Python Numpy Tutorial 22/07/2016, 11:03 am

CS231n Convolutional Neural Networks for Visual

Recognition

Python Numpy Tutorial

This tutorial was contributed by Justin Johnson.

We will use the Python programming language for all assignments in this course. Python is
a great general-purpose programming language on its own, but with the help of a few
popular libraries (numpy, scipy, matplotlib) it becomes a powerful environment for scientiAc
computing.

We expect that many of you will have some experience with Python and numpy; for the rest
of you, this section will serve as a quick crash course both on the Python programming
language and on the use of Python for scientiAc computing.

Some of you may have previous knowledge in Matlab, in which case we also recommend
the numpy for Matlab users page.

You can also And an IPython notebook version of this tutorial here created by Volodymyr
Kuleshov and Isaac Caswell for CS 228.

Table of contents:

Python
Basic data types
Containers
Lists
Dictionaries
Sets
Tuples
Functions
Classes
Numpy
Arrays

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Array indexing
Datatypes
Array math
Broadcasting
SciPy
Image operations
MATLAB Ales
Distance between points
Matplotlib
Plotting
Subplots
Images

Python
Python is a high-level, dynamically typed multiparadigm programming language. Python
code is often said to be almost like pseudocode, since it allows you to express very
powerful ideas in very few lines of code while being very readable. As an example, here is an
implementation of the classic quicksort algorithm in Python:

def quicksort(arr):
if len(arr) <= 1:
return arr
pivot = arr[len(arr) / 2]
left = [x for x in arr if x < pivot]
middle = [x for x in arr if x == pivot]
right = [x for x in arr if x > pivot]
return quicksort(left) + middle + quicksort(right)

print quicksort([3,6,8,10,1,2,1])
# Prints "[1, 1, 2, 3, 6, 8, 10]"

Python versions
There are currently two different supported versions of Python, 2.7 and 3.4. Somewhat
confusingly, Python 3.0 introduced many backwards-incompatible changes to the language,
so code written for 2.7 may not work under 3.4 and vice versa. For this class all code will

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use Python 2.7.

You can check your Python version at the command line by running python --version .

Basic data types

Like most languages, Python has a number of basic types including integers, [oats,
booleans, and strings. These data types behave in ways that are familiar from other
programming languages.

Numbers: Integers and [oats work as you would expect from other languages:

x = 3
print type(x) # Prints "<type 'int'>"
print x # Prints "3"
print x + 1 # Addition; prints "4"
print x - 1 # Subtraction; prints "2"
print x * 2 # Multiplication; prints "6"
print x ** 2 # Exponentiation; prints "9"
x += 1
print x # Prints "4"
x *= 2
print x # Prints "8"
y = 2.5
print type(y) # Prints "<type 'float'>"
print y, y + 1, y * 2, y ** 2 # Prints "2.5 3.5 5.0 6.25"

Note that unlike many languages, Python does not have unary increment ( x++ ) or
decrement ( x-- ) operators.

Python also has built-in types for long integers and complex numbers; you can And all of the
details in the documentation.

Booleans: Python implements all of the usual operators for Boolean logic, but uses English
words rather than symbols ( && , || , etc.):

t = True
f = False
print type(t) # Prints "<type 'bool'>"

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print t and f # Logical AND; prints "False"

print t or f # Logical OR; prints "True"
print not t # Logical NOT; prints "False"
print t != f # Logical XOR; prints "True"

Strings: Python has great support for strings:

hello = 'hello' # String literals can use single quotes

world = "world" # or double quotes; it does not matter.
print hello # Prints "hello"
print len(hello) # String length; prints "5"
hw = hello + ' ' + world # String concatenation
print hw # prints "hello world"
hw12 = '%s %s %d' % (hello, world, 12) # sprintf style string formatting
print hw12 # prints "hello world 12"

String objects have a bunch of useful methods; for example:

s = "hello"
print s.capitalize() # Capitalize a string; prints "Hello"
print s.upper() # Convert a string to uppercase; prints "HELLO"
print s.rjust(7) # Right-justify a string, padding with spaces; prints
print s.center(7) # Center a string, padding with spaces; prints " hello
print s.replace('l', '(ell)') # Replace all instances of one substring with
# prints "he(ell)(ell)o"
print ' world '.strip() # Strip leading and trailing whitespace; prints "w

You can And a list of all string methods in the documentation.

