Dimensions of fractals: boxing count, hausdorff, packing in R^n space - math

Question

I would like to calculate dimensions of fractal written as a n-dimensional array of 0s and 1s. It includes boxing count, hausdorff and packing dimension.
I have only idea how to code boxing count dimensions (just counting 1's in n-dimensional matrix and then use this formula:
boxing_count=-log(v)/log(n);
where n-number of 1's and n-space dimension (R^n)
This approach simulate counting minimal resolution boxes 1 x 1 x ... x 1 so numerical it is like limit eps->0. What do you think about this solution?
Do you have any idea (or maybe code) for calculating hausdorff or packing dimension?

Answer 1

The Hausdorff and packing dimension are purely mathematical tools based in measure theory. They have wonderful properties in that context but are not well suited for experimentation. In short, there is no reason to expect that you can estimate their values based on a single matrix approximation to some set.
Box counting dimension, by contrast, is well suited for numerical investigation. Specifically, let N(e) denote the number of squares of side length e required to cover your fractal set. As you seem to know, the box counting dimension of your set is the limit as e->0 of
log(N(e))/log(1/e)
However, I don't think that just choosing the smallest available value of e is generally a good idea. The standard interpretation in the physics literature, as I understand it, is to presume that the relationship between N(e) and e should be maintained over a broad range of values. A standard way to compute the box-counting dimension is compute N(e) for some choices of e chosen from a sequence that tends geometrically to zero. We then fit a line to the points in a log-log plot of N(e) versus 1/e The box-counting dimension should be approximately the slope of that line.
Example
As a concrete example, the following Python code generates a binary matrix that describes a fractal structure.
import numpy as np
size = 1024
first_row = np.zeros(size, dtype=int)
first_row[int(size/2)-1] = 1
rows = np.zeros((int(size/2),size),dtype=int)
rows[0] = first_row
for i in range(1,int(size/2)):
rows[i] = (np.roll(rows[i-1],-1) + rows[i-1] + np.roll(rows[i-1],1)) % 2
m = int(np.log(size)/np.log(2))
rows = rows[0:2**(m-1),0:2**m]
We can view the fractal structure by simply interpreting each 1 as a black pixel and each zero as white pixel.
import matplotlib.pyplot as plt
plt.matshow(rows, cmap = plt.cm.binary)
This matrix makes a nice test since it can be shown that there is an actual limiting object whose fractal dimension is log(1+sqrt(5))/log(2) or approximately 1.694, yet it's complicated enough to make the box counting estimate a little tricky.
Now, this matrix is 512 rows by 1024 columns; it decomposes naturally into 2 matrices that are 512 by 512. Each of those decomposes naturally into 4 matrices that are 256 by 256, etc. For each such decomposition, we need to count the number of sub matrices that have at least one non-zero element. We can perform this analysis as follows:
cnts = []
for lev in range(m):
block_size = 2**lev
cnt = 0
for j in range(int(size/(2*block_size))):
for i in range(int(size/block_size)):
cnt = cnt + rows[j*block_size:(j+1)*block_size, i*block_size:(i+1)*block_size].any()
cnts.append(cnt)
data = np.array([(2**(m-(k+1)),cnts[k]) for k in range(m)])
data
# Out:
# array([[ 512, 45568],
# [ 256, 22784],
# [ 128, 7040],
# [ 64, 2176],
# [ 32, 672],
# [ 16, 208],
# [ 8, 64],
# [ 4, 20],
# [ 2, 6],
# [ 1, 2]])
Now, your idea is to simply compute log(45568)/log(512) or approximately 1.7195, which is not too bad. I'm recommending that we examine a log-log plot of this data.
xs = np.log(data[:,0])
ys = np.log(data[:,1])
plt.plot(xs,ys, 'o')
This indeed looks close to linear, indicating that we might expect our box-counting technique to work reasonably well. First, though, it might be reasonable to exclude the one point that appears to be an outlier. In fact, that's one of the desirable characteristics of this approach. Here's how to do so:
plt.plot(xs,ys, 'o')
xs = xs[1:]
ys = ys[1:]
A = np.vstack([xs, np.ones(len(xs))]).T
m,b = np.linalg.lstsq(A, ys)[0]
def line(x): return m*x+b
ys = line(xs)
plt.plot(xs,ys)
m
# Out: 1.6902585379630133
Well, the result looks pretty good. In particular, this is a definitive example that this approach can work better than the simple idea of using just one data point. In fairness, though, it's not hard to find examples where the simple approach works better. Also, this set is regular enough that we get some nice results. Generally, one can't really expect box-counting computations to be too reliable.