@c Copyright (C) 2007-2019 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} describe additional methods.

@DOCSTRING(interp1)

There are some important differences between the various interpolation
methods.  The @qcode{"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
@qcode{"spline"} method are generally smoother.  If the function to be
interpolated is in fact smooth, then @qcode{"spline"} will give excellent
results.  However, if the function to be evaluated is in some manner
discontinuous, then @qcode{"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
@center @image{interpderiv1,4in}
@caption{Comparison of @qcode{"pchip"} and @qcode{"spline"} interpolation methods for a
step function}
@end float

@float Figure,fig:interpderiv2
@center @image{interpderiv2,4in}
@caption{Comparison of the second derivative of the @qcode{"pchip"} and @qcode{"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.  First,
the function signal is assumed to be periodic, and so non-periodic
signals will be poorly represented at the edges.  Second, 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

@noindent
@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
@center @image{interpft,4in}
@caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data}
@end float
@end ifnotinfo

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

@DOCSTRING(spline)

@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
multi-dimensional 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
@center @image{interpn,4in}
@caption{Demonstration of the use of @code{interpn}}
@end float
@end ifnotinfo
