I do not need a complete answer but just some advice.
I have a sparse matrix of points in a volume. I know a surface passing by these points exists and this surface is mostly flat and relatively smooth with some small harmonic contents. See the two images below (this surface of the first one is a bit too complicated and de second one is a bit too simple).
I am looking for a method (Laplacian surfaces / Splines / etc.) to fit my points to the best surface of minimal order. Actually I am expecting do describe my surface with less than 30 coefficients.
Any advice of the method I can use (with Matlab or Mathematica) ? If I can get a working example it would be awesome.
Ok, It seems my expectations were too high. In fact, it is very difficult to fit a 2D surface without a model. I recently tried with a pretty simple model:
$ f(x,y) = a_0 + a_1\cdot x^3 + a_2\cdot x^2y + a_3\cdot xy + a_4\cdot y^2x a_5\cdot y^3 + b_1 \sin(b_2\cdot x) + b_3 \cos(b_4\cdot x) + c_1 \sin(c_2\cdot x) + c_3 \cos(c_4 \cdot x)$
With this one I can fit my surface with a least squares approach and normal equations. The result is not bad but it would be better to find a "smart" algorithm that can build the output function with an iterative method. I still haven't find anything like this.