Containers
Python includes several built-in container types: lists, dictionaries, sets, and tuples.

Lists

A list is the Python equivalent of an array, but is resizeable and can contain elements of
different types:

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xs = [3, 1, 2] # Create a list

print xs, xs[2] # Prints "[3, 1, 2] 2"
print xs[-1] # Negative indices count from the end of the list; prints "
xs[2] = 'foo' # Lists can contain elements of different types
print xs # Prints "[3, 1, 'foo']"
xs.append('bar') # Add a new element to the end of the list
print xs # Prints "[3, 1, 'foo', 'bar']"
x = xs.pop() # Remove and return the last element of the list
print x, xs # Prints "bar [3, 1, 'foo']"

As usual, you can And all the gory details about lists in the documentation.

Slicing: In addition to accessing list elements one at a time, Python provides concise
syntax to access sublists; this is known as slicing:

nums = range(5) # range is a built-in function that creates a list of int

print nums # Prints "[0, 1, 2, 3, 4]"
print nums[2:4] # Get a slice from index 2 to 4 (exclusive); prints "[2,
print nums[2:] # Get a slice from index 2 to the end; prints "[2, 3, 4]"
print nums[:2] # Get a slice from the start to index 2 (exclusive); prin
print nums[:] # Get a slice of the whole list; prints ["0, 1, 2, 3, 4]"
print nums[:-1] # Slice indices can be negative; prints ["0, 1, 2, 3]"
nums[2:4] = [8, 9] # Assign a new sublist to a slice
print nums # Prints "[0, 1, 8, 9, 4]"

We will see slicing again in the context of numpy arrays.

Loops: You can loop over the elements of a list like this:

animals = ['cat', 'dog', 'monkey']

for animal in animals:
print animal
# Prints "cat", "dog", "monkey", each on its own line.

If you want access to the index of each element within the body of a loop, use the built-in
enumerate function:

animals = ['cat', 'dog', 'monkey']

for idx, animal in enumerate(animals):

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print '#%d: %s' % (idx + 1, animal)

# Prints "#1: cat", "#2: dog", "#3: monkey", each on its own line

List comprehensions: When programming, frequently we want to transform one type of

data into another. As a simple example, consider the following code that computes square
numbers:

nums = [0, 1, 2, 3, 4]
squares = []
for x in nums:
squares.append(x ** 2)
print squares # Prints [0, 1, 4, 9, 16]

You can make this code simpler using a list comprehension

comprehension:

nums = [0, 1, 2, 3, 4]
squares = [x ** 2 for x in nums]
print squares # Prints [0, 1, 4, 9, 16]

List comprehensions can also contain conditions:

nums = [0, 1, 2, 3, 4]
even_squares = [x ** 2 for x in nums if x % 2 == 0]
print even_squares # Prints "[0, 4, 16]"

Dictionaries
A dictionary stores (key, value) pairs, similar to a Map in Java or an object in Javascript.
You can use it like this:

d = {'cat': 'cute', 'dog': 'furry'} # Create a new dictionary with some dat
print d['cat'] # Get an entry from a dictionary; prints "cute"
print 'cat' in d # Check if a dictionary has a given key; prints "True"
d['fish'] = 'wet' # Set an entry in a dictionary
print d['fish'] # Prints "wet"
# print d['monkey'] # KeyError: 'monkey' not a key of d
print d.get('monkey', 'N/A') # Get an element with a default; prints "N/A"

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print d.get('fish', 'N/A') # Get an element with a default; prints "wet"

del d['fish'] # Remove an element from a dictionary
print d.get('fish', 'N/A') # "fish" is no longer a key; prints "N/A"

You can And all you need to know about dictionaries in the documentation.

