DIP-CHAPTERs

Digital Image Processing
Gopal Krishan Prajapat

Image Processing Books
• Gonzalez, R. C. and Woods, R. E., "Digital
Image Processing", Prentice Hall.
• Jain, A. K., "Fundamentals of Digital Image
Processing", PHI Learning, 1
st
Ed.
• Bernd, J., "Digital Image Processing", Springer,
6
th
Ed.
• Burger, W. and Burge, M. J., "Principles of
Digital Image Processing", Springer
• Scherzer, O., " Handbook of Mathematical
Methods in Imaging", Springer
Sunday, December 18, 2022 2

Why we need Image Processing?
• Improvement of pictorial information for
human perception
• Image processing for autonomus machine
applications
• Efficient storage and transmission

What is digital image processing?
• An image may be defined as a two dimensional
function f(x,y), where ‘x’ and ‘y’ are spatial(plane)
coordinates and the amplitude of ‘f’ at any pair of
coordinates (x,y) is called the intensity or gray
level of the image at that point.
• When x,y, and the amplitude values of ‘f’ are all
finite, descrete quantities we call the image a
digital image.
• The field of digital image processing refers to
processing digital images by means of digital
computers.

What is digital image processing?
(Cont…)

Image Processing Applications
• Automobile driver assistance
– Lane departure warning
– Adaptive cruise control
– Obstacle warning
• Digital Photography
– Image Enhancement
– Compression
– Color manipulation
– Image editing
– Digital cameras
• Sports analysis
– sports refereeing and commentary
– 3D visualization and tracking sports actions

Image Processing Applications(Cont…)
• Film and Video
– Editing
– Special effects
• Image Database
– Content based image retrieval
– visual search of products
– Face recognition
• Industrial Automation and Inspection
– vision-guided robotics
– Inspection systems
• Medical and Biomedical
– Surgical assistance
– Sensor fusion
– Vision based diagnosis
• Astronomy
– Astronomical Image Enhancement
– Chemical/Spectral Analysis

Image Processing Applications(Cont...)
• Arial Photography
– Image Enhancement
– Missile Guidance
– Geological Mapping
• Robotics
– Autonomous Vehicles
• Security and Safety
– Biometry verification (face, iris)
– Surveillance (fences, swimming pools)
• Military
– Tracking and localizing
– Detection
– Missile guidance
• Traffic and Road Monitoring
– Traffic monitoring
– Adaptive traffic lights

Brief History of IP
• In 1920s, submarine cables were used to transmit
digitized newspaper pictures between London &
New York – using Bartlane cable picture
transmission System.
• Specialized printing equipments(eg. Telegraphic
printer) used to code the picture for cable
transmission and its reproduction on the
receiving end.
• In 1921, printing procedure was changed to
photographic reproduction from tapes perforated
at telegraph receiving terminals.
• This improved both tonal quality & resolution.

Brief History of IP(Cont…)

Brief History of IP(Cont…)
• Bartlane system was capable of coding 5 distinct
brightness levels. This was increased to 15 levels
by 1929.
• Improvement of processing techniques continued
for next 35 years .
• In 1964 computer processing techniques were
used to improve the pictures of moon tranmitted
by ranger 7 at Jet Propulsion Laboratory.
• This was the basis of modern Image Processing
techniques.

Image Processing Steps

Components of IP System

Image Acquisition Process

Image Sensing and Acquisition

Image Sensing and Acquisition(Cont…)
• Image acquisition using a single sensor

Image Sensing and Acquisition(Cont…)
• Using sensor strips

Image Representation
x
y
IMAGE
An image is a 2-D light intensity function F(X,Y).
F(X,Y) = R(X,Y)* I(X,Y) , where
R(X,Y) = Reflectivity of the surface of the corresponding image point.
I(X,Y) = Intensity of the incident light.
A digital image F(X,Y) is discretized both in spatial coordinates and brightness.
It can be considered as a matrix whose row, column indices specify a point in
the image & the element value identifies gray level value at that point known
as pixel or pels.

