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Scattered Data Modelling Using Radial Basis Functions

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Tutorials on Multiresolution in Geometric Modelling

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Radial basis functions provide powerful meshless methods for multivariate interpolation from scattered data in arbitrary space dimension. This tutorial first explains the basic features of radial basis function interpolation, including its optimality properties and available error estimates, before critical aspects concerning its stability are discussed. The remainder of this contribution is then devoted to related techniques in scattered data modelling other than plain interpolation. To this end, least squares approximation and multiresolution techniques are explained, and recent progress concerning scattered data filtering is reported.

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References

  1. K. Ball. Eigenvalues of Euclidean distance matrices. J. Approx. Theory 68, 1992, 74–82.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Ball, N. Sivakumar, and J. D. Ward. On the sensitivity of radial basis interpolation to minimal distance separation. J. Approx. Theory 8, 1992, 401–426.

    MathSciNet  MATH  Google Scholar 

  3. Å. Bjørck and G. H. Golub. Iterative refinement of linear least squares solutions by Householder transformation. BIT 7, 1967, 322–337.

    Article  Google Scholar 

  4. M. D. Buhmann. Radial basis functions. Acta Numerica, 2000, 1–38.

    Google Scholar 

  5. Y. D. Burago and V. A. Zalgaller. Geometric Inequalities. Springer, Berlin, 1988.

    MATH  Google Scholar 

  6. J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer, New York, 1993.

    Book  MATH  Google Scholar 

  7. J. Duchon. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. R.A.I.R.O. Analyse Numériques 10, 1976, 5–12.

    MathSciNet  Google Scholar 

  8. J. Duchon. Splines minimizing rotation-invariant semi-nor ms in Sobolev spaces. Constructive Theory of Functions of Several Variables, W. Schempp and K. Zeller (eds.), Springer, Berlin, 1977, 85–100.

    Chapter  Google Scholar 

  9. J. Duchon. Sur l’erreur d’interpolation des fonctions de plusieurs variables par les D m-splines. R.A.I.R.O. Analyse Numériques 12, 1978, 325–334.

    MathSciNet  MATH  Google Scholar 

  10. N. Dyn. Interpolation of scattered data by radial functions. Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. I. Utreras (eds.), Academic Press, New York, 1987, 47–61.

    Google Scholar 

  11. N. Dyn. Interpolation and approximation by radial and related functions. Approximation Theory VI, Vol. 1, C. K. Chui, L. L. Schumaker, and J. D. Ward (eds.), Academic Press, New York, 1989, 211–234.

    Google Scholar 

  12. M. S. Floater and A. Iske. Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math. 73, 1996, 65–78.

    Article  MathSciNet  MATH  Google Scholar 

  13. M.S. Floater and A. Iske. Thinning algorithms for scattered data interpolation. BIT 38, 1998, 705–720.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Gotsman, S. Gumhold, and L. Kobbelt. Simplification and Compression of 3D Meshes. This volume.

    Google Scholar 

  15. K. Guo, S. Hu, and X. Sun. Conditionally positive definite functions and Laplace-Stieltjes integrals. J. Approx. Theory 74, 1993, 249–265.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. L. Hardy. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1971, 1905–1915.

    Article  Google Scholar 

  17. D.S. Hochbaum. Approximation Algorithms for NP-hard Problems. PWS Publishing Company, Boston, 1997.

    Google Scholar 

  18. A. Iske. Charakterisierung bedingt positiv deûniter Funktionen für multivariate Interpolationsmethoden mit radialen Basisfunktionen. Dissertation, Universität Göttingen, 1994.

    Google Scholar 

  19. A. Iske. Characterization of function spaces associated with conditionally positive definite functions. Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche, and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, 1995, 265–270.

    Google Scholar 

  20. A. Iske. Reconstruction of functions from generalized Hermite-Birkhoff data. Approximation Theory VIII, Vol. 1: Approximation and Interpolation, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, 257–264.

    Google Scholar 

  21. A. Iske. Hierarchical scattered data filtering for multilevel interpolation schemes. Mathematical Methods for Curves and Surfaces: Oslo 2000, T. Lyche and L. L. Schumaker (eds), Vanderbilt University Press, Nashville, 2001, 211–220.

    Google Scholar 

  22. A. Iske. Progressive scattered data filtering. Preprint, Technische Universität München, 2002.

    Google Scholar 

  23. A. Iske. Optimal distribution of centers for radial basis function methods. Preprint, Technische Universität München, 2000.

    Google Scholar 

  24. H. Jung. Über die kleinste Kugel, die eine räumliche Figur einschließt. J. Reine Angew. Math. 123, 1901, 241–257.

    MATH  Google Scholar 

  25. C. L. Lawson and R. J. Hanson. Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs, N.J., 1974.

    MATH  Google Scholar 

  26. W. R. Madych and S. A. Nelson. Multivariate interpolation: a variational theory. Manuscript, 1983.

    Google Scholar 

  27. W. R. Madych and S. A. Nelson. Multivariate interpolation and conditionally positive definite functions. Approx. Theory Appl. 4, 1988, 77–89.

