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@c Copyright (C) 2007-2013 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.

@c interp1 scripts/general/interp1.m
@anchor{XREFinterp1}
@deftypefn  {Function File} {@var{yi} =} interp1 (@var{x}, @var{y}, @var{xi})
@deftypefnx {Function File} {@var{yi} =} interp1 (@var{y}, @var{xi})
@deftypefnx {Function File} {@var{yi} =} interp1 (@dots{}, @var{method})
@deftypefnx {Function File} {@var{yi} =} interp1 (@dots{}, @var{extrap})
@deftypefnx {Function File} {@var{pp} =} interp1 (@dots{}, "pp")

One-dimensional interpolation.  Interpolates to determine the value of
@var{yi} at the points, @var{xi}.  If not specified, @var{x} is taken
to be the indices of @var{y}.  If @var{y} is a matrix or an N-dimensional
array, the interpolation is performed on each column of @var{y}.

Method is one of:

@table @asis
@item @qcode{"nearest"}
Return the nearest neighbor.

@item @qcode{"linear"}
Linear interpolation from nearest neighbors

@item @qcode{"pchip"}
Piecewise cubic Hermite interpolating polynomial

@item @qcode{"cubic"}
Cubic interpolation (same as @code{pchip})

@item @qcode{"spline"}
Cubic spline interpolation---smooth first and second derivatives
throughout the curve
@end table

Appending '*' to the start of the above method forces @code{interp1}
to assume that @var{x} is uniformly spaced, and only @code{@var{x}(1)}
and @code{@var{x}(2)} are referenced.  This is usually faster,
and is never slower.  The default method is @qcode{"linear"}.

If @var{extrap} is the string @qcode{"extrap"}, then extrapolate values
beyond the endpoints.  If @var{extrap} is a number, replace values beyond
the endpoints with that number.  If @var{extrap} is missing, assume NA.

If the string argument @qcode{"pp"} is specified, then @var{xi} should not be
supplied and @code{interp1} returns the piecewise polynomial that
can later be used with @code{ppval} to evaluate the interpolation.
There is an equivalence, such that @code{ppval (interp1 (@var{x},
@var{y}, @var{method}, @qcode{"pp"}), @var{xi}) == interp1 (@var{x}, @var{y},
@var{xi}, @var{method}, @qcode{"extrap"})}.

Duplicate points in @var{x} specify a discontinuous interpolant.  There
may be at most 2 consecutive points with the same value.
If @var{x} is increasing, the default discontinuous interpolant is
right-continuous.  If @var{x} is decreasing, the default discontinuous
interpolant is left-continuous.
The continuity condition of the interpolant may be specified by using
the options, @qcode{"-left"} or @qcode{"-right"}, to select a left-continuous
or right-continuous interpolant, respectively.
Discontinuous interpolation is only allowed for @qcode{"nearest"} and
@qcode{"linear"} methods; in all other cases, the @var{x}-values must be
unique.

An example of the use of @code{interp1} is

@example
@group
xf = [0:0.05:10];
yf = sin (2*pi*xf/5);
xp = [0:10];
yp = sin (2*pi*xp/5);
lin = interp1 (xp, yp, xf);
spl = interp1 (xp, yp, xf, "spline");
cub = interp1 (xp, yp, xf, "cubic");
near = interp1 (xp, yp, xf, "nearest");
plot (xf, yf, "r", xf, lin, "g", xf, spl, "b",
      xf, cub, "c", xf, near, "m", xp, yp, "r*");
legend ("original", "linear", "spline", "cubic", "nearest");
@end group
@end example

@seealso{@ref{XREFinterpft,,interpft}}
@end deftypefn


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.

@c interpft scripts/general/interpft.m
@anchor{XREFinterpft}
@deftypefn  {Function File} {} interpft (@var{x}, @var{n})
@deftypefnx {Function File} {} interpft (@var{x}, @var{n}, @var{dim})

Fourier interpolation.  If @var{x} is a vector, then @var{x} is
resampled with @var{n} points.  The data in @var{x} is assumed to be
equispaced.  If @var{x} is a matrix or an N-dimensional array, the
interpolation is performed on each column of @var{x}.  If @var{dim} is
specified, then interpolate along the dimension @var{dim}.

@code{interpft} assumes that the interpolated function is periodic,
and so assumptions are made about the endpoints of the interpolation.

@seealso{@ref{XREFinterp1,,interp1}}
@end deftypefn


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

@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.

@c spline scripts/polynomial/spline.m
@anchor{XREFspline}
@deftypefn  {Function File} {@var{pp} =} spline (@var{x}, @var{y})
@deftypefnx {Function File} {@var{yi} =} spline (@var{x}, @var{y}, @var{xi})
Return the cubic spline interpolant of points @var{x} and @var{y}.

When called with two arguments, return the piecewise polynomial @var{pp}
that may be used with @code{ppval} to evaluate the polynomial at specific
points.  When called with a third input argument, @code{spline} evaluates
the spline at the points @var{xi}.  The third calling form @code{spline
(@var{x}, @var{y}, @var{xi})} is equivalent to @code{ppval (spline
(@var{x}, @var{y}), @var{xi})}.

