rbf_interp_2d

rbf_interp_2d, a Python code 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)||)

where the weights w have been precomputed by solving

        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)

Each uses a "scale factor" r0, whose value is recommended to be greater than the minimal distance between points, and rather less than the maximal distance. Changing the value of r0 changes the shape of the interpolant function.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

rbf_interp_2d is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

rbf_interp_2d_test

padua, a Python code which returns the points and weights for Padua sets, useful for interpolation in 2D.

pwl_interp_2d, a Python code which evaluates a piecewise linear interpolant to data defined on a regular 2d grid.

r8lib, a Python code which contains many utility routines, using double precision real (r8) arithmetic.

rbf_interp_1d, a Python code which defines and evaluates radial basis function (RBF) interpolants to 1d data.

test_interp_2d, a Python code which defines test problems for interpolation of data z(x,y) of a 2d argument.

Reference:

Richard Franke,
Scattered Data Interpolation: Tests of Some Methods,
Mathematics of Computation,
Volume 38, Number 157, January 1982, pages 181-200.

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.

Last modified on 01 July 2018.