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!