Multi-dimensional Interpolation (GNU Octave (version 5.2.0))

29.2 Multi-dimensional Interpolation

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

If called without 가로, 세로, and just a single reference data matrix z, the 2-D region 가로 = 1:columns (z), 세로 = 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".

같이 보기: interp1, interp3, interpn, meshgrid.

vi = interp3 (가로, 세로, 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 가로, 세로, z, v to determine vi at the coordinates xi, yi, zi. The reference data 가로, 세로, z can be matrices, as returned by meshgrid, in which case the sizes of 가로, 세로, z, and v must be equal. If 가로, 세로, z are vectors describing a cubic grid then length (가로) == columns (v), length (세로) == rows (v), and length (z) == size (v, 3). In either case the input data must be strictly monotonic.

If called without 가로, 세로, z, and just a single reference data matrix v, the 3-D region 가로 = 1:columns (v), 세로 = 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".

같이 보기: interp1, interp2, interpn, meshgrid.

vi = interpn (가로1, x2, …, v, 세로1, y2, …)

vi = interpn (v, 세로1, 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 가로1, x2, …, 가로n. The parameters 가로1, x2, …, 가로n are either n-dimensional arrays of the same size as the array v in the "ndgrid" format or vectors. The parameters 세로1, etc. respect a similar format to 가로1, etc., and they represent the points at which the array vi is interpolated.

If 가로1, …, 가로n 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".

같이 보기: interp1, interp2, interp3, spline, ndgrid.

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. The result of this code can be seen in Figure 29.4.

Figure 29.4: Demonstration of the use of interpn