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Numerical Algorithms

, Volume 9, Issue 2, pp 263–276 | Cite as

Nonnegative surface fitting with Powell-Sabin splines

  • Karin Willemans
  • Paul Dierckx
Article

Abstract

Algorithms are presented for fitting a nonnegative Powell-Sabin spline to a set of scattered data. Existing necessary and sufficient nonnegativity conditions for a quadratic polynomial on a triangle are used to compose a set of necessary and sufficient nonnegativity constraints for the PS-spline. The PS-spline is expressed as a linear combination of locally supported basis functions, of which the Bernstein-Bézier representation is considered to improve the efficiency. Numerical examples illustrate the profit of nonnegative surface fitting with Powell-Sabin splines.

Keywords

Conforming triangulations Bézier ordinates Powell-Sabin splines nonnegativity 

AMS(MOS) subject classification

41A15 41A29 65D07 

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References

  1. [1]
    G. Chang and T.W. Sederberg, Nonnegative quadratic Bézier triangular patches, Comp. Aided Geom. Design 11 (1994) 113–116.CrossRefGoogle Scholar
  2. [2]
    P. Dierckx,Curve and Surface Fitting with Splines, Monographs on Numerical Analysis (Clarendon Press, Oxford, 1993).Google Scholar
  3. [3]
    P. Dierckx, S. Van Leemput and T. Vermeire, Algorithms for surface fitting using Powell-Sabin splines, IMA J. Numer. Anal. 12 (1992) 271–299.Google Scholar
  4. [4]
    G. Farin, Triangular Bernstein-Bézier patches, Comp. Aided Geom. Design 3 (1986) 83–127.CrossRefGoogle Scholar
  5. [5]
    R.H.J. Gmelig-Meyling and P.R. Pfluger, On the dimension of the spline spaceS 21(Δ) in special cases, in:Multivariate Approximation Theory III, eds. W. Schempp and K. Zeller (Birkhäuser, Basel, 1985) pp. 180–190.Google Scholar
  6. [6]
    C.A. Micchelli and A. Pinkus, Some remarks on nonnegative polynomials on polyhedra, in:Probability, Statistics, and Mathematics, eds. T. Anderson, K. Alhreya and D. Iglehart (Academic Press, New York, 1989) pp. 163–185.Google Scholar
  7. [7]
    B. Mulansky and J.W. Schmidt, NonnegativeC 1 interpolation of scattered data using Powell-Sabin splines, Technical report, Technische Universität Dresden (March 1993).Google Scholar
  8. [8]
    E. Nadler, Nonnegativity of bivariate quadratic functions on a triangle, Comp. Aided Geom. Design 9 (1992) 195–205.CrossRefGoogle Scholar
  9. [9]
    The NAG Fortran Library Manual-Mark 15 (The Numerical Algorithms group, Oxford, 1991).Google Scholar
  10. [10]
    C.C. Paige and M.A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8 (1982) 43–71.CrossRefGoogle Scholar
  11. [11]
    C.C. Paige and M.A. Saunders, Algorithm 583 LSQR: Sparse linear equations and least squares problems, ACM Trans. Math. Software 8 (1982) 195–209.CrossRefGoogle Scholar
  12. [12]
    M.J.D. Powell and M.A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977) 316–325.CrossRefGoogle Scholar
  13. [13]
    J.W. Schmidt, Positive, monotone, and S-convexC 1-interpolation on rectangular grids, Computing 48 (1992) 363–371.Google Scholar
  14. [14]
    L.L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, in:Multivariate Approximation Theory, eds. W. Schempp and K. Zeller (Birkhäuser, Basel, 1979). pp. 396–412.Google Scholar
  15. [15]
    L.L. Schumaker, Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mt. J. Math. 14 (1984) 251–264.Google Scholar
  16. [16]
    L.L. Schumaker, Numerical aspects of spaces of piecewise polynomials on triangulations, in:Algorithms for Approximation, eds. J.C. Mason and M.G. Cox (Clarendon Press, Oxford, 1987) pp. 373–406.Google Scholar
  17. [17]
    K. Willemans and P. Dierckx, Surface fitting with boundary conditions by means of Powell-Sabin splines, Technical report, K.U. Leuven, Dept. Comp. Science, Report TW 199 (October 1993).Google Scholar
  18. [18]
    K. Willemans and P. Dierckx, Surface fitting using convex Powell-Sabin splines, J. Comp. Appl. Math. (1994).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Karin Willemans
    • 1
  • Paul Dierckx
    • 1
  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenLeuvenBelgium

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