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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.

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Authors and Affiliations

  1. Department of Mathematics, Naval Postgraduate School, 93943, Monterey, CA, USA

    Richard Franke

  2. FB Informatik, Universität Kaiserslautern, 67653, Kaiserlautern, Germany

    Hans Hagen

  3. Department of Computer Science, Arizona State University, 85287, Tempe, AZ, USA

    Gregory M. Nielson

Authors
  1. Richard Franke
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  2. Hans Hagen
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  3. Gregory M. Nielson
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Additional information

This work was performed while the first and last authors were visiting Universität Kaiserslautern and were supported by DFG Grant Ka 477/13-1. The first author was also supported by Direct Research Funds from the Naval Postgraduate School.

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Franke, R., Hagen, H. & Nielson, G.M. Least squares surface approximation to scattered data using multiquadratic functions. Adv Comput Math 2, 81–99 (1994). https://doi.org/10.1007/BF02519037

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