Full disclosure: I am grad student in geosciences teaching myself about fractals. Since I've been learning on the fly, my knowledge is pretty spotty.
Here is what I know about lacunarity:
- It quantifies the distribution of gaps in a fractal.
- There are a lot of ways to calculate lacunarity, and which method you use seems to depend on your discipline.
I've been using software to get the lacunarities of a bunch of 2D binary images, and it uses a box-counting algorithm to calculate lacunarity with the formula $\lambda_{\epsilon,g} = (\sigma/\mu)^{2}_{\epsilon,g}$ where $\sigma$ and $\mu$ are the standard deviation and mean for how many pixels are in each box of size $\epsilon$ for a particular grid orientation $g$. And then it does the usual graph of $\ln\lambda$ vs $\ln\epsilon$, pulls the slope out: bam, donezo.
My values all seem to be between 0.2 and 0.5, and my question is: would the $L$ values around 0.5 be high in the grand scheme of things? I've seen that lacunarity (if calculated such that $L=0$ indicates a perfectly homogeneous shape) can get higher than 1 for a 2D shape, but surely not 2, right? Or am I just assuming it can't get higher than 2 because the dimension of these images can't get higher than 2, and that's a bad assumption to make?
Here's a couple (1, 2) of those images so you have an idea of what I'm looking at.