Loops: It is easy to iterate over the keys in a dictionary:

d = {'person': 2, 'cat': 4, 'spider': 8}

for animal in d:
legs = d[animal]
print 'A %s has %d legs' % (animal, legs)
# Prints "A person has 2 legs", "A spider has 8 legs", "A cat has 4 legs"

If you want access to keys and their corresponding values, use the iteritems method:

d = {'person': 2, 'cat': 4, 'spider': 8}

for animal, legs in d.iteritems():
print 'A %s has %d legs' % (animal, legs)
# Prints "A person has 2 legs", "A spider has 8 legs", "A cat has 4 legs"

Dictionary comprehensions: These are similar to list comprehensions, but allow you to
easily construct dictionaries. For example:

nums = [0, 1, 2, 3, 4]
even_num_to_square = {x: x ** 2 for x in nums if x % 2 == 0}
print even_num_to_square # Prints "{0: 0, 2: 4, 4: 16}"

Sets
A set is an unordered collection of distinct elements. As a simple example, consider the
following:

animals = {'cat', 'dog'}

print 'cat' in animals # Check if an element is in a set; prints "True"
print 'fish' in animals # prints "False"
animals.add('fish') # Add an element to a set

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print 'fish' in animals # Prints "True"

print len(animals) # Number of elements in a set; prints "3"
animals.add('cat') # Adding an element that is already in the set does
print len(animals) # Prints "3"
animals.remove('cat') # Remove an element from a set
print len(animals) # Prints "2"

As usual, everything you want to know about sets can be found in the documentation.

Loops: Iterating over a set has the same syntax as iterating over a list; however since sets
are unordered, you cannot make assumptions about the order in which you visit the
elements of the set:

animals = {'cat', 'dog', 'fish'}

for idx, animal in enumerate(animals):
print '#%d: %s' % (idx + 1, animal)
# Prints "#1: fish", "#2: dog", "#3: cat"

Set comprehensions: Like lists and dictionaries, we can easily construct sets using set
comprehensions:

from math import sqrt

nums = {int(sqrt(x)) for x in range(30)}
print nums # Prints "set([0, 1, 2, 3, 4, 5])"

Tuples

A tuple is an (immutable) ordered list of values. A tuple is in many ways similar to a list; one
of the most important differences is that tuples can be used as keys in dictionaries and as
elements of sets, while lists cannot. Here is a trivial example:

d = {(x, x + 1): x for x in range(10)} # Create a dictionary with tuple key

t = (5, 6) # Create a tuple
print type(t) # Prints "<type 'tuple'>"
print d[t] # Prints "5"
print d[(1, 2)] # Prints "1"

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The documentation has more information about tuples.

Functions
Python functions are deAned using the def keyword. For example:

def sign(x):
if x > 0:
return 'positive'
elif x < 0:
return 'negative'
else:
return 'zero'

for x in [-1, 0, 1]:

print sign(x)
# Prints "negative", "zero", "positive"

We will often deAne functions to take optional keyword arguments, like this:

def hello(name, loud=False):

if loud:
print 'HELLO, %s!' % name.upper()
else:
print 'Hello, %s' % name

hello('Bob') # Prints "Hello, Bob"

hello('Fred', loud=True) # Prints "HELLO, FRED!"

There is a lot more information about Python functions in the documentation.

Classes
The syntax for deAning classes in Python is straightforward:

class Greeter(object):

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# Constructor
def __init__(self, name):
self.name = name # Create an instance variable

# Instance method
def greet(self, loud=False):
if loud:
print 'HELLO, %s!' % self.name.upper()
else:
print 'Hello, %s' % self.name

g = Greeter('Fred') # Construct an instance of the Greeter class

g.greet() # Call an instance method; prints "Hello, Fred"
g.greet(loud=True) # Call an instance method; prints "HELLO, FRED!"

You can read a lot more about Python classes in the documentation.

Numpy
Numpy is the core library for scientiAc computing in Python. It provides a high-performance
multidimensional array object, and tools for working with these arrays. If you are already
familiar with MATLAB, you might And this tutorial useful to get started with Numpy.

Arrays
A numpy array is a grid of values, all of the same type, and is indexed by a tuple of
nonnegative integers. The number of dimensions is the rank of the array; the shape of an
array is a tuple of integers giving the size of the array along each dimension.

