octave: Multi-dimensional Interpolation
29.2 Multi-dimensional Interpolation
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There are three multi-dimensional interpolation functions in Octave,
with similar capabilities. Methods using Delaunay tessellation are
described in 
Interpolation on Scattered Data.
-- : ZI = interp2 (X, Y, Z, XI, YI)
-- : ZI = interp2 (Z, XI, YI)
-- : ZI = interp2 (Z, N)
-- : ZI = interp2 (Z)
-- : ZI = interp2 (..., METHOD)
-- : ZI = interp2 (..., METHOD, EXTRAP)
Two-dimensional interpolation.
Interpolate reference data X, Y, Z to determine ZI at the
coordinates XI, YI. The reference data X, Y can be matrices, as
returned by ‘meshgrid’, in which case the sizes of X, Y, and Z must
be equal. If X, Y are vectors describing a grid then ‘length (X)
== columns (Z)’ and ‘length (Y) == rows (Z)’. In either case the
input data must be strictly monotonic.
If called without X, Y, and just a single reference data matrix Z,
the 2-D region ‘X = 1:columns (Z), Y = 1:rows (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 Z and a refinement
value N, then perform interpolation over a grid where each original
interval has been recursively subdivided N times. This results in
‘2^N-1’ additional points for every interval in the original grid.
If N is omitted a value of 1 is used. As an example, the interval
[0,1] with ‘N==2’ results in a refined interval with points at [0,
1/4, 1/2, 3/4, 1].
The interpolation METHOD is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.
"pchip"
Piecewise cubic Hermite interpolating
polynomial—shape-preserving interpolation with smooth first
derivative.
"cubic"
Cubic interpolation (same as "pchip").
"spline"
Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
EXTRAP is a scalar number. It replaces values beyond the endpoints
with EXTRAP. Note that if EXTRAP is used, METHOD must be specified
as well. If EXTRAP is omitted and the METHOD is "spline", then the
extrapolated values of the "spline" are used. Otherwise the
default EXTRAP value for any other METHOD is "NA".
See also: interp1 XREFinterp1, interp3 XREFinterp3,
interpn XREFinterpn, meshgrid XREFmeshgrid.
-- : VI = interp3 (X, Y, Z, V, XI, YI, ZI)
-- : VI = interp3 (V, XI, YI, ZI)
-- : VI = interp3 (V, N)
-- : VI = interp3 (V)
-- : VI = interp3 (..., METHOD)
-- : VI = interp3 (..., METHOD, EXTRAPVAL)
Three-dimensional interpolation.
Interpolate reference data X, Y, Z, V to determine VI at the
coordinates XI, YI, ZI. The reference data X, Y, Z can be
matrices, as returned by ‘meshgrid’, in which case the sizes of X,
Y, Z, and V must be equal. If X, Y, Z are vectors describing a
cubic grid then ‘length (X) == columns (V)’, ‘length (Y) == rows
(V)’, and ‘length (Z) == size (V, 3)’. In either case the input
data must be strictly monotonic.
If called without X, Y, Z, and just a single reference data matrix
V, the 3-D region ‘X = 1:columns (V), Y = 1:rows (V), Z = 1:size
(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 V and a refinement
value N, then perform interpolation over a 3-D grid where each
original interval has been recursively subdivided N times. This
results in ‘2^N-1’ additional points for every interval in the
original grid. If N is omitted a value of 1 is used. As an
example, the interval [0,1] with ‘N==2’ results in a refined
interval with points at [0, 1/4, 1/2, 3/4, 1].
The interpolation METHOD is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.
"cubic"
Piecewise cubic Hermite interpolating
polynomial—shape-preserving interpolation with smooth first
derivative (not implemented yet).
"spline"
Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
EXTRAPVAL is a scalar number. It replaces values beyond the
endpoints with EXTRAPVAL. Note that if EXTRAPVAL is used, METHOD
must be specified as well. If EXTRAPVAL is omitted and the METHOD
is "spline", then the extrapolated values of the "spline" are used.
Otherwise the default EXTRAPVAL value for any other METHOD is "NA".
See also: interp1 XREFinterp1, interp2 XREFinterp2,
interpn XREFinterpn, meshgrid XREFmeshgrid.
-- : VI = interpn (X1, X2, ..., V, Y1, Y2, ...)
-- : VI = interpn (V, Y1, Y2, ...)
-- : VI = interpn (V, M)
-- : VI = interpn (V)
-- : VI = interpn (..., METHOD)
-- : VI = interpn (..., METHOD, EXTRAPVAL)
Perform N-dimensional interpolation, where N is at least two.
Each element of the N-dimensional array V represents a value at a
location given by the parameters X1, X2, ..., XN. The parameters
X1, X2, ..., XN are either N-dimensional arrays of the same size as
the array V in the "ndgrid" format or vectors. The parameters Y1,
etc. respect a similar format to X1, etc., and they represent the
points at which the array VI is interpolated.
If X1, ..., XN are omitted, they are assumed to be ‘x1 = 1 : size
(V, 1)’, etc. If M is specified, then the interpolation adds a
point half way between each of the interpolation points. This
process is performed M times. If only V is specified, then M is
assumed to be ‘1’.
The interpolation METHOD is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.
"pchip"
Piecewise cubic Hermite interpolating
polynomial—shape-preserving interpolation with smooth first
derivative (not implemented yet).
"cubic"
Cubic interpolation (same as "pchip" [not implemented yet]).
"spline"
Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
The default method is "linear".
EXTRAPVAL is a scalar number. It replaces values beyond the
endpoints with EXTRAPVAL. Note that if EXTRAPVAL is used, METHOD
must be specified as well. If EXTRAPVAL is omitted and the METHOD
is "spline", then the extrapolated values of the "spline" are used.
Otherwise the default EXTRAPVAL value for any other METHOD is "NA".
See also: interp1 XREFinterp1, interp2 XREFinterp2,
DONTPRINTYET interp3 XREFinterp3, spline XREFspline, *notendgrid:
DONTPRINTYET interp3 XREFinterp3, spline XREFspline, ndgrid
XREFndgrid.
A significant difference between ‘interpn’ and the other two
multi-dimensional interpolation functions is the fashion in which the
dimensions are treated. For ‘interp2’ and ‘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 ‘interpn’ effectively reverses the ’x’ and ’y’ dimensions. Consider
the example,
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,:,:)));
where ‘vi’ and ‘vi2’ are identical. The reversal of the dimensions is
treated in the ‘meshgrid’ and ‘ndgrid’ functions respectively.
Info Catalog
octave: One-dimensional Interpolation
octave: Interpolation