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1Chapter 1: Introduction
Unit 1
1 INTRODUCTION
Unit Structure
1.0 Objectives
1.1 Origin
1.2 Digital Image Processing
1.3 Applications and Examples of Digital Image Processing
1.4 Fundamental steps in Digital Image Processing
1.5 Components of Digital Image Processing
1.6 Elements of Visual Perception
1.7 Image Sensing and Acquisition
1.8 Image Sampling and Quantization
1.9 Basic Relationship between pixels
1.10 Summary
1.11 Review Questions
1.12 References
1.0 Objectives
1. To enhance and improve the information of the picture for better interpretation and understanding
2. Processing the acquired images for efficient data storage, transmission over
network and extraction of right information from the picture.
1.1 Origin
The first industry to use digital images was the newspaper industry. The pictures
were sent by submarine cable between London and New York .
1.2 Digital Image Processing
The human eye -brain mechanism produces the best imaging system. An image is
an object or visual which one sees. It is a 2 -dimensional function of a 3 -dimensional munotes.in

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world that surrounds us. Basically, images are 2 -D light intensity function f(x, y)
where x an d y are spatial or plane co -ordinates and the amplitude at any co -
ordinates pair (x, y) is defined as the intensity or gray level of the image at that
point .
If x, y and the intensity values are all finite and discrete, then the image is known
as a digita l image.
The digital image is composed of a finite number of elements which has a particular
location and value. These elements are called picture elements or pixels or pels.
Digital Image Processing is modification and enhancement of images by applying
various filtering and enhancement techniques for perceiving better visual information and perform various analysis on the images.
Image processing accepts images as inputs and generates a modified image as
output for human perception or the modified image may provide useful information.
1.3 Applications and Examples of Digital Image Processing
To develop a basic understanding of the breadth and length of the applications, the
image processing applications are categorized according to their sources.
The principal source of energy for images is the electromagnetic energy spectrum.
It also includes acoustic, ultrasonic, electronic etc.
Synthetic images used for modelling and visualization are generated in computer.
The most common applications of digital image processing are
1. Gamma -ray Imaging
Imaging based on gamma rays are mostly for nuclear medicine, astronomical
observations. In this type of imaging, images are produced from the emissions collected from gamma -ray detectors.
2. X-ray Imaging
X-ray imaging is used mainly for medical diagnostics and industrial imaging.
It is also used for astronomical applications.
3. Ultraviolet Imaging
It is used for lithography, industrial inspection, microscopy, lasers, biological
imaging and astronomical observations.
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4. Wide applications include
a. Remote Sensing
b. Light microscopy
c. Astronomy
d. Weather observation and prediction
e. Visual inspection of manufactured goods
f. Traffic monitoring and surveillance
g. License plate character recognition
h. Currency recognition
i. Finger -print and face recognition
j. Radar imaging to explore inaccessible regions of the Earth’s surface.
k. Mineral and oil exploration
l. Ultrasound imaging of foetus
The list of applications is limited only for writing here, the applications of
digital image processing is wide and the scope is large.
1.4 Fundamental Steps in Digital Image Processing

Fig:1.1 Image Courtesy://Digital Image Processing -Gonzalez
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1. Image acquisition
The image is captured by a sensor like camera or any analog device and
digitized if the output of the camera or sensor is not already in digital form,
using analogue -to-digital convert er. It involves pre -processing of images.
2. Image Enhancement
The process of manipulating an image so that the result is more suitable than the original for specific applications. The idea behind enhancement techniques is to bring out details that are hidden, or simple to highlight certain
features of interest in an image
3. Image Restoration
Deals with improving the appearance of an image.
Based on mathematical or probabilistic models of image degradation.
4. Colour Image Processing
Use the colour of the image to extract features of interest in an image.
Understand the basics concepts of color models and color processing in
digital domain.
5. Wavelets
Wavelets are the foundation of representing images in various degrees of
resolution. It is used for image data compr ession and for representation of
images in smaller regions.
6. Compression
Deals with various techniques used for reducing the storage required to save
an image in digital form or the bandwidth required to transmit the images.
7. Morphological Processing
Deals with tools for extracting image components that are useful in the
representation and description of shape. In this step, there would be a
transition from processes that output images, to processes that output image
attribut es.
8. Segmentation
Segmentation partitions an image into its constituent parts or objects.
Autonomous segmentation is the most difficult tasks in digital image
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5Chapter 1: Introduction
The more accurate the segmentation, the better automated object classification.
9. Feature extraction
It consists of feature detection and feature description. Feature detection refers to finding a feature in an image, region or boundary. Feature description assigns quantitative attributes to the detected features.
10. Image pattern classifica tion
The process that assigns a label to an object based on its feature descriptors.
There are various classification algorithms like correlation, Bayes classifiers
to identify and predict the class label for the object.
1.5 Components of an Image Processing Systems