We can initialize numpy arrays from nested Python lists, and access elements using square
brackets:

import numpy as np

a = np.array([1, 2, 3]) # Create a rank 1 array

print type(a) # Prints "<type 'numpy.ndarray'>"
print a.shape # Prints "(3,)"
print a[0], a[1], a[2] # Prints "1 2 3"

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a[0] = 5 # Change an element of the array

print a # Prints "[5, 2, 3]"

b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 array

print b.shape # Prints "(2, 3)"
print b[0, 0], b[0, 1], b[1, 0] # Prints "1 2 4"

Numpy also provides many functions to create arrays:

import numpy as np

a = np.zeros((2,2)) # Create an array of all zeros

print a # Prints "[[ 0. 0.]
# [ 0. 0.]]"

b = np.ones((1,2)) # Create an array of all ones

print b # Prints "[[ 1. 1.]]"

c = np.full((2,2), 7) # Create a constant array

print c # Prints "[[ 7. 7.]
# [ 7. 7.]]"

d = np.eye(2) # Create a 2x2 identity matrix

print d # Prints "[[ 1. 0.]
# [ 0. 1.]]"

e = np.random.random((2,2)) # Create an array filled with random values

print e # Might print "[[ 0.91940167 0.08143941]
# [ 0.68744134 0.87236687]]"

You can read about other methods of array creation in the documentation.

Array indexing
Numpy offers several ways to index into arrays.

Slicing: Similar to Python lists, numpy arrays can be sliced. Since arrays may be
multidimensional, you must specify a slice for each dimension of the array:

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import numpy as np

# Create the following rank 2 array with shape (3, 4)

# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])

# Use slicing to pull out the subarray consisting of the first 2 rows
# and columns 1 and 2; b is the following array of shape (2, 2):
# [[2 3]
# [6 7]]
b = a[:2, 1:3]

# A slice of an array is a view into the same data, so modifying it

# will modify the original array.
print a[0, 1] # Prints "2"
b[0, 0] = 77 # b[0, 0] is the same piece of data as a[0, 1]
print a[0, 1] # Prints "77"

You can also mix integer indexing with slice indexing. However, doing so will yield an array
of lower rank than the original array. Note that this is quite different from the way that
MATLAB handles array slicing:

import numpy as np

# Create the following rank 2 array with shape (3, 4)

# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])

# Two ways of accessing the data in the middle row of the array.
# Mixing integer indexing with slices yields an array of lower rank,
# while using only slices yields an array of the same rank as the
# original array:
row_r1 = a[1, :] # Rank 1 view of the second row of a
row_r2 = a[1:2, :] # Rank 2 view of the second row of a
print row_r1, row_r1.shape # Prints "[5 6 7 8] (4,)"
print row_r2, row_r2.shape # Prints "[[5 6 7 8]] (1, 4)"

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# We can make the same distinction when accessing columns of an array:

col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print col_r1, col_r1.shape # Prints "[ 2 6 10] (3,)"
print col_r2, col_r2.shape # Prints "[[ 2]
# [ 6]
# [10]] (3, 1)"

Integer array indexing: When you index into numpy arrays using slicing, the resulting
array view will always be a subarray of the original array. In contrast, integer array indexing
allows you to construct arbitrary arrays using the data from another array. Here is an
example:

import numpy as np

a = np.array([[1,2], [3, 4], [5, 6]])

# An example of integer array indexing.

# The returned array will have shape (3,) and
print a[[0, 1, 2], [0, 1, 0]] # Prints "[1 4 5]"

# The above example of integer array indexing is equivalent to this:

print np.array([a[0, 0], a[1, 1], a[2, 0]]) # Prints "[1 4 5]"

# When using integer array indexing, you can reuse the same
# element from the source array:
print a[[0, 0], [1, 1]] # Prints "[2 2]"

# Equivalent to the previous integer array indexing example

print np.array([a[0, 1], a[0, 1]]) # Prints "[2 2]"

One useful trick with integer array indexing is selecting or mutating one element from each
row of a matrix:

import numpy as np

# Create a new array from which we will select elements

a = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])

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print a # prints "array([[ 1, 2, 3],

# [ 4, 5, 6],
# [ 7, 8, 9],
# [10, 11, 12]])"

# Create an array of indices

b = np.array([0, 2, 0, 1])

# Select one element from each row of a using the indices in b

print a[np.arange(4), b] # Prints "[ 1 6 7 11]"

# Mutate one element from each row of a using the indices in b

a[np.arange(4), b] += 10

print a # prints "array([[11, 2, 3],

# [ 4, 5, 16],
# [17, 8, 9],
# [10, 21, 12]])

Boolean array indexing: Boolean array indexing lets you pick out arbitrary elements of an
array. Frequently this type of indexing is used to select the elements of an array that satisfy
some condition. Here is an example:

import numpy as np

a = np.array([[1,2], [3, 4], [5, 6]])

bool_idx = (a > 2) # Find the elements of a that are bigger than 2;

# this returns a numpy array of Booleans of the same
# shape as a, where each slot of bool_idx tells
# whether that element of a is > 2.

print bool_idx # Prints "[[False False]

# [ True True]
# [ True True]]"

# We use boolean array indexing to construct a rank 1 array

# consisting of the elements of a corresponding to the True values
# of bool_idx

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print a[bool_idx] # Prints "[3 4 5 6]"

# We can do all of the above in a single concise statement:

print a[a > 2] # Prints "[3 4 5 6]"

For brevity we have left out a lot of details about numpy array indexing; if you want to know
more you should read the documentation.