Image Representation (Cont..)
(0,0) (0,1) ... (0, 1)
(1,0) (1,1) ... (1, 1)
( , )
... ... ... ...
( 1,0) ( 1,1) ... ( 1, 1)
f f f N
f f f N
f x y
f M f M f M N
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Image Representation in Matrix form

( , ) ( , ) ( , )
( , ): intensity at the point ( , )
( , ): illumination at the point ( , )
(the amount of source illumination incident on the scene)
( , ): reflectance/transmissivity
f x y i x y r x y
f x y x y
i x y x y
r x y

at the point ( , )
(the amount of illumination reflected/transmitted by the object)
where 0 < ( , ) < and 0 < ( , ) < 1
x y
i x y r x y


• By theory of real numbers :
Between any two given points there are infinite
number of points.
• Now by this theory :
An image should be represented by infinite
number of points.
Each such image point may contain one of the
infinitely many possible intensity/color values
needing infinite number of bits.
Obviously such a representation is not possible in
any digital computer.

Image Sampling and Quantization
• By above slides we came to know that we need to
find some other way to represent an image in
digital format.
• So we will consider some discrete set of points
known as grid and in each rectangular grid
consider intensity of a particular point. This
process is known as sampling.
• Image representation by 2-d finite matrix –
Sampling
• Each matrix element represented by one of the
finite set of discrete values - Quantization

Image Sampling and Quantization

Colour Image Processing
• Why we need CIP when we get information from
black and white image itself?
1. Colour is a very powerful descriptor & using the
colour information we can extract the objects of
interest from an image very easily which is not so easy
in some cases using black & white pr simple gray level
image.
2. Human eyes can distinguish between thousands of
colours & colour shades whereas when we talk about
only black and white image or gray scale image we
can distinguish only about dozens of intensity
distinguishness or different gray levels.

Color Image processing(Cont…)
• The color that human perceive in an object =
the light reflected from the object
Illumination source
scene
reflection
Humen eye

Colour Image Processing(Cont...)
• In CIP there are 2 major areas:
1.FULL CIP : Image which are acquired by full colour
TV camera or by full color scanner, than, you find
that all the colour you perceive they are present in
the images.
2.PSEUDO CIP : Is a problem where we try to assign
certain colours to a range of gray levels. Pseudo CIP is
mostly used for human interpretation.
So here it is very difficult to distinguish between two
ranges which are very nearer to each other or gray
intensity value are very near to each other.

• Problem with CIP
Interpretation of color from human eye is a
psycophisological problem and we have not yet
been fully understand what is the mechanism by
which we really interpret a color.

• In 1666 Isacc Newton discover color spectrum
by optical prism.

• We can perceive the color depending on the nature of light
which is reflected by the object surface.
• Spectrum of light or spectrum of energy in the visible range that
we are able to perceive a color(400 nm to 700 nm)

• Attribute of Light
Achromatic Light : A light which has no color
component i.e., the only attribute which
describes that particular light is the intensity of
the light.
Chromatic Light : Contain color component.
• 3 quantities that describe the quality of light:
Radiance
Luminance
Brightness

• Radiance : Total amount of energy which comes
out of a light (Unit : watts)
• Luminance : Amount of energy that is perceived
by an observer (Unit : Lumens)
• Brightness : It is a subjective thing. Practically we
can’t measure brightness.
We have 3 primary colors:
Red
Blue
Green

• Newton discovered 7 different color but only 3
colors i.e., red, green and blue are the primary
colors. Why?
Because by mixing these 3 colors in some proportion
we can get all other colors.
There are around 6-7 millions cone cells in our eyes
which are responsible for color sensations.
Around 65% cone cells are sensitive to red color.
Around 33% cone cells are sensitive to green color.
Around 2% cone cells are sensitive to blue color.

• According to CIE standard
Red have wavelength : 700 nm
Green have wavelength : 546.1 nm
Blue have wavelength : 435.6 nm
But, practically :
Red is sensitive to 450 nm to 700 nm
Green is sensitive to 400 nm to 650 nm
Blue is sensitive to 400 nm to 550 nm

• Note : In practical no single wavelength can
specify any particular color.
• By spectrum color also we can see that there is
no clear cut boundaries between any two color.
• One color slowly or smoothly get merged into
another color i.e., there is no clear cut boundary
between transition of color in spectrum.
• So, we can say a band of color give red, green
and blue color sensation respectively.

