RBF_INTERP_2D
Radial Basis Function Interpolation in 2D

RBF_INTERP_2D is a Python library which defines and evaluates radial basis function (RBF) interpolants to 2D data.

A radial basis interpolant is a useful, but expensive, technique for definining a smooth function which interpolates a set of function values specified at an arbitrary set of data points.

Given nd multidimensional points xd with function values fd, and a basis function phi(r), the form of the interpolant is

       f(x) = sum ( 1 <= i <= nd ) w(i) * phi(||x-xd(i)||)
      

        sum ( 1 <= i <= nd ) w(i) * phi(||xd(j)-xd(i)||) = fd(j)
      

Although the technique is generally applied in a multidimensional setting, in this directory we look specifically at the case involving 2D data. This allows us to easily plot and compare the various results.

Four families of radial basis functions are provided.

• phi1(r) = sqrt ( r^2 + r0^2 ) (multiquadric)
• phi2(r) = 1 / sqrt ( r^2 + r0^2 ) (inverse multiquadric)
• phi3(r) = r^2 * log ( r / r0 ) (thin plate spline)
• phi4(r) = exp ( -0.5 r^2 / r0^2 ) (gaussian)

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

RBF_INTERP_2D is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

PADUA, a Python library which returns the points and weights for Padu sets, useful for interpolation in 2D.

PWL_INTERP_2D, a Python library which evaluates a piecewise linear interpolant to data defined on a regular 2D grid.

R8LIB, a Python library which contains many utility routines, using double precision real (R8) arithmetic.

RBF_INTERP_1D, a Python library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

TEST_INTERP_2D, a Python library which defines test problems for interpolation of data z(x,y) of a 2D argument.

Reference:

1. Richard Franke,
   Scattered Data Interpolation: Tests of Some Methods,
   Mathematics of Computation,
   Volume 38, Number 157, January 1982, pages 181-200.
2. William Press, Brian Flannery, Saul Teukolsky, William Vetterling,
   Numerical Recipes in FORTRAN: The Art of Scientific Computing,
   Third Edition,
   Cambridge University Press, 2007,
   ISBN13: 978-0-521-88068-8,
   LC: QA297.N866.

Source Code:

• rbf_interp_2d.py, the source code.
• test_interp_2d.py, defines some 2D interpolation test problems.
• rbf_interp_2d.sh, runs the tests.
• rbf_interp_2d.txt, the output file.

The test program makes a number of plots.

