.. Copyright (C) Brad Miller, David Ranum, Jeffrey Elkner, Peter Wentworth, Allen B. Downey, Chris

Meyers, and Dario Mitchell. Permission is granted to copy, distribute

and/or modify this document under the terms of the GNU Free Documentation

License, Version 1.3 or any later version published by the Free Software

Foundation; with Invariant Sections being Forward, Prefaces, and

Contributor List, no Front-Cover Texts, and no Back-Cover Texts. A copy of

the license is included in the section entitled "GNU Free Documentation

License".

.. shortname:: MoreAboutIteration

.. description:: This module has more information about iteration and while loops

Iteration Revisited

===================

.. index:: iteration, assignment, assignment statement, reassignment

.. index::

single: statement; assignment

Computers are often used to automate repetitive tasks. Repeating identical or

similar tasks without making errors is something that computers do well and

people do poorly.

Repeated execution of a sequence of statements is called **iteration**. Because

iteration is so common, Python provides several language features to make it

easier. We've already seen the ``for`` statement in Chapter 3. This is a very common

form of iteration in Python. In this chapter

we are also going to look at the ``while`` statement --- another way to have your

program do iteration.

.. admonition:: Scratch Editor

.. actex:: scratch_7_1

.. index:: for loop

The ``for`` loop revisited

--------------------------

Recall that the ``for`` loop processes each item in a list. Each item in

turn is (re-)assigned to the loop variable, and the body of the loop is executed.

We saw this example in an earlier chapter.

.. activecode:: ch07_for1

for f in ["Joe", "Amy", "Brad", "Angelina", "Zuki", "Thandi", "Paris"]:

invitation = "Hi " + f + ". Please come to my party on Saturday!"

print(invitation)

We have also seen iteration paired with the update idea to form the accumulator pattern. For example, to compute

the sum of the first n integers, we could create a for loop using the ``range`` to produce the numbers 1 thru n.

Using the accumulator pattern, we can start with a running total and on each iteration, add the current value of the loop

variable. A function to compute this sum is shown below.

.. activecode:: ch07_summation

def sumTo(aBound):

theSum = 0

for aNumber in range(1, aBound+1):

theSum = theSum + aNumber

return theSum

print(sumTo(4))

print(sumTo(1000))

To review, the variable ``theSum`` is called the accumulator. It is initialized to zero before we start the loop. The loop variable, ``aNumber`` will take on the values produced by the ``range(1,aBound+1)`` function call. Note that this produces all the integers starting from 1 up to the value of ``aBound``. If we had not added 1 to ``aBound``, the range would have stopped one value short since ``range`` does not include the upper bound.

The assignment statement, ``theSum = theSum + aNumber``, updates ``theSum`` each time thru the loop. This accumulates the running total. Finally, we return the value of the accumulator.

.. admonition:: Scratch Editor

.. actex:: scratch_7_2

The ``while`` Statement

-----------------------

.. video:: whileloop

:controls:

:thumb: ../_static/whileloop.png

http://knuth.luther.edu/~pythonworks/thinkcsVideos/whileloop.mov

http://knuth.luther.edu/~pythonworks/thinkcsVideos/whileloop.webm

There is another Python statement that can also be used to build an iteration. It is called the ``while`` statement.

The ``while`` statement provides a much more general mechanism for iterating. Similar to the ``if`` statement, it uses

a boolean expression to control the flow of execution. The body of while will be repeated as long as the controlling boolean expression evaluates to ``True``.

The following figure shows the flow of control.

.. image:: Figures/while_flow.png

We can use the ``while`` loop to create any type of iteration we wish, including anything that we have previously done with a ``for`` loop. For example, the program in the previous section could be rewritten using ``while``.

Instead of relying on the ``range`` function to produce the numbers for our summation, we will need to produce them ourselves. To to this, we will create a variable called ``aNumber`` and initialize it to 1, the first number in the summation. Every iteration will add ``aNumber`` to the running total until all the values have been used.

In order to control the iteration, we must create a boolean expression that evaluates to ``True`` as long as we want to keep adding values to our running total. In this case, as long as ``aNumber`` is less than or equal to the bound, we should keep going.

Here is a new version of the summation program that uses a while statement.

.. activecode:: ch07_while1

def sumTo(aBound):

""" Return the sum of 1+2+3 ... n """

theSum = 0

aNumber = 1

while aNumber <= aBound:

theSum = theSum + aNumber

aNumber = aNumber + 1

return theSum

print(sumTo(4))

print(sumTo(1000))

You can almost read the ``while`` statement as if it were in natural language. It means,

while ``aNumber`` is less than or equal to ``aBound``, continue executing the body of the loop. Within

the body, each time, update ``theSum`` using the accumulator pattern and increment ``aNumber``. After the body of the loop, we go back up to the condition of the ``while`` and reevaluate it. When ``aNumber`` becomes greater than ``aBound``, the condition fails and flow of control continues to the ``return`` statement.

