Summary
We describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of “basis functions” which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ℤ2), the thin plate spline, and the multiquadric case give comparably equal and good results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
deBoor, C., Höllig, K., Riemenschneider, S.D. (1985): Bivariate cardinal interpolation by splines on a three direction mesh. Illinois J. Math.29, 533–566
Buhmann, M.D., Powell, M.J.D. (1988): Radial basis function interpolation on an infinite regular grid. In: Cox, M., Mason, J., eds., Algorithms for Approximation II. Clarendon Press, Oxford, pp. 146–169
Chui, C.K. (1988): Multivariate Splines, CBMS-NSF Reg. Conf. Series in Applied Math., Vol. 54. SIAM, Philadelphia
Chui, C.K., Stöckler, J., Ward, J.D. (1991): Invertibility of shifted box spline interpolation operators. SIAM J. Math. Anal.22, 543–553
Chui, C.K., Stöckler, J., Ward, J.D.: A Faber series approach to cardinal interpolation. CAT Report #208, Texas A & M University. Math. Comput. (to appear)
Dyn, N. (1989): Interpolation and approximation by radial and related functions. In: Chui, C.K., Schumaker, L.L., Ward, J.D., eds., Approximation Theory VI. Academic Press, New York, pp. 211–234
Franke, R. (1982): Scattered data interpolation: Test of some methods. Math. Comput.38, 181–200
Höllig, K. (1986): Box splines. In: Chui, C.K., Schumaker, L.L., Ward, J.D., eds., Approximation Theory V. Academic Press, New York, pp. 71–95
Jetter, K. (1987): A short survey on cardinal interpolation by box splines. In: Chui, C.K., Schumaker, L.L., Utreras, F.I., eds., Topics in Multivariate Approximation. Academic Press, New York, pp. 125–139
Jetter, K., Koch, P. (1989): Methoden der Fourier-Transformation bei der kardinalen Interpolation periodischer Daten. In: Chui, C.K., Schempp, W., Zeller, K., eds., Multivariate Approximation IV, pp. 201–208. Birkhäuser, Basel, pp. 201–208
Jetter, K., Riemenschneider, S.D. (1987): Cardinal interpolation, submodules, and the 4-direction mesh. Constr. Approx.3, 169–188
Jetter, K., Riemenschneider, S.D., Sivakumar, N. (1991): On Schoenberg's exponential Euler spline curves. Proc. Roy. Soc. Edinburgh Sect.118A, 21–33
Madych, W. (1989): Cardinal interpolation with polyharmonic splines. In: Chui, C.K., Schempp, W., Zeller, K., eds., Multivariate Approximation IV. Birkhäuser, Basel, pp. 241–248
ter Morsche, H. (1987): Attenuation factors and multivariate periodic spline interpolation. In: Chui, C.K., Schumaker, L.L., Utreras, F.I., eds., Topics in Multivariate Approximation. Academic Press, New York, pp. 165–174
Rabut, C. (1990): B-splines polyharmoniques cardinales: interpolation, quasi-interpolation, filtrage. Thesis, Université Paul Sabatier de Toulouse
Riemenschneider, S.D. (1989): Multivariate cardinal interpolation. In: Chui, C.K., Schumaker, L.L., Ward, J.D., eds., Approximation Theory IV. Academic Press, New York, pp. 561–580
Sivakumar, N. (1990): On bivariate cardinal interpolation by shifted splines on a three-direction mesh. J. Approx. Theory61, 178–193
Stöckler, J. (1988): Interpolation mit mehrdimensionalen Bernoulli-Splines und periodischen Box-Splines. Thesis, University of Duisburg
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jetter, K., Stöckler, J. Algorithms for cardinal interpolation using box splines and radial basis functions. Numer. Math. 60, 97–114 (1991). https://doi.org/10.1007/BF01385716
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01385716