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Copy pathDSA_generator_and_box_counting.py
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DSA_generator_and_box_counting.py
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"""
Created on Sat Oct 5 12:11:44 2022
Code used to generate undulating surfaces using the diamond-square algorithm (DSA). A box-counting method is then used to determine the fractal dimension of the surfaces generated.
Fractal dimension, surface roughness (a parameter used within the DSA), root-mean square of the bacteria heights and kurtosis are all plotted together.
@author: Harry Lipscomb
"""
import random
import numpy as np
import matplotlib.pyplot as plt
def main():
k = 5 # size of box parameter
n = 2**k +1
rs = np.arange(0, 9, 0.5)
repeats = 1
show=False
def fixed( d, i, j, v, offsets ):
# For fixed bdries, all cells are valid. Define n so as to allow the
# usual lower bound inclusive, upper bound exclusive indexing.
n = d.shape[0]
res, k = 0, 0
for p, q in offsets:
pp, qq = i + p*v, j + q*v
if 0 <= pp < n and 0 <= qq < n:
res += d[pp, qq]
k += 1.0
return res/k
def periodic( d, i, j, v, offsets ):
# For periodic bdries, the last row/col mirrors the first row/col.
# Hence the effective square size is (n-1)x(n-1). Redefine n accordingly!
n = d.shape[0] - 1
res = 0
for p, q in offsets:
res += d[(i + p*v)%n, (j + q*v)%n]
return res/4.0
def single_diamond_square_step( d, w, s, avg ):
# w is the dist from one "new" cell to the next
# v is the dist from a "new" cell to the nbs to average over
n = d.shape[0]
v = w//2
# offsets:
diamond = [ (-1,-1), (-1,1), (1,1), (1,-1) ]
square = [ (-1,0), (0,-1), (1,0), (0,1) ]
# (i,j) are always the coords of the "new" cell
# Diamond Step
for i in range( v, n, w ):
for j in range( v, n, w ):
d[i, j] = avg( d, i, j, v, diamond ) + random.uniform(-s,s)
# Square Step, rows
for i in range( v, n, w ):
for j in range( 0, n, w ):
d[i, j] = avg( d, i, j, v, square ) + random.uniform(-s,s)
# Square Step, cols
for i in range( 0, n, w ):
for j in range( v, n, w ):
d[i, j] = avg( d, i, j, v, square ) + random.uniform(-s,s)
def make_terrain( n, r, bdry ):
# Returns an n-by-n landscape using the Diamond-Square algorithm, using
# roughness delta ds (0..1). bdry is an averaging fct, including the
# bdry conditions: fixed() or periodic(). n must be 1+2**k, k integer.
d = np.zeros( n*n ).reshape( n, n )
w, s = n-1, r
while w > 1:
single_diamond_square_step( d, w, s, bdry )
w //= 2
return d
def delete_row_and_column(terrain):
return terrain[0:-1, 0:-1]
def box_lengths_of_interest(n):
boxes = np.array([])
for value in np.arange(2, n):
box = 2**value
boxes = np.append(boxes, box)
return boxes
def check_box(box_size, z, terrain_of_interest):
if box_size != 1:
for value in terrain_of_interest:
for zs in value:
if z>=zs>(z-box_size):
return 1
return 0
else:
if z>=terrain_of_interest[0]>(z-box_size):
return 1
return 0
def boxes_overlapping_surface(box_sizes, terrain):
boxes_overlapping = np.array([])
length_terrain = len(terrain[:,0])
for box_size in box_sizes:
boxes_contained_for_box_size = 0
x = 0
y = 0
z = length_terrain
while x < length_terrain:
while y < length_terrain:
while z>=(-3*length_terrain):
terrain_of_interest = terrain[x:int(x+box_size), y:int(y+box_size)]
is_contained = check_box(box_size, z, terrain_of_interest)
boxes_contained_for_box_size += is_contained
z -= box_size/10
y += box_size
y = int(y)
z = 3*length_terrain
x += box_size
x = int(x)
y = int(0)
boxes_overlapping = np.append(boxes_overlapping, boxes_contained_for_box_size)
return boxes_overlapping
def plot_results(x, y, r, show):
parameters = np.polyfit(x,y,1)
if show:
fit_line = parameters[0]*x + parameters[1]
plt.figure()
plt.title(r"$log(N)$ vs $log(r)$ for $r={}$".format(r))
plt.ylabel('$log(N)$')
