Identify circles in Image and make list of all diameters

Question

I have this SEM image:

I would like to make a list of all diameters of each circle.

Can I somehow let Mathematic identify the circles (approx.) and extract the diameters ?

Answer 1

Binarize does a good job of separating the circles from the background, so I wouldn't use anything more fancy:

img = Import["https://i.stack.imgur.com/WMekK.png"];    
bin = DeleteSmallComponents@Binarize[img];

and once you have a binarized image, ComponentMeasurements is your friend: it searches for connected components in a binary image and performs measurements on them:

components = 
 ComponentMeasurements[
  bin, {"BoundingDiskCenter", "BoundingDiskRadius", "Area", 
   "FilledCircularity"}, #Area > 100 && #FilledCircularity > 0.5 &]

returns a list like:

{1 -> {{799.443, 645.185}, 111.547, 20099.9, 0.670784}, 2 -> {{352.465, 642.549}, 114.035, 22007.8, 0.6883}, 3 -> {{123.306, 402.687}, 110.381, 22336., 0.683878}, 4 -> {{571.642, 401.299}, 111.803, 22602.6, 0.679659}, 5 -> {{341.824, 170.178}, 114.181, 26117.5, 0.70231}, 6 -> {{784.029, 169.859}, 112.099, 22142.8, 0.685281}}

i.e. for every component an element index -> {center, radius, area, filled circularity}

We can use replacement rules to turn these components to Circles:

HighlightImage[bin, {Thick, 
  components /. (index_ -> {center_, radius_, __}) :> 
    Circle[center, radius]}]

Instead of the bounding disk center/radius, you could also use:

• Centroid gives the center mass of the bright pixels
• CaliperLength/CaliperWidth measure the largest/smallest diameter

Other measurements to distinguish between circles and other objects (instead of or in addition to FilledCircularity) include:

• Eccentricity is the eccentricity of the best-fit ellipse (0 for a circle)
• CaliperElongation measures 1 - the ratio of largest/smallest diameter (0 for a circle)

You'll have to play with these a little to find what works best with your data.

Answer 2

I made some changes to the code proposed by swish in the comments and I think it worked.

markschulze = 
ImageResize[Import["https://i.stack.imgur.com/WMekK.png"], 500]
markschulzeedges = Binarize[markschulze, .75]~Blur~3
ParallelMap[
Image@Divide[
 ListConvolve[#, ImageData@markschulzeedges, 
  Ceiling[(Length@#)/2]], Total[#, 2]] &, 
Map[Function[{r}, 
  DiskMatrix[r] - ArrayPad[DiskMatrix[r - 1], 1]][#] &, 
Range[14, 18, 1]]];
HoughCircleDetection[image_Image, radiusmin_Integer: 1, 
radiusmax_Integer: 40, edgedetectradius_Integer: 10, 
minfitvalue_Real: .25, radiusstep_Integer: 1, 
minhoughvoxels_Integer: 4] := 
Module[{edgeimage, hough3dbin, hough3dbinlabels, coords, arraydim}, 
edgeimage = 
SelectComponents[
 DeleteBorderComponents[
  EdgeDetect[image, edgedetectradius, Method -> "Sobel"]], 
 "EnclosingComponentCount", # == 0 &];
hough3dbin = 
DeleteSmallComponents[
 Image3D[ParallelMap[
   Binarize[
     Image@Divide[
       ListConvolve[#, ImageData@edgeimage, 
        Ceiling[(Length@#)/2]], Total[#, 2]], minfitvalue] &, 
   Map[
    Function[{r}, 
       DiskMatrix[r] - ArrayPad[DiskMatrix[r - 1], 1]][#] &, 
    Range[radiusmin, radiusmax, radiusstep]]]], minhoughvoxels];
hough3dbinlabels = MorphologicalComponents[hough3dbin];
coords = 
ParallelMap[Round[Mean[Position[hough3dbinlabels, #]]] &, 
 Sort[Rest@Tally@Flatten@hough3dbinlabels, #1[[2]] > #2[[2]] &][[
  All, 1]]];
 arraydim = Rest@Dimensions[hough3dbinlabels];
 Print["Radii: ", radiusmin + coords[[All, 1]] - 1];
 ParallelMap[
 Function[{level, offx, offy}, 
   ImageMultiply[image, 
    Image@ArrayPad[
      DiskMatrix[
       radiusmin + level - 1], {{offx - radiusmin - level, 
        First@arraydim - offx - radiusmin - level + 1}, {offy - 
         radiusmin - level, 
        Last@arraydim - offy - radiusmin - level + 1}}]]][
  Sequence @@ #] &, coords]];

Show[ImageApply[Plus, HoughCircleDetection[markschulzeedges, 40, 80]],
ImageSize -> 200]

Radii: {52,52,52,51,53,53}

it would be better if you cropped the images first

you must always check the boundaries of the search (in the last line of the code)
for images like yours I set them between 40-80

here is another example with different circles (set 10-100)

Radii: {39,63,19,40,36,29,52}

of course these numbers are scaled so you will have to find the correct scale