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 computer code and data files described and made available on this web page are 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 a Python version.

Related Data and Programs:

padua, a Python code which returns the points and weights for Padu 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:

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:

The test program makes a number of plots.

Last modified on 01 July 2018.