The same program in codelens will allow you to observe the flow of execution.

.. codelens:: ch07_while2

def sumTo(aBound):

""" Return the sum of 1+2+3 ... n """

theSum = 0

aNumber = 1

while aNumber <= aBound:

theSum = theSum + aNumber

aNumber = aNumber + 1

return theSum

print(sumTo(4))

.. note:: The names of the variables have been chosen to help readability.

More formally, here is the flow of execution for a ``while`` statement:

#. Evaluate the condition, yielding ``False`` or ``True``.

#. If the condition is ``False``, exit the ``while`` statement and continue

execution at the next statement.

#. If the condition is ``True``, execute each of the statements in the body and

then go back to step 1.

The body consists of all of the statements below the header with the same

indentation.

This type of flow is called a **loop** because the third step loops back around

to the top. Notice that if the condition is ``False`` the first time through the

loop, the statements inside the loop are never executed.

The body of the loop should change the value of one or more variables so that

eventually the condition becomes ``False`` and the loop terminates. Otherwise the

loop will repeat forever. This is called an **infinite loop**.

An endless

source of amusement for computer scientists is the observation that the

directions on shampoo, lather, rinse, repeat, are an infinite loop.

In the case shown above, we can prove that the loop terminates because we

know that the value of ``n`` is finite, and we can see that the value of ``v``

increments each time through the loop, so eventually it will have to exceed ``n``. In

other cases, it is not so easy to tell.

.. note::

Introduction of the while statement causes us to think about the types of iteration we have seen. The ``for`` statement will always iterate through a sequence of values like the list of names for the party or the list of numbers created by ``range``. Since we know that it will iterate once for each value in the collection, it is often said that a ``for`` loop creates a

**definite iteration** because we definitely know how many times we are going to iterate. On the other

hand, the ``while`` statement is dependent on a condition that needs to evaluate to ``False`` in order

for the loop to terminate. Since we do not necessarily know when this will happen, it creates what we

call **indefinite iteration**. Indefinite iteration simply means that we don't know how many times we will repeat but eventually the condition controlling the iteration will fail and the iteration will stop. (Unless we have an infinite loop which is of course a problem)

What you will notice here is that the ``while`` loop is more work for

you --- the programmer --- than the equivalent ``for`` loop. When using a ``while``

loop you have to control the loop variable yourself. You give it an initial value, test

for completion, and then make sure you change something in the body so that the loop

terminates.

So why have two kinds of loop if ``for`` looks easier? This next example shows an indefinite iteration where

we need the extra power that we get from the ``while`` loop.

.. admonition:: Scratch Editor

.. actex:: scratch_7_3

**Check your understanding**

.. mchoicemf:: test_question7_2_1

:answer_a: True

:answer_b: False

:correct: a

:feedback_a: Although the while loop uses a different syntax, it is just as powerful as a for-loop and often more flexible.

:feedback_b: Often a for-loop is more natural and convenient for a task, but that same task can always be expressed using a while loop.

True or False: You can rewrite any for-loop as a while-loop.

.. mchoicemf:: test_question7_2_2

:answer_a: n starts at 10 and is incremented by 1 each time through the loop, so it will always be positive

:answer_b: answer starts at 1 and is incremented by n each time, so it will always be positive

:answer_c: You cannot compare n to 0 in while loop. You must compare it to another variable.

:answer_d: In the while loop body, we must set n False, and this code does not do that.

:correct: a

:feedback_a: The loop will run as long as n is positive. In this case, we can see that n will never become non-positive.

:feedback_b: While it is true that answer will always be positive, answer is not considered in the loop condition.

:feedback_c: It is perfectly valid to compare n to 0. Though indirectly, this is what causes the infinite loop.

:feedback_d: The loop condition must become False for the loop to terminate, but n by itself is not the condition in this case.

The following code contains an infinite loop. Which is the best explanation for why the loop does not terminate?

<pre>

n = 10

answer = 1

while ( n > 0 ):

answer = answer + n

n = n + 1

print answer

</pre>

Randomly Walking Turtles

------------------------

Suppose we want to entertain ourselves by watching a turtle wander around

randomly inside the screen. When we run the program we want the turtle and

program to behave in the following way:

#. The turtle begins in the center of the screen.

#. Flip a coin. If its heads then turn to the left 90 degrees. If its tails

then turn to the right 90 degrees.

