Catmull-Rom splines can be easily generalized to any number of dimensions.
The cubic Hermite spline article will remind you that for some 4-vector which is a function of x alone, where is the value at of the function to be interpolated.
Rewrite this approximation as
This formula can be directly generalized to N dimensions:[1]
Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.
In regards to efficiency, the general formula can in fact be computed as a composition of successive -type operations for any type of tensor product splines, as explained in the tricubic interpolation article.
However, the fact remains that if there are terms in the 1-dimensional -like summation, then there will be terms in the -dimensional summation.
Irregular grid (scattered data)
Schemes defined for scattered data on an irregular grid are more general.
They should all work on a regular grid, typically reducing to another known method.