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<html>
<head>
<title>
RBF_INTERP_2D - Radial Basis Function Interpolation in 2D
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
RBF_INTERP_2D <br> Radial Basis Function Interpolation in 2D
</h1>
<hr>
<p>
<b>RBF_INTERP_2D</b>
is a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
</p>
<p>
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.
</p>
<p>
Given nd multidimensional points xd with function values fd, and a
basis function phi(r), the form of the interpolant is
<pre>
f(x) = sum ( 1 <= i <= nd ) w(i) * phi(||x-xd(i)||)
</pre>
where the weights w have been precomputed by solving
<pre>
sum ( 1 <= i <= nd ) w(i) * phi(||xd(j)-xd(i)||) = fd(j)
</pre>
</p>
<p>
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.
</p>
<p>
Four families of radial basis functions are provided.
<ul>
<li>
phi1(r) = sqrt ( r^2 + r0^2 ) (multiquadric)
</li>
<li>
phi2(r) = 1 / sqrt ( r^2 + r0^2 ) (inverse multiquadric)
</li>
<li>
phi3(r) = r^2 * log ( r / r0 ) (thin plate spline)
</li>
<li>
phi4(r) = exp ( -0.5 r^2 / r0^2 ) (gaussian)
</li>
</ul>
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.
<p>
<p>
<b>RBF_INTERP_2D</b> needs access to the R8LIB library. The test program
also needs access to the TEST_INTERP_2D library.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>RBF_INTERP_2D</b> is available in
<a href = "../../c_src/rbf_interp_2d/rbf_interp_2d.html">a C version</a> and
<a href = "../../cpp_src/rbf_interp_2d/rbf_interp_2d.html">a C++ version</a> and
<a href = "../../f77_src/rbf_interp_2d/rbf_interp_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/rbf_interp_2d/rbf_interp_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/rbf_interp_2d/rbf_interp_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/lagrange_interp_2d/lagrange_interp_2d.html">
LAGRANGE_INTERP_2D</a>,
a C++ library 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).
</p>
<p>
<a href = "../../cpp_src/padua/padua.html">
PADUA</a>,
a C++ library which
returns the points and weights for Padu sets, useful for interpolation
in 2D. GNUPLOT is used to plot the points.
</p>
<p>
<a href = "../../cpp_src/pwl_interp_2d/pwl_interp_2d.html">
PWL_INTERP_2D</a>,
a C++ library which
evaluates a piecewise linear interpolant to data defined on
a regular 2D grid.
</p>
<p>
<a href = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines, using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_1d/rbf_interp_1d.html">
RBF_INTERP_1D</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 1D data.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_nd/rbf_interp_nd.html">
RBF_INTERP_ND</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
</p>
<p>
<a href = "../../cpp_src/shepard_interp_2d/shepard_interp_2d.html">
SHEPARD_INTERP_2D</a>,
a C++ library which
defines and evaluates Shepard interpolants to 2D data,
based on inverse distance weighting.
</p>
<p>
<a href = "../../cpp_src/test_interp_2d/test_interp_2d.html">
TEST_INTERP_2D</a>,
a C++ library which
defines test problems for interpolation of data z(x,y) of a 2D argument.
</p>
<p>
<a href = "../../cpp_src/toms886/toms886.html">
TOMS886</a>,
a C++ library 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 C++ version of ACM TOMS algorithm 886.
</p>
<p>
<a href = "../../cpp_src/vandermonde_interp_2d/vandermonde_interp_2d.html">
VANDERMONDE_INTERP_2D</a>,
a C++ library 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.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Richard Franke,<br>
Scattered Data Interpolation: Tests of Some Methods,<br>
Mathematics of Computation,<br>
Volume 38, Number 157, January 1982, pages 181-200.
</li>
<li>
William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
Numerical Recipes in FORTRAN: The Art of Scientific Computing,<br>
Third Edition,<br>
Cambridge University Press, 2007,<br>
ISBN13: 978-0-521-88068-8,<br>
LC: QA297.N866.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "rbf_interp_2d.cpp">rbf_interp_2d.cpp</a>, the source code.
</li>
<li>
<a href = "rbf_interp_2d.hpp">rbf_interp_2d.hpp</a>, the source code.
</li>
<li>
<a href = "rbf_interp_2d.sh">rbf_interp_2d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "rbf_interp_2d_prb.cpp">rbf_interp_2d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "rbf_interp_2d_prb.sh">rbf_interp_2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "rbf_interp_2d_prb_output.txt">rbf_interp_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DNRM2</b> returns the euclidean norm of a vector.
</li>
<li>
<b>DROT</b> applies a plane rotation.
</li>
<li>
<b>DROTG</b> constructs a Givens plane rotation.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>DSVDC</b> computes the singular value decomposition of a real rectangular matrix.
</li>
<li>
<b>DSWAP</b> interchanges two vectors.
</li>
<li>
<b>PHI1</b> evaluates the multiquadric radial basis function.
</li>
<li>
<b>PHI2</b> evaluates the inverse multiquadric radial basis function.
</li>
<li>
<b>PHI3</b> evaluates the thin-plate spline radial basis function.
</li>
<li>
<b>PHI4</b> evaluates the gaussian radial basis function.
</li>
<li>
<b>R8MAT_SOLVE_SVD</b> solves a linear system A*x=b using the SVD.
</li>
<li>
<b>RBF_INTERP</b> evaluates a radial basis function interpolant.
</li>
<li>
<b>RBF_WEIGHT</b> computes weights for radial basis function interpolation.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 06 October 2012.
</i>
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