• Mixing of Primary color generates the secondary
colors i.e.,
 RED+BLUE=Magenta
 GREEN+BLUE = Cyan
 RED+GREEN = yellow
• Here red, green and blue are the primary color
and magenta, cyan and yellow are the secondary
color.
• Pigments : The primary color of pigment is
defined as wavelength which are absorbed by the
pigment and it reflect other wavelength.

• Primary color of light should be opposite of primary color of
pigment i.e., magenta , cyan and yellow are primary color of
pigment.
• If we mix red, green and blue color in appropriate proportion
we get white light and similarly when we mix magenta, cyan
and yellow we get black color.

• For hardware i.e., camera, printer, display
device, scanner this above concept of color is
used i.e., concept of primary color
component.
• But when we perceive a color for human
beings we don’t think that how much
red,green and blue components are mixed in
that particular color.
• So the way by which we human differentiate
or recognize or distinguish color are :
Brightness, Hue and Saturation.

• Spectrum colors are not diluted i.e., spectrum
colors are fully saturated . It means no white
light or white component are added to it.
• Example: Pink is not spectrum color.
Red + white = pink
Here red is fully saturated.
• So, Hue+Saturation indicates chromaticity of
light and Brightness gives some sentation of
intensity.

• Brightness : Achromatic notion of Intensity.
• Hue : It represents the dominant wavelength
present in a mixture of colors.
• Saturation : eg., when we say color is red i.e.,
we may have various shades of red. So
saturation indicates what is the purity of red
i.e., what is the amount of light which has
been mixed to that particular color to make it
a diluted one.

• The amount of red, green and blue
component is needed to get another color
component is known as tristimulus.
• Tristimulus = (X,Y,Z)
• Chromatic cofficient for red = X/(X+Y+Z) , for
green = Y/(X+Y+Z) , for blue = Z/(X+Y+Z).
• Here X+Y+Z=1
• So any color can be specified by its chromatic
cofficient or a color can be specified by a
chromaticity diagram.

• Here Z = 1-(X+Y) , In chromaticity diagram around the
boundary we have all the color of the spectrum colors and
point of equal energy is : white color.

• Color Models : A coordinate system within
which a specified color will be represented by
a single point.
• RGB , CMY , CMYK : Hardware oriented
• HSI : Hue , Saturation and Intensity :
Application oriented / Perception oriented
• In HSI model : I part gives you gray scale
information. H & S taken together gives us
chromatic information.

• RGB Color Model : Here a color model is represented
by 3 primary colors i.e., red , green and blue.
• In RGB color model we can have 224 different color
combinations but practically 216 different colors can
be represented by RGB model.
• RGB color model is based on Cartesian coordinate
system.
• This is an additive color model
• Active displays, such as computer monitors and
television sets, emit combinations of red, green and
blue light.

• RGB Color Model

• RGB Color Model
• RGB 24-bit color cube is shown below

• RGB example:
Original Green Band Blue Band
Red Band

• CMY Color Model : secondary colors of light, or
primary colors of pigments & Used to generate
hardcopy output
Source: www.hp.com
Passive displays, such as colour inkjet printers, absorb light instead of
emitting it. Combinations of cyan, magenta and yellow inks are used. This
is a subtractive colour model.

• Equal proportion of CMY gives a muddy black
color i.e., it is not a pure black color. So, to get
pure black color with CMY another
component is also specified known as Black
component i.e., we get CMYK model.
• In CMYK “K” is the black component.
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• HSI Color Model (Based on human perception of
colors )
• H = What is the dominant specified color present in a
particular color. It is a subjective measure of color.
• S = How much a pure spectrum color is really diluted
by mixing white color to it i.e., Mixing more “white”
with a color reduces its saturation.
If we mix white color in different proportion with a
color we get different shades of that color.
• I = Chromatic notation of brightness of black and
white image i.e., the brightness or darkness of an
object.