The variable @var{x} must be a vector of length @var{n}.  @var{y} can be
either a vector or array.  If @var{y} is a vector it must have a length of
either @var{n} or @code{@var{n} + 2}.  If the length of @var{y} is
@var{n}, then the "not-a-knot" end condition is used.  If the length of
@var{y} is @code{@var{n} + 2}, then the first and last values of the
vector @var{y} are the values of the first derivative of the cubic spline
at the endpoints.

If @var{y} is an array, then the size of @var{y} must have the form
@tex
$$[s_1, s_2, \cdots, s_k, n]$$
@end tex
@ifnottex
@code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]}
@end ifnottex
or
@tex
$$[s_1, s_2, \cdots, s_k, n + 2].$$
@end tex
@ifnottex
@code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n} + 2]}.
@end ifnottex
The array is reshaped internally to a matrix where the leading
dimension is given by
@tex
$$s_1 s_2 \cdots s_k$$
@end tex
@ifnottex
@code{@var{s1} * @var{s2} * @dots{} * @var{sk}}
@end ifnottex
and each row of this matrix is then treated separately.  Note that this
is exactly opposite to @code{interp1} but is done for @sc{matlab}
compatibility.

@seealso{@ref{XREFpchip,,pchip}, @ref{XREFppval,,ppval}, @ref{XREFmkpp,,mkpp}, @ref{XREFunmkpp,,unmkpp}}
@end deftypefn


@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}.

@c interp2 scripts/general/interp2.m
@anchor{XREFinterp2}
@deftypefn  {Function File} {@var{zi} =} interp2 (@var{x}, @var{y}, @var{z}, @var{xi}, @var{yi})
@deftypefnx {Function File} {@var{zi} =} interp2 (@var{Z}, @var{xi}, @var{yi})
@deftypefnx {Function File} {@var{zi} =} interp2 (@var{Z}, @var{n})
@deftypefnx {Function File} {@var{zi} =} interp2 (@dots{}, @var{method})
@deftypefnx {Function File} {@var{zi} =} interp2 (@dots{}, @var{method}, @var{extrapval})

Two-dimensional interpolation.  @var{x}, @var{y} and @var{z} describe a
surface function.  If @var{x} and @var{y} are vectors their length
must correspondent to the size of @var{z}.  @var{x} and @var{y} must be
monotonic.  If they are matrices they must have the @code{meshgrid}
format.

@table @code
@item interp2 (@var{x}, @var{y}, @var{Z}, @var{xi}, @var{yi}, @dots{})
Returns a matrix corresponding to the points described by the
matrices @var{xi}, @var{yi}.

If the last argument is a string, the interpolation method can
be specified.  The method can be @qcode{"linear"}, @qcode{"nearest"} or
@qcode{"cubic"}.  If it is omitted @qcode{"linear"} interpolation is
assumed.

@item interp2 (@var{z}, @var{xi}, @var{yi})
Assumes @code{@var{x} = 1:rows (@var{z})} and @code{@var{y} =
1:columns (@var{z})}

@item interp2 (@var{z}, @var{n})
Interleaves the matrix @var{z} n-times.  If @var{n} is omitted a value
of @code{@var{n} = 1} is assumed.
@end table

The variable @var{method} defines the method to use for the
interpolation.  It can take one of the following values

@table @asis
@item @qcode{"nearest"}
Return the nearest neighbor.

@item @qcode{"linear"}
Linear interpolation from nearest neighbors.

@item @qcode{"pchip"}
Piecewise cubic Hermite interpolating polynomial.

@item @qcode{"cubic"}
Cubic interpolation from four nearest neighbors.

@item @qcode{"spline"}
Cubic spline interpolation---smooth first and second derivatives
throughout the curve.
@end table

If a scalar value @var{extrapval} is defined as the final value, then
values outside the mesh as set to this value.  Note that in this case
@var{method} must be defined as well.  If @var{extrapval} is not
defined then NA is assumed.

@seealso{@ref{XREFinterp1,,interp1}}
@end deftypefn


@c interp3 scripts/general/interp3.m
@anchor{XREFinterp3}
@deftypefn  {Function File} {@var{vi} =} interp3 (@var{x}, @var{y}, @var{z}, @var{v}, @var{xi}, @var{yi}, @var{zi})
@deftypefnx {Function File} {@var{vi} =} interp3 (@var{v}, @var{xi}, @var{yi}, @var{zi})
@deftypefnx {Function File} {@var{vi} =} interp3 (@var{v}, @var{m})
@deftypefnx {Function File} {@var{vi} =} interp3 (@var{v})
@deftypefnx {Function File} {@var{vi} =} interp3 (@dots{}, @var{method})
@deftypefnx {Function File} {@var{vi} =} interp3 (@dots{}, @var{method}, @var{extrapval})

Perform 3-dimensional interpolation.  Each element of the 3-dimensional
array @var{v} represents a value at a location given by the parameters
@var{x}, @var{y}, and @var{z}.  The parameters @var{x}, @var{x}, and
@var{z} are either 3-dimensional arrays of the same size as the array
@var{v} in the @qcode{"meshgrid"} format or vectors.  The parameters
@var{xi}, etc. respect a similar format to @var{x}, etc., and they
represent the points at which the array @var{vi} is interpolated.

