Transcript Slide 1

Chapter 8
while loop syntax
while <condition>
<statement>;
<statement>;
. . .
end;
Loops
Example 1: We wrote a function insert that inserts A(k)
in the array A(1:k – 1). Now we will write the function
insertionsort to call insert iteratively.
function insert(A, k)
% assume k > 1 and A(1:k-1)is sorted
temp = A(k); j = k-1;
while (j > 0)
if A(j) > temp
A(j+1) = A(j); j = j – 1;
else
break;
end;
end;
A(j+1) = temp;
Insertion sorting
function B = insertionSort(A)
j = 2;
while (j <= length(A))
B = insert(A, j); j = j + 1;
end;
Comparison between insertion sorting and selection sorting
• In the best case, insertion sorting is much faster than
selection sorting.
• On average, insertion sorting performs half as many
comparisons as selection sorting.
• Selection sorting is much better than insertion sorting in the
number of data movements.
for loop syntax
for j = 1:k
<statement sequence>
end
An example involving image processing
Median filter: Suppose an image is corrupted by a random
noise that randomly changes a small number (say 10%) of
pixels to arbitrary value. How can we remove this noise?
Sample input and output:
Median filter design
We should avoid trying to determine which pixels are
corrupted. (too difficult.)
Instead, replace the content of each pixel by some combined
value of the neighbors.
Good pixel: likely surrounded by nearly identical values, so
replacement does not hurt.
Bad pixel: most (or all) neighbors are good so replacement
removes the noise!
How to combine the neighbor values?
• mean filter
median filter
Median filter implementation
function out = medianFilter(img)
[row, col, num]= size(img);
for i = 2:row - 1
for j = 2:col-1
lst1 = sort([img(i-1,j-1,1),
img(i, j-1,1), img(i,j,1), img(i,
img(i+1,j,1), img(i+1, j+1,1)]);
out(i,j,1) = lst1(5);
lst1 = sort([img(i-1,j-1,2),
img(i, j-1,2), img(i,j,2), img(i,
img(i+1,j,2), img(i+1, j+1,2)]);
out(i,j,2) = lst1(5);
lst1 = sort([img(i-1,j-1,3),
img(i, j-1,3), img(i,j,3), img(i,
img(i+1,j,3), img(i+1, j+1,3)]);
out(i,j,3) = lst1(5);
end;
end;
img(i-1,j,1), img(i-1,j+1,1),
j+1,1), img(i+1, j-1,1),
img(i-1,j,2), img(i-1,j+1,2),
j+1,2),img(i+1, j-1,2),
img(i-1,j,3), img(i-1,j+1,3),
j+1,3), img(i+1, j-1,3),
Simulating the outcome of a probability game
Players A and B play the following game. A fair coin is tossed
until one of the two patterns HTH or THT occurs. If the first
pattern occurs, A wins else B wins.
What is the probability that A wins the game?
There are two ways to solve this problem:
• Use Markov chain (can find the answer exactly. Will be
studied later.)
• Simulation (simulate the experiment a large number of
times and record the number of wins by each player)
Simulation based solution
Outer loop: repeat the game a large number of times, say
100000.
Inner loop: each iteration represents one game.
Outline of solution:
countA = 0; countB = 0;
for j = 1: 1000000
while the game is not over
toss coin once; update the last three bits;
end;
update the winner count;
end;
We will write three functions:
toss()  produces 1 or 2 with probability 0.5
tossgame()  produces 1 (2) if A (B) is the winner
of one randomly simulated game
repeatTossGame(n)  repeat the tossgame n
times, find a = the number of times A wins and
output the ratio a/n
Recursive functions
Before we conclude this chapter, we will discuss recursive
functions, those that can call themselves.
We have examples of functions that call other functions, e.g.
insertionsort calling insert etc.
If f(n) is defined in terms of f(n – 1), as for example, in the case
of factorial, why not let f call itself?
n! = n x (n – 1)!
Or in matlab:
fact(n) = n .* fact(n – 1)
Rules for recursive functions
1. there should be exit from recursion. (i.e., there
should be some conditional branch path in which
there is no recursive call). Such cases are called the
base cases.
2. recursive calls should make towards base case
(usually by calling itself with smaller input values).
3. Recursion may be more time or memory consuming
so should be careful with their use.
Example 1: Write a recursive function to compute n!
function out = fact(n)
if n <= 1 out = 1;
else out = n .* fact(n-1);
end;
Example 2: Write a recursive function in Matlab to perform
binary search.
Assume array A is sorted in ascending order.
Search(A, low, high, x) will return the largest t such
that A(t) <= x.
Pre-condition: A(low) <= x <= A(high)
Thus low <= t <= high.