• HSI Color Model
H
dominant
wavelength
S
purity
% white
I
Intensity

• HSI Color Model
RGB -> HSI model

• HSI Color Model

• Pseudo-color Image Processing
Pseudo-Color = False Color
Assign colors to gray values based on a specified
criterion
For human visualization and interpretation of
gray-scale events
Intensity slicing
Gray level to color transformations

• Pseudo-color Image Processing(cont…)
 Here first consider an intensity image to be a 3D plane.
 Place a plane which is parallel to XY plane(it will slice the
plane into two different hubs).
 We can assign different color on two different sides of
the plane i.e., any pixel whose intensity level is above
the plane will be coded with one color and any pixel
below the plane will be coded with the other.
 Level that lie on the plane itself may be arbitrarily
assigned one of the two colors.

 Geometric interpretation of the intensity slicing
technique

 Let we have total ‘L’ number of intensity values: 0 to (L-1)
 L0 corresponds to black [ f(x , y) = 0]
 LL-1 corresponds to white [ f(x , y) = L-1]
 Suppose ‘P’ number of planes perpendicular to the
intensity axis i.e., they are parallel to the image plane and
these planes will be placed at the intensity values given
by L1,L2,L3,………,LP.
 Where , 0< P < L-1.

• Intensity slicing
 The P planes partition the gray scale(intensity)
into (P+1) intervals, V1,V2,V3,………,VP+1.
 Color assigned to location (x,y) is given by the
relation
f(x , y) = Ck if f(x , y) ∈ Vk

• Intensity slicing Numerical

• Intensity slicing
 Give ROI(region of interest) one color and rest part
other color
 Keep ROI as it is and rest assign one color
 Keep rest as it is and give ROI one color

• Intensity slicing Example

• Pseudo-coloring is also used from gray to color
image transformation.
• Gray level to color transformation

• Gray level to color transformation
fR(X,Y) = f(x,y)
fG(X,Y) = 0.33f(x,y)
fB(X,Y) = 0.11f(x,y)
 Combining these 3 planes we get the pseudo
color image.
 Application of Pseudo CIP : Machine using at
railways and airport for bag checking.

• Gray level to color transformation Example

Image Enhancement
• Intensity Transformation Functions
• Enhancing an image provides better contrast and a more
detailed image as compare to non enhanced image.
Image enhancement has very applications. It is used to
enhance medical images, images captured in remote
sensing, images from satellite e.t.c
• The transformation function has been given below
s = T ( r )
• where r is the pixels of the input image and s is the pixels
of the output image. T is a transformation function that
maps each value of r to each value of s.

Image Enhancement(Cont…)
• Image enhancement can be done through gray
level transformations which are discussed below.
• There are three basic gray level transformation.
• Linear
• Logarithmic
• Power – law

• Linear Transformation
 Linear transformation includes simple identity and negative
transformation.
 Identity transition is shown by a straight line. In this transition,
each value of the input image is directly mapped to each other
value of output image. That results in the same input image
and output image. And hence is called identity transformation.
• Negative Transformation
 The second linear transformation is negative transformation,
which is invert of identity transformation. In negative
transformation, each value of the input image is subtracted
from the L-1 and mapped onto the output image.

• Negative Transformation
s = (L – 1) – r
s = 255 – r

• Logarithmic Transformations
 The log transformations can be defined by this
formula
s = c log(r + 1).
 Where s and r are the pixel values of the output
and the input image and c is a constant. The value
1 is added to each of the pixel value of the input
image because if there is a pixel intensity of 0 in
the image, then log (0) is equal to infinity. So 1 is
added, to make the minimum value at least 1.

• Logarithmic Transformations
 In log transformation we decrease the dynamic range of a
particular intensity i.e., here intensity of the pixels are
increased which we require to get more information. The
maximum information is contained in the center pixel.
 Log transformation is mainly applied in frequency domain.

• Logarithmic Transformation

• Power – Law transformations
• This symbol γ is called gamma, due to which this
transformation is also known as gamma
transformation.
s = crγ, c,γ –positive constants
curve the grayscale components either to brighten
the intensity (when γ < 1) or darken the intensity
(when γ > 1).

