This lesson is being piloted (Beta version)

## Overview

Teaching: 30 min
Exercises: 20 min
Questions
• How can I do whole-array operations with arrays of different shape?

Objectives
• Understand the rules around broadcasting arrays of different sizes

We know that when you add two arrays of the same size in Numpy that they are added element-wise:

import numpy as np

A = np.arange(100).reshape((10, 10))
B = np.arange(10, 20, 0.1).reshape((10, 10))

A + B
array([[ 10. ,  11.1,  12.2,  13.3,  14.4,  15.5,  16.6,  17.7,  18.8, 19.9],
[ 21. ,  22.1,  23.2,  24.3,  25.4,  26.5,  27.6,  28.7,  29.8, 30.9],
[ 32. ,  33.1,  34.2,  35.3,  36.4,  37.5,  38.6,  39.7,  40.8, 41.9],
[ 43. ,  44.1,  45.2,  46.3,  47.4,  48.5,  49.6,  50.7,  51.8, 52.9],
[ 54. ,  55.1,  56.2,  57.3,  58.4,  59.5,  60.6,  61.7,  62.8, 63.9],
[ 65. ,  66.1,  67.2,  68.3,  69.4,  70.5,  71.6,  72.7,  73.8, 74.9],
[ 76. ,  77.1,  78.2,  79.3,  80.4,  81.5,  82.6,  83.7,  84.8, 85.9],
[ 87. ,  88.1,  89.2,  90.3,  91.4,  92.5,  93.6,  94.7,  95.8, 96.9],
[ 98. ,  99.1, 100.2, 101.3, 102.4, 103.5, 104.6, 105.7, 106.8, 107.9],
[109. , 110.1, 111.2, 112.3, 113.4, 114.5, 115.6, 116.7, 117.8, 118.9]])

What happens when we want to operate using something of a different size? We know that we can use a scalar number:

B + 100
array([[110. , 110.1, 110.2, 110.3, 110.4, 110.5, 110.6, 110.7, 110.8, 110.9],
[111. , 111.1, 111.2, 111.3, 111.4, 111.5, 111.6, 111.7, 111.8, 111.9],
[112. , 112.1, 112.2, 112.3, 112.4, 112.5, 112.6, 112.7, 112.8, 112.9],
[113. , 113.1, 113.2, 113.3, 113.4, 113.5, 113.6, 113.7, 113.8, 113.9],
[114. , 114.1, 114.2, 114.3, 114.4, 114.5, 114.6, 114.7, 114.8, 114.9],
[115. , 115.1, 115.2, 115.3, 115.4, 115.5, 115.6, 115.7, 115.8, 115.9],
[116. , 116.1, 116.2, 116.3, 116.4, 116.5, 116.6, 116.7, 116.8, 116.9],
[117. , 117.1, 117.2, 117.3, 117.4, 117.5, 117.6, 117.7, 117.8, 117.9],
[118. , 118.1, 118.2, 118.3, 118.4, 118.5, 118.6, 118.7, 118.8, 118.9],
[119. , 119.1, 119.2, 119.3, 119.4, 119.5, 119.6, 119.7, 119.8, 119.9]])

In fact, this is a special case of a more general Numpy feature called broadcasting. When you try to operate on two arrays of different shapes, Numpy will try to duplicate the array along any missing dimensions in order to make the shapes match. For example:

values = np.arange(16).reshape((4, 4))
column_weights = np.arange(0, 400, 100)

values * column_weights
array([[   0,  100,  400,  900],
[   0,  500, 1200, 2100],
[   0,  900, 2000, 3300],
[   0, 1300, 2800, 4500]])

The column_weights array is expanded out to apply to every row. (In reality, no copying of data happens; Numpy efficiently implements the memory accesses so that it looks as if the copy occurred.)

What about if we wanted to weight the rows rather than the columns?

row_weights = np.arange(0, 40, 10).reshape((4, 1))
values * row_weights
array([[  0,   0,   0,   0],
[ 40,  50,  60,  70],
[160, 180, 200, 220],
[360, 390, 420, 450]])

We can now start to see a pattern emerging. Looking at the shapes of these arrays:

print("values:", values.shape)
print("column_weights:", column_weights.shape)
print("row_weights:", row_weights.shape)
values: (4, 4)
column_weights: (4,)
row_weights: (4, 1)

Recall that the right-most index of an ndarray is the most local, i.e. for a 2D array it will be the index that scans across a row, while the second index from the right moves down a column. (Alternatively, the rightmost index controls which column is being considered, while the next controls which row is being considered.)

