How many boundary conditions possibilities are there?  It doesn't seem to me that there are that many choices.

Take a look at www.maths.lth.se/na/courses/FMN081/FMN081-06/lecture11.pdf, which I've attached.  They define just 4 different end conditions including "natural" and "not-a-knot" which is what they claim Matlab uses.  The first part of the algorithm for interior points might always be the same and it is only adjustment for the boundary conditions which changes with "natural" and "cubic".



(file #48834)

Matlab's description of "natural" and "cubic" isn't much help.  You might also take a look at the documentation for interp1.  In that case, "cubic" is the same as "pchip", which uses Hermite polynomials rather than just ordinary cubic polynomials.

As a guess, it may be that the Delaunay triangulation is used to quickly locate the 4 nearest points in the original source data to the query point and then construct a cubic spline.  This could still be a win computationally, when compared to interp2 (..., "spline"), because splines would only be created for regions with query points.  In the case of interp2, it creates a spline for every single gap between data points.

after some searching, the report:

From: J.C. Chan, J. Ma, F. Canters "A comparison of superresolution reconstruction methods for multiangle CHRIS/Proba images", SPIE Image and Signal Processing for Remote Sensing XIV. Vol. 7109, 2008
"For the interpolation, the griddata function in
Matlab is used with the option of triangle-based cubic interpolation. This interpolation function is based on the Quickhull algorithm described in [21]"

[21] Barber, C. B., Dobkin, D.P. and Huhdanpaa, H.T., “The Quickhull Algorithm for Convex Hulls”, ACM
Transactions on Mathematical Software, 22(4), 469-483 (1996)

so, progress.

minor update: in another griddata bug (bug #57323), the pending patch updates the help text to indicate that Octave only supports the options linear and nearest, and a 'not yet implemented' error appears if methods v4, cubic, or natural are called.

This bug is still present in Octave 4.2.0.

The function griddata.m claims to support the method "cubic" in its help text, but trying to use it results in an error message: "error: griddata: cubic interpolation not yet implemented".

A simple script to reproduce this:

xy = gallery('uniformdata',[100 2],0);
x = xy(:,1);
y = xy(:,2);
v = exp(-x.^2-y.^2);
[xq,yq] = meshgrid(-2:.5:2, -2:.5:2);
vq = griddata(x,y,v,xq,yq, "cubic");

Using the 3.4.3 binary distribution for Windows, I receive the error:

griddata: cubic interpolation not yet implemented

when using the following:

griddata(x,y,z,xx,yy, "cubic");

According to the documentation, "cubic" is a valid parameter.