# rbf_interp_2d

rbf_interp_2d, a MATLAB 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.

### 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:

lagrange_interp_2d, a MATLAB code which defines and evaluates the lagrange polynomial p(x,y) which interpolates a set of data depending on a 2d argument that was evaluated on a product grid, so that p(x(i),y(j)) = z(i,j).

padua, a MATLAB code which returns the points and weights for padu sets, useful for interpolation in 2d. MATLAB graphics are used to plot the points.

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

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

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

rbf_interp_nd, a MATLAB code which defines and evaluates radial basis function (rbf) interpolants to multidimensional data.

shepard_interp_2d, a MATLAB code which defines and evaluates shepard interpolants to 2d data, which are based on inverse distance weighting.

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

toms886, a MATLAB code which defines the padua points for interpolation in a 2d region, including the rectangle, triangle, and ellipse, by marco caliari, stefano de marchi, marco vianello. this is a MATLAB version of acm toms algorithm 886.

vandermonde_interp_2d, a MATLAB code which finds a polynomial interpolant to data z(x,y) of a 2d argument by setting up and solving a linear system for the polynomial coefficients, involving the vandermonde matrix.

### 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.