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Radial Basis Functions Interpolation

Author: - 28 June 2018

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This is a set of Matlab functions to interpolate scattered data with Radial Basis Functions (RBF).

Getting Started

y = RBFinterp(xs, ys, x, RBFtype, R)

interpolates to find y, the values of the function y=f(x) at the points x.

Xs must be a matrix of size [N,Dx], with N the number of data points and Dx the dimension of the points in xs and x.

Ys must be a matrix of size [N,Dy], with N the number of known values at points in Xs, and Dy the dimension of the y values.

X must be a matrix of size [M,Dx], with M the number of query points.

RBFtype specifies the radial basis functions (RBF) to be used.

  • The available global support RBFs are:
    • ‘R1’ - linear spline
    • ‘R3’ - cubic spline
    • ‘TPS2’ - thin plate spline
    • ‘Q’ - quadric
    • ‘MQ’ - multiquadric
    • ‘IMQ’ - inverse multiquadric
    • ‘IQ’ - inverse quadric
    • ‘GS’ - Gauss
RBF name Abbreviation
Linear spline R1
Cubic splie R3
Thin plate spline TPS2
Quadric Q
Multiquadric MQ
Inverse multiquadric IMQ
Inverse quadric IQ
Gauss GS
  • The available compact support RBFs are (see Wendland H., Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger. PhD thesis, Göttingen, Georg-August-Universität zu Göttingen, Diss, 1996):
    • ‘CP_C0’
    • ‘CP_C2’
    • ‘CP_C4’
    • ‘CP_C6’
    • ‘CTPS_C0’
    • ‘CTPS_C1’
    • ‘CTPS_C2a’
    • ‘CTPS_C2b’

Compact support functions have the form

RBF name

R is either the support radius for the compact support RBFs or a parameter to make the distance values dimensionless for the global support RBFs.

[fPar, M] = RBFparam(xs, ys, RBFtype, R)

returns the weights in the RBF summation and the polynomial coefficients in a column vector fPar by solving a linear system

[y] = RBFeval(xs, x, fPar, RBFtype, R)

returns the values of the interpolation weighted function at points x by performing the matrix-vector product

Running the example

An example case can be run just by typing in the Matlab command line

test

References

  • Beckert, Armin and Wendland, Holger. Multivariate interpolation for fluid-structure-interaction problems using radial basis functions. Aerospace Science and Technology, 5 (2), p. 125-134, 2001.
  • Wendland, Holger. Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger}. PhD thesis, Göttingen, Georg-August-Universität zu Göttingen, Diss, 1996.
  • De Boer, A and Van der Schoot, MS and Bijl, Hester. Mesh deformation based on radial basis function interpolation. Computers & structures, 85 (11-14), p. 784-795, 2007.
  • Biancolini, Marco Evangelos. Fast Radial Basis Functions for Engineering Applications. Springer, 2018.