NumPy / SciPy Quiz

Please do explore beyond the problems given, and feel free to ask questions at any time.

If you need a quick refresher on NumPy, have a look at this introduction. Also see the SciPy lectures.

If you run out of exercises, there are 100 more here!

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import numpy as np

%matplotlib inline
import matplotlib.pyplot as plt

import scipy

The NumPy array

Create a 3x4 array containing the numbers from 0 through 11 (using np.arange and np.reshape), and call it x.

Predict whether x changes after executing the following:

y = x[2]
y[0] = 3

Compute the sums of the columns of x, using np.sum.

Construct the array x = np.array([0, 1, 2, 3], dtype=np.uint8). Predict the value of x - 1.

Broadcasting

Consider two arrays, x = np.arange(5); y = np.array([0.5, 1.5]).
Construct a 5x2 matrix such that A[i, j] = x[i] * y[j], without using for-loops.

Given a list of XYZ-coordinates, p,
```
[[1.0, 2.0, 10],
 [3.0, 4.0, 20],
 [5.0, 6.0, 30],
 [7.0, 8.0, 40]]
```
Normalise each coordinate by dividing with its Z (3rd) element. For example, the first row becomes:
```
[1/10, 2/10, 10/10]
```
Hint: extract the last column into a variable z, and then change its dimensions so that p / z works.

Indexing

Create a 3x3 ndarray, A = np.array([[0, 1, 2], [1, 1, 3], [2, 3, 2]]). Find an indexing expression for extracting the diagonal elements, i.e.
delems = A[...]

Generate a 10 x 3 array of random numbers (all between 0 and 1). From each row, pick the number closest to 0.75. Make use of np.abs and np.argmax to find the column j which contains the closest element in each row.

Predict and verify the shape of the following slicing operation.

x = np.empty((10, 8, 6))

idx0 = np.zeros((3, 8)).astype(int)
idx1 = np.zeros((3, 1)).astype(int)
idx2 = np.zeros((1, 1)).astype(int)

x[idx0, idx1, idx2]

Optimization

Consider the Rosenbrock test function. You can visualize it by executing the following cell:

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# From https://commons.wikimedia.org/wiki/File:Rosenbrock_function.svg

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.colors import LogNorm
import matplotlib.pyplot as plt
import numpy as np

def rosenbrock(x, y):
    return (1 - x)**2 + 100 * (y - x**2)**2

fig = plt.figure()
ax = Axes3D(fig, azim = -128, elev = 43)
s = .05
X = np.arange(-2, 2.+s, s)
Y = np.arange(-1, 3.+s, s)
X, Y = np.meshgrid(X, Y)
Z = rosenbrock(X, Y)
# ax.plot_surface(X, Y, Z, rstride = 1, cstride = 1, norm = LogNorm(), cmap = cm.jet)
# Without using `` linewidth=0, edgecolor='none' '', the code may produce a graph with wide black edges, which 
# will make the surface look much darker than the one illustrated in the figure above.
ax.plot_surface(X, Y, Z, rstride = 1, cstride = 1, norm = LogNorm(), cmap = cm.viridis, linewidth=0, edgecolor='none')

# Set the axis limits so that they are the same as in the figure above.
ax.set_xlim([-2, 2.0])                                                       
ax.set_ylim([-1, 3.0])                                                       
ax.set_zlim([0, 2500]) 

plt.xlabel("x")
plt.ylabel("y");

Find the minimum of the test function rosenbrock defined above. Use scipy.optimize.minimize, and the following template:

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from scipy import optimize

def cost_f(p):
    x, y = p
    return rosenbrock(x, y)

Plotting

Generate some random data using:

np.random.seed(0)

mu = 200
sigma = 25
n_bins = 50
x = np.random.normal(mu, sigma, size=100)

Now, try to reproduce this plot:

ndimage

Use scipy.ndimage.distance_transform_edt to calculate the distance map of the image generated in the next cell:

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from skimage import draw

N = 200
S = 10

np.random.seed(0)
image = np.ones((N, N), dtype=bool)

coords = np.random.randint(low=0, high=N, size=4 * S).reshape(-1, 4)

for line in coords:
    image[draw.line(*line)] = 0

Display the distance map; can you interpret what you see?

interpolation

An image is sampled T times by the code below. Take the coordinates and values of the samples, and try to reconstruct the original image. Hint: Use scipy.interpolate.griddata, and provide it with the coordinates of the full image, generated using np.indices.

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from skimage import data, color, filters
image = color.rgb2gray(data.astronaut())
image = filters.gaussian(image, sigma=2)

M, N = image.shape

T = 5000  # nr of randomly drawn samples
coords = np.column_stack([
    np.random.randint(0, M, size=T),
    np.random.randint(0, N, size=T)
])


# Use `coords` and `samples` to reconstruct the image
coords = tuple(coords.T)
samples = image[coords]

We can visualize the data below, to see how densely the image was sampled:

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# Visualize sampling

sampled_image = np.zeros_like(image)
sampled_image[coords] = image[coords]

f, (ax0, ax1) = plt.subplots(1, 2, figsize=(20, 10))
ax0.imshow(image, cmap='gray')
ax1.imshow(sampled_image, cmap='gray');