2d dft of a scattered signal
Without knowing what you want to get, I will assume you are interested in periodicities of the elevated dot patterns. They can give you access to roughness measures, texture attributes.

Visually, 2D periodicities are apparent from your data point $(x,y)$ locations. At different scales though: beside a potential periodic base pattern, little streaks of 5 to 10 dots, closely spaced, can be observed. 

The situation will depend on how to want to treat the data. If you treat it as an image, putting white values (or whatever) between dots, you are already doing interpolation. In that case, a true 2D DFT would take a separable 1D DFT on both rows AND columns. So, you definitely need to venture into 2D, unless you will only  see horizontal frequencies; look at the last line of your image: there are 8 or 9 periods only, you are missing out the small slope.

I   cannot propose data interpolation without knowledge of the underlying process, and the $z$ value. My take is that you ought to try with true 2D non-uniform DFT first. Interpolation causes smearing.

In recent question (what is $1/f$ property of the visual world?), Frequency Synthesis of Landscapes was evoked, and a parametric spectrum modeling (eg decay in $1/f^p$) can provide you with parameters to compare different surfaces. For some spectral shapes, this can be naturally done in a Fourier domain. There should be related techniques under the field of "Surface roughness analysis".

My basic thinking for a procedure is the following: your different scans of surfaces may have different $(x,y)$ locations, and sizes. So it is not easy to compare their properties directly in the space domain, unless you regrid, interpolate, etc. which can cause harm. Non regular Fourier transforms can regularize the outcomes: from any two scattered grids, you can cast Fourier coefficients to the same 2D regular frequency grid, and infer global parameters from them: peaks, moments, parametric fits, etc. 

So you can check NUFFT (NFFT, USFFT) Software and the related keywords:


  algorithms have been developed to overcome this limitation - generally
  referred to as non-uniform FFTs (NUFFT), non-equispaced FFTs (NFFT) or
  unequally-spaced FFTs (USFFT)


with a related paper: Accelerating the Nonuniform Fast Fourier Transform, by Leslie Greengard and June-Yub Lee. Apparently there are Matlab versions too: 
NUFFT, NFFT, USFFT.

If this is not enough, you can try (lifting) wavelet transforms on 2D irregular grids, to catch the proper scale and local details. You could mesh it with MeshLab. And then use lifting wavelets for a multiresolution vision.