So, schematically, the first multiplication has the form:

column_weights: [  0, 100, 200, 300]
*    *    *    *
arrayvalues:    [[ 0,   1,   2,   3],
[ 4,   5,   6,   7],
[ 8,   9,  10,  11],
[12,  13,  14,  15]]

while the second has the form:

row_weights             values
[[  0 ]    *  [[ 0,   1,   2,   3],
[ 10 ]    *   [ 4,   5,   6,   7],
[ 20 ]    *   [ 8,   9,  10,  11],
[ 30 ]]   *   [12,  13,  14,  15]]

To go into more detail: indices are matched up from right-to-left, and can only match if they are equal to each other, or to 1. If the indices don’t match, then an error is raised. (Numpy doesn’t try to match other indices!) Any additional indices (if one array has a higher dimension than the other) are taken to be 1.

In the first example above, the 4 of the rightmost (column) index of values matches the 4 of the only index of column_weights; this array is then broadcast across to match every element of the row index of values. In the second example, the 4 of the column index of values matches up with the 1 of the second index of row_weights, and so this is broadcast to match every column; the second 4 from the row index matches the 4 from the row index of row_weights.

Looking now at a rectangular array:

rectangular_values = np.arange(6).reshape((3, 2))
two_vector = np.asarray([1, 10])
rectangular_values * two_vector
array([[ 0, 10],
[ 2, 30],
[ 4, 50]])

In this case, the rightmost index of rectangular_values matches the size of the two_vector array, and so broadcasting is done over the leftmost row index of rectangular_values. Conversely

three_vector = np.asarray([1, 10, 100])
rectangular_values * three_vector

raises an error:

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-14-4ce8501f6c65> in <module>
1 three_vector = np.asarray([1, 10, 100])
----> 2 rectangular_values * three_vector

ValueError: operands could not be broadcast together with shapes (3,2) (3,)

The rightmost index of rectangular_values (2) doesn’t equal that of three_vector (3), and neither is 1, so the broadcasting fails. Numpy doesn’t try matching with the leftmost index of rectangular_values!

If we wanted Numpy to broadcast to this element, we would instead need to reshape so that the indices were correct:

three_vector_column = np.asarray([1, 10, 100]).reshape((3, 1))
rectangular_values * three_vector_column
array([[  0,   1],
[ 20,  30],
[400, 500]])

This generalises readily to more than two dimensions. Work from right to left index by index, and the indices must either be equal to each other, or one must be 1.

values_4d = np.arange(120).reshape((2, 3, 4, 5))
values_4d * row_weights
array([[[[   0,    0,    0,    0,    0],
[  50,   60,   70,   80,   90],
[ 200,  220,  240,  260,  280],
[ 450,  480,  510,  540,  570]],

[[   0,    0,    0,    0,    0],
[ 250,  260,  270,  280,  290],
[ 600,  620,  640,  660,  680],
[1050, 1080, 1110, 1140, 1170]],

[[   0,    0,    0,    0,    0],
[ 450,  460,  470,  480,  490],
[1000, 1020, 1040, 1060, 1080],
[1650, 1680, 1710, 1740, 1770]]],

[[[   0,    0,    0,    0,    0],
[ 650,  660,  670,  680,  690],
[1400, 1420, 1440, 1460, 1480],
[2250, 2280, 2310, 2340, 2370]],

[[   0,    0,    0,    0,    0],
[ 850,  860,  870,  880,  890],
[1800, 1820, 1840, 1860, 1880],
[2850, 2880, 2910, 2940, 2970]],

[[   0,    0,    0,    0,    0],
[1050, 1060, 1070, 1080, 1090],
[2200, 2220, 2240, 2260, 2280],
[3450, 3480, 3510, 3540, 3570]]]])

If we want to work on different axes, then as an alternative to reshape, we can also use expand_dims. For example, to use a $2 \times 3$ array to work on the leftmost two columns of values_4d:

matrix_weights = np.expand_dims(np.expand_dims(
[[0, 2, 0], [1, 0, 3]], axis=2), axis=3
)
values_4d * matrix_weights
array([[[[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0]],

[[ 40,  42,  44,  46,  48],
[ 50,  52,  54,  56,  58],
[ 60,  62,  64,  66,  68],
[ 70,  72,  74,  76,  78]],

[[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0]]],

[[[ 60,  61,  62,  63,  64],
[ 65,  66,  67,  68,  69],
[ 70,  71,  72,  73,  74],
[ 75,  76,  77,  78,  79]],

[[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0],
[  0,   0,   0,   0,   0]],

[[300, 303, 306, 309, 312],
[315, 318, 321, 324, 327],
[330, 333, 336, 339, 342],
[345, 348, 351, 354, 357]]]])

Another way to do this is to slice the array, and use np.newaxis:

matrix_weights = np.asarray(
[[0, 2, 0], [1, 0, 3]]
)[:, :, np.newaxis, np.newaxis]
values_4d * matrix_weights

which gives the same output as the previous version.