If @var{x}, @var{y}, @var{z} are omitted, they are assumed to be
@code{x = 1 : size (@var{v}, 2)}, @code{y = 1 : size (@var{v}, 1)} and
@code{z = 1 : size (@var{v}, 3)}.  If @var{m} is specified, then
the interpolation adds a point half way between each of the interpolation
points.  This process is performed @var{m} times.  If only @var{v} is
specified, then @var{m} is assumed to be @code{1}.

Method is one of:

@table @asis
@item @qcode{"nearest"}
Return the nearest neighbor.

@item @qcode{"linear"}
Linear interpolation from nearest neighbors.

@item @qcode{"cubic"}
Cubic interpolation from four nearest neighbors (not implemented yet).

@item @qcode{"spline"}
Cubic spline interpolation---smooth first and second derivatives
throughout the curve.
@end table

The default method is @qcode{"linear"}.

If @var{extrap} is the string @qcode{"extrap"}, then extrapolate values
beyond the endpoints.  If @var{extrap} is a number, replace values beyond
the endpoints with that number.  If @var{extrap} is missing, assume NA.
@seealso{@ref{XREFinterp1,,interp1}, @ref{XREFinterp2,,interp2}, @ref{XREFspline,,spline}, @ref{XREFmeshgrid,,meshgrid}}
@end deftypefn


@c interpn scripts/general/interpn.m
@anchor{XREFinterpn}
@deftypefn  {Function File} {@var{vi} =} interpn (@var{x1}, @var{x2}, @dots{}, @var{v}, @var{y1}, @var{y2}, @dots{})
@deftypefnx {Function File} {@var{vi} =} interpn (@var{v}, @var{y1}, @var{y2}, @dots{})
@deftypefnx {Function File} {@var{vi} =} interpn (@var{v}, @var{m})
@deftypefnx {Function File} {@var{vi} =} interpn (@var{v})
@deftypefnx {Function File} {@var{vi} =} interpn (@dots{}, @var{method})
@deftypefnx {Function File} {@var{vi} =} interpn (@dots{}, @var{method}, @var{extrapval})

Perform @var{n}-dimensional interpolation, where @var{n} is at least two.
Each element of the @var{n}-dimensional array @var{v} represents a value
at a location given by the parameters @var{x1}, @var{x2}, @dots{}, @var{xn}.
The parameters @var{x1}, @var{x2}, @dots{}, @var{xn} are either
@var{n}-dimensional arrays of the same size as the array @var{v} in
the @qcode{"ndgrid"} format or vectors.  The parameters @var{y1}, etc.
respect a similar format to @var{x1}, etc., and they represent the points
at which the array @var{vi} is interpolated.

If @var{x1}, @dots{}, @var{xn} are omitted, they are assumed to be
@code{x1 = 1 : size (@var{v}, 1)}, etc.  If @var{m} is specified, then
the interpolation adds a point half way between each of the interpolation
points.  This process is performed @var{m} times.  If only @var{v} is
specified, then @var{m} is assumed to be @code{1}.

Method is one of:

@table @asis
@item @qcode{"nearest"}
Return the nearest neighbor.

@item @qcode{"linear"}
Linear interpolation from nearest neighbors.

@item @qcode{"cubic"}
Cubic interpolation from four nearest neighbors (not implemented yet).

@item @qcode{"spline"}
Cubic spline interpolation---smooth first and second derivatives
throughout the curve.
@end table

The default method is @qcode{"linear"}.

If @var{extrapval} is the scalar value, use it to replace the values
beyond the endpoints with that number.  If @var{extrapval} is missing,
assume NA.
@seealso{@ref{XREFinterp1,,interp1}, @ref{XREFinterp2,,interp2}, @ref{XREFspline,,spline}, @ref{XREFndgrid,,ndgrid}}
@end deftypefn


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

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

@c bicubic scripts/general/bicubic.m
@anchor{XREFbicubic}
@deftypefn {Function File} {@var{zi} =} bicubic (@var{x}, @var{y}, @var{z}, @var{xi}, @var{yi}, @var{extrapval})

Return a matrix @var{zi} corresponding to the bicubic
interpolations at @var{xi} and @var{yi} of the data supplied
as @var{x}, @var{y} and @var{z}.  Points outside the grid are set
to @var{extrapval}.

See @url{http://wiki.woodpecker.org.cn/moin/Octave/Bicubic}
for further information.
@seealso{@ref{XREFinterp2,,interp2}}
@end deftypefn

