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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  {} {@var{yi} =} interp1 (@var{x}, @var{y}, @var{xi})
@deftypefnx {} {@var{yi} =} interp1 (@var{y}, @var{xi})
@deftypefnx {} {@var{yi} =} interp1 (@dots{}, @var{method})
@deftypefnx {} {@var{yi} =} interp1 (@dots{}, @var{extrap})
@deftypefnx {} {@var{yi} =} interp1 (@dots{}, "left")
@deftypefnx {} {@var{yi} =} interp1 (@dots{}, "right")
@deftypefnx {} {@var{pp} =} interp1 (@dots{}, "pp")

One-dimensional interpolation.

Interpolate input data 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}
(@code{1:length (@var{y})}).  If @var{y} is a matrix or an N-dimensional
array, the interpolation is performed on each column of @var{y}.

The interpolation @var{method} is one of:

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

@item @qcode{"previous"}
Return the previous neighbor.

@item @qcode{"next"}
Return the next neighbor.

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

@item @qcode{"pchip"}
Piecewise cubic Hermite interpolating polynomial---shape-preserving
interpolation with smooth first derivative.

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

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

Adding '*' to the start of any method above 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 using the current @var{method}.  If @var{extrap} is a
number, then replace values beyond the endpoints with that number.  When
unspecified, @var{extrap} defaults to @code{NA}.

If the string argument @qcode{"pp"} is specified, then @var{xi} should not
be supplied and @code{interp1} returns a piecewise polynomial object.  This
object 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);
near = interp1 (xp, yp, xf, "nearest");
pch = interp1 (xp, yp, xf, "pchip");
spl = interp1 (xp, yp, xf, "spline");
plot (xf,yf,"r", xf,near,"g", xf,lin,"b", xf,pch,"c", xf,spl,"m",
      xp,yp,"r*");
legend ("original", "nearest", "linear", "pchip", "spline");
@end group
@end example

@xseealso{@ref{XREFpchip,,pchip}, @ref{XREFspline,,spline}, @ref{XREFinterpft,,interpft}, @ref{XREFinterp2,,interp2}, @ref{XREFinterp3,,interp3}, @ref{XREFinterpn,,interpn}}
@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  {} {} interpft (@var{x}, @var{n})
@deftypefnx {} {} 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.
@xseealso{@ref{XREFinterp1,,interp1}}
@end deftypefn


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.

@c spline scripts/polynomial/spline.m
@anchor{XREFspline}
@deftypefn  {} {@var{pp} =} spline (@var{x}, @var{y})
@deftypefnx {} {@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 @qcode{"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 the opposite of @code{interp1} but is done for @sc{matlab}
compatibility.

@xseealso{@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  {} {@var{zi} =} interp2 (@var{x}, @var{y}, @var{z}, @var{xi}, @var{yi})
@deftypefnx {} {@var{zi} =} interp2 (@var{z}, @var{xi}, @var{yi})
@deftypefnx {} {@var{zi} =} interp2 (@var{z}, @var{n})
@deftypefnx {} {@var{zi} =} interp2 (@var{z})
@deftypefnx {} {@var{zi} =} interp2 (@dots{}, @var{method})
@deftypefnx {} {@var{zi} =} interp2 (@dots{}, @var{method}, @var{extrap})

Two-dimensional interpolation.

Interpolate reference data @var{x}, @var{y}, @var{z} to determine @var{zi}
at the coordinates @var{xi}, @var{yi}.  The reference data @var{x}, @var{y}
can be matrices, as returned by @code{meshgrid}, in which case the sizes of
@var{x}, @var{y}, and @var{z} must be equal.  If @var{x}, @var{y} are
vectors describing a grid then @code{length (@var{x}) == columns (@var{z})}
and @code{length (@var{y}) == rows (@var{z})}.  In either case the input
data must be strictly monotonic.

If called without @var{x}, @var{y}, and just a single reference data matrix
@var{z}, the 2-D region
@code{@var{x} = 1:columns (@var{z}), @var{y} = 1:rows (@var{z})} is assumed.
This saves memory if the grid is regular and the distance between points is
not important.

