# Comparing lacunarity vs. normalized lacunarity

Background: I'm computing lacunarity plots of binary images for a project. The lacunarity plot is basically the plot of lacunarity on the y axis with box sizes on the x axis. For reference. Lacunarity is another measure for denseness, when the fractal dimension of two objects is the same.

Fractals are figures where fractal dimension and lacunarity are highly relevant. In general, the fractal dimension and lacunarity are inversely related. High fractal dimension of a figure implies low lacunarity and vice versa.

So, in the example of a triangular koch curve (KC1) and square koch curve ( KC2), KC2 has higher fractal dimension than KC1. Hence, the lacunarity of KC1 should be higher than that of KC2.

I created binary images of KC1 and KC2, and computed the lacunarities of these two curves, using the box gliding algorithm. As expected, lacunarity plot of KC2 was below that of KC1, as you can see in the image below. Red corresponds to KC1, blue corresponds to KC2.

But, when I normalized the lacunarity curves and plotted them on the same graph, the lacunarity plot for KC2 is above that of KC1, which shouldn't happen. (Normalizing meaning dividing lacunarity[r] by lacunarity1, for all r, where r = box size).

I don't understand why normalizing the lacunarity plots is giving out an unexpected result. What exactly am I doing wrong? The paper I've referred to above, uses normalized lacunarity for its research, hence I assume that normalized lacunarity should also work to measure denseness.