    MathSciNet  MATH  Google Scholar 

  28. W. R. Madych and S. A. Nelson. Multivariate interpolation and conditionally positive definite functions II. Math. Comp. 54, 1990, 211–230.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Meinguet. Multivariate interpolation at arbitrary points made simple. Z. Angew. Math. Phys. 30, 1979, 292–304.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Meinguet. An intrinsic approach to multivariate spline interpolation at arbitrary points. Polynomial and Spline Approximations, N. B. Sahney (ed.), Reidel, Dordrecht, 1979, 163–190.

    Google Scholar 

  31. J. Meinguet. Surface spline interpolation: basic theory and computational aspects. Approximation Theory and Spline Functions, S. P. Singh, J. H. Bury, and B. Watson (eds.), Reidel, Dordrecht, 1984, 127–142.

    Chapter  Google Scholar 

  32. C. A. Micchelli. Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 1986, 11–22.

    Article  MathSciNet  MATH  Google Scholar 

  33. C A. Micchelli, T. J. Rivlin, and S. Winograd. Optimal recovery of smooth functions. Numer. Math. 260, 1976, 191–200.

    Article  MathSciNet  Google Scholar 

  34. F. J. Narcowich and J. D. Ward. Norms of inverses and conditions numbers for matrices associated with scattered data. J. Approx. Theory 64, 1991, 69–94.

    Article  MathSciNet  MATH  Google Scholar 

  35. F. J. Narcowich and J. D. Ward. Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory 69, 1992, 84–109.

    Article  MathSciNet  MATH  Google Scholar 

  36. M. J. D. Powell. The theory of radial basis function approximation in 1990. Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Basis Functions, W. A. Light (ed.), Clarendon Press, Oxford, 1992, 105–210.

    Google Scholar 

  37. F. P. Preparata and M. I. Shamos Computational Geometry, 2nd edition. Springer, New York, 1988.

    Google Scholar 

  38. R. A. Rankin. The closest packing of spherical caps in n dimensions. Proc. Glasgow Math. Assoc. 2, 1955, 139–144.

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Schaback. Creating surfaces from scattered data using radial basis functions. Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche, and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, 1995, 477–496.

    Google Scholar 

  40. R. Schaback. Error estimates and condition numbers for radial basis function interpolation. Advances in Comp. Math. 3, 1995, 251–264.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Schaback. Multivariate interpolation and approximation by translates of a basis function. Approximation Theory VIII, Vol. 1: Approximation and Interpolation, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, 491–514.

    Google Scholar 

  42. R. Schaback. Stability of radial basis function interpolants. Preprint, Universität Göttingen, 2001.

    Google Scholar 

  43. R. Schaback and H. Wendland. Inverse and saturation theorems for radial basis function interpolation. Math. Comp. 71, 2002, 669–681.

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Schaback and H. Wendland. Characterization and construction of radial basis functions. Multivariate Approximation and Applications, N. Dyn, D. Leviatan, D. Levin, and A. Pinkus (eds.), Cambridge University Press, Cambridge, 2001, 1–24.

    Chapter  Google Scholar 

  45. I. J. Schoenberg. Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 1938, 522–536.

    Article  MathSciNet  Google Scholar 

  46. I. J. Schoenberg. Metric spaces and completely monotone functions. Math. Ann. 39, 1938, 811–841.

    Article  MathSciNet  Google Scholar 

  47. H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Comp. Math. 4, 1995, 389–396.

    Article  MathSciNet  MATH  Google Scholar 

  48. Z. Wu. Multivariate compactly supported positive definite radial functions. Advances in Comp. Math. 4, 1995, 283–292.

    Article  MATH  Google Scholar 

  49. Z. Wu and R. Schaback. Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 1993, 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  50. F. Zeilfelder. Scattered data fitting with bivariate splines. This volume.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Munich, Germany

    Armin Iske

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  1. Armin Iske
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Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, 80290, München, Germany

    Armin Iske

  2. SINTEF Applied Mathematics, Postboks 124, Blindern, 0314, Oslo, Norway

    Ewald Quak  & Michael S. Floater  & 

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Iske, A. (2002). Scattered Data Modelling Using Radial Basis Functions. In: Iske, A., Quak, E., Floater, M.S. (eds) Tutorials on Multiresolution in Geometric Modelling. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04388-2_9

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  • DOI: https://doi.org/10.1007/978-3-662-04388-2_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07819-4

  • Online ISBN: 978-3-662-04388-2

  • eBook Packages: Springer Book Archive

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