How to quantify translational symmetry? I'm trying to study phase transitions and I'm trying to find a way to classify regions of space based on their "crystallinity". 
I'm working with 3D coordinates, but I'll present the problem in 2D coordinates for visualization purposes.
I have a set of 2D points that look like this:

As you can see, there is  a region of high-order that gradually transitions to a region with low order. Here order is defined as something that has some sort of "translational symmetry".
What I'd like to find out is to classify those region based on their local order. In a certain sense I'd like to have a function that, given a point in space $f(x, y)$ has value 0 if the region is not ordered, 1 if it is completely ordered, and something in between when there is partial order, invariant on scale and rotation.
 A: If your transition from high order to low order occurs slowly over a large number of lattice periods, then you can try doing a discrete Fourier transform of the area of interest taking some windowing function to focus your the computation to area of interest. Then if you get relatively sharp peaks (up to spectral leakage of the window you chose), the crystallinity is higher than if you get wide spectrum.
This of course won't give you a clear 0 for unordered and 1 for ordered, but I think you won't find any definite instrument for this because the final verdict on orderliness measure depends on the application. Take this as a possible starting point of search for better ways.
Also note that this doesn't actually measure translational symmetry as it may give high crystallinity verdict for quasicrystals, which don't have any translational symmetry. Maybe to actually measure translational symmetry you need to look for some definite number of peaks in windowed spectrum (like 6 peaks for 3D crystal).
