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Vector Field Generative Art with Perlin noise¶
Hello world
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from noise import pnoise2
octaves = 5
factor = 3.0
num_lines = 600
num_steps = 50
step_sizes = 0.5 * (np.arange(10) + 1.0)
seed = 42 # set random seed for reproducibility
random_state = np.random.RandomState(seed)
# resolution
h, w = 200, 200
y, x = np.ogrid[:h, :w]
X, Y = np.broadcast_arrays(x, y)
The function pnoise2
has range [-1, 1]. However, the outputs tend mostly
to be centered around 0. Let’s blow up the range by some factor
and
squash it through the \(\tanh\) function so that it is still in the
desired range.
Z = np.tanh(factor * np.vectorize(pnoise2)(x/w, y/h, octaves=octaves)) # range [-1, 1]
theta = np.pi * Z # range [-pi, pi]
Sparsify the grid so we can later draw intelligible quiver plots.
w_factor = h_factor = 10
x_sparse = x[..., ::w_factor]
y_sparse = y[::h_factor]
X_sparse = X[::h_factor, ::w_factor]
Y_sparse = Y[::h_factor, ::w_factor]
theta_sparse = theta[::w_factor, ::w_factor]
We use use Perlin noise at every grid point as the angle (in radians).
fig, ax = plt.subplots(figsize=(10, 8))
ax.set_title(r"angle $\theta$ (rad)")
contours = ax.pcolormesh(X, Y, theta, cmap="twilight")
fig.colorbar(contours, ax=ax)
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$y$")
plt.show()
![angle $\theta$ (rad)](../../_images/sphx_glr_plot_perlin_noise_001.png)
Out:
/usr/src/app/examples/misc/plot_perlin_noise.py:60: MatplotlibDeprecationWarning: shading='flat' when X and Y have the same dimensions as C is deprecated since 3.3. Either specify the corners of the quadrilaterals with X and Y, or pass shading='auto', 'nearest' or 'gouraud', or set rcParams['pcolor.shading']. This will become an error two minor releases later.
contours = ax.pcolormesh(X, Y, theta, cmap="twilight")
For sanity-check, we view the numerical values of the angles normalized by \(\pi\) on the heatmap.
data = pd.DataFrame(theta_sparse / np.pi,
index=y_sparse.squeeze(axis=1),
columns=x_sparse.squeeze(axis=0))
fig, ax = plt.subplots(figsize=(10, 8))
ax.set_title(r"angle $\theta / \pi$ (rad)")
sns.heatmap(data, annot=True, fmt=".2f", cmap="twilight", ax=ax)
ax.invert_yaxis()
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$y$")
plt.show()
![angle $\theta / \pi$ (rad)](../../_images/sphx_glr_plot_perlin_noise_002.png)
Flow field¶
dx = np.cos(theta_sparse)
dy = np.sin(theta_sparse)
fig, ax = plt.subplots(figsize=(10, 8))
ax.quiver(x_sparse, y_sparse, dx, dy)
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$y$")
plt.show()
![plot perlin noise](../../_images/sphx_glr_plot_perlin_noise_003.png)
Path¶
# TODO: `w`, `h` arguments
def line(x, y, x_lim, y_lim, step_fn, num_steps, step_size=1.0):
x_min, x_max = x_lim
y_min, y_max = y_lim
if not (num_steps and x_min <= x < x_max and y_min <= y < y_max):
return [], []
dx, dy = step_fn(x, y, step_size=step_size)
xs, ys = line(x=x+dx, y=y+dy, x_lim=x_lim, y_lim=y_lim, step_fn=step_fn,
num_steps=num_steps-1, step_size=step_size)
xs.append(x)
ys.append(y)
return xs, ys
def step(x, y, step_size):
j, i = int(x), int(y)
dx = step_size * np.cos(theta[i, j])
dy = step_size * np.sin(theta[i, j])
return dx, dy
x = y = 25
xs, ys = line(x, y, x_lim=(0, w), y_lim=(0, h), step_fn=step,
num_steps=num_steps, step_size=5.0)
_xs = np.asarray(xs[::-1])
_ys = np.asarray(ys[::-1])
fig, ax = plt.subplots(figsize=(10, 8))
ax.quiver(_xs[:-1], _ys[:-1], _xs[1:] - _xs[:-1], _ys[1:] - _ys[:-1],
scale_units='xy', angles='xy', scale=1.0, width=3e-3, color='r')
ax.quiver(x_sparse, y_sparse, dx, dy)
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$y$")
plt.show()
![plot perlin noise](../../_images/sphx_glr_plot_perlin_noise_004.png)
dfs = []
for step_size in step_sizes:
for lineno in range(num_lines):
# sample starting point uniformly at random
x = w * random_state.rand()
y = h * random_state.rand()
xs, ys = line(x, y, x_lim=(0, w), y_lim=(0, h), step_fn=step,
num_steps=num_steps, step_size=step_size)
df = pd.DataFrame(dict(lineno=lineno, stepsize=step_size, x=xs, y=ys))
dfs.append(df)
data = pd.concat(dfs, axis="index", sort=True)
data
fig, ax = plt.subplots(figsize=(10, 8))
sns.lineplot(x='x', y='y', hue='stepsize', units='lineno', estimator=None,
sort=False, palette='Spectral', legend=None, linewidth=1.0,
alpha=0.4, data=data, ax=ax)
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$y$")
plt.show()
![plot perlin noise](../../_images/sphx_glr_plot_perlin_noise_005.png)
Total running time of the script: ( 0 minutes 38.293 seconds)