Fig 1.2: Components of a general -purpose image processing systems
(Image courtesy:// Digital Image Processing -Fourth Edition)
1. Image Sensors
Two subsystems are required to acquire digital images.
The first is the physical device that is sensitive to the energy radiated by the
object we wish to image (Sensor). The second, called a digitizer, is a device
for converting the output of the physical sensing device into digital form. The
digitizer converts the electrical outputs t o digital data.
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2. Specialized Image Processing Hardware
It usually consists of the digitizer, and the hardware that performs other
primitive operations, such as an arithmetic logic unit (ALU), which performs
arithmetic and logical operations in parallel on entire images. This type of
hardware is called a s frontend subsystem, and its most distinguishing
characteristic is speed. In other words, this unit performs functions that
require fast data throughputs that the typical main computer cannot handle.
3. Computer
The computer in an image processing system is a general -purpose computer
and can range from a PC to a supercomputer.
4. Image Processing Software
The software performs tasks with the help of specialized modules. There are
many software’s available commercially.
5. Mass Storage
Digital storage for image processing applications basically is divided into
three principal categories.
a) Short -term storage for use during processing
b) On-line storage for fast recall
c) Archival storage which is characterized by infrequent access.
Storage is measured in bytes (Kbytes, Mbytes, Gbytes and Tbytes)
6. Image Displays
Monitors are driven by the outputs of the image and graphics display cards
that are an integral part of a computer system. There are variants in monitor
displays.
7. Hardcopy devices
Used for recording images, include laser printers, film cameras, heat -
sensitive devices, inkjet units and digital units, such as optical and CD -ROM
disks.
8. Networking and cloud communication
Transmission bandwidth has improved due to opt ical fibre and other cloud
technologies.
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7Chapter 1: Introduction
1.6 Elements of Visual Perception
Image processing applications produce images that are to be viewed by human
observers.
The Elements of Visual Perception are
1) Structure of the Human eye
2) Image formation in the eye
3) Brightness adaptation and discrimination
1) Structure of the Human eye

Fig 1.3: Simplified diagram of the human eye. (Image Courtesy: nptel)
The first part of the visual system is the eye. Its form is nearly spherical and
its diameter is approximately 20 mm. Its outer cover consists of the ‘cornea'
and ‘sclera'
The cornea is a tough transparent tissue in the front part of the eye. The sclera
is an opaque membrane, which is continuous with cornea and covers the
remainder of the eye. Direct ly below the sclera lies the “choroids”, which has
many blood vessels. At its anterior extreme lies the iris diaphragm. The light
enters in the eye through the central opening of the iris, whose diameter varies
from 2mm to 8mm, according to the illuminatio n conditions. Behind the iris
is the “lens” which consists of concentric layers of fibrous cells and contains
up to 60 to 70% of water. Its operation is similar to that of the man -made
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optical lenses. It focuses the light on the “retina” which is the inner most
membrane of the eye.
Retina has two kinds of photoreceptors: cones and rods. The cones are highly
sensitive to color. Their number is 6 -7 million and they are mainly located at
the central part of the retina. Each cone is connected to one nerve end.
Cone vision is the photopic or bright light vision. Rods serve to view the
general picture of the vision field. They are sensitive to low levels of
illumination and cannot discriminate colors. This is the scotopic or dim -light
vision. Their number is 75 to 150 million and they are distributed over the
retinal surface. Several rods are connected to a single nerve end. This fact
and their large spatial distribution explain their low resolution.
Both cones and rods transform light to electric stimulus, which is carried
through the optical nerve to the human brain for the high -level image
processing and perception.
2) Image formation in the eye
x In the human eye, the distance between the centr e of the lens and the
imaging sensor is fixed
x Lens in the eye is flexible
x Shape controlled by muscles
x To focus on distance objects – Muscles flatten lens
x To focus on close objects – Muscles allow lens to thicken
3) Brightness Adaptation and Discrimination
x Digital Images are displayed as a discrete set of intensi ty
x Eye’s ability to discriminate intensities at a given adaptation level is an
important consideration when displaying images .
x Range of brightness's that can be discriminated simultaneously is small
in comparison to total adaptation range.
x For a given set of conditions the current sensitivity level of the visual
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Fig 1.4: Range of subjective brightness adaptation level
Weber Ratio
The ability of the eye to discriminate between changes in light intensity at any
specific adaptation level.
7KHTXDQWLW\ǻ,F, ǻ,FLVWKHLQFUHPHQWRILOOXPLQDWLR n discriminable 50% of the
time with background illumination I which is constant, is called the Weber Ratio.
Small weber ratio - good brightness discrimination
Large weber ratio -poor brightness discrimination
Visual Perception
Perceived brightness is not a simple function of intensity
Mach bands - Visual system tends to undershoot or overshoot around the boundaries
of regions with different intensities.
Simultan eous Contrast - regions perceived interest does not depend on intensity.
Optical illusions - Eyes fill in non -existing information or perceives geometry
properties of objects in an incorrect manner.
1.7 Image Sensing and Acquisition
Images are generated by t he combination of illumination of source and the
reflection or absorption of energy from that source by element of the scene being
imaged.
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There are three principal sensor arrangements that can be used to transform incident
energy into digital images.
Incoming energy is transformed into a voltage pulse by input electric power and
sensor response where a digital quantity is obtained by digitizing the response.
1.71. Image Acquisition using a single sensing element

