# ----------------------------------------------------------------------------- # 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). # ----------------------------------------------------------------------------- from PIL import Image 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) plt.imshow(Z) plt.show() # 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] fname = 'sierpinski.png' I = Image.open(fname).convert("L") I = np.asarray(I)/255

_ = plt.hist(I.ravel()) # 看一下分布

print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I, 0.05)) print("Haussdorf dimension (theoretical): ", (np.log(3)/np.log(2))) print(f"min {I[:,:].min()} and max {I[:,:].max()} in the array/image")

import numpy as np import matplotlib.pyplot as plt 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) plt.imshow(Z) plt.show() # 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] special = [ 'https://upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Peano_Sierpinski_carpet_4.svg/440px-Peano_Sierpinski_carpet_4.svg.png', 'https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Quadratic_Koch_2.svg/2880px-Quadratic_Koch_2.svg.png', 'https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Quadratic_Koch.svg/300px-Quadratic_Koch.svg.png' ] images = [ 'https://upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Peano_Sierpinski_carpet_4.svg/440px-Peano_Sierpinski_carpet_4.svg.png', 'https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Sierpinski_carpet_3.svg/244px-Sierpinski_carpet_3.svg.png', 'https://upload.wikimedia.org/wikipedia/commons/f/f0/Flocke.PNG', 'https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Quadratic_Koch_2.svg/2880px-Quadratic_Koch_2.svg.png', 'https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Quadratic_Koch.svg/300px-Quadratic_Koch.svg.png' ] dimensions = [(np.log(3)/np.log(2)), (np.log(3)/np.log(2)), (np.log(4)/np.log(3)), 1.46, 1.5] for dim, im in zip(dimensions, images): # load im x = plt.imread(im) if im in special: print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(x[:,:,3], threshold=0.2)) print("theoretical: ", dim) print(f"min {x[:,:,3].min()} and max {x[:,:,3].max()} in the array/image") else: print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(x[:,:,1],threshold=0.2)) print("theoretical: ", dim) print(f"min {x[:,:,1].min()} and max {x[:,:,1].max()} in the array/image")