• Variation in the value of γ varies the enhancement
of the images. Different display devices / monitors
have their own gamma correction, that’s why they
display their image at different intensity.
• This type of transformation is used for enhancing
images for different type of display devices. The
gamma of different display devices is different.
For example Gamma of CRT lies in between of 1.8
to 2.5, that means the image displayed on CRT is
dark.

 Gamma Correction
 Different camera or video recorder devices do not
correctly capture luminance. (they are not linear)
Different display devices (monitor, phone screen, TV) do
not display luminance correctly neither. So, one needs to
correct them, therefore the gamma correction function
is needed. Gamma correction function is used to correct
image's luminance.
s=cr^γ
s=cr^(1/2.5)

• Piecewise-Linear Transformation Functions
 Three types:
 Contrast Stretching
 Intensity Level Slicing
 Bit-Plane Slicing

• Contrast stretching
 Aims increase the dynamic range of the gray
levels in the image being processed.
 Contrast stretching is a process that expands the
range of intensity levels in a image so that it
spans the full intensity range of the recording
medium or display device.
 Contrast-stretching transformations increase the
contrast between the darks and the lights

 The locations of (r1,s1) and (r2,s2) control the shape of
the transformation function.
– If r1= s1 and r2= s2 the transformation is a linear
function and produces no changes.
– If r1=r2, s1=0 and s2=L-1, the transformation becomes
a thresholding function that creates a binary image.
– Intermediate values of (r1,s1) and (r2,s2) produce
various degrees of spread in the gray levels of the
output image, thus affecting its contrast.
– Generally, r1≤r2 and s1≤s2 is assumed.

Thresholding function

• Intensity-level slicing
 Highlighting a specific range of gray levels in an
image.
 One way is to display a high value for all gray levels
in the range of interest and a low value for all
other gray levels (binary image).
 The second approach is to brighten the desired
range of gray levels but preserve the background
and gray-level tonalities in the image

• Intensity Level Slicing

• Bit-Plane Slicing
• To highlight the contribution made to the total
image appearance by specific bits.
– i.e. Assuming that each pixel is represented by 8 bits,
the image is composed of 8 1-bit planes.
– Plane 0 contains the least significant bit and plane 7
contains the most significant bit.
– Only the higher order bits (top four) contain visually
significant data. The other bit planes contribute the
more subtle details.
– Plane 7 corresponds exactly with an image thresholded
at gray level 128.

• Bit-Plane Slicing

• Histogram Processing
 Two Types : (a). Histogram Stretching (b). Histogram
equalization
 Histogram Stretching
 Contrast is the difference between maximum and
minimum pixel intensity.
 Pictorial view to represent the distribution of pixel which
tell frequency of pixel.

• The histogram of digital image with gray values
is the discrete function
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The shape of a histogram provides useful information for
contrast enhancement.

Dark image
Bright image

Low contrast image
High contrast image

• Histogram Stretching

• Histogram Stretching (cont…)
• In the above example (0,8) is smin and smax respectively and
rmin = 0 , rmax = 4 is given.
• S-0 = (8 – 0) / (4 – 0) * (r – 0)
• s=(8/4) r
• S= 2r
• Now we have a relation between r and s.
• So get different values of ‘s’ for given rmin to rmax .

• Histogram Stretching (cont…)

• Histogram Equalization
– Recalculate the picture gray levels to make the
distribution more equalized
– Used widely in image editing tools and computer
vision algorithms
– Can also be applied to color images

Objective of histogram equalization
• We want to find T(r) so that
Ps(s) is a flat line.
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normalized histogram function.
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ʃPss ds=ʃprr dr =1
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 So, e.q. (1)
 The transformation function T(r) for histogram
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 Differentiate w.r.t ‘r’ :
 As we know, , SO,
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1
(
)
(
0
0
0



















