Weighted bicubic spline interpolation to rapidly varying data

ABSTRACT

The weighted bicubic spline that is a C1 piecewise bicubic interpolant to three-dimensional gridded data is introduced. This is a generalization of the univariate weighted spline, developed by Salkauskas, in that a weighted minimization problem is solved. The minimization problem solved is a weighted version of the problem that the natural bicubic spline and Gordon's spline-blended interpolants minimize. The surface is represented as a piecewise bicubic Hermite interpolant whose derivatives are the solution of a linear system of equations. For computer-aided-design applications, the shape of the surface is controlled by weighting the variation over the individual patches, whereas many other shape-control methods weight the discrete data points. A method for selecting the weights is presented so that the weighted bicubic spline effectively solves the important and often difficult problem of interpolating rapidly varying data.

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INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

General Terms:
Algorithms