Datatypes
Every numpy array is a grid of elements of the same type. Numpy provides a large set of
numeric datatypes that you can use to construct arrays. Numpy tries to guess a datatype
when you create an array, but functions that construct arrays usually also include an
optional argument to explicitly specify the datatype. Here is an example:

import numpy as np

x = np.array([1, 2]) # Let numpy choose the datatype

print x.dtype # Prints "int64"

x = np.array([1.0, 2.0]) # Let numpy choose the datatype

print x.dtype # Prints "float64"

x = np.array([1, 2], dtype=np.int64) # Force a particular datatype

print x.dtype # Prints "int64"

You can read all about numpy datatypes in the documentation.

Array math
Basic mathematical functions operate elementwise on arrays, and are available both as
operator overloads and as functions in the numpy module:

import numpy as np

x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)

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# Elementwise sum; both produce the array

# [[ 6.0 8.0]
# [10.0 12.0]]
print x + y
print np.add(x, y)

# Elementwise difference; both produce the array

# [[-4.0 -4.0]
# [-4.0 -4.0]]
print x - y
print np.subtract(x, y)

# Elementwise product; both produce the array

# [[ 5.0 12.0]
# [21.0 32.0]]
print x * y
print np.multiply(x, y)

# Elementwise division; both produce the array

# [[ 0.2 0.33333333]
# [ 0.42857143 0.5 ]]
print x / y
print np.divide(x, y)

# Elementwise square root; produces the array

# [[ 1. 1.41421356]
# [ 1.73205081 2. ]]
print np.sqrt(x)

Note that unlike MATLAB, * is elementwise multiplication, not matrix multiplication. We

instead use the dot function to compute inner products of vectors, to multiply a vector by
a matrix, and to multiply matrices. dot is available both as a function in the numpy
module and as an instance method of array objects:

import numpy as np

x = np.array([[1,2],[3,4]])
y = np.array([[5,6],[7,8]])

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v = np.array([9,10])
w = np.array([11, 12])

# Inner product of vectors; both produce 219

print v.dot(w)
print np.dot(v, w)

# Matrix / vector product; both produce the rank 1 array [29 67]
print x.dot(v)
print np.dot(x, v)

# Matrix / matrix product; both produce the rank 2 array

# [[19 22]
# [43 50]]
print x.dot(y)
print np.dot(x, y)

Numpy provides many useful functions for performing computations on arrays; one of the
most useful is sum :

import numpy as np

x = np.array([[1,2],[3,4]])

print np.sum(x) # Compute sum of all elements; prints "10"

print np.sum(x, axis=0) # Compute sum of each column; prints "[4 6]"
print np.sum(x, axis=1) # Compute sum of each row; prints "[3 7]"

You can And the full list of mathematical functions provided by numpy in the
documentation.

Apart from computing mathematical functions using arrays, we frequently need to reshape
or otherwise manipulate data in arrays. The simplest example of this type of operation is
transposing a matrix; to transpose a matrix, simply use the T attribute of an array object:

import numpy as np

x = np.array([[1,2], [3,4]])
print x # Prints "[[1 2]

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# [3 4]]"
print x.T # Prints "[[1 3]
# [2 4]]"

# Note that taking the transpose of a rank 1 array does nothing:

v = np.array([1,2,3])
print v # Prints "[1 2 3]"
print v.T # Prints "[1 2 3]"

Numpy provides many more functions for manipulating arrays; you can see the full list in
the documentation.

Broadcasting
Broadcasting is a powerful mechanism that allows numpy to work with arrays of different
shapes when performing arithmetic operations. Frequently we have a smaller array and a
larger array, and we want to use the smaller array multiple times to perform some operation
on the larger array.