L
k
n
MN
L
s
MN
n
r
p
r
P
L
r
T
s
dw
w
p
L
r
T
s
k
j
j
k
k
k
k
j
j
r
k
k
r
r
126

Histogram Equalization - Example
• Let f be an image with size 64x64 pixels and L=8 and let f has the intensity
distribution as shown in the table
p r(rk )=nk/MN
nk
rk
0.19
790
0
0.25
1023
2
0.21
850
1
0.16
656
3
0.08
329
4
0.06
245
5
0.03
122
6
0.02
81
7
.
00
.
7
,
86
.
6
,
65
.
6
,
23
.
6
,
67
.
5
,
55
.
4
08
.
3
))
(
)
(
(
7
)
(
7
)
(
33
.
1
)
(
7
)
(
7
)
(
7
6
5
4
3
2
1
0
1
0
1
1
0
0
0
0
0



















s
s
s
s
s
s
r
p
r
p
r
p
r
T
s
r
p
r
p
r
T
s
r
r
j
j
r
r
j
j
r
round the values to the nearest integer

Filtering
• Image filtering is used to:
 Remove noise
 Sharpen contrast
 Highlight contours
 Detect edges
 Image filters can be classified as linear or nonlinear.
 Linear filters are also know as convolution filters as
they can be represented using a matrix multiplication.
 Thresholding and image equalisation are examples of
nonlinear operations, as is the median filter.

Filtering(cont…)
• There are two types of processing:
• Point Processing (eg. Histogram equalization)
• Mask Processing
 Two types of filtering methods:
• Smoothing
Linear (Average Filter) and Non-Linear (Median
Filter)
• Sharpening
Laplacian
Gradient

Filtering(Cont…)
output image

Filtering(Cont…)
• Correlation [ 1-D & 2-D]
• Convolution [ 1-D & 2-D]
• In correlation we use weight to get output image and for
applying convolution we just rotate the weight 180
degree.
• Eg. Weight
• After 180 degree rotation
• After 180 degree rotation
1 2 3
3 2 1
1 2 3
4 5 6
7 8 9
9 8 7
6 5 4
3 2 1

Filtering(Cont…)
• 1-D Correlation
• I =
• W=
• Output =
• For convolution just rotate mask 180 degree.
1 2 3 4
1 2 3
[(2*1)+(3*2)]
8
[(1*1)+(2*2)+
(3*3)]
14
[(1*2)+(2*3)+
(3*4)]
20
[(1*3)+(4*2)]
11

Filtering(Cont…)
• A filtering method is linear when the output is a
weighted sum of the input pixels. Eg. Average filter
• Methods that do not satisfy the above property are
called non-linear. Eg. Median filter
• Average (or mean) filtering is a method of ‘smoothing’
images by reducing the amount of intensity variation
between neighbouring pixels.
• The average filter works by moving through the image
pixel by pixel, replacing each value with the average value
of neighbouring pixels, including itself.

Filtering(Cont…)
• Average filter mask (2-D):

Filtering(Cont…)

Filtering(Cont…)
• When we apply average filter noise is removed
but blurring is introduced and to remove blurring
we use weighted filter.

Filtering(Cont…)
• Median Filter (non-linear filter)
• Very effective in removing salt and pepper or impulsive noise
while preserving image detail
• Disadvantages: computational complexity, non linear filter
• The median filter works by moving through the image pixel by
pixel, replacing each value with the median value of
neighbouring pixels.
• The pattern of neighbours is called the "window", which
slides, pixel by pixel over the entire image 2 pixel, over the
entire image.
• The median is calculated by first sorting all the pixel values
from the window into numerical order, and then replacing the
pixel being considered with the middle (median) pixel value.

Filtering(Cont…)
• Median Filter Example:

Filtering(Cont…)

Filtering(Cont…)
• From left to right: the results of a 3 x 3, 5 x 5 and 7 x 7 median filter

Filtering(Cont…)
 Sharpening(high pass filter) is performed by noting only the
gray level changes in the image that is the differentiation.
• Sharpening is used for edge detection ,line detection, point
detection and it also highlight changes.
 Operation of Image Differentiation
• Enhance edges and discontinuities (magnitude of output gray
level >>0)
• De-emphasize areas with slowly varying gray-level values
(output gray level: 0)
 Mathematical Basis of Filtering for Image Sharpening
• First-order and second-order derivatives
• Approximation in discrete-space domain
• Implementation by mask filtering

Filtering(Cont…)
 Common sharpening filters:
• Gradient (1st order derivative)
• Laplacian (2nd order derivative)
• Taking the derivative of an image results in sharpening
the image.
• The derivative of an image (i.e., 2D function) can be
computed using the gradient.