1.7.2 Image Acquisition using Line sensor

1.7.3 Image Acquisition using Array sensor

Image Acquisition using sensor strips
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11Chapter 1: Introduction
The most common sensor is the photodiode constructed of silicon materials and
output voltage waveform proportional to light. Using a filter in front of the sensor
improves its selectively.
In order to generate a 2 -D image using a single sensor, there have to be relative
displacements in both the x - and y -directions between the sensor and the area to be
imaged. An arrangement used in high precision scanning, where a film negative is
mounted onto a drum w hose mechanical rotation provides displacement in one
dimension. The single sensor is mounted on a lead screw that provides motion in
the perpendicular direction. Since mechanical motion can be controlled with high
precision, this method is an inexpensive (but slow) way to obtain high -resolution
images.
1.8 Image Sampling and Quantization
The output of most sensors is a continuous voltage waveform whose amplitude and
spatial behaviour are related to the physical phenomenon being sensed. To create a
digital image, it is essential to convert the continuous sensed data into digital form.
This involves two processes: sampling and quantization.
1.1.1 Image Sampling
An image may be continuous with respect to the x and y coordinates and also in
amplitude. To convert it into digital form we have to sample the function in both
coordinates and in amplitudes.
Digitalizing the coordinate values is called sampling.
Digita lizing the amplitude values is called quantization.
There is a continuous image along the line segment AB. To sample this function,
we take equally spaced samples along line AB. The location of each sample is given
by a vertical tick back (mark) in the bot tom part. The samples are shown as block
squares superimposed on function the set of these discrete locations gives the
sampled function.
In order to form a digital image, the gray level values must also be converted
(quantized) into discrete quantities. S o, we divide the gray level scale into eight
discrete levels ranging from black to white. The vertical tick mark assigns the
specific value assigned to each of the eight level values. The continuous gray levels
are quantized simply by assigning one of the eight discrete gray levels to each
sample. The assignment it made depending on the vertical proximity of a simple to
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Fig 1.5: Sampling


