Spatial spectrum of EEG data: non-uniform DFT?
QUESTION 1: Do you think an nonuniform DFT makes sense for such an electrode grid?


The answer to that specific question is yes.


  QUESTION 2: The reason I would like to used the nonuniform DFT is that I would like to extract the most parsimonious frequency representation. ...


With this, I suppose that you are after the most parsimonious frequency representation in the spatial domain. In other words, the answer to the question, "How many waves and at what frequencies and amplitudes compose the "surfface" of the electric field potential that is detected by the electrodes at the scalp". If that is accurate (?), it will be a useful reference later on.


  ... For the non-uniform case it is unclear to me, how many non-interpolated coefficients one can extract and what these coefficients are. Any idea?


In the case of a type 2 NUDFT, it is still $\frac{N}{2}$ discrete frequencies.

In the non-iterative cases, there is an implicit interpolation step anyway in order to be able to then use the Fast Fourier Transform, whether this is done via oversampling or "gridding". As is mentioned in "Using NFFT 3 – a software library for various nonequispaced fast Fourier transforms" (page 15): 

"The chief idea of the NFFT algorithm is to use standard FFTs in combination with an approximation scheme that is based on a window function $\phi$. This function needs to be mutually well localized in time/spatial and frequency domain."


  QUESTION 3: I was checking the NFFT implementation, however I am confused by the fact that the grid points must be constrained to the [-1/2 1/2] grid. Is it enough if I scale down my grid to this range?


I cannot find that specific reference but this probably refers to normalised frequency. If you notice (1) from here, the only thing that changes between the uniform and non-uniform Discrete Fourier Transform (DFT) is the substitution of the time variable with a sequence. In other words, the discrete sinusoids of the DFT (the $k$s) still "run" at the frequencies they would run anyway but you take extra care to correlate the acquired signal samples at the right phase. That is, take their relative timestamps into account, rather than simply putting them one next to the other as if they were acquired at equi distant sample intervals.


  QUESTION 4: The electrodes are actually placed on the scalp surface. The grid below is a 2-D representation of the 3D scalp surface. Is there a way to compute a nonuniform DFT directly on the 3D grid?


Yes, with the non-uniform spherical version of the FFT. The brain is not a sphere though.

Now, let me suggest a different approach to achieve what you are trying to do.

The reason I asked you "...how do you envisage this "spatial spectrum"..." is because given a $N \times M$ matrix $X$ of acquired data over $N$ samples and $M$ electrodes, there are $N$ different "spatial spectra" that you can obtain. This is because, time and space are two different variables and obtaining the spatial spectrum means that you obtain the two dimensional Fourier Transform over all stations ("theses") at the same time instance. So, you freeze time and you obtain the spatial spectrum of *the electric potential measured across the scalp".

The next natural thought here is to say "that's not a big deal I will integrate over time and possibly over frequency as well and therefore will be extracting the spatial spectrum of the strongest wave activity across the scalp". Which leads us to things like filtering at a brain wave rhythm and then evaluating the distribution of those waves across the scalp and then, by that integration step, we essentially "freeze" activity over a unit of time and that allows us to obtain a spatial spectrum.

Eventually, you would be getting closer and closer to something that a technique like LORETA gives you for free.

LORETA estimates a small number of sources that are responsible for the electric activity across the scalp and projects that activity back to the scalp to form a potential map. So, essentially, if you were to use LORETA to estimate the distribution of the field you also get a nicely interpolated potential from the small number of sources that the low number of electrodes can estimate and you can then do a two dimensional FFT on that "reconstruction".

For more information you might want to have a closer look at this.

Hope this helps.