Filtering(Cont…)
 Gradient (rotation variant or non-isotropic)
or
Sensitive to
vertical
edges
Sensitive to
horizontal
edges

Filtering(Cont…)
 Gradient
Kernels used in prewitt edge detection

Filtering(Cont…)
• Laplacian
Original Mask
, C = +1 or C= -1

Filtering(Cont…)
• Laplacian(rotation invariant or isotropic)
(b)Extended
Laplacian
mask to
increase
sharpness
and it covers
diagonal
also, so ,
provide good
results.

Image Transforms
• Many times, image processing tasks are best
performed in a domain other than the spatial
domain.
• Key steps
(1) Transform the image
(2) Carry the task(s) in the transformed domain.
(3) Apply inverse transform to return to the spatial
domain.

Math Review - Complex numbers
• Real numbers:
1
-5.2

• Complex numbers
4.2 + 3.7i
9.4447 – 6.7i
-5.2 (-5.2 + 0i)
1


i
We often denote in EE i by j

Math Review - Complex numbers
• Complex numbers
4.2 + 3.7i
9.4447 – 6.7i
-5.2 (-5.2 + 0i)
• General Form
Z = a + bi
Re(Z) = a
Im(Z) = b
• Amplitude
A = | Z | = √(a2 + b2)
• Phase
 =  Z = tan-1(b/a)
Real and imaginary parts

Math Review – Complex Numbers
• Polar Coordinate
Z = a + bi
• Amplitude
A = √(a2 + b2)
• Phase
 = tan-1(b/a)
a
b


A


Math Review – Complex Numbers and
Cosine Waves
• Cosine wave has three properties
– Frequency
– Amplitude
– Phase
• Complex number has two properties
– Amplitude
– Wave
• Complex numbers to represent cosine waves at varying frequency
– Frequency 1: Z1 = 5 +2i
– Frequency 2: Z2 = -3 + 4i
– Frequency 3: Z3 = 1.3 – 1.6i
Simple but great idea !!

Fourier Transforms & its Properties
• Jean Baptiste Joseph Fourier (1768-1830)
• Had crazy idea (1807):
• Any periodic function can be
rewritten as a weighted sum of
Sines and Cosines of different
frequencies.
• Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and other
big wigs
– Not translated into English
until 1878!
• But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering

• In image processing:
– Instead of time domain: spatial domain (normal image
space)
– frequency domain: space in which each image value at
image position F represents the amount that the
intensity values in image I vary over a specific distance
related to F

• Fourier Transforms & Inverse Fourier Transforms

• As we deal with 2-d discrete images so we need to
discuss 2-D discrete Fourier Transforms.

• Inverse F.T

• If the image is represented as square array i.e.,
M=N than F.T and I.F.T is given by equation:

• Separability property

• Periodicity: The DFT and its inverse are periodic wit
period N.

• Scaling: If a signal is multiply by a scalar quantity ‘a’
than its Fourier Transformation is also multiplied by
same scalar quantity ‘a’.

• Distributivity:
but …

• Average:
Average:
F(u,v) at u=0, v=0:
So:

Frequency Domain Filters
• Low pass filter: it allows low frequency range signal to pass as
output.(Useful for noise suppression)
• High pass filter: it allows high pass frequency range to pass as
output. (Useful for edge detection)
• D(u,v) is distance of (u , v) in frequency domain from the origin of
the frequency rectangle.
• Do implies that all signals lies in this range i.e., D(u,v)<= Do all
low pass frequency to pass to the output and rest are not allowed
to pass as output.

Frequency Domain Filters
• In the above example, for the same cut of
frequency the blurring is more in Ideal low
pass filter than in butterworth filter and as cut
of frequency increases the number of
undesired lines are increased in Ideal low pass
filter than in butterworth filter.

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