#. Take 50 steps forward.

#. If the turtle has moved outside the screen then stop, otherwise go back to

step 2 and repeat.

Notice that we cannot predict how many times the turtle will need to flip the

coin before it wanders out of the screen, so we can't use a for loop in this

case. In fact, although very unlikely, this program might never end,

that is why we call this indefinite iteration.

So based on the problem description above, we can outline a program as follows:

.. sourcecode:: python

create a window and a turtle

while the turtle is still in the window:

generate a random number between 0 and 1

if the number == 0 (heads):

turn left

else:

turn right

move the turtle forward 50

Now, probably the only thing that seems a bit confusing to you is the part

about whether or not the turtle is still in the screen. But this is the nice

thing about programming, we can delay the tough stuff and get *something* in

our program working right away. The way we are going to do this is to

delegate the work of deciding whether the turtle is still in the screen or

not to a boolean function. Lets call this boolean function ``isInScreen`` We

can write a very simple version of this boolean function by having

it always return ``True``, or by having it decide randomly,

the point is to have it do something simple so that we can focus on the parts

we already know how to do well and get them working. Since having it always

return true would not be a good idea we will write our version to decide

randomly. Lets say that there is a 90% chance the turtle is still in the

window and 10% that the turtle has escaped.

.. activecode:: iter_randwalk1

import random

import turtle

def isInScreen(w,t):

if random.random() > 0.1:

return True

else:

return False

t = turtle.Turtle()

wn = turtle.Screen()

t.shape('turtle')

while isInScreen(wn,t):

coin = random.randrange(0,2)

if coin == 0: # heads

t.left(90)

else: # tails

t.right(90)

t.forward(50)

wn.exitonclick()

Now we have a working program that draws a random walk of our turtle that has

a 90% chance of staying on the screen. We are in a good position,

because a large part of our program is working and we can focus on the next

bit of work -- deciding whether the turtle is inside the screen boundaries or

not.

We can find out the width and the height of the screen using the

``window_width`` and ``window_height`` methods of the screen object.

However, remember that the turtle starts at position 0,0 in the middle of the

screen. So we never want the turtle to go farther right than width/2 or

farther left than negative width/2. We never want the turtle to go further

up than height/2 or further down than negative height/2. Once we know what

the boundaries are we can use some conditionals to check the turtle position

against the boundaries and return ``False`` if the turtle is outside or

``True`` if the turtle is inside.

Once we have computed our boundaries we can get the current position of the

turtle and then use conditionals to decide. Here is one implementation:

.. sourcecode:: python

def isInScreen(wn,t):

leftBound = - wn.window_width()/2

rightBound = wn.window_width()/2

topBound = wn.window_height()/2

bottomBound = -wn.window_height()/2

turtleX = t.xcor()

turtleY = t.ycor()

stillIn = True

if turtleX > rightBound or turtleX < leftBound:

stillIn = False

if turtleY > topBound or turtleY < bottomBound:

stillIn = False

return stillIn

There are lots of ways that the conditional could be written. In this case

we have given ``stillIn`` the default value of ``True`` and use two ``if``

statements

to set the value to ``False``. You could rewrite this to use nested

conditionals or ``elif`` statements and set ``stillIn`` to ``True`` in an else

clause.

Here is the full version of our random walk program.

.. activecode:: iter_randwalk2

import random

import turtle

def isInScreen(w,t):

leftBound = - w.window_width()/2

rightBound = w.window_width()/2

topBound = w.window_height()/2

bottomBound = -w.window_height()/2

turtleX = t.xcor()

turtleY = t.ycor()

stillIn = True

if turtleX > rightBound or turtleX < leftBound:

stillIn = False

if turtleY > topBound or turtleY < bottomBound:

stillIn = False

return stillIn

t = turtle.Turtle()

wn = turtle.Screen()

t.shape('turtle')

while isInScreen(wn,t):

coin = random.randrange(0,2)

if coin == 0:

t.left(90)

else:

t.right(90)

t.forward(50)

wn.exitonclick()

We could have written this program without using a boolean function,

You might try to rewrite it using a complex condition on the while statement,

but using a boolean function makes the program much more readable and easier

to understand. It also gives us another tool to use if this was a

larger program and we needed to have a check for whether the turtle

was still in the screen in another part of the program. Another advantage is

that if you ever need to write a similar program, you can reuse this function

with confidence the next time you need it. Breaking up this

program into a couple of parts is another example of functional decomposition.