Fig 1.6: Quantization



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Example

Fig 1.7 a) Continuous image projected onto a sensor array
b) Result of sampling and quantization
Digital Image
A digital image f [m, n] described in a 2D discrete space is derived from an
analogue image f (x.y) in a 2D continuous space through a sampling process that is
frequently referred to as digitization.
The 2D continuous image f (x, y) is divided into N rows and M columns. The
intersection of a row and a column is termed a pixel. The value assigned to the integer coordinates [m, n] with {m=0,1, 2…, M -1}and {n=0,1, 2..., N -1}is F
[m, n].
1.9 Basic Relationship between pixels
Pixel Relationships
Neighbours of a pixel
A pixel p at coordinate (x, y) has four horizontal and vertical neighbour whose
coordinate can be given by
(x+1, y) (x -1, y) ( x, y + 1) (x, y -1)
This set of pixels is called the 4 -neighbours of p and is denoted by n4(p), Each
pixel is at a unit distance from (x, y) and some of the neighbours of P lie outside
the digital image or (x, y) is on the border of the image .
The four-diagonal neighbour of P have co -ordinates
(x+1, y+1), (x+1, y+1), (x-1, y+1), (x-1, y-1)
And are denoted by nd(p) these points, together with the 4 -neighbours are called
8 – neighbours of P denoted by n8(p) 
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Adjacency
Let V be the set of gray–level values used to define adjacency in a binary image, if
V= {1} we are referencing to adjacency of pixel with value. Three types of
adjacencies occur
4-Adjacency – two-pixel P and Q with value from V are 4 –adjacency if A is in the
set N4(P)
8- Adjacency – two-pixel P and Q with value from V are 8 –adjacency if A is in the
set N8(P)
M-adjacency –two-pixel P and Q with value from V are m– adjacency if
x Q is in N4 (p) or
x Q is in Nd (q) and the set N4(p) ŀ N4(q) has no pixel whose values
are from V
x Distance measures
For pixel p, q and z with coordinate (x, y), (s, t) and (v, w) respectively D is a
distance function or metric if
'>ST@•2^ D [p.q] = O iff p=q} D
[p.q] = D [p.q] and
'>ST@•2^ D [p.q] + D (q, z)
The Euclidean Distance between p and is defined as De ( p, q) = I y – t I
The D4 Education Distance b etween p and is defined as De ( p, q) = I y – t I
1.10 Summary
In this chapter, we learned about the basic concepts in digital image processing .
The various components comprise and contribute to image acquisition. Image
perception, brightness, and adaptation. Sampling and quantization technique to
discretize image. We also learned about the basic relationship between pixels
1.11 Review Questions
1. Explain Digital Image Processing briefly?
2. Describe various methods of image sensing and acquisition.
3. Explain pixel and its relationships with its neighbourhood pixels.
4. Write a short note on Image Sampling and Quantization.
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1.12 References for in detail study
1. www.ImageProcessingplace.com
2. https://nptel.ac.in/courses/106/105/106105032/
3. https://nptel.ac.in/courses/117/105/117105079/
4. All Images courtesy from Digital Image Processing by Gonzalez and Woods,
4th Edition
5. Figure 1.1 adapted from
https://www.bharathuniv.ac.in/coll eges1/downloads/courseware_ece/notes/BEC0
07%20%20 -Digital%20image%20processing.pdf

™™™
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Unit 1
2 INTENSITY TRANSFORMATIONS
AND SPATIAL FILTERING
Unit Structure
2.0 Objectives
2.1 Introduction
2.2 Basics Intensity Transformation functions
2.3 Histogram Processing
2.4 Fundamentals of spatial filtering
2.4.1 Smoothing (Low pass) Spatial filters
2.4.2 Sharpening (High pass) Spatial filters
2.4.3 Using LAPLACIAN - Second derivative for image sharpening
2.4.4 Unsharp masking and high boost filtering
2.5 High pass, Band reject and band pass filters
2.6 Spatial Intensity transformation and filtering methods using fuzzy techniques
2.7 Summary
2.8 Review Questions
2.9 References
2.0 Objective s
1. To understand the meaning of spatial domain and apply techniques for
intensity -based transformations
2. To know the concept of the histogram and use it for image enhancement
3. To understand the mechanics of spatial filtering and use combination methods or enh ancing image
4. Use fuzzy techniques to perform spatial filtering methods
2.1 Introduction
The principal objective of enhancement is to process an image so that the result is
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1. Spatial domain methods
2. Frequency domain methods
The chapter mainly focusses on Spatial domain methods. All methods and
techniques covered here are for spatial domain.
The term spatial domain refers to the image plane itself and approaches in these
categories are based on direct manipulation of pixel in an image.
Spatial domain process is denoted by the expression
g (x, y) =T [f (x, y)]
f (x, y) - input image, T - operator on f, defined over some neighbourhoo d of
f (x, y)
g (x, y) -processed image
The neighbourhood of a point (x, y) can be explained by using as square or
rectangular sub image area centred at (x, y).

