Interpolation in quadtrees/octrees

Question

I'm looking for an interpolation algorithm for quadtrees and octrees that is derived from bi(tri)linear or bi(tri)cubic interpolation. I'm mostly interested in the case where:

• the interpolant is globally continuous (or continuous together with the first derivative),
• we have interpolated values at nodes (not centers) of squares/cubes.

I can find a lot of related results but the algorithm for this particular case eludes me. Could anyone help me?

Answer 1

OK, I've found one possible method. The key is to have interpolation nodes placed only in points with all four (eight) edges (+ boundaries), e.g.

x---o---x---x---x
|       |   |   |
|       |   |   |
|       |   |   |
o       o---x---x
|       |   |   |
|       |   |   |
|       |   |   |
x---o---x---o---x
|       |       |
|       |       |
|       |       |
o       o       o
|       |       |
|       |       |
|       |       |
x---o---x---o---x

You have nodes at x and not at o. I'll wait for other options.