• p01_data.png, the data for problem p01 with a linear interpolant.
• p01_phi1_poly.png, the data for problem p01 with a PHI1 RBF interpolant.
• p01_phi2_poly.png, the data for problem p01 with a PHI2 RBF interpolant.
• p01_phi3_poly.png, the data for problem p01 with a PHI3 RBF interpolant.
• p01_phi4_poly.png, the data for problem p01 with a PHI4 RBF interpolant.
• p02_data.png, the data for problem p02 with a linear interpolant.
• p02_phi1_poly.png, the data for problem p02 with a PHI1 RBF interpolant.
• p02_phi2_poly.png, the data for problem p02 with a PHI2 RBF interpolant.
• p02_phi3_poly.png, the data for problem p02 with a PHI3 RBF interpolant.
• p02_phi4_poly.png, the data for problem p02 with a PHI4 RBF interpolant.
• p03_data.png, the data for problem p03 with a linear interpolant.
• p03_phi1_poly.png, the data for problem p03 with a PHI1 RBF interpolant.
• p03_phi2_poly.png, the data for problem p03 with a PHI2 RBF interpolant.
• p03_phi3_poly.png, the data for problem p03 with a PHI3 RBF interpolant.
• p03_phi4_poly.png, the data for problem p03 with a PHI4 RBF interpolant.
• p04_data.png, the data for problem p04 with a linear interpolant.
• p04_phi1_poly.png, the data for problem p04 with a PHI1 RBF interpolant.
• p04_phi2_poly.png, the data for problem p04 with a PHI2 RBF interpolant.
• p04_phi3_poly.png, the data for problem p04 with a PHI3 RBF interpolant.
• p04_phi4_poly.png, the data for problem p04 with a PHI4 RBF interpolant.
• p05_data.png, the data for problem p05 with a linear interpolant.
• p05_phi1_poly.png, the data for problem p05 with a PHI1 RBF interpolant.
• p05_phi2_poly.png, the data for problem p05 with a PHI2 RBF interpolant.
• p05_phi3_poly.png, the data for problem p05 with a PHI3 RBF interpolant.
• p05_phi4_poly.png, the data for problem p05 with a PHI4 RBF interpolant.
• p06_data.png, the data for problem p06 with a linear interpolant.
• p06_phi1_poly.png, the data for problem p06 with a PHI1 RBF interpolant.
• p06_phi2_poly.png, the data for problem p06 with a PHI2 RBF interpolant.
• p06_phi3_poly.png, the data for problem p06 with a PHI3 RBF interpolant.
• p06_phi4_poly.png, the data for problem p06 with a PHI4 RBF interpolant.
• p07_data.png, the data for problem p07 with a linear interpolant.
• p07_phi1_poly.png, the data for problem p07 with a PHI1 RBF interpolant.
• p07_phi2_poly.png, the data for problem p07 with a PHI2 RBF interpolant.
• p07_phi3_poly.png, the data for problem p07 with a PHI3 RBF interpolant.
• p07_phi4_poly.png, the data for problem p07 with a PHI4 RBF interpolant.
• p08_data.png, the data for problem p08 with a linear interpolant.
• p08_phi1_poly.png, the data for problem p08 with a PHI1 RBF interpolant.
• p08_phi2_poly.png, the data for problem p08 with a PHI2 RBF interpolant.
• p08_phi3_poly.png, the data for problem p08 with a PHI3 RBF interpolant.
• p08_phi4_poly.png, the data for problem p08 with a PHI4 RBF interpolant.
• p09_data.png, the data for problem p09 with a linear interpolant.
• p09_phi1_poly.png, the data for problem p09 with a PHI1 RBF interpolant.
• p09_phi2_poly.png, the data for problem p09 with a PHI2 RBF interpolant.
• p09_phi3_poly.png, the data for problem p09 with a PHI3 RBF interpolant.
• p09_phi4_poly.png, the data for problem p09 with a PHI4 RBF interpolant.
• p10_data.png, the data for problem p10 with a linear interpolant.
• p10_phi1_poly.png, the data for problem p10 with a PHI1 RBF interpolant.
• p10_phi2_poly.png, the data for problem p10 with a PHI2 RBF interpolant.
• p10_phi3_poly.png, the data for problem p10 with a PHI3 RBF interpolant.
• p10_phi4_poly.png, the data for problem p10 with a PHI4 RBF interpolant.
• p11_data.png, the data for problem p11 with a linear interpolant.
• p11_phi1_poly.png, the data for problem p11 with a PHI1 RBF interpolant.
• p11_phi2_poly.png, the data for problem p11 with a PHI2 RBF interpolant.
• p11_phi3_poly.png, the data for problem p11 with a PHI3 RBF interpolant.
• p11_phi4_poly.png, the data for problem p11 with a PHI4 RBF interpolant.
• p12_data.png, the data for problem p12 with a linear interpolant.
• p12_phi1_poly.png, the data for problem p12 with a PHI1 RBF interpolant.
• p12_phi2_poly.png, the data for problem p12 with a PHI2 RBF interpolant.
• p12_phi3_poly.png, the data for problem p12 with a PHI3 RBF interpolant.
• p12_phi4_poly.png, the data for problem p12 with a PHI4 RBF interpolant.
• p13_data.png, the data for problem p13 with a linear interpolant.
• p13_phi1_poly.png, the data for problem p13 with a PHI1 RBF interpolant.
• p13_phi2_poly.png, the data for problem p13 with a PHI2 RBF interpolant.
• p13_phi3_poly.png, the data for problem p13 with a PHI3 RBF interpolant.
• p13_phi4_poly.png, the data for problem p13 with a PHI4 RBF interpolant.

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