.. admonition:: Scratch Editor

.. actex:: scratch_7_4

.. index:: 3n + 1 sequence

**Check your understanding**

.. mchoicemf:: test_question7_3_1

:answer_a: a for-loop or a while-loop

:answer_b: only a for-loop

:answer_c: only a while-loop

:correct: a

:feedback_a: Although you do not know how many iterations you loop will run before the program starts running, once you have chosen your random integer, Python knows exactly how many iterations the loop will run, so either a for-loop or a while-loop will work.

:feedback_b: As you learned in section 7.2, a while-loop can always be used for anything a for-loop can be used for.

:feedback_c: Although you do not know how many iterations you loop will run before the program starts running, once you have chosen your random integer, Python knows exactly how many iterations the loop will run, so this is an example of definite iteration.

Which type of loop can be used to perform the following iteration: You choose a positive integer at random and then print the numbers from 1 up to and including the selected integer.

.. mchoicemf:: test_question7_3_2

:answer_a: Returns True if the turtle is still on the screen and False if the turtle is no longer on the screen.

:answer_b: Uses a while loop to move the turtle randomly until it goes off the screen.

:answer_c: Turns the turtle right or left at random and moves the turtle forward 50.

:answer_d: Calculates and returns the position of the turtle in the window.

:correct: a

:feedback_a: The isInScreen function computes the boolean test of whether the turtle is still in the window. It makes the condition of the while loop in the main part of the code simpler.

:feedback_b: The isInScreen function does not contain a while-loop. That loop is outside the isInScreen function.

:feedback_c: The isInScreen function does not move the turtle.

:feedback_d: While the isInScreen function does use the size of the window and position of the turtle, it does not return the turtle position.

In the random walk program in this section, what does the isInScreen function do?

The 3n + 1 Sequence

-------------------

As another example of indefinite iteration, let's look at a sequence that has fascinated mathematicians for many years.

The rule for creating the sequence is to start from

some given ``n``, and to generate

the next term of the sequence from ``n``, either by halving ``n``,

whenever ``n`` is even, or else by multiplying it by three and adding 1 when it is odd. The sequence

terminates when ``n`` reaches 1.

This Python function captures that algorithm. Try running this program several times supplying different values for n.

.. activecode:: ch07_indef1

def seq3np1(n):

""" Print the 3n+1 sequence from n, terminating when it reaches 1."""

while n != 1:

print(n)

if n % 2 == 0: # n is even

n = n // 2

else: # n is odd

n = n * 3 + 1

print(n) # the last print is 1

seq3np1(3)

The condition for this loop is ``n != 1``. The loop will continue running until

``n == 1`` (which will make the condition false).

Each time through the loop, the program prints the value of ``n`` and then

checks whether it is even or odd using the remainder operator. If it is even, the value of ``n`` is divided

by 2 using integer division. If it is odd, the value is replaced by ``n * 3 + 1``.

Try some other examples.

Since ``n`` sometimes increases and sometimes decreases, there is no obvious

proof that ``n`` will ever reach 1, or that the program terminates. For some

particular values of ``n``, we can prove termination. For example, if the

starting value is a power of two, then the value of ``n`` will be even each

time through the loop until it reaches 1.

You might like to have some fun and see if you can find a small starting

number that needs more than a hundred steps before it terminates.

.. admonition:: Lab

* `Experimenting with the 3n+1 Sequence <../Labs/sequencelab.html>`_ In this guided lab exercise we will try to learn more about this sequence.

Particular values aside, the interesting question is whether we can prove that

this sequence terminates for *all* values of ``n``. So far, no one has been able

to prove it *or* disprove it!

Think carefully about what would be needed for a proof or disproof of the hypothesis

*"All positive integers will eventually converge to 1"*. With fast computers we have

been able to test every integer up to very large values, and so far, they all

eventually end up at 1. But this doesn't mean that there might not be some

as-yet untested number which does not reduce to 1.

You'll notice that if you don't stop when you reach one, the sequence gets into

its own loop: 1, 4, 2, 1, 4, 2, 1, 4, and so on. One possibility is that there might

be other cycles that we just haven't found.

.. admonition:: Choosing between ``for`` and ``while``

Use a ``for`` loop if you know the maximum number of times that you'll

need to execute the body. For example, if you're traversing a list of elements,

or can formulate a suitable call to ``range``, then choose the ``for`` loop.

So any problem like "iterate this weather model run for 1000 cycles", or "search this

list of words", "check all integers up to 10000 to see which are prime" suggest that a ``for`` loop is best.

By contrast, if you are required to repeat some computation until some condition is

met, as we did in this 3n + 1 problem, you'll need a ``while`` loop.

As we noted before, the first case is called **definite iteration** --- we have some definite bounds for

what is needed. The latter case is called **indefinite iteration** --- we are not sure

how many iterations we'll need --- we cannot even establish an upper bound!