For example, suppose that we want to add a constant vector to each row of a matrix. We
could do it like this:

import numpy as np

# We will add the vector v to each row of the matrix x,

# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = np.empty_like(x) # Create an empty matrix with the same shape as x

# Add the vector v to each row of the matrix x with an explicit loop
for i in range(4):
y[i, :] = x[i, :] + v

# Now y is the following

# [[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]

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print y

This works; however when the matrix x is very large, computing an explicit loop in Python
could be slow. Note that adding the vector v to each row of the matrix x is equivalent to
forming a matrix vv by stacking multiple copies of v vertically, then performing
elementwise summation of x and vv . We could implement this approach like this:

import numpy as np

# We will add the vector v to each row of the matrix x,

# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each other
print vv # Prints "[[1 0 1]
# [1 0 1]
# [1 0 1]
# [1 0 1]]"
y = x + vv # Add x and vv elementwise
print y # Prints "[[ 2 2 4
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"

Numpy broadcasting allows us to perform this computation without actually creating

multiple copies of v . Consider this version, using broadcasting:

import numpy as np

# We will add the vector v to each row of the matrix x,

# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = x + v # Add v to each row of x using broadcasting
print y # Prints "[[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"

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The line y = x + v works even though x has shape (4, 3) and v has shape
(3,) due to broadcasting; this line works as if v actually had shape (4, 3) , where
each row was a copy of v , and the sum was performed elementwise.

Broadcasting two arrays together follows these rules:

1. If the arrays do not have the same rank, prepend the shape of the lower rank array
with 1s until both shapes have the same length.
2. The two arrays are said to be compatible in a dimension if they have the same size in
the dimension, or if one of the arrays has size 1 in that dimension.
3. The arrays can be broadcast together if they are compatible in all dimensions.
4. After broadcasting, each array behaves as if it had shape equal to the elementwise
maximum of shapes of the two input arrays.
5. In any dimension where one array had size 1 and the other array had size greater than
1, the Arst array behaves as if it were copied along that dimension

If this explanation does not make sense, try reading the explanation from the
documentation or this explanation.

Functions that support broadcasting are known as universal functions. You can And the list
of all universal functions in the documentation.

Here are some applications of broadcasting:

import numpy as np

# Compute outer product of vectors

v = np.array([1,2,3]) # v has shape (3,)
w = np.array([4,5]) # w has shape (2,)
# To compute an outer product, we first reshape v to be a column
# vector of shape (3, 1); we can then broadcast it against w to yield
# an output of shape (3, 2), which is the outer product of v and w:
# [[ 4 5]
# [ 8 10]
# [12 15]]
print np.reshape(v, (3, 1)) * w

# Add a vector to each row of a matrix

x = np.array([[1,2,3], [4,5,6]])
# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),

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# giving the following matrix:

# [[2 4 6]
# [5 7 9]]
print x + v

# Add a vector to each column of a matrix

# x has shape (2, 3) and w has shape (2,).
# If we transpose x then it has shape (3, 2) and can be broadcast
# against w to yield a result of shape (3, 2); transposing this result
# yields the final result of shape (2, 3) which is the matrix x with
# the vector w added to each column. Gives the following matrix:
# [[ 5 6 7]
# [ 9 10 11]]
print (x.T + w).T
# Another solution is to reshape w to be a row vector of shape (2, 1);
# we can then broadcast it directly against x to produce the same
# output.
print x + np.reshape(w, (2, 1))

# Multiply a matrix by a constant:

# x has shape (2, 3). Numpy treats scalars as arrays of shape ();
# these can be broadcast together to shape (2, 3), producing the
# following array:
# [[ 2 4 6]
# [ 8 10 12]]
print x * 2

Broadcasting typically makes your code more concise and faster, so you should strive to
use it where possible.

Numpy Documentation
This brief overview has touched on many of the important things that you need to know
about numpy, but is far from complete. Check out the numpy reference to And out much
more about numpy.

SciPy

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Numpy provides a high-performance multidimensional array and basic tools to compute

with and manipulate these arrays. SciPy builds on this, and provides a large number of
functions that operate on numpy arrays and are useful for different types of scientiAc and
engineering applications.

The best way to get familiar with SciPy is to browse the documentation. We will highlight
some parts of SciPy that you might And useful for this class.

