vision, vol. 31, pp. 59–75. Springer, The Netherlands (2006)
Noakes, L., Kozera, R.: More-or-less uniform sampling and lengths of curves. Quar. Appl. Maths 61(3), 475–484 (2003)
Noakes, L., Kozera, R.: Nonlinearities and noise reduction in 3-source photometric stereo. J. Math. Imag. Vis. 18(3), 119–127 (2003)
Noakes, L., Kozera, R.: 2D leap-frog algorithm for optimal surface reconstruction. In: Latecki, M.J. (ed.) SPIE 1999. Vision Geometry VIII, vol. 3811, pp. 317–328. Society of Industrial and Applied Mathematics, Bellingham (1999)
Lee, E.T.Y.: Corners, cusps, and parameterization: variations on a theorem of Epstein. SIAM J. Numer. Anal. 29, 553–565 (1992)
Lee, E.T.Y.: Choosing nodes in parametric curve interpolation. Comput. Aided Geom. Des. 21, 363–370 (1989)
Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin (1997)
Prokopenya, A.N.: Hamiltonian normalization in the restricted many-body problem by computer algebra methods. Program. Comput. Softw. 38(3), 156–166 (2012)
Rababah, A.: High order approximation methods for curves. Comput. Aided Geom. Des. 12, 89–102 (1995)
Schaback, R.: Optimal geometric Hermite interpolation of curves. In: Dæhlen, M., Lyche, T., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 1–12. Vanderbilt University Press, Nashville (1998)
Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Champaign (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kozera, R., Noakes, L. (2017). Non-linearity and Non-convexity in Optimal Knots Selection for Sparse Reduced Data. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-66320-3_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66319-7
Online ISBN: 978-3-319-66320-3
eBook Packages: Computer ScienceComputer Science (R0)