.. There are also some great visualization tools becoming available to help you

.. trace and understand small fragments of Python code. The one we recommend is at

.. http://netserv.ict.ru.ac.za/python3_viz

.. admonition:: Scratch Editor

.. actex:: scratch_7_5

.. index::

single: Newton's method

**Check your understanding**

.. mchoicemf:: test_question7_4_1

:answer_a: Yes.

:answer_b: No.

:answer_c: No one knows.

:correct: c

:feedback_a: The 3n+1 sequence has not been proven to terminate for all values of n.

:feedback_b: It has not been disproven that the 3n+1 sequence will terminate for all values of n. In other words, there might be some value for n such that this sequence does not terminate. We just have not found it yet.

:feedback_c: That this sequence terminates for all values of n has not been proven or disproven so no one knows whether the while loop will always terminate or not.

Consider the code that prints the 3n+1 sequence in ActiveCode box 6. Will the while loop in this code always terminate for any value of n?

Newton's Method

---------------

Loops are often used in programs that compute numerical results by starting

with an approximate answer and iteratively improving it.

For example, one way of computing square roots is Newton's method. Suppose

that you want to know the square root of ``n``. If you start with almost any

approximation, you can compute a better approximation with the following

formula:

.. sourcecode:: python

better = 1/2 * (approx + n/approx)

Execute this algorithm a few times using your calculator. Can you

see why each iteration brings your estimate a little closer? One of the amazing

properties of this particular algorithm is how quickly it converges to an accurate

answer.

The following implementation of Newton's method requires two parameters. The first is the

value whose square root will be approximated. The second is the number of times to iterate the

calculation yielding a better result.

.. activecode:: chp07_newtonsdef

def newtonSqrt(n, howmany):

approx = 0.5 * n

for i in range(howmany):

betterapprox = 0.5 * (approx + n/approx)

approx = betterapprox

return betterapprox

print(newtonSqrt(10,3))

print(newtonSqrt(10,5))

print(newtonSqrt(10,10))

You may have noticed that the second and third calls to ``newtonSqrt`` in the previous example both returned the same value for the square root of 10. Using 10 iterations instead of 5 did not improve the the value. In general, Newton's algorithm will eventually reach a point where the new approximation is no better than the previous. At that point, we could simply stop.

In other words, by repeatedly applying this formula until the better approximation gets close

enough to the previous one, we can write a function for computing the square root that uses the number of iterations necessary and no more.

This implementation, shown in codelens,

uses a ``while`` condition to execute until the approximation is no longer changing. Each time thru the loop we compute a "better" approximation using the formula described earlier. As long as the "better" is different, we try again. Step thru the program and watch the approximations get closer and closer.

.. codelens:: chp07_newtonswhile

def newtonSqrt(n):

approx = 0.5 * n

better = 0.5 * (approx + n/approx)

while better != approx:

approx = better

better = 0.5 * (approx + n/approx)

return approx

print(newtonSqrt(10))

.. note::

The ``while`` statement shown above uses comparison of two floating point numbers in the condition. Since floating point numbers are themselves approximation of real numbers in mathematics, it is often

better to compare for a result that is within some small threshold of the value you are looking for.

.. index:: algorithm

.. admonition:: Scratch Editor

.. actex:: scratch_7_6

Algorithms Revisited

--------------------

Newton's method is an example of an **algorithm**: it is a mechanical process

for solving a category of problems (in this case, computing square roots).

It is not easy to define an algorithm. It might help to start with something

that is not an algorithm. When you learned to multiply single-digit numbers,

you probably memorized the multiplication table. In effect, you memorized 100

specific solutions. That kind of knowledge is not algorithmic.

But if you were lazy, you probably cheated by learning a few tricks. For

example, to find the product of n and 9, you can write n - 1 as the first digit

and 10 - n as the second digit. This trick is a general solution for

multiplying any single-digit number by 9. That's an algorithm!

Similarly, the techniques you learned for addition with carrying, subtraction

with borrowing, and long division are all algorithms. One of the

characteristics of algorithms is that they do not require any intelligence to

carry out. They are mechanical processes in which each step follows from the

last according to a simple set of rules.

On the other hand, understanding that hard problems can be solved by step-by-step

algorithmic processess is one of the major simplifying breakthroughs that has

had enormous benefits. So while the execution of the algorithm

may be boring and may require no intelligence, algorithmic or computational

thinking is having a vast impact. It is the process of designing algorithms that is interesting,

intellectually challenging, and a central part of what we call programming.

