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28.2 Multidimensional Interpolation
There are three multidimensional interpolation functions in Octave, with similar capabilities. Methods using Delaunay tessellation are described in Interpolation on Scattered Data.
 Function File: zi = interp2 (x, y, z, xi, yi)
 Function File: zi = interp2 (Z, xi, yi)
 Function File: zi = interp2 (Z, n)
 Function File: zi = interp2 (…, method)
 Function File: zi = interp2 (…, method, extrapval)
Twodimensional interpolation. x, y and z describe a surface function. If x and y are vectors their length must correspondent to the size of z. x and y must be monotonic. If they are matrices they must have the
meshgrid
format.
interp2 (x, y, Z, xi, yi, …)
Returns a matrix corresponding to the points described by the matrices xi, 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.

interp2 (z, xi, yi)
Assumes
x = 1:rows (z)
andy = 1:columns (z)

interp2 (z, n)
Interleaves the matrix z ntimes. If n is omitted a value of
n = 1
is assumed.
The variable method defines the method to use for the interpolation. It can take one of the following values
 'nearest'
Return the nearest neighbor.
 'linear'
Linear interpolation from nearest neighbors.
 'pchip'
Piecewise cubic hermite interpolating polynomial (not implemented yet).
 'cubic'
Cubic interpolation from four nearest neighbors.
 'spline'
Cubic spline interpolation–smooth first and second derivatives throughout the curve.
If a scalar value extrapval is defined as the final value, then values outside the mesh as set to this value. Note that in this case method must be defined as well. If extrapval is not defined then NA is assumed.
See also: interp1.

 Function File: vi = interp3 (x, y,z, v, xi, yi, zi)
 Function File: vi = interp3 (v, xi, yi, zi)
 Function File: vi = interp3 (v, m)
 Function File: vi = interp3 (v)
 Function File: vi = interp3 (…, method)
 Function File: vi = interp3 (…, method, extrapval)
Perform 3dimensional interpolation. Each element of the 3dimensional array v represents a value at a location given by the parameters x, y, and z. The parameters x, x, and z are either 3dimensional arrays of the same size as the array v in the 'meshgrid' format or vectors. The parameters xi, etc. respect a similar format to x, etc., and they represent the points at which the array vi is interpolated.
If x, y, z are omitted, they are assumed to be
x = 1 : size (v, 2)
,y = 1 : size (v, 1)
andz = 1 : size (v, 3)
. 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 be1
.Method is one of:
 'nearest'
Return the nearest neighbor.
 'linear'
Linear interpolation from nearest neighbors.
 'cubic'
Cubic interpolation from four nearest neighbors (not implemented yet).
 'spline'
Cubic spline interpolation–smooth first and second derivatives throughout the curve.
The default method is 'linear'.
If extrap is the string 'extrap', then extrapolate values beyond the endpoints. If extrap is a number, replace values beyond the endpoints with that number. If extrap is missing, assume NA.
 Function File: vi = interpn (x1, x2, …, v, y1, y2, …)
 Function File: vi = interpn (v, y1, y2, …)
 Function File: vi = interpn (v, m)
 Function File: vi = interpn (v)
 Function File: vi = interpn (…, method)
 Function File: vi = interpn (…, method, extrapval)
Perform ndimensional interpolation, where n is at least two. Each element of the ndimensional array v represents a value at a location given by the parameters x1, x2, …, xn. The parameters x1, x2, …, xn are either ndimensional 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 be1
.Method is one of:
 'nearest'
Return the nearest neighbor.
 'linear'
Linear interpolation from nearest neighbors.
 'cubic'
Cubic interpolation from four nearest neighbors (not implemented yet).
 'spline'
Cubic spline interpolation–smooth first and second derivatives throughout the curve.
The default method is 'linear'.
If extrapval is the scalar value, use it to replace the values beyond the endpoints with that number. If extrapval is missing, assume NA.
A significant difference between interpn
and the other two
multidimensional 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.
The result of this code can be seen in fig:interpn.
Figure 28.4: Demonstration of the use of interpn
In additional the support function bicubic
that underlies the
cubic interpolation of interp2
function can be called directly.
 Function File: zi = bicubic (x, y, z, xi, yi, extrapval)
Return a matrix zi corresponding to the bicubic interpolations at xi and yi of the data supplied as x, y and z. Points outside the grid are set to extrapval.
See http://wiki.woodpecker.org.cn/moin/Octave/Bicubic for further information.
See also: interp2.
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