Visualizing the Mandelbrot set doesn't have anything to do with machine learning, but it makes for a fun example of how one can use TensorFlow for general mathematics. This is actually a pretty naive implementation of the visualization, but it makes the point. (We may end up providing a more elaborate implementation down the line to produce more truly beautiful images.)
Note: This tutorial was originally prepared as an IPython notebook.
We'll need a few imports to get started.
# Import libraries for simulation import tensorflow as tf import numpy as np # Imports for visualization import PIL.Image from io import BytesIO from IPython.display import Image, display
Now we'll define a function to actually display the image once we have iteration counts.
def DisplayFractal(a, fmt='jpeg'): """Display an array of iteration counts as a colorful picture of a fractal.""" a_cyclic = (6.28*a/20.0).reshape(list(a.shape)+) img = np.concatenate([10+20*np.cos(a_cyclic), 30+50*np.sin(a_cyclic), 155-80*np.cos(a_cyclic)], 2) img[a==a.max()] = 0 a = img a = np.uint8(np.clip(a, 0, 255)) f = BytesIO() PIL.Image.fromarray(a).save(f, fmt) display(Image(data=f.getvalue()))
Session and Variable Initialization
For playing around like this, we often use an interactive session, but a regular session would work as well.
sess = tf.InteractiveSession()
It's handy that we can freely mix NumPy and TensorFlow.
# Use NumPy to create a 2D array of complex numbers on [-2,2]x[-2,2] Y, X = np.mgrid[-1.3:1.3:0.005, -2:1:0.005] Z = X+1j*Y
Now we define and initialize TensorFlow tensors.
xs = tf.constant(Z.astype(np.complex64)) zs = tf.Variable(xs) ns = tf.Variable(tf.zeros_like(xs, tf.float32))
TensorFlow requires that you explicitly initialize variables before using them.
Defining and Running the Computation
Now we specify more of the computation...
# Compute the new values of z: z^2 + x zs_ = zs*zs + xs # Have we diverged with this new value? not_diverged = tf.complex_abs(zs_) < 4 # Operation to update the zs and the iteration count. # # Note: We keep computing zs after they diverge! This # is very wasteful! There are better, if a little # less simple, ways to do this. # step = tf.group( zs.assign(zs_), ns.assign_add(tf.cast(not_diverged, tf.float32)) )
... and run it for a couple hundred steps
for i in range(200): step.run()
Let's see what we've got.