Fractal dimension computing

fractal-dimension.py

# -----------------------------------------------------------------------------
# From https://en.wikipedia.org/wiki/Minkowski–Bouligand_dimension:
#
# In fractal geometry, the Minkowski–Bouligand dimension, also known as
# Minkowski dimension or box-counting dimension, is a way of determining the
# fractal dimension of a set S in a Euclidean space Rn, or more generally in a
# metric space (X, d).
# -----------------------------------------------------------------------------
import scipy.misc
import numpy as np


def fractal_dimension(Z, threshold=0.9):

    # Only for 2d image
    assert(len(Z.shape) == 2)

    # From https://github.com/rougier/numpy-100 (#87)
    def boxcount(Z, k):
        S = np.add.reduceat(
            np.add.reduceat(Z, np.arange(0, Z.shape[0], k), axis=0),
                               np.arange(0, Z.shape[1], k), axis=1)

        # We count non-empty (0) and non-full boxes (k*k)
        return len(np.where((S > 0) & (S < k*k))[0])


    # Transform Z into a binary array
    Z = (Z < threshold)

    # Minimal dimension of image
    p = min(Z.shape)

    # Greatest power of 2 less than or equal to p
    n = 2**np.floor(np.log(p)/np.log(2))

    # Extract the exponent
    n = int(np.log(n)/np.log(2))

    # Build successive box sizes (from 2**n down to 2**1)
    sizes = 2**np.arange(n, 1, -1)

    # Actual box counting with decreasing size
    counts = []
    for size in sizes:
        counts.append(boxcount(Z, size))

    # Fit the successive log(sizes) with log (counts)
    coeffs = np.polyfit(np.log(sizes), np.log(counts), 1)
    return -coeffs[0]

I = scipy.misc.imread("sierpinski.png")/256.0
print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I))
print("Haussdorf dimension (theoretical):        ", (np.log(3)/np.log(2)))

There is one problem when I input the figure data(array), it always the three-dimension, could you tell me how to obtain the 2d array?

@Yutianzhang888

There is one problem when I input the figure data(array), it always the three-dimension, could you tell me how to obtain the 2d array?

You need to select a single channel of the image.

Hello, Nice contribution.
I have a question.
Why do you use threshold?
Why threshold has 0.9 value?
Thank you in advance.

Hi, I also don't know the meaning of the threshold (0.9), and now do you understand the answer?

The threshold is used to convert the image to binary. Usually images have 8-bit color depths, which map to 256 values for each pixel. Binary image has only 2 shades - black or white.

Hey guys! May be, I understand something wrong, but my PyCharm writes:

AttributeError: module 'scipy.misc' has no attribute 'imread'

I am getting this error message when executing this programme in PyCharm.
1)"AttributeError: module 'scipy.misc' has no attribute 'imread'"
2) "AttributeError: module 'scipy' has no attribute 'misc'"

@akashi @Nikita7416, dpatikas above said "Please note that scipy.misc.imread has been discontinued after scipy v.1.2.0. It is suggested to use imageio.imread instead."

I used "imageio.imread" but the problem seems to be not resolved.

Here what I get after running the program

Traceback (most recent call last):
File "C:\Users\Anup Shahi\Documents\pythonProject\venv\lib\site-packages\IPython\core\interactiveshell.py", line 3418, in run_code
exec(code_obj, self.user_global_ns, self.user_ns)
File "", line 1, in
runfile('C:/Users/Anup Shahi/Documents/pythonProject/FD.py', wdir='C:/Users/Anup Shahi/Documents/pythonProject')
File "C:\Program Files\JetBrains\PyCharm 2020.3\plugins\python\helpers\pydev_pydev_bundle\pydev_umd.py", line 197, in runfile
pydev_imports.execfile(filename, global_vars, local_vars) # execute the script
File "C:\Program Files\JetBrains\PyCharm 2020.3\plugins\python\helpers\pydev_pydev_imps_pydev_execfile.py", line 18, in execfile
exec(compile(contents+"\n", file, 'exec'), glob, loc)
File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 51, in
print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I))
File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 14, in fractal_dimension
assert(len(Z.shape) == 2)
AssertionError

I used "imageio.imread" but the problem seems to be not resolved.

Here what I get after running the program

Traceback (most recent call last): File "C:\Users\Anup Shahi\Documents\pythonProject\venv\lib\site-packages\IPython\core\interactiveshell.py", line 3418, in run_code exec(code_obj, self.user_global_ns, self.user_ns) File "", line 1, in runfile('C:/Users/Anup Shahi/Documents/pythonProject/FD.py', wdir='C:/Users/Anup Shahi/Documents/pythonProject') File "C:\Program Files\JetBrains\PyCharm 2020.3\plugins\python\helpers\pydev_pydev_bundle\pydev_umd.py", line 197, in runfile pydev_imports.execfile(filename, global_vars, local_vars) # execute the script File "C:\Program Files\JetBrains\PyCharm 2020.3\plugins\python\helpers\pydev_pydev_imps_pydev_execfile.py", line 18, in execfile exec(compile(contents+"\n", file, 'exec'), glob, loc) File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 51, in print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I)) File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 14, in fractal_dimension assert(len(Z.shape) == 2) AssertionError

Try

import cv2
img = cv2.imread("sierpinski.png", 0)/256.0
print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(img, 0.9))

Thank you for sharing this.

I believe we can reduce

    # Greatest power of 2 less than or equal to p
    n = 2**np.floor(np.log(p)/np.log(2))

    # Extract the exponent
    n = int(np.log(n)/np.log(2))

to

    # largest n for which 2^n is less than or equal p
    n = int(np.floor(np.log2(p)))

sorry, want to ask if this can be used on iris recognition or is there any other version?
btw the image is

thanks before and really appreciate it