Fig 2.1 A 3 x 3 neighbourhood about a point (x0, y0) in an image. The
neighbourhood is moved from pixel to pixel in the image to generate an output
image
The center of sub image is moved from pixel to pixel starting at the top left corner.
The operator T is ap plied to each location (x, y) to find the output g at that location.
The process utilizes only the pixel in the area of the image spanned by the
neighbourhood.
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2.2 Basic Intensity Transformation Functions
It is the simplest form of the transformation when the neighbourhood is of size 1 x
1. In this case g depends only on the value of f at (x, y) and T becomes an intensity
or gray level transformation function of the form
S=T(r)
r-Denotes the gray level of f (x, y)
s-Denotes the gray level of g (x, y) at a ny point (x, y)
Because enhancement at any point in an image deepens only on the gray level at
that point, technique in this category is referred to as point processing.
2.2.1 Point Processing
Contract stretching -It produces an image of higher contrast than the original one.
The operation is performed by darkening the levels below m and brightening the
levels above m in the original image.

Fig: 2.2 Contrast Stretching
In this technique the value of r below m are compressed by the transformation
function into a narrow range of s towards black . The opposite effect takes place
for the values of r above m.
Thresholding function – It is a limiting case where T(r) produces a two levels
binary image.
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The values below m are transformed as black and above m are transformed as
white.

Fig 2.3 Thresholding function
2.2.2 Basic Intensity Level Transformation
These are the simplest image enhancement techniques
2.2.2.1. Image Negative – The negative of in image with gray level in the range
[0, L -1] is obtained by using the negative transformation.
The expression of the transformation is s= L-1-r
Reverting the intensity levels of an image in this manner produces the equivalent
of a photographic negative. This type of processing is practically suited for
enhancing white or gray details embedded in dark regions of an image especially
when the black areas are dominant in size.

Fig 2.4 Basic intensity transformation functions
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2.2.2.2 Log transformations -
The general form of log transform is S=c log(1+R)
Where R >= 0
This transformation maps a narrow range of gray level values in the input image
into a wider range of output gray levels. The opposite is true for higher values of
input levels. These transformations are used to expand the values of dark pixels in
an image while compressing the higher -level values. The opposite is true for
inverse log transformation.
The log transformation function has an important characteristic that it compresses
the dynamic range of images with large variations in pixel values.
E.g.- Fourier spectrum

Fig 2.5 a) Fourier Spectrum b) Log transformed Fourier spectrum
2.2.2.3 Power law transformation
Power law transformation has the basic function
S= c r ஔ
Where c and ஔare positive constants.
Power law curves with fractional values of ஔ map a narrow range of dark input
values into a wider range of output values, with the opposite being true for higher
values of input gray levels. We may get various curves by varying values of ஔ


Fig: 2.6 Shape of the curve formed by the gamma equation
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A variety of devices used for image capture, printing and display respond according
to a power law. The process used to correct this power law response phenomenon
is called gamma correction.
For E.g. -CRT devices have intensity to voltage response that is a power function
Gamma correction is important if displaying an image accurately on a computer
screen is of concern. Images that are not corrected properly can look either
bleached out or too dark.
Color phenomenon also uses this concept of gamma correction. It is becoming
more popular due to use of images over the internet.
It is important in general purpose contract manipulation.
To make an image black we use ஔ!and ஔ <1 for white image.
2.2.2.4 Piece wise Linear transformation functions -
The principal advantage of piecewise linear functions is that these functions can
be arbitrarily complex. But their specification requires considerably more user
input
Contrast Stretching -
It is the simplest piecewise linear transformation function.
We may have various low contrast images and that might result due to various
reasons such as lack of illumination, problem in imaging sensor or wrong setting
of lens aperture during image acquisition.
The idea behind contrast stretching is to increase the dynamic range of gray levels
in the i mage being processed.

