Do you use ImageJ to estimate the fractal dimension? Then the fractal dimension is actually the fractal box dimension. The fractal box dimension measures how the object fills the "space". If your object becomes larger, the fractal dimension increases...

Do you use fractal analysis with ImageJ(FracLac) and/or another software? May you send 2 captured pictures of your process at different transformation time A and B?

Yes. I am using FracLac plugin in ImageJ to measure the fractal box dimension. I am attaching two images captured at different times.

The Fd for 2D images represents the degree of 1D line-like (Fd=1.0) to 2D plane-like (Fd=2.0) characteristic of an object. For an infinite boundary Fd is not directly related to area. When using images, however, we have boundary limits (the image dimensions). The maximum area an object can possess is the area of the image. This is a 2D plane, thus as your area increases toward the full image area the Fd will necessarily approach 2.0.

What you did not explain in your question was the object segmentation. When using FracLac, segmentation of appropriate objects is important. Are your objects of interest a single object that changes shape boundaries or multiple objects increasing in number between phase A and phase B? Assuming multiple objects, try ensuring the segmentation separates them from each other and obtain the Fd for each object (average Fd for all objects) rather than the Fd of all objects as one single entity. The difference between these approaches is the first is a single particle with varying perimeter to area ratio while the second is a composite of all particles making a lattice. The lattice will necessarily fill in and lead to the problem noted above.

If you can, please post an example image for further help. Scratch that, just noticed the posted time.zip file. I see multiple objects that should each have their own Fd. The average Fd of all objects in each phase should be used rather than the Fd of the composite object. Let us know which one you used.

Thanks Kristopher. I have calculated Fd of all objects as one single entity. As the number of domains as well as area of each domain of phase B increase with time, how can I focus on a single object ? I mean how to do segmentation.

The fractal dimension of the 2D images is not a direct measure of (only) the degree of filling the picture area (as is the ratio between the positive area and the total area). The Fd reflects also the irregularity of the inner boundaries. In my experience on pictures similar with yours the log-log plot displays clear distinctions between at least two domains, with distinct Fd: a structural one and a textural one. The fractal character of the real object you analyse is usually limited to a domain of scales, on which the linear correlation defining the Fd is good to excellent. I think you should select the scale domains and make consistent comparisons. If the dynamic of the process is related to the inner boundaries, to their length and/or irregularity, I suppose Fd could be related to changes in that dynamics. Scarce inner boundaries make, I think, the fractal analysis hardly relevant.

FracLac has the ability to scan ROIs from the ROI manager versus the entire image. After setting up your desired scan type, use the button in the FracLac Panel title ROIs. This will output Fd per ROI.

Assuming images are binarized as provided in sample. To load the ROIs into the manager, use Analyze > Analyze Particles... Set the size option to 0-Infinity and circularity to 0.00-1.00 to ensure all objects are segmented. Show nothing, and only check "Add to Manager". Remember to clear the ROI manager prior to running this command. This will isolate your objects into ROIs and store them in the ROI manager. Verify the segmentation and handle as appropriate, i.e., watershed or other separation method may be necessary.

With the ROIs stored, run the ROI scan type in FracLac. Warning on accuracy of box counting method. Box counting method requires scaling the object to determine perimeter at scale steps. If your object is too small, then scaling will be limited and lead to an erroneous Fd value. I recommend that in addition to Fd from FracLac, you also use Analyze > Set Measurements... and select Area and Perimeter. Then using the ROI Manager deselect all ROIs and run the measure command. This will measure all ROIs, join this data back with the FracLac Fd values to use a screening variables.

Using your sample images, I found 29 objects in Phase A and 41 in Phase B. After screening to 1.0

What about lacunarity ? I should follow the same procedure. Thank you, Kristopher.

Yes same procedure. Isolate objects into separate ROIs then use the scan ROIs button of FracLac. The output will have will have Fd and Lacunarity.

For more help and info on FracLac see the guide http://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm

Use the side bar menu to navigate to more detailed info on Fractal Analysis, FraLac tutorials, etc. Tutorial page is here http://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Tutorials.htm

Thanks.

I would suggest caution in relying heavily at this stage on ROIs. They are very useful when rational and well defined selection criteria are already defined. When „fishing” for correlations with an abstract measure as Fd, ROIs can be a source for bias, for subjectivity.

Hi Liviu. In your first post you said ".. the log-log plot displays clear distinctions between at least two domains..". log-log plot of what ?

Fd itself doen't mean anything without log-log plot. Saling range can be of the same importance as Fd

Hi Dmitry. Do you mean log(box-number)-vs-log(box-dimension) plot?

There are a lot of methods for measuring Fd. And you will never measure real Fd. In fact you can use "a kind of Fd measured using ... method" - it can be enough for some purposes. In any method you will have a log-log plot. For my experience - this plot is the most valuable thing in fractal analysis. First of all you should find a straight segment for al least one order of X.

I'm not sure that box counting method is good for your case. As for me - I would prefer Korcak method (not sure anout spelling). I gouss in your case it would be a straight part (power low) finishing with exponential decay (due to boundary). Enlarging your area will enlarge scaling range (move this transition point from power to exp).

I attach log-log plots for the two pictures, illustrating the scale domains with distinct Fds. Labels are quite self-explanatory. I used FracLab.

I have a question: your fractal objects are with black or with white? It is very important to know this in fractal analysis, because without Benoit software, almost all fractal analysis softwares consider background to be white and fractal object to be black. If fractal object in your situation are with white please revert your image with imageJ - Edit - Invert. After invert, with new inverted images, make again with FracLac, your fractal analysis.

I suggest to you, as also reported by Dmitry, to analyze the scaling behavior of the measures. In the book of Paul Meakin " Fractals, scaling and growth far from equilibrium (Cambridge University Press)" you may find useful comments. From a physical point of view it is important to define the lower and upper limits of the cut-offs. Try to make yourself a code, I think it's easy.

Thank all of you.

There exists a program (source) named "fd3" you can find on the web.

It calculates fractal dimension for n-dimensional objects.

You can try to trace how it sets limits for the cut-offs.