Some of the things that people do naturally, without difficulty or conscious

thought, are the hardest to express algorithmically. Understanding natural

language is a good example. We all do it, but so far no one has been able to

explain *how* we do it, at least not in the form of a step-by-step mechanical

algorithm.

.. index:: table, logarithm, Intel, Pentium, escape sequence, tab, newline,

cursor

.. admonition:: Scratch Editor

.. actex:: scratch_7_7

Simple Tables

-------------

One of the things loops are good for is generating tabular data. Before

computers were readily available, people had to calculate logarithms, sines and

cosines, and other mathematical functions by hand. To make that easier,

mathematics books contained long tables listing the values of these functions.

Creating the tables was slow and boring, and they tended to be full of errors.

When computers appeared on the scene, one of the initial reactions was, *"This is

great! We can use the computers to generate the tables, so there will be no

errors."* That turned out to be true (mostly) but shortsighted. Soon thereafter,

computers and calculators were so pervasive that the tables became obsolete.

Well, almost. For some operations, computers use tables of values to get an

approximate answer and then perform computations to improve the approximation.

In some cases, there have been errors in the underlying tables, most famously

in the table the Intel Pentium processor chip used to perform floating-point division.

Although a power of 2 table is not as useful as it once was, it still makes a good

example of iteration. The following program outputs a sequence of values in the

left column and 2 raised to the power of that value in the right column:

.. activecode:: ch07_table1

print("n",'\t',"2**n") #table column headings

print("---",'\t',"-----")

for x in range(13): # generate values for columns

print(x, '\t', 2**x)

The string ``'\t'`` represents a **tab character**. The backslash character in

``'\t'`` indicates the beginning of an **escape sequence**. Escape sequences

are used to represent invisible characters like tabs and newlines. The sequence

``\n`` represents a **newline**.

An escape sequence can appear anywhere in a string. In this example, the tab

escape sequence is the only thing in the string. How do you think you represent

a backslash in a string?

As characters and strings are displayed on the screen, an invisible marker

called the **cursor** keeps track of where the next character will go. After a

``print`` function, the cursor normally goes to the beginning of the next

line.

The tab character shifts the cursor to the right until it reaches one of the

tab stops. Tabs are useful for making columns of text line up, as in the output

of the previous program.

Because of the tab characters between the columns, the position of the second

column does not depend on the number of digits in the first column.

.. admonition:: Scratch Editor

.. actex:: scratch_7_8

.. index::

single: local variable

single: variable; local

**Check your understanding**

.. mchoicemf:: test_question7_7_1

:answer_a: A tab will line up items in a second column, regardless of how many characters were in the first column, while spaces will not.

:answer_b: There is no difference

:answer_c: A tab is wider than a sequence of spaces

:answer_d: You must use tabs for creating tables. You cannot use spaces.

:correct: a

:feedback_a: Assuming the size of the first column is less than the size of the tab width.

:feedback_b: Tabs and spaces will sometimes make output appear visually different.

:feedback_c: A tab has a pre-defined width that is equal to a given number of spaces.

:feedback_d: You may use spaces to create tables. The columns might look jagged, or they might not, depending on the width of the items in each column.

What is the difference between a tab (\t) and a sequence of spaces?

2-Dimensional Iteration: Image Processing

-----------------------------------------

Two dimensional tables have both rows and columns. You have probably seen many tables like this if you have used a

spreadsheet program. Another object that is organized in rows and columns is a digital image. In this section we will

explore how iteration allows us to manipulate these images.

A **digital image** is a finite collection of small, discrete picture elements called **pixels**. These pixels are organized in a two-dimensional grid. Each pixel represents the smallest amount of picture information that is

available. Sometimes these pixels appear as small "dots".

Each image (grid of pixels) has its own width and its own height. The width is the number of columns and the height is the number of rows. We can name the pixels in the grid by using the column number and row number. However, it is very important to remember

that computer scientists like to start counting with 0! This means that if there are 20 rows, they will be named 0,1,2, and so on thru 19. This will be very useful later when we iterate using range.

In the figure below, the pixel of interest is found at column **c** and row **r**.

.. image:: Figures/image.png

The RGB Color Model

^^^^^^^^^^^^^^^^^^^

Each pixel of the image will represent a single color. The specific color depends on a formula that mixes various amounts

of three basic colors: red, green, and blue. This technique for creating color is known as the **RGB Color Model**.

The amount of each color, sometimes called the **intensity** of the color, allows us to have very fine control over the

resulting color.

The minimum intensity value for a basic color is 0. For example if the red intensity is 0, then there is no red in the pixel. The maximum

intensity is 255. This means that there are actually 256 different amounts of intensity for each basic color. Since there

are three basic colors, that means that you can create 256\ :sup:`3` distinct colors using the RGB Color Model.