Fig 2.7 Piecewise linear transformation function
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The location of points (r1, s1) and (r 2, s2) control the shape of the curve
a) If r1=r2 and s 1=s2, the transformation is a linear function that deduces no
change in gray levels.
b) If r1=s1, s1=0 , and s2=L -1, then the transformation become a
thresholding function that creates a binary image
c) Intermediate values of (r1, s1) and (r2, s2) produce various degrees of
spread in the gray value of the output image thus effecting its contract.
*HQHUDOO\U”UDQGV”VVRWKDWWKHIXQFWLRQLVVLQJOHYDOXHGDQG
monotonically increasing
Gray Level Slicing - Highlighting a specific range of gray levels in an image is
often desirable
For example, when enhancing features such as masses of water i n satellite image
and enhancing flaws in x - ray images.
There are two ways
1) This method is to display a high value for all gray level in the range of
interest and a low value for all other gray level
2) Second method is to brighten the desired ranges of gray levels but
preserve the background and gray level tonalities in the image

Fig 2.8 a) Highlights the range [A, B] b) Highlights the range [A, B] and
preserves background details
2.2.2.5 Bit Plane Slicing
It is important to highlight the contribution made to the total image appearance by
specific bits. Suppose that each pixel is represented by 8 bits.
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Imagine that an image is composed of eight 1 -bit planes ranging from bit plane 0
for the least significant bit to bit plane 7 for the most significant bit. In terms of
8-bit bytes, plane 0 contains all the lowest order bits in the image and plane 7
contains all the high order bits.

Fig 2.9 Bit -planes of an image
2.3 Histogram Processing
The histogram of a digital image with gray levels in the range [0, L -1] is a
discrete function of the form H(rk)=nk
where rk is the kth gray level and nk is the number of pixels in the image having
the level rk.
A normalized histogram is given by the equation
P(rk)=nk/n for k=0,1, 2, . . ., L -1
P(rk) gives the estimate of the probability of occurrence of gray level rk. The sum
of all components of a normalized histogram is equal to 1.
The histogram plots are simple plots of H(rk)=nk versus rk.

Fig 2.10 Four image types and their corresponding histograms. (a) dark; (b) light; (c) low contrast; d) high contrast
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In the dark image the components of the histogram are concentrated on the low
(dark) side of the gray scale. In case of bright image, the histogram components are
biased towards the high side of the gray scale.
The histogram of a low contrast image will be narrow and will be centered towards
the middle of the gray scale.
The components of the histogram in the high contrast image cover a broad range of
the gray scale. The net effect of this will be an image that shows a great deal of
gray levels details and has high dynamic range.
2.3.1 Histogram Equalization
Histogram equalization is a com mon technique for enhancing the appearance of
images . Suppose we have an image which is predominantly dark. Then its
histogram would be skewed towards the lower end of the grey scale and all the
image detail are compressed into the dark end of the histogram. If we could ‘stretch
out’ the grey levels at the dark end to produce a more uniformly distributed
histogram then the image would become much clearer.
Let there be a continuous function with r being gray levels of the image to be
enhanced.
The ra nge of r is [0, 1] with r=0 repressing black and r=1 representing white.
The transformation function is of the form.
S=T(r) where 0It produces a level s for every pixel value r in the original image. the
transformation function is assumed to fulfill t wo condition T(r)) is single valued
and monotonically increasing in the interval 0The transformation function should be single valued so that the inverse transformations should exist. Monotonically increasing condition preserves the
incre asing order from black to white in the output image. The second conditions
guarantee that the output gray levels will be in the same range as the input levels.
The gray levels of the image may be viewed as random variables in the interval
[0.1]
The most fundamental descriptor of a random variable is its probability density
function (PDF) Pr(r) and Ps(s) denote the probability density functions of random
variables r and s respectively. Basic results from an elementary probability theory
state that if Pr(r) a nd Tr are known and T -1 (s) satisfies conditions (a), then the
probability density function Ps(s) of the transformed variable s is given by the
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Thus, the PDF of the transformed variable s is the determined by the gray levels
PDF of the input image and by the chosen transformations function
A transformation function of a particular importance in image processing
This is the cumulative distribution function of r
Using this definition of T we see that the derivative of s with respect to r is
Substituting it back in the expression for Ps we may get

An important point here is that Tr depends on Pr(r) but the resulting Ps(s) always
is uniform, and independent of th e form of P(r).
For discrete values we deal with probability and summations instead of probability
density functions and integrals.
The probability of occurrence of gray levels rk in an image as approximated
Pr(r)=nk/N
N is the total number of the pixels i n an image. nk is the number of the pixels that
have gray level rk. L is the total number of possible gray levels in the image. The
discrete transformation function is given by

Thus, a processed image is obtained by mapping each pixel with levels rk in t he
input image into a corresponding pixel with level sk in the output image.
A plot of Pