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@c Copyright (C) 2007-2012 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 further methods.

@c interp1 scripts/general/interp1.m
@anchor{doc-interp1}
@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.  Interpolate @var{y}, defined at the
points @var{x}, at the points @var{xi}.  The sample points @var{x}
must be monotonic.  If not specified, @var{x} is taken to be the
indices of @var{y}.  If @var{y} is an array, treat the columns
of @var{y} separately.

Method is one of:

@table @asis
@item 'nearest'
Return the nearest neighbor.

@item 'linear'
Linear interpolation from nearest neighbors

@item 'pchip'
Piecewise cubic Hermite interpolating polynomial

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

@item '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 'linear'.

If @var{extrap} is the string '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 '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}, 'pp'), @var{xi}) == interp1 (@var{x}, @var{y},
@var{xi}, @var{method}, 'extrap')}.

Duplicate points in @var{x} specify a discontinuous interpolant.  There
should be at most 2 consecutive points with the same value.
The discontinuous interpolant is right-continuous if @var{x} is increasing,
left-continuous if it is decreasing.
Discontinuities are (currently) only allowed for "nearest" and "linear"
methods; in all other cases, @var{x} must be strictly monotonic.

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{doc-interpft,,interpft}}
@end deftypefn


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
@center @image{interpderiv1,4in}
@caption{Comparison of 'pchip' and '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 'pchip' and 'spline' 
interpolation methods for a step function}
@end float
@end ifnotinfo

A simplified version of @code{interp1} that performs only linear
interpolation is available in @code{interp1q}.  This argument is slightly
faster than @code{interp1} as to performs little error checking.

@c interp1q scripts/general/interp1q.m
@anchor{doc-interp1q}
@deftypefn {Function File} {@var{yi} =} interp1q (@var{x}, @var{y}, @var{xi})
One-dimensional linear interpolation without error checking.
Interpolates @var{y}, defined at the points @var{x}, at the points
@var{xi}.  The sample points @var{x} must be a strictly monotonically
increasing column vector.  If @var{y} is an array, treat the columns
of @var{y} separately.  If @var{y} is a vector, it must be a column
vector of the same length as @var{x}.

Values of @var{xi} beyond the endpoints of the interpolation result
in NA being returned.

Note that the error checking is only a significant portion of the
execution time of this @code{interp1} if the size of the input arguments
is relatively small.  Therefore, the benefit of using @code{interp1q}
is relatively small.
@seealso{@ref{doc-interp1,,interp1}}
@end deftypefn


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{doc-interpft}
@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 an array, then operate along each column of
the array separately.  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{doc-interp1,,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 additional the support function @code{spline} and @code{lookup} that
underlie the @code{interp1} function can be called directly.
@ref{Finding Elements and Checking Conditions}

@c spline scripts/polynomial/spline.m
@anchor{doc-spline}
@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{doc-pchip,,pchip}, @ref{doc-ppval,,ppval}, @ref{doc-mkpp,,mkpp}, @ref{doc-unmkpp,,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{doc-interp2}
@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 'linear', 'nearest' or 'cubic'.
If it is omitted '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 'nearest'
Return the nearest neighbor.

@item 'linear'
Linear interpolation from nearest neighbors.

@item 'pchip'
Piecewise cubic Hermite interpolating polynomial.

@item 'cubic'
Cubic interpolation from four nearest neighbors.

@item '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{doc-interp1,,interp1}}
@end deftypefn


@c interp3 scripts/general/interp3.m
@anchor{doc-interp3}
@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 '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 'nearest'
Return the nearest neighbor.

@item 'linear'
Linear interpolation from nearest neighbors.

@item 'cubic'
Cubic interpolation from four nearest neighbors (not implemented yet).

@item 'spline'
Cubic spline interpolation---smooth first and second derivatives
throughout the curve.
@end table

The default method is 'linear'.

If @var{extrap} is the string '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{doc-interp1,,interp1}, @ref{doc-interp2,,interp2}, @ref{doc-spline,,spline}, @ref{doc-meshgrid,,meshgrid}}
@end deftypefn


@c interpn scripts/general/interpn.m
@anchor{doc-interpn}
@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 '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 'nearest'
Return the nearest neighbor.

@item 'linear'
Linear interpolation from nearest neighbors.

@item 'cubic'
Cubic interpolation from four nearest neighbors (not implemented yet).

@item 'spline'
Cubic spline interpolation---smooth first and second derivatives
throughout the curve.
@end table

The default method is '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{doc-interp1,,interp1}, @ref{doc-interp2,,interp2}, @ref{doc-spline,,spline}, @ref{doc-ndgrid,,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{doc-bicubic}
@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{doc-interp2,,interp2}}
@end deftypefn

