Confusion regarding lacunarity of koch curves with same fractal dimension but different orders I'm trying to use the concept of lacunarity to classify images, as per how a few research papers have done it. To verify whether my equations and implementation are correct, I'm trying to verify the output of my program - for images with same fractal dimension, lacunarity is inversely proportional to the denseness of an image - denser the image, lower is its lacunarity.

For example, if I plot two triangular Koch curves, one with order 4 and another with order 6(order meaning iterations), both would have the same fractal dimension. Then you would expect the image with order 4 to have comparatively higher lacunarity since its less dense. 

Now, after computing the lacunarity, there are two possible plots you can get - non-normalized and normalized lacunarity. When I plot the non-normalized lacunarity, I get the correct output - the order 4 curve has higher lacunarity. Whereas when I plot the normalized lacunarity, the plots exchange their positions and the order 6 curve ends up having higher lacunarity which seems incorrect. 

The research papers that I'm referring are only using normalized lacunarity in their applications. I'm not sure where am I making an error. 

Lacunarity is defined as follows -
For a m x n binary image, you define box size r, where r ranges from, for example, from 1 to 100.
Lacunarity is defined for every r. For a given r, it is computed as the ratio of (A+B)/B
Where A = Variance of X
B = Square of the mean of X    

Where X = no. of 1s within a box of size r x r. So you compute mean and variance over all possible boxes of size r x r in the m x n matrix. Then r is varied from 1 to 100 for example, and hence lacunarity value is computed for 100 values of r, after which you can plot lacunarity. 

Normalized lacunarity is obtained by dividing Lacunarity[r] by Lacunarity[1] for all r. 

So I need help with 2 questions:
[1] Am I making an error in the concept of lacunarity being inversely proportional to denseness, or is it that I'm making some mistake while plotting normalized lacunarity itself?

[2] Is there a way I can verify that I am implementing lacunarity calculation correctly? For example, if there are standard images whose lacunarity plots are well known. I can plot lacunarity for those standard images and verify that I am getting a similar lacunarity plot.
I think this is a conceptual issue. Your images are not fractals, they are approximations to fractals. Lacunarity is a property of the fractal curve, not of its approximation. The Koch curve has a specific lacunarity.

Your program computes a measure that is inspired by lacunarity, from a given approximation to a fractal. This measure determines what fraction of the image can be considered a hole at a given scale.

Your normalized lacunarity I hadn’t seen before. But it normalizes the curve by the value at r=1, the smallest possible scale in the image. Obviously at this scale the difference between your two images is the largest, since at a larger scale they should be identical (the difference is in the fine details). By normalizing by this value, it is expected that the ratio elsewhere, where the differences are smaller, will reverse.