This document provides an overview of mathematical techniques for image processing. It discusses grayscale and color images, and how they can be represented as matrices. Methods for analyzing image data statistically, such as histograms and measures of center and spread, are presented. Working with images in MATLAB is demonstrated, including functions for reading, displaying, and extracting color channels of images. Different color models like RGB and YCbCr are described.

1. Mathematical Techniques
of Image Processing
Master of Applied Mathematics
Department of Mathematics
College of Science - University of Baghdad
Saad Al-Momen
1

3. Grayscale Digital Images
01
Working with Images in MATLAB
02
Images and Statistical Description of Quantitative Data
 Image Histograms
 Measures of Center and Spread
03
Color Images and Color Spaces
04

4. Grayscale Digital
Images
01

5. • A gray scale digital image is just an
M  N matrix of discrete pixel values.
Grayscale Digital Images
𝐴 =
𝑎1,1 𝑎1,2
𝑎2,1 𝑎2,2
⋯
𝑎1,𝑁−1 𝑎1,𝑁
𝑎2,𝑁−1 𝑎2,𝑁
⋮ ⋱ ⋮
𝑎𝑀−1,1 𝑎𝑀−1,2
𝑎𝑀,1 𝑎𝑀,2
⋯
𝑎𝑀−1,𝑁−1 𝑎𝑀−1,𝑁
𝑎𝑀,𝑁−1 𝑎𝑀,𝑁
• The maximal light intensity is
represented by the pixel value of 255
• The absence of light is represented
by the pixel value 0
• The shades of gray thus represented
by the values between 0 and 255
223 111
127 255
159 47
63 143
191 79
95 175
207 15
31 239
⇔

6. Working with Images in
MATLAB
02

7. MATLAB
Implementation
• The MATLAB function imread is used to
read the image from a graphics file of any
of a large number of common formats
>> A=imread(‘trees.tif’);
• To know the size of the image we use the
MATLAB function size
>> [M,N]=size(A)
For example, if it return the values M=258
and N=350, that indicates that the image has
dimensions 258350.
• Entering the MATLAB function
>> class(A)
into the command line returns the value
“uint8”, which tells us that the pixel values are
represented by 8-bit unsigned integers,
providing for 28 = 256 possible shades of
gray.

8. MATLAB
Implementation
• The MATLAB function imshow is used to
display the image that we are already read
>> imshow(A)
where A is the image matrix in the “uint8”
type obtained by executing the imread
command.

9. MATLAB
Implementation
• If we just use the function imshow in its
basic form, all the pixels whose values
exceed 1.0 will be interpreted as white
and displayed accordingly.
• However, in order to properly display an
image whose matrix is of the data type
“double”, we need to modify our usage of
the imshow function by supplying the range
of the pixel values. For example, we can
use
>> imshow(A,[0,255])
• If we want MATLAB to perform floating-
point arithmetic on pixels values, we must
convert the matrix type from “uint8” to
“double” by means of the command
>> A=double(A)

10. MATLAB
Implementation
• The MATLAB function subplot enables us to
fit several images or other graphics into the
same figure. Executing the command
>> subplot(m,n,p)
divides the current figure into an mn grid
and prepares to display an object in the
position of that grid specified by p.
A=imread('coins.png');
B=imread('trees.tif');
subplot(1,2,1);
imshow(A);
subplot(1,2,2);
imshow(B);
12
1 2

11. Images and Statistical
Description of Quantitative Data
03

12. • A matrix of a digital image may considered as a type of statistical data,
and we can analyze it with the help of statistical methods.
Image Histograms
• A common way to summarize quantitative data is by constructing
frequency distributions.
 Calculate the range of the dataset (the difference between the largest and
the smallest values)
 Decide on the number of bins (the number of subintervals to divide the data
into)
The subintervals are always adjacent to each other and are usually selected
to be of equal width.
 Count the frequencies (the numbers of data values that fall within each
subinterval).
 Dividing frequencies by the size of the data set (the total number of data
points) gives us relative frequencies.
 These bins, together with the frequencies, constitute a frequency distribution

13. The frequency distribution of the data set {0, 2, 4, 4, 4, 4, 4, 6, 7, 10}
using 4 bins is
Example

14. • An image histogram in its simplest form is a graphical representation of
the number of times each pixel value (that is, each gray level) occurs in
the image.
• By default, an image histogram has 256 bins.
Image Histograms

15. MATLAB
Implementation
• In order to display an image histogram, we
can use the MATLAB function imhist. For
example, typing
>> imhist(A);
into the command window produces a
histogram consisting of as many as 256
bars (one for each possible gray level),
where as typing in
>> imhist(A,16);
results in the histogram of the image A
using 16 bins.