Here are the red, green and blue intensities for some common colors. Note that "Black" is represented by a pixel having

no basic color. On the other hand, "White" has maximum values for all three basic color components.

======= ======= ======= =======

Color Red Green Blue

======= ======= ======= =======

Red 255 0 0

Green 0 255 0

Blue 0 0 255

White 255 255 255

Black 0 0 0

Yellow 255 255 0

Magenta 255 0 255

======= ======= ======= =======

In order to manipulate an image, we need to be able to access individual pixels. This capability is provided by

a module called **image**. The image module defines two classes: ``Image`` and ``Pixel``.

Each Pixel object has three attributes: the red intensity, the green intensity, and the blue intensity. A pixel provides three methods

that allow us to ask for the intensity values. They are called ``getRed``, ``getGreen``, and ``getBlue``. In addition, we can ask a

pixel to change an intensity value using its ``setRed``, ``setGreen``, and ``setBlue`` methods.

============ ================ ===============================================

Method Name Example Explanation

============ ================ ===============================================

Pixel(r,g,b) Pixel(20,100,50) Create a new pixel with 20 red, 100 green, and 50 blue.

getRed() r = p.getRed() Return the red component intensity.

getGreen() r = p.getGreen() Return the green component intensity.

getBlue() r = p.getBlue() Return the blue component intensity.

setRed() p.setRed(100) Set the red component intensity to 100.

setGreen() p.setGreen(45) Set the green component intensity to 45.

setBlue() p.setBlue(156) Set the blue component intensity to 156.

============ ================ ===============================================

In the example below, we first create a pixel with 45 units of red, 76 units of green, and 200 units of blue.

We then print the current amount of red, change the amount of red, and finally, set the amount of blue to be

the same as the current amount of green.

.. activecode:: pixelex1a

import image

p = image.Pixel(45,76,200)

print(p.getRed())

p.setRed(66)

print(p.getRed())

p.setBlue(p.getGreen())

print(p.getGreen(), p.getBlue())

**Check your understanding**

.. mchoicemf:: test_question7_8_1_1

:answer_a: Dark red

:answer_b: Light red

:answer_c: Dark green

:answer_d: Light green

:correct: a

:feedback_a: Because all three values are close to 0, the color will be dark. But because the red value is higher than the other two, the color will appear red.

:feedback_b: The closer the values are to 0, the darker the color will appear.

:feedback_c: The first value in RGB is the red value. The second is the green. This color has no green in it.

:feedback_d: The first value in RGB is the red value. The second is the green. This color has no green in it.

If you have a pixel whose RGB value is (20, 0, 0), what color will this pixel appear to be?

Image Objects

^^^^^^^^^^^^^

To access the pixels in a real image, we need to first create an ``Image`` object. Image objects can be created in two

ways. First, an Image object can be made from the

files that store digital images. The image object has an attribute corresponding to the width, the height, and the

collection of pixels in the image.

It is also possible to create an Image object that is "empty". An ``EmptyImage`` has a width and a height. However, the

pixel collection consists of only "White" pixels.

We can ask an image object to return its size using the ``getWidth`` and ``getHeight`` methods. We can also get a pixel from a particular location in the image using ``getPixel`` and change the pixel at

a particular location using ``setPixel``.

The Image class is shown below. Note that the first two entries show how to create image objects. The parameters are

different depending on whether you are using an image file or creating an empty image.

=================== =============================== ==================================================

Method Name Example Explanation

=================== =============================== ==================================================

Image(filename) img = image.Image("cy.png") Create an Image object from the file cy.png.

EmptyImage() img = image.EmptyImage(100,200) Create an Image object that has all "White" pixels

getWidth() w = img.getWidth() Return the width of the image in pixels.

getHeight() h = img.getHeight() Return the height of the image in pixels.

getPixel(col,row) p = img.getPixel(35,86) Return the pixel at column 35, row 86d.

setPixel(col,row,p) img.setPixel(100,50,mp) Set the pixel at column 100, row 50 to be mp.

=================== =============================== ==================================================

Consider the image shown below. Assume that the image is stored in a file called "luther.jpg". Line 2 opens the

file and uses the contents to create an image object that is referred to by ``img``. Once we have an image object,

we can use the methods described above to access information about the image or to get a specific pixel and check

on its basic color intensities.

.. raw:: html

<img src="../_static/LutherBellPic.jpg" id="luther.jpg">

.. activecode:: pixelex1

import image

img = image.Image("luther.jpg")

print(img.getWidth())

print(img.getHeight())

p = img.getPixel(45,55)

print(p.getRed(), p.getGreen(), p.getBlue())

When you run the program you can see that the image has a width of 400 pixels and a height of 244 pixels. Also, the

pixel at column 45, row 55, has RGB values of 165, 161, and 158. Try a few other pixel locations by changing the ``getPixel`` arguments and rerunning the program.