Finally, sometimes you will need to rearrange the order of your axes to let broadcasting work. This can be done with swapaxes; for example to operate on the first two indices of values_4d with the rectangular_values array we created earlier:

matrix_weights = np.expand_dims(np.expand_dims(
rectangular_values, axis=2), axis=3
).swapaxes(0, 1)
values_4d * matrix_weights

## Outer products

In the solution of Mandelbrot example in the previous episode, I used np.meshgrid to set up the initial c values. Specifically, the lines (paraphrased slightly):

real_range = np.linspace(xmin, xmax, width)
imaginary_range = np.linspace(ymin, ymax, height)
real_parts, imaginary_parts = np.meshgrid(
real_range, 1j * imaginary_range, sparse=False
)
c = real_parts + imaginary_parts

Rewrite the third and fourth lines to use broadcasting rather than np.meshgrid.

## Solution

real_range = np.linspace(xmin, xmax, width)
imaginary_range = np.linspace(ymin, ymax, height)
real_range + 1j * imaginary_range[:, np.newaxis]

## Array broadcasts for image manipulation

Images can be represented in Numpy as three-dimensional arrays: the first two axes are the vertical and horizontal pixel co-ordinates, and the third axis gives the colour channels (red, green, blue, and optionally transparency, also known as alpha).

The imageio library, included with Anaconda, provides the imread function which reads an image into an array with this format. (This functionality was previously provided by scipy.misc.)

Read in a JPEG image from your hard drive with the following code

%matplotlib inline
import imageio
from matplotlib import pyplot as plt
image = imageio.imread('cat.jpg') / 256
plt.imshow(image)
plt.show()

(Replace cat.jpg with the name of your image!)

Use broadcasts to apply the following transformations:

1. Suppress the blue channel to zero, and reduce the intensity of the green channel by half.
2. Make the image fade to black from left to right. (I.e. all colour channels are multiplied by 1 at the left edge, zero at the right edge, and a number between 0 and 1 in between.)
3. Make the image fade to white from top to bottom.
4. Use a variant of the peaked_function in the previous episode to multiply all colour channels of the image.
5. Do the same, but applying it only to the red channel.

## Solution

1. Here we vary with the last index, and broadcast to fill the rest:

plt.imshow(image * [1, 0.5, 0])
plt.show()

(You may need to add an extra 1 at the end of the list if your image has an alpha channel.)

2. Here we are varying the second-to-last index, so need to add an extra axis

image2 = image * np.linspace(1, 0, image.shape[1])[:, np.newaxis]
plt.imshow(image2)
plt.show()

3. To fade to white, we need to multiply the difference from 1 rather than from zero, and now we are using the leftmost index:

image3 = 1 - (1 - image) * np.linspace(
1, 0, image.shape[0]
)[:, np.newaxis, np.newaxis]
plt.imshow(image3)
plt.show()

4. Now we are varying two indices. We can use np.linspace instead of np.arange to get an array of the correct size.

x_range = np.linspace(-30, 30, image.shape[1])
y_range = np.linspace(-30, 30, image.shape[0])
x_values, y_values = np.meshgrid(x_range, y_range, sparse=False)
peaked_function = (np.sin(x_values**2 + y_values**2) /
(x_values**2 + y_values**2) ** 0.25)

#scale the output to be between 0 and 1.0, not doing this causes errors on some systems
peaked_function = peaked_function / 2
peaked_function = peaked_function + 0.5

peaked_function = peaked_function[:, :, np.newaxis]
plt.imshow(peaked_function * image)
plt.show()

5. Now rather than broadcasting when multiplying the image, we need to broadcast to generate the mask in the first place. First expand the rightmost axis out to three components, then make the green and blue components multiply by 1 to not affect those channels.

peaked_function_5 = peaked_function * [1, 0, 0] + [0, 1, 1]
plt.imshow(peaked_function_5 * image)
plt.show()

If you instead read in an image with an alpha channel, some extra work would be needed to work out what to do with that, since it doesn’t represent colour data in the way that the other three channels do.

## Key Points

• Numpy automatically expands smaller arrays to match the shape of larger ones

• Axes are read right to left, and must be either the same size or size 1

• Where one array has more dimensions, the smaller array is interpreted as having size 1 on the additional axes