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I just finished a small, demo exercise on fractal dimension of a plant by using MATLAB and box-count method. There were two different treatments. A plant treated with a specific hormone and a plant without treated with anything.

Above there are the initial pictures of these plants [treated and untreated].

Treated Untreated

Here are the logarithmic plots for these two treatments

Treated-log Untreated-log

Finally i got the derivatives of ln(N)/ln(R) and as you can see :

Treated-slope Untreated-slope

So after all these photos the fractal dimension of treated plant was 1.8853 and the dimension of the untreated was 1.9322

So i would like to understand what these two number expressing. Can we get a conclusion about these two different treatments ?

Also in the derivative plots. What the (kind-of) lineal part of the function means ? Can we say that in these regions our image acts like a fractal?

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When the log-log plot of $n(r)$ is approximately linear, this means that the number of boxes covering the set grows as some power of $r$, that is $n(r)\approx Cr^{-d}$. This is an indication of the set having box-counting (Minkowski) dimension equal to $d$. Yes, you can say that in this range of scales $r$ the image looks like a fractal.

The linearity appears to fail at very small scale, but this is to be expected as you hit the resolution limit of the image. No physical object can be uniformly fractal at all scales: something has to change at very small scales (quantum mechanics) or very large scales (cosmology).

Unfortunately, there is no easy way to interpret the results in biological sense. Does greater value of dimension mean better health of the plant, or is it the other way around? We don't know.

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