Initially, low = 1, high = size(A)
Recursive binary search program
function out = search(A, low, high, x)
% returns the largest t s.t. A(t) <= x where
low <= t <= high
% A is assumed to be sorted
if high - low == 1
if A(high) == x
out = high;
else
out = low;
end;
else
mid = floor((low + high)/2);
if A(mid) == x out = mid;
elseif A(mid) < x
out = search(A, mid + 1, high, x);
else out = search(A, low, mid - 1, x);
end;
end;
A recursive image tiling problem
Given an input image as in the left-side, we want to
produce the image on the right-side.
Solution to the recursive image tiling problem
Observation 1: 3rd
quadrant has the input
image, but shrunk into
quarter.
We need to write a
function shrink that
shrinks the image to
quarter size.
Quadrant 1 of the output is the shrunk version of the whole output.
Equivalently, if F is the function that outputs the image on the right when given
as input on the left, then F applied to the image on the 3rd quadrant of the
output is what goes into the 1st quadrant of the output.
Quadrants 2 and 4 are now easy to determine. They can be obtained by
copying a part of the image in quadrant 1.
Some exercises in Chapter 8 and solutions
Exercise 8.1.
The input is a two-column matrix. The first column gives the number and the
second column gives the number of repetitions. You are to create an output
vector that matches these numbers.
For example:
>> repeat([2 3; 3 1; 4 2]
ans =
[222344]
Exercise 8.1.
>> repeat([2 3; 3 1; 4 2]
ans =
[222344]
function out = repeat1(m)
out = [];
[r, c] = size(m);
for i = 1 : r %Go through each row
for j = 1 : m(i, 2)
%Repeat the number of times specified in the second column
out(end + 1) = m(i, 1);
end
end
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes as
input a boolean vector of 1’s and 0’s.
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes as
input a boolean vector of 1’s and 0’s.
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes as
input a boolean vector of 1’s and 0’s.
>> A = [9, 11, 8, 6, 11, 12];
>> myfind(A, 11)
ans =
[ 0 1 0 0 1 0]
Exercise 8.4. Write a function myfind, that mimics the
behavior of the built-in function find. That is, it takes as
input a boolean vector of 1’s and 0’s.
function out = myfind(A, x)
out = [];
for j = 1:length(A)
if A(j) == x
out = [out, j];
end;
Exercise 8.8
Write a function intoBits to take an integer number as
input and output a string of 0’s and 1’s representing the
number in base 2.
>> intoBits(12)
ans =
1100
Exercise 8.11
Write functions minAll, minCol and minRow that find the
overall minimum, column minima and row minima of a
matrix A.
Exercise 8.11
Write functions minAll, minCol and minRow that find the
overall minimum, column minima and row minima of a
matrix A.
function res = minAll(m)
[r, c] = size(m);
res = m(1, 1); %Set up the first temporary minimum
for i = 1 : r %Going through each row
for j = 1 : c %Going throuch each column
if res > m(i, j) %Updating if necessary
res = m(i, j)
end
end
end
Exercise 8.11
Write functions minAll, minCol and minRow that find the
overall minimum, column minima and row minima of a
matrix A.
function res = minRow(m)
[r, c] = size(m);
res = zeros(1, r); %Set up res
for i = 1 : r %Going through each row
res(i) = m(i, 1);
%Set up the first temporary minimum for row i
for j = 2 : c %Going throuch each column
if res(i) > m(i, j) %Updating if necessary
res(i) = m(i, j);
end
end
end
Project – entropy computation
The concept of entropy is fundamental to information
and coding theory – which deals with efficient ways to
add redundancy to the data so that the receiver can
recover the message even if some of data is corrupted
by the channel.
Information theory was developed by Claude
Shannon.
Watch movie about Shannon in UCTV.
http://www.uctv.tv/search-details.aspx?showID=6090
probability distribution of individual letters
The basic idea of information theory is that if the input has less
randomness, it can be more easily predicted and hence it can be
compressed to greater degree.
Suppose there are k distinct symbols in a text.
Let P(j-th symbol) = pj.
Entropy is defined as:
E= –
Sj pj log pj
Goal is to compute the entropy of a given text.
Example: Given a three-letter alphabet containing E, Z and _
with frequencies as follows:
E
Z
_
0.5 0.25 0.25
Entropy is given by:
E = -0.5 log2 0.5 – 0.25 log2 0.25 – 0.25 log2 0.25
Using matlab, we can compute E as:
>> p = [0.5 0.25 0.25]
p =
0.5000
0.2500
>> -sum(p.*log2(p))
ans =
1.5000
0.2500
Project: Write a program in Matlab that computes the entropy
of English.
•Step 1: collect a corpus of text. count the frequency of letters
for the corpus.
• text will be stored in multiple files
• program should calculate the frequency of letters across
all the files.
•Step 2: compute entropy from the frequency table.