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Numerical Algorithms

, Volume 5, Issue 11, pp 577–587 | Cite as

Scattered data interpolation using minimum energy Powell-Sabin elements and data dependent triangulations

  • Shmuel Rippa
Geometric Modelling

Abstract

A popular approach for obtaining surfaces interpolating to scattered data is to define the interpolant in a piecewise manner over a triangulation with vertices at the data points. In most cases, the interpolant cannot be uniquely determined from the prescribed function values since it belongs to a space of functions of dimension greater than the number of data points. Thus, additional parameters are needed to define an interpolant and have to be estimated somehow from the available data. It is intuitively clear that the quality of approximation by the interpolant depends on the choice of the triangulation and on the method used to provide the additional parameters. In this paper we suggest basing the selection of the triangulation and the computation of the additional parameters on the idea of minimizing a given cost functional measuring the quality of the interpolant. We present a scheme that iteratively updates the triangulation and computes values of the additional parameters so that the quality of the interpolant, as measured by the cost functional, improves from iteration to iteration. This method is discussed and tested numerically using an energy functional and Powell-Sabin twelve split interpolants.

Keywords

Minimum Energy Additional Parameter Popular Approach Scattered Data Data Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    P. Alfeld, Derivative generation from multivariate scattered data by functional minimization, Comp. Aided Gem. Des. 2 (1985) 281–296.Google Scholar
  2. [2]
    R.E. Barnhill, Representation and approximation of surfaces, in:Mathematical Software III, ed. J.R. Rice (Academic Press, New York, 1977) pp. 68–119.Google Scholar
  3. [3]
    G. Baszenski and L.L. Schumaker, Use of simulated annealing to construct triangular facet surfaces, in:Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, 1991) pp. 27–32.Google Scholar
  4. [4]
    A.K. Cline and R.J. Renka, A triangle-basedC 1 interpolation method, Rocky Mt. J. Math. 14 (1984) 223–237.Google Scholar
  5. [5]
    N. Dyn, D. Levin and S. Rippa, Data dependent triangulations for piecewise linear interpolation, IMA J. Numer. Anal. 10 (1990) 137–154.Google Scholar
  6. [6]
    N. Dyn, D. Levin and S. Rippa, Algorithms for the construction of data dependent triangulations,Algorithms for Approximation II, eds. J.C. Mason and M.G. Cox (Chapman and Hall, London, 1990) pp. 185–192.Google Scholar
  7. [7]
    N. Dyn, I. Goren and S. Rippa, Transforming triangulations in polygonal domains, Report DAMTPNA/13(1991).Google Scholar
  8. [8]
    G. Farin, Triangular Bernstein-Bézier patches, Comp. Aided Geom. Des. 3 (1986) 83–127.Google Scholar
  9. [9]
    R. Franke, Scattered data interpolation: tests of some methods, Math. Comp. 38 (1982) 181–200.Google Scholar
  10. [10]
    C.L. Lawson, Transforming triangulations, Discr. Math. 3 (1972) 365–372.Google Scholar
  11. [11]
    C.L. Lawson, Software forC 1 interpolation, in:Mathematical Software III, ed. J.R. Rice (Academic Press, New York, 1977) pp. 161–194.Google Scholar
  12. [12]
    G.M. Nielson and R. Franke, Surface construction based upon triangulations, in:Surfaces in Computer Aided Design, eds. R.E. Barnhill and W. Boehm (North-Holland, Amsterdam, 1983) pp. 163–177.Google Scholar
  13. [13]
    M.J.D. Powell and M.A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977) 316–325.Google Scholar
  14. [14]
    E. Quak and L.L. Schumaker, Cubic spline fitting using data dependent triangulations, Comp. Aided Geom. Des. 7 (1990) 293–301.Google Scholar
  15. [15]
    S. Rippa, Long, thin triangles can be good for linear interpolation, SIAM J. Numer. Anal. 29 (1992) 257–270.Google Scholar
  16. [16]
    L.L. Schumaker, Fitting surfaces to scattered data, in:Approximation Theory II, eds. G.G. Lorents, C.K. Chui and L.L. Schumaker (Academic Press, 1976) pp. 203–268.Google Scholar
  17. [17]
    L.L. Schumaker, Numerical aspects of spaces of piecewise polynomials on triangulations, in:Algorithm for Approximation, eds. J.C. Mason and M.G. Cox (Clarendon Press, Oxford, 1987) pp. 373–406.Google Scholar
  18. [18]
    S.E. Stead, Estimation of gradients from scattered data, Rocky Mt. J. Math. 14 (1984) 265–279.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Shmuel Rippa
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsCambridgeEngland

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