**Check your understanding**

.. mchoicemf:: test_question7_8_2_1

:answer_a: 149 132 122

:answer_b: 183 179 170

:answer_c: 165 161 158

:answer_d: 201 104 115

:correct: a

:feedback_a: Yes, the RGB values are 149 132 122 at row 100 and column 30.

:feedback_b: These are the values for the pixel at row 30, column 100. Get the values for row 100 and column 30 with p = img.getPixel(100,30).

:feedback_c: These are the values from the original example (row 45, column 55). Get the values for row 100 and column 30 with p = img.getPixel(100,30).

:feedback_d: These are simply made-up values that may or may not appear in the image. Get the values for row 100 and column 30 with p = img.getPixel(100,30).

In the example in ActiveCode box 10, what are the RGB values of the pixel at row 100, column 30?

Image Processing and Nested Iteration

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

**Image processing** refers to the ability to manipulate the individual pixels in a digital image. In order to process

all of the pixels, we need to be able to systematically visit all of the rows and columns in the image. The best way

to do this is to use **nested iteration**.

Nested iteration simply means that we will place one iteration construct inside of another. We will call these two

iterations the **outer iteration** and the **inner iteration**.

To see how this works, consider the simple iteration below.

.. sourcecode:: python

for i in range(5):

print(i)

We have seen this enough times to know that the value of ``i`` will be 0, then 1, then 2, and so on up to 4.

The ``print`` will be performed once for each pass.

However, the body of the loop can contain any statements including another iteration (another ``for`` statement). For example,

.. sourcecode:: python

for i in range(5):

for j in range(3):

print(i,j)

The ``for i`` iteration is the `outer iteration` and the ``for j`` iteration is the `inner iteration`. Each pass thru

the outer iteration will result in the complete processing of the inner iteration from beginning to end. This means that

the output from this nested iteration will show that for each value of ``i``, all values of ``j`` will occur.

Here is the same example in activecode. Try it. Note that the value of ``i`` stays the same while the value of ``j`` changes. The inner iteration, in effect, is moving faster than the outer iteration.

.. activecode:: nested1

for i in range(5):

for j in range(3):

print(i,j)

Another way to see this in more detail is to examine the behavior with codelens. Step thru the iterations to see the

flow of control as it occurs with the nested iteration. Again, for every value of ``i``, all of the values of ``j`` will occur. You can see that the inner iteration completes before going on to the next pass of the outer iteration.

.. codelens:: nested2

for i in range(5):

for j in range(3):

print(i,j)

Our goal with image processing is to visit each pixel. We will use an iteration to process each `row`. Within that iteration, we will use a nested iteration to process each `column`. The result is a nested iteration, similar to the one

seen above, where the outer ``for`` loop processes the rows, from 0 up to but not including the height of the image.

The inner ``for`` loop will process each column of a row, again from 0 up to but not including the width of the image.

The resulting code will look like the following. We are now free to do anything we wish to each pixel in the image.

.. sourcecode:: python

for col in range(img.getWidth()):

for row in range(img.getHeight()):

#do something with the pixel at position (col,row)

One of the easiest image processing algorithms will create what is known as a **negative** image. A negative image simply means that

each pixel will be the `opposite` of what it was originally. But what does opposite mean?

In the RGB color model, we can consider the opposite of the red component as the difference between the original red

and 255. For example, if the original red component was 50, then the opposite, or negative red value would be

``255-50`` or 205. In other words, pixels with alot of red will have negatives with little red and pixels with little red will have negatives with alot. We do the same for the blue and green as well.

The program below implements this algorithm using the previous image. Run it to see the resulting negative image. Note that there is alot of processing taking place and this may take a few seconds to complete. In addition, here are two other images that you can use. Change the name of the file in the ``image.Image()`` call to see how these images look as negatives. Also, note that there is an ``exitonclick`` method call at the very end which will close the window when you click on it. This will allow you to "clear the screen" before drawing the next negative.

.. raw:: html

<img src="../_static/cy.png" id="cy.png">

cy.png

.. raw:: html

<img src="../_static/goldygopher.png" id="goldygopher.png">

goldygopher.png

.. activecode:: acimg_1

import image

img = image.Image("luther.jpg")

newimg = image.EmptyImage(img.getWidth(),img.getHeight())

win = image.ImageWin()

for col in range(img.getWidth()):

for row in range(img.getHeight()):