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I have recently started learning how to use Mathematica's brilliant image processing tools, and the image I've taken as first example is the following (source):

So far I've tried to first use LocalAdaptiveBinarize on the image and then use the MorphologicalGraph for the graph mapping but the results appear quite off since the resulting graph has about $40000$ vertices, whereas we have about $310$ particles in the image. The ideal mapping would be to map each particle to a vertex (and edges between particles in contact) and study the structure of the configuration as a graph.

s2 = MaxDetect@
 LocalAdaptiveBinarize[img, 8, PerformanceGoal -> "Quality"]
gvertex = MorphologicalGraph[s2, VertexCoordinates -> Automatic]

Binarized version:

Trying without the Binarization yields somewhat better results but the resulting graph still has little to do with the image.

• Is there a way to process the image such that the particles can be more accurately detected? In other words, how should one process such particle based images (where typically like here the particles can be assumed to be spheres) in order to detect the particles positions before invoking MorphologicalGraph?




• Finally, given we perform the graph mapping, how do assess how close the mapping has been? In other words, other than the basic checks of looking at vertex counts, how can we draw a close comparison between the result and the original image?

Here is one approach:

Starting with your binarized image:

img = Import["https://i.stack.imgur.com/GAghg.png"]

The basic idea is to use the fact that the borders between particles seem to be nicely separated from the partciles themselves.

Next, we use MorphologicalComponents and SelectComponents to get the background:

bgImg = SelectComponents[MorphologicalComponents[ColorNegate[img], 0.99], Large] //
 Unitize //
 Colorize[#1, ColorRules -> 1 -> White] &

Next, some cleaning:

procImg = bgImg //
 Dilation[#, 2] & //
 Closing[#, DiskMatrix@6] & //
 ColorNegate

Now we can apply MorphologicalComponents to get the individual particles, and then we use ArrayFilter with Max to grow them together:

comps = procImg //
 ImagePad[#, -2] & //
 MorphologicalComponents[#, 0.5, CornerNeighbors -> False] & //
 ArrayFilter[Max, #, 7] &;
Colorize@comps

The last step is to use ComponentMeasurements with "Neightbours" and "Centroid" to build the graph:

ComponentMeasurements[comps, "Neighbors", "Centroid", "PropertyComponentAssociation"] //
 Graph[
 DeleteDuplicates[Sort /@ Join @@ Thread /@ KeyValueMap[UndirectedEdge]@#Neighbors],
 VertexCoordinates -> Normal@#Centroid,
 VertexSize -> 0.7,
 VertexStyle -> Yellow,
 EdgeStyle -> Directive[Yellow, Thick],
 PlotRange -> Transpose@0, 0, ImageDimensions@img,
 Prolog -> Inset[ImageMultiply[img, 0.7], Automatic, Automatic, Scaled@1]
 ] &