If called with a single reference data matrix @var{z} and a refinement
value @var{n}, then perform interpolation over a grid where each original
interval has been recursively subdivided @var{n} times.  This results in
@code{2^@var{n}-1} additional points for every interval in the original
grid.  If @var{n} is omitted a value of 1 is used.  As an example, the
interval [0,1] with @code{@var{n}==2} results in a refined interval with
points at [0, 1/4, 1/2, 3/4, 1].

The interpolation @var{method} is one of:

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

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

@item @qcode{"pchip"}
Piecewise cubic Hermite interpolating polynomial---shape-preserving
interpolation with smooth first derivative.

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

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

@var{extrap} is a scalar number.  It replaces values beyond the endpoints
with @var{extrap}.  Note that if @var{extrap} is used, @var{method} must
be specified as well.  If @var{extrap} is omitted and the @var{method} is
@qcode{"spline"}, then the extrapolated values of the @qcode{"spline"} are
used.  Otherwise the default @var{extrap} value for any other @var{method}
is @qcode{"NA"}.
@xseealso{@ref{XREFinterp1,,interp1}, @ref{XREFinterp3,,interp3}, @ref{XREFinterpn,,interpn}, @ref{XREFmeshgrid,,meshgrid}}
@end deftypefn


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

Three-dimensional interpolation.

Interpolate reference data @var{x}, @var{y}, @var{z}, @var{v} to determine
@var{vi} at the coordinates @var{xi}, @var{yi}, @var{zi}.  The reference
data @var{x}, @var{y}, @var{z} can be matrices, as returned by
@code{meshgrid}, in which case the sizes of @var{x}, @var{y}, @var{z}, and
@var{v} must be equal.  If @var{x}, @var{y}, @var{z} are vectors describing
a cubic grid then @code{length (@var{x}) == columns (@var{v})},
@code{length (@var{y}) == rows (@var{v})}, and
@code{length (@var{z}) == size (@var{v}, 3)}.  In either case the input
data must be strictly monotonic.

If called without @var{x}, @var{y}, @var{z}, and just a single reference
data matrix @var{v}, the 3-D region
@code{@var{x} = 1:columns (@var{v}), @var{y} = 1:rows (@var{v}),
@var{z} = 1:size (@var{v}, 3)} is assumed.
This saves memory if the grid is regular and the distance between points is
not important.

If called with a single reference data matrix @var{v} and a refinement
value @var{n}, then perform interpolation over a 3-D grid where each
original interval has been recursively subdivided @var{n} times.  This
results in @code{2^@var{n}-1} additional points for every interval in the
original grid.  If @var{n} is omitted a value of 1 is used.  As an
example, the interval [0,1] with @code{@var{n}==2} results in a refined
interval with points at [0, 1/4, 1/2, 3/4, 1].

The interpolation @var{method} is one of:

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

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

@item @qcode{"cubic"}
Piecewise cubic Hermite interpolating polynomial---shape-preserving
interpolation with smooth first derivative (not implemented yet).

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

@var{extrapval} is a scalar number.  It replaces values beyond the endpoints
with @var{extrapval}.  Note that if @var{extrapval} is used, @var{method}
must be specified as well.  If @var{extrapval} is omitted and the
@var{method} is @qcode{"spline"}, then the extrapolated values of the
@qcode{"spline"} are used.  Otherwise the default @var{extrapval} value for
any other @var{method} is @qcode{"NA"}.
@xseealso{@ref{XREFinterp1,,interp1}, @ref{XREFinterp2,,interp2}, @ref{XREFinterpn,,interpn}, @ref{XREFmeshgrid,,meshgrid}}
@end deftypefn


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

The interpolation @var{method} is one of:

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

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

@item @qcode{"pchip"}
Piecewise cubic Hermite interpolating polynomial---shape-preserving
interpolation with smooth first derivative (not implemented yet).

@item @qcode{"cubic"}
Cubic interpolation (same as @qcode{"pchip"} [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"}.

@var{extrapval} is a scalar number.  It replaces values beyond the endpoints
with @var{extrapval}.  Note that if @var{extrapval} is used, @var{method}
must be specified as well.  If @var{extrapval} is omitted and the
@var{method} is @qcode{"spline"}, then the extrapolated values of the
@qcode{"spline"} are used.  Otherwise the default @var{extrapval} value for
any other @var{method} is @qcode{"NA"}.
@xseealso{@ref{XREFinterp1,,interp1}, @ref{XREFinterp2,,interp2}, @ref{XREFinterp3,,interp3}, @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
