@c Copyright (C) 2007 John W. Eaton
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@node Interpolation
@chapter Interpolation

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

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

Octave supports several methods for one-dimensional interpolation, most
of which are described in this section.  @ref{Polynomial Interpolation}
and @ref{Interpolation on Scattered Data} describes further methods.

@DOCSTRING(interp1)

There are some important differences between the various interpolation
methods. The 'spline' method enforces that both the first and second
derivatives of the interpolated values have a continuous derivative,
whereas the other methods do not. This means that the results of the
'spline' method are generally smoother. If the function to be
interpolated is in fact smooth, then 'spline' will give excellent
results. However, if the function to be evaluated is in some manner
discontinuous, then 'pchip' interpolation might give better results.

This can be demonstrated by the code

@example
@group
t = -2:2;
dt = 1;
ti =-2:0.025:2;
dti = 0.025;
y = sign(t);
ys = interp1(t,y,ti,'spline');
yp = interp1(t,y,ti,'pchip');
ddys = diff(diff(ys)./dti)./dti;
ddyp = diff(diff(yp)./dti)./dti;
figure(1);
plot (ti, ys,'r-', ti, yp,'g-');
legend('spline','pchip',4);
figure(2);
plot (ti, ddys,'r+', ti, ddyp,'g*');
legend('spline','pchip');
@end group
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:interpderiv1} and
@ref{fig:interpderiv2}.

@float Figure,fig:interpderiv1
@image{interpderiv1,8cm}
@caption{Comparison of 'phcip' and 'spline' interpolation methods for a 
step function}
@end float

@float Figure,fig:interpderiv2
@image{interpderiv2,8cm}
@caption{Comparison of the second derivate of the 'phcip' and 'spline' 
interpolation methods for a step function}
@end float
@end ifnotinfo

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 non 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

@ifinfo
which demonstrates the poor behavior of Fourier interpolation for non
periodic functions.
@end ifinfo
@ifnotinfo
which demonstrates the poor behavior of Fourier interpolation for non
periodic functions, as can be seen in @ref{fig:interpft}.

@float Figure,fig:interpft
@image{interpft,8cm}
@caption{Comparison of @code{interp1} and @code{interpft} for non
periodic data}
@end float
@end ifnotinfo

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 functions in Octave, with
similar capabilities.  Methods using Delaunay tessellation are described
in @ref{Interpolation on Scattered Data}.

@DOCSTRING(interp2)

@DOCSTRING(interp3)

@DOCSTRING(interpn)

A significant difference between @code{interpn} and the other two
multidimensional interpolation functions 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 of 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 = -1:0.1:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3(x, y, z, v, xxi, yyi, zzi, 'spline');
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn(x, y, z, v, xxi, yyi, zzi, 'spline');
mesh (zi, yi, squeeze (vi2(1,:,:)));
@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.
@ifnotinfo
The result of this code can be seen in @ref{fig:interpn}.

@float Figure,fig:interpn
@image{interpn,8cm}
@caption{Demonstration of the use of @code{interpn}}
@end float
@end ifnotinfo

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

@DOCSTRING(bicubic)