Image operations
SciPy provides some basic functions to work with images. For example, it has functions to
read images from disk into numpy arrays, to write numpy arrays to disk as images, and to
resize images. Here is a simple example that showcases these functions:

from scipy.misc import imread, imsave, imresize

# Read an JPEG image into a numpy array

img = imread('assets/cat.jpg')
print img.dtype, img.shape # Prints "uint8 (400, 248, 3)"

# We can tint the image by scaling each of the color channels

# by a different scalar constant. The image has shape (400, 248, 3);
# we multiply it by the array [1, 0.95, 0.9] of shape (3,);
# numpy broadcasting means that this leaves the red channel unchanged,
# and multiplies the green and blue channels by 0.95 and 0.9
# respectively.
img_tinted = img * [1, 0.95, 0.9]

# Resize the tinted image to be 300 by 300 pixels.

img_tinted = imresize(img_tinted, (300, 300))

# Write the tinted image back to disk

imsave('assets/cat_tinted.jpg', img_tinted)

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Left: The original image. Right: The tinted and resized image.

MATLAB Ales
The functions scipy.io.loadmat and scipy.io.savemat allow you to read and write
MATLAB Ales. You can read about them in the documentation.

Distance between points

SciPy deAnes some useful functions for computing distances between sets of points.

The function scipy.spatial.distance.pdist computes the distance between all

pairs of points in a given set:

import numpy as np
from scipy.spatial.distance import pdist, squareform

# Create the following array where each row is a point in 2D space:

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# [[0 1]
# [1 0]
# [2 0]]
x = np.array([[0, 1], [1, 0], [2, 0]])
print x

# Compute the Euclidean distance between all rows of x.

# d[i, j] is the Euclidean distance between x[i, :] and x[j, :],
# and d is the following array:
# [[ 0. 1.41421356 2.23606798]
# [ 1.41421356 0. 1. ]
# [ 2.23606798 1. 0. ]]
d = squareform(pdist(x, 'euclidean'))
print d

You can read all the details about this function in the documentation.

A similar function ( scipy.spatial.distance.cdist ) computes the distance between

all pairs across two sets of points; you can read about it in the documentation.

Matplotlib
Matplotlib is a plotting library. In this section give a brief introduction to the
matplotlib.pyplot module, which provides a plotting system similar to that of
MATLAB.

Plotting
The most important function in matplotlib is plot , which allows you to plot 2D data. Here
is a simple example:

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on a sine curve

x = np.arange(0, 3 * np.pi, 0.1)
y = np.sin(x)

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# Plot the points using matplotlib

plt.plot(x, y)
plt.show() # You must call plt.show() to make graphics appear.

Running this code produces the following plot:

With just a little bit of extra work we can easily plot multiple lines at once, and add a title,
legend, and axis labels:

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on sine and cosine curves
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
y_cos = np.cos(x)

# Plot the points using matplotlib

plt.plot(x, y_sin)
plt.plot(x, y_cos)
plt.xlabel('x axis label')
plt.ylabel('y axis label')
plt.title('Sine and Cosine')
plt.legend(['Sine', 'Cosine'])
plt.show()

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You can read much more about the plot function in the documentation.

Subplots
You can plot different things in the same Agure using the subplot function. Here is an
example:

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on sine and cosine curves
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
y_cos = np.cos(x)

# Set up a subplot grid that has height 2 and width 1,

# and set the first such subplot as active.
plt.subplot(2, 1, 1)

# Make the first plot

plt.plot(x, y_sin)
plt.title('Sine')

# Set the second subplot as active, and make the second plot.

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plt.subplot(2, 1, 2)
plt.plot(x, y_cos)
plt.title('Cosine')

# Show the figure.

plt.show()

You can read much more about the subplot function in the documentation.

Images
You can use the imshow function to show images. Here is an example:

import numpy as np
from scipy.misc import imread, imresize
import matplotlib.pyplot as plt

img = imread('assets/cat.jpg')
img_tinted = img * [1, 0.95, 0.9]

# Show the original image

plt.subplot(1, 2, 1)
plt.imshow(img)

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# Show the tinted image

plt.subplot(1, 2, 2)

# A slight gotcha with imshow is that it might give strange results

# if presented with data that is not uint8. To work around this, we
# explicitly cast the image to uint8 before displaying it.
plt.imshow(np.uint8(img_tinted))
plt.show()

cs231n
cs231n
karpathy@cs.stanford.edu

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