# Least squares surface approximation to scattered data using multiquadratic functions

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## Abstract

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.

## Keywords

Radial Basis Function Greedy Algorithm Parent Function Scattered Data Parent Surface## Preview

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## References

- [1]M.D. Buhmann and C.A. Micchelli, Multiquadric interpolation improved, Comp. Math. Appl. 24(1992)21–26.MATHMathSciNetCrossRefGoogle Scholar
- [2]R.E. Carlson and T.A. Foley, The parameter
*R*^{2}in multiquadric interpolation, Comp. Math. Appl. 21(1991)29–42.MATHMathSciNetCrossRefGoogle Scholar - [3]J. Duchon, Splines minimizing rotation-invariant seminorms in Sobolev spaces, in:
*Multivariate Approximation Theory*, ed. W. Schempp and K. Zeller (Birkhäuser, 1979) pp. 85–100.Google Scholar - [4]R. Franke and G.M. Nielson, Scattered data interpolation and applications: A tutorial and survey, in:
*Geometric Modelling: Methods and Their Applications*, ed. H. Hagen and D. Roller (Springer, 1991) pp. 131–160.Google Scholar - [5]R. Franke, Scattered data interpolation: Tests of some methods, Math. Comp. 38(1982)181–200.MATHMathSciNetCrossRefGoogle Scholar
- [6]R. Franke, Thin plate splines with tension, Comp. Aided Geom. Design 2(1985)87–95.MATHMathSciNetCrossRefGoogle Scholar
- [7]R. Franke, Approximation of scattered data for meteorological applications, in:
*Multivariate Approximation and Interpolation*, ed. W. Haussman and K. Jetter (Birkhäuser, 1990) pp. 107–120.Google Scholar - [8]F. Girosi, On some extensions of radial basis functions and their applications in artificial intelligence, Comp. Math. Appl. 24(1992)61–80.MATHMathSciNetCrossRefGoogle Scholar
- [9]R.L. Harder and R.N. Desmarais, Interpolation using surface splines, J. Aircraft 9(1972)189–191.Google Scholar
- [10]R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76(1971)1905–1915.CrossRefGoogle Scholar
- [11]R.L. Hardy, Theory and applications of the multiquadric-biharmonic method. Comp. Math. Appl. 19(1990)163–208.MATHMathSciNetCrossRefGoogle Scholar
- [12]H. Hagen and G. Schulze, Variational principles in curve and surface design, in:
*Geometric Modelling: Methods and Applications*, ed. Hagen and Roller (Springer, 1991) pp. 161–184.Google Scholar - [13]E.J. Kansa and R.E. Carlson, Improved accuracy of multiquadric interpolation using variable shape parameters, Comp. Math. Appl. 24(1992)99–120.MATHMathSciNetCrossRefGoogle Scholar
- [14]C.A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Const. Approx. 2(1986)11–22.MATHMathSciNetCrossRefGoogle Scholar
- [15]J. McMahon and R. Franke, Knot selection for least squares thin plate splines, SIAM J. Sci. Stat. Comput. 13(1992)484–498.MATHMathSciNetCrossRefGoogle Scholar
- [16]G.M. Nielson and T. Foley, A survey of applications of an affine invariant norm, in:
*Mathematical Methods in CAGD*, ed. T. Lyche and L. Schumaker (Academic Press, 1989) pp. 445–467.Google Scholar - [17]G.M. Nielson, A first-order blending method for triangles based upon cubic interpolation, Int. J. Numer. Meth. Eng. 15(1978)308–318.MathSciNetCrossRefGoogle Scholar
- [18]M.J.D. Powell, Wavelets, subdivision algorithms and radial basis functions, in:
*Advances in Numerical Analysis*, Vol. 2, ed. W.A. Light (Clarendon Press, 1992) pp. 105–210.Google Scholar - [19]K. Salkauskas, Moving least squares interpolation with thin-plate splines and radial basis functions, Comp. Math. Appl. 24(1992)177–186.MATHMathSciNetCrossRefGoogle Scholar
- [20]A.E. Tarwater, A parameter study of Hardy’s multiquadric method for scattered data interpolation, Report TR UCRL-53670, Lawrence Livermore National Laboratory (1985).Google Scholar
- [21]D.J. Woods, Report 85-5, Dept. Math. Sciences, Rice University (1985).Google Scholar