@c Copyright (C) 2007 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Interpolation
@chapter Interpolation

@menu
* One-dimensional Interpolation::
* Multi-dimensional Interpolation::
@end menu

@node One-dimensional Interpolation
@section One-dimensional Interpolation

@DOCSTRING(interp1)

Fourier interpolation, is a resampling technique where a signal is
converted to the frequency domain, padded with zeros and then
reconverted to the time domain.

@DOCSTRING(interpft)

There are two significant limitations on Fourier interpolation. Firstly,
the function signal is assumed to be periodic, and so no periodic
signals will be poorly represented at the edges. Secondly, both the
signal and its interpolation are required to be sampled at equispaced
points. An example of the use of @code{interpft} is

@example
@group
t = 0 : 0.3 : pi; dt = t(2)-t(1);
n = length (t); k = 100;
ti = t(1) + [0 : k-1]*dt*n/k;
y = sin (4*t + 0.3) .* cos (3*t - 0.1);
yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1);
plot (ti, yp, 'g', ti, interp1(t, y, ti, 'spline'), 'b', ...
      ti, interpft (y, k), 'c', t, y, 'r+');
legend ('sin(4t+0.3)cos(3t-0.1','spline','interpft','data');
@end group
@end example

which demonstrates the poor behavior of Fourier interpolation for non
periodic functions.

In additional the support function @code{spline} and @code{lookup} that
underlie the @code{interp1} function can be called directly.

@DOCSTRING(spline)

The @code{lookup} is used by other interpolation function to identify
the points of the original data that are closest to the current point
of interest.

@DOCSTRING(lookup)

@node Multi-dimensional Interpolation
@section Multi-dimensional Interpolation

There are three multi-dimensional interpolation function in Octave, with
similar capabilities.

@DOCSTRING(interp2)

@DOCSTRING(interp3)

@DOCSTRING(interpn)

A significant difference between @code{interpn} and the other two
multidimensional interpolation function is the fashion in which the
dimensions are treated. For @code{interp2} and @code{interp3}, the 'y'
axis is considered to be the columns of the matrix, whereas the 'x'
axis corresponds to the rows the the array. As Octave indexes arrays in
column major order, the first dimension of any array is the columns, and
so @code{interpn} effectively reverses the 'x' and 'y' dimensions. 
Consider the example

@example
@group
x = y = z = -1:1;
f = @@(x,y,z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f (xx,yy,zz);
xi = yi = zi = -1:0.5:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3(x, y, z, v, xxi, yyi, zzi);
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn(x, y, z, v, xxi, yyi, zzi);
@end group
@end example

@noindent
where @code{vi} and @code{vi2} are identical. The reversal of the
dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions
respectively.

In additional the support function @code{bicubic} that underlies the
cubic interpolation of @code{interp2} function can be called directly.

@DOCSTRING(bicubic)