Bicubic interpolation for multiple dimensions?

Question

There is a Wikipedia article on bicubic interpolation fro 2D functions.

Is it possible to generate the same/similar matrices for ND function interpolation?

Answer 1

Normally you would do this by creating two implicit functions and then interpolating between them. For the case of 2D shapes, all the data points are constrained within two planes, one for each shape. These planes are then placed parallel to one another in 3D (N+1)D where N=2.

Answer 2

Yes, this is called multivariate polynomial interpolation. The set of all possible multivariate monomials, $x^i y^j z^k ...$, is a basis. Their values at the sample points make up the interpolation matrix. There is a condition on the points location for its invertibility, but if your data is on a more or less regular grid, for example, the solution (for the coefficients) is unique.