plt.xlabel('$log(r)$')
plt.scatter(x, y)
plt.plot(x, fit_line, '--')
plt.show()
return parameters
def rms(terrain):
average_line = np.average(terrain)
shapes = np.shape(terrain)
rows = shapes[0]
columns = shapes[1]
rms = 0
for row in np.arange(0, rows, 1):
for column in np.arange(0, columns, 1):
rms += np.sqrt((terrain[row, column] - average_line)**2)
rms = rms/(rows*columns)
print(rms)
return rms
def kurtosis(terrain, rms):
average_line = np.average(terrain)
shapes = np.shape(terrain)
rows = shapes[0]
columns = shapes[1]
kurtosis_number = 0
for row in np.arange(0, rows, 1):
for column in np.arange(0, columns, 1):
kurtosis_number += np.sqrt((terrain[row, column] - average_line)**4)
kurtosis_number = kurtosis_number/(rows*columns)
kurtosis_number = kurtosis_number/(rms**4)
return kurtosis_number
def sub_main(n, r, fixed, show=False):
terrain = make_terrain(n, r, fixed)
terrain = delete_row_and_column(terrain)
box_sizes = box_lengths_of_interest(k)
rms_value = rms(terrain)
kurtosis_value = kurtosis(terrain, rms_value)
boxes_overlapping = boxes_overlapping_surface(box_sizes, terrain)
x = np.log(1/box_sizes)
y = np.log(boxes_overlapping)
parameters = plot_results(x, y, r, show)
return parameters[0], rms_value, kurtosis_value
r_values = np.array([])
D_means = np.array([])
D_Stds = np.array([])
rms_means = np.array([])
rms_Stds = np.array([])
kurtosis_means = np.array([])
kurtosis_Stds = np.array([])
for r in rs:
r_values = np.append(r_values, r)
i = 0
D_values = np.array([])
rms_array = np.array([])
kurtosis_array = np.array([])
while i<=(repeats-1):
D, rms_value, kurtosis_value = sub_main(n, r, fixed, show)
D_values = np.append(D_values, D)
i += 1
rms_array = np.append(rms_array, rms_value)
kurtosis_array = np.append(kurtosis_array, kurtosis_value)
D_value = np.average(D_values)
D_Std = np.std(D_values)
D_means = np.append(D_means, D_value)
D_Stds = np.append(D_Stds, D_Std)
rms_value = np.average(rms_array)
rms_Std = np.std(rms_array)
rms_means = np.append(rms_means, rms_value)
rms_Stds = np.append(rms_Stds, rms_Std)
kurtosis_value = np.average(kurtosis_array)
kurtosis_Std = np.std(kurtosis_array)
kurtosis_means = np.append(kurtosis_means, kurtosis_value)
kurtosis_Stds = np.append(kurtosis_Stds, kurtosis_Std)
print('Completed r = {}'.format(r))
parameters = np.polyfit(r_values, D_means, 1)
#best_fit_line = parameters[0]*r_values + parameters[1]
plt.figure(figsize=(8,6))
plt.ylabel('$D$', size=18)
plt.xlabel('$r$', size=18)
plt.title("Fractal Dimension against roughness parameter")
plt.errorbar(r_values, D_means, yerr=D_Stds, color='k', marker='o', markersize=4)
#plt.plot(r_values, best_fit_line, '--')
plt.xticks(size=16)
plt.yticks(size=16)
plt.savefig('r vs D for DSA', dpi=900)
plt.legend()
plt.show()
plt.figure(figsize=(8,6))
plt.ylabel('$R_{rms}$', size=18)
plt.xlabel('$r$', size=18)
plt.title('Root Mean Square vs Roughness parameter $r$', size=14)
plt.errorbar(r_values, rms_means, yerr=rms_Stds, color='k', marker='o', markersize=4)
#plt.plot(r_values, best_fit_line, '--')
plt.xticks(size=16)
plt.yticks(size=16)
plt.savefig('RMS vs r for DSA', dpi=900)
plt.show()
plt.figure(figsize=(8,6))
plt.ylabel('$R_{ku}$', size=18)
plt.xlabel('$r$', size=18)
plt.title('Kurtosis vs Roughness parameter $r$', size=14)
plt.errorbar(r_values, kurtosis_means, yerr=kurtosis_Stds, color='k', marker='o', markersize=4)
#plt.plot(r_values, best_fit_line, '--')
plt.xticks(size=16)
plt.yticks(size=16)
plt.savefig('kurtosis vs r for DSA', dpi=900)
plt.legend()
plt.show()
for index, r in enumerate(r_values):
print("r: {}, D: {}, Unc_D: {}, rms: {}, Unc_rms {}, kurtosis: {}, Unc_kurtosis {} ".format(r_values[index], D_means[index], D_Stds[index],
rms_means[index], rms_Stds[index],
kurtosis_means[index], kurtosis_Stds[index]))