Fraclac box counting - fractal dimension of aggregate nanoparticles

Hi there!

I am trying to estimate the fractal dimension of nanoaggregates, e.g. soot particles, using Fraclac box counting. But before calculating the Db for the actual particles I would like to test the algorithm with some simple test images.

Playing around with test images, such as a circle contour, I get Db = 1.0, as expected. But when I calculate the Db for a filled circle or a square I get Db = [1.7…1.8] and not 2.0, as I would expect.

contour Db = 1.0
square Db = 1.78
Untitled Db = 1.71
All images were first binarized and then analyzed as monofractals. Tried many different options, such as minimum box size, number of positions for grids – no change. When I visualize the grid (“draw grids”), it looks fine.

Did someone encounter this problem, any thoughts?

Thanks!

I would contact the author of this plugin directly, see:

https://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm

You can get erroneous values of D if the range of scales is not set right.
Fractal analysis works when you consider orders of magnitude of scale over which your measurements are done, not just any arbitrary scales, and specially when the objects are too small in relation to the pixel size as the object itself gets too close to the largest grid sizes.
In addition to contact the author, you could post the log-log plots obtained so we cab have an idea of what is going on.

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Thank you for your answers!

I have played a bit more with the minimum box size (default settings, 1% of the image size, 2% image size, 5% image size) but nothing changed. @gabriel, this is what you meant, right?

Below are the default settings and the results I got including the log-log plots.

image

image

Hopefully, that is helpful.

I think your plots are too crowded at the right hand side (the coarse scales, where the object is not really filling all boxes in the expected fashion).
Spacing the plot logarithmically, e.g.

run("Fractal Box Count...", "box=2,4,8,16 black");

might help reducing that problem.
If I do that I get the mass dimension of a disc diameter=226 as D=1.9368
If I floodfill the image (theoretically D=2.0) I get D=1.9847 which I would think it is pretty good. If the filled image size is a power of 2, with the boxes range above I get the expected D=2.0, obviously.
As mentioned before, with small objects, the computation of D will be only approximate as you are not spanning “orders of magnitude” of scale, which is what you need to define a fractal object.
Other methods to get more accurate measures include retaining the smallest number of boxes, after a series of rounds grids placed at random positions to estimate the minimal cover.
Hope this helps.

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The main problem was with the maximum box size. It was simply too large, which made the log-log plot too crowded and biased the result. Exactly as you noted. Thank you @gabriel!