Dear Denis, I think I have an answer for your picture. I just applied these lines of code to the picture you posted for the fungi pores. First, I imported your image as imfungi. Then apply ImageMesh

regBinIm = ImageMesh[imfungi]

Now, count the number of 2D mesh cells for each kind of polygon having from 3 up to 13 sides, according to the definition of Voronoi entropy

numedgepoly = 
 Table[Length@
   Cases[MeshCells[regBinIm, 2], 
    Polygon[Flatten@Table[{_}, {i}]]], {i, 3, 13}]

You can also get the total number of polygons in your picture by applying Total to the previously obtained list. You can also check that it's equal to the number of ConnectedMeshComponents obtained from ImageMesh region. I get a total number of 913 polygons. These are the final data to compute Voronoi entropy

 kdata=numedgepoly/Total[numedgepoly]
{0,207/913,410/913,245/913,47/913,4/913,0,0,0,0,0}

Then delete elements equal to zero (i.e. polygons with higher number of edges or 3 edges) in order to suppress indeterminate error message from Log function with

DeleteCases[kdata,0]
{207/913,410/913,245/913,47/913,4/913}

and finally you can compute the Voronoi entropy, actually the form of Shannon entropy with

-Total@Map[#*Log[#]&,DeleteCases[kdata,0]]//N

which gives the result 1.22549, a little above the number you inferred from simulations. This can also be refined by a better image manipulation at the start. I hope this can be a helpful strategy for other pictures!

Dear Jessica,
This parameter (and your function) will be very useful to me in the study that will be published soon.
It’s about the distribution of the pores of the mushrooms (Tinder Fungi).
Here’s a fragment of it's Voronoi Mesh :

I think it’s like Poisson Disk with Voronoi Entropy $1.0 \le VE \le 1.2$, but not sure exactly.
I’m comparing distributions by different parameters, and definitely add yours.
There should be a post in these groups within a week, and I’ll be glad if you to take a look (-_-)

-- you have earned Featured Contributor Badge Your exceptional post has been selected for our editorial column Staff Picks http://wolfr.am/StaffPicks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!

Take a look at the [Experimental] function HardcorePointProcess.

proc = HardcorePointProcess[30, 0.3, 2];
reg = Rectangle[{0, 0}, {10, 10}];
pts = RandomPointConfiguration[proc, reg];
Show[VoronoiMesh[pts[[1, 1]]], PlotRange -> {{0, 10}, {0, 10}}]