16. MATLAB
Implementation
A=imread('img01.jpg');
B=imread('img02.jpg');
C=imread('img03.jpg');
A=rgb2gray(A);
B=rgb2gray(B);
C=rgb2gray(C);
subplot(3,2,1);
imshow(A,[0,255]);
subplot(3,2,2);
imhist(A);
subplot(3,2,3);
imshow(B,[0,255]);
subplot(3,2,4);
imhist(B);
subplot(3,2,5);
imshow(C,[0,255]);
subplot(3,2,6);
imhist(C);
Example

17. Consider a somewhat unrealistic example of an opulent $10,000,000
mansion in the middle of a modest community of 49 other houses,
each worth about $200,000. The mean property value in the
community is
𝑦 =
$10,000,000+$200,000+⋯+$200,000
50
= $369,000 per house
which, unfortunately, is representative of neither the opulent mansion
nor of the rest of the properties. It just would not be reasonable to
claim that the average house in the community is worth $396,000.
The obvious conclusion is that if the data are skewed or contain
outliers, the mean is not a very useful measure of center of the
dataset. Instead, in such circumstances, the other measure of
average - the median – is usually used.
Measures of Center and Spread
Mean & Median

18. For example, in order to calculate
the median of the even-size data
set
{5, 1, 4, 8, 2, 6, 11, 72, 4, 7, 1,
9}
We first rearrange it in the
ascending order as
{1, 1, 2, 4, 4, 5, 6, 7, 8,
9, 11, 72}
and then observe that the average
of the two middle entries of the
list is
m =(5 +6)/2= 5.5
Which is precisely the median of
Measures of Center and Spread
Mean & Median
In order to calculate the median of
the odd-size data set
{5, 1, 4, 8, 2, 6, 17, 72, 4, 7, 10}
We first rewrite it as
{1, 2, 4, 4, 5, 6, 7, 8, 10,
17, 72}
and then observe that the middle
entry is 6, which is precisely the
median of these data.

19. We note that the
median property value
in the real estate
example we
considered earlier is
$200,000, and it is
quite reasonable to
state that the average
house in the
community is worth
that amount

20. • We could define the mainstream of a data set as its middle half –
the part of the data that falls between the thresholds for the lower
quarter and the upper quarter.
• These thresholds are called quartiles.
• The first quartile 𝑄1 is the median of the lower half of the data set.
• The third quartile 𝑄3 is the median of the upper half.
• The second quartile 𝑄2 is the median of the entire data set.
• The interquartile range
𝐼𝑄𝑅 = 𝑄3 − 𝑄1
is a meaningful measure of spread of data.
Measures of Center and Spread
The Mainstream

21. In connection with the quartiles and the interquartile range, we mention
one more concept that might prove useful in image analysis is – that
of an outlier –a value that is vastly different from most of the rest of
the data.
There exist several technical definitions of the term outlier. One of
them is the Tukey fences definition, defines an outlier to be any value
outside the range
[𝑄1 − 1.5 × 𝐼𝑄𝑅, 𝑄3 + 1.5 × 𝐼𝑄𝑅]
and a strong outlier to be any value outside the range
[𝑄1 − 3 × 𝐼𝑄𝑅, 𝑄3 + 3 × 𝐼𝑄𝑅]
Measures of Center and Spread
The Outlier

22. To find the quartiles of the familiar data set
{5, 1, 4, 8, 2, 6, 17, 72, 4, 7, 1, 10}
we first rearrange it in the ascending order as
{1, 1, 2, 4, 4, 5, 6, 7, 8, 10, 17, 72}
and then observe that 𝑄1 = 3 and 𝑄3 = 9 as the medians of the lower
and upper halves respectively.
𝐼𝑄𝑅 = 𝑄3 − 𝑄1 = 6
The value 72 is a strong outlier because it is larger than the upper Tukey
fence
𝑄3 + 3 × 𝐼𝑄𝑅 = 9 + 3  6 = 27
whereas the value 17 is not an outlier because it falls within the Tukey
fences
[𝑄1 − 1.5 × 𝐼𝑄𝑅, 𝑄3 + 1.5 × 𝐼𝑄𝑅] = [3 − 1.5  6, 9 + 1.5  6] = [−6, 18]
Example

23. Is there any outliers in the following image
222 115
127 251
159 47
63 143
191 79
95 175
207 25
31 249
First we reshape the 2-D matrix representing the image into a 1-D list values
{222, 115, 159, 47, 127, 251, 63, 143, 191, 79, 207, 25, 95, 175, 31, 249}
which, after rearranging, becomes
{25, 31, 47, 63, 79, 95, 115, 127, 143, 159, 175, 191, 207,
222, 249, 251}
and we can look up the values
𝑄1 = 71, 𝑄2 = 135, 𝑎𝑛𝑑 𝑄3 = 199
𝐼𝑄𝑅 = 199 – 71 = 128
[𝑄1 − 1.5 × 𝐼𝑄𝑅, 𝑄3 + 1.5 × 𝐼𝑄𝑅] = [71 − 1.5  128, 199 + 1.5  128] = [−121, 391]
There are no outliers because all the values fall within this interval.
Example

24. MATLAB
Implementation
A=imread('cameraman.tif');
[M,N]=size(A);
A=reshape(A,[1,M*N]);
Q1=prctile(A,25)
Q3=prctile(A,75)
IQR=Q3-Q1
LowerBound=Q1-1.5*IQR
UpperBound=Q3+1.5*IQR

25. Quartiles provide thresholds for breaking data sets up into
quarters. But this is not the only possible useful way to
subdivide data.
More generally, the 𝑘𝑡ℎ percentile, denoted by 𝑃𝑘, is a value
with the property that 𝑘% of the data are less than or equal
to it.

26. Color Images and Color
Spaces
04

27. • A color image is a stack of three
matrices which contain information on
the three color channels. The exact
composition of those color channels
differs from model to model.
• We will focus on just two commonly
used color models -the RGB model
and the YCbCr model.
• In the RGB color model, the three
primary colors -Red, Green, and
Blue- are added together, which, at
least in theory, enables us to
reproduce as many as 2563 =
16,777,216 synthetic colors.
Color Images and Color Spaces

28. MATLAB
Implementation
imshow(A);
% Showing the channels in gray scale
Red=A(:,:,1); Green=A(:,:,2); Blue=A(:,:,3);
figure;
subplot(1,3,1); imshow(Red); title('(a)The red channel');
subplot(1,3,2); imshow(Green); title('(b)The green
channel');
subplot(1,3,3); imshow(Blue); title('(c)The blue
channel');

29. The RGB color model is most commonly used for displaying images, but it
is not very efficient for storage and transmission due to its inherent
redundancy. The reason is that we perceive the brightness and color
separately and are much more sensitive to small distortions in brightness
than to those in color. In most cases, we can readily sacrifice the latter for
the sake of increasing the speed of transmission.
For that reason, several color models that separate light intensity and color
into different channels have been developed. One of the most commonly
used models of this kind is YCbCr. The luminance channel Y contains the
information on the light intensity and is stored with high resolution, whereas
the two chrominance channels-Cb and Cr-represent the difference from pure
Blue and pure Red and can be compress data very high compression ratio.
Color Images and Color Spaces

30. MATLAB
Implementation
A=imread('FlagOfSeychelles.jpg’); B=rgb2ycbcr(A);
imshow(B);
% Showing the channels in gray scale
Y=B(:,:,1); Cb=B(:,:,2); Cr=B(:,:,3);
figure;
subplot(1,3,1); imshow(Y); title('(a)The Y channel');
subplot(1,3,2); imshow(Cb); title('(b)The Cb channel');
subplot(1,3,3); imshow(Cr); title('(c)The Cr channel');

31. THANK YOU
Any questions?
You can find me at:
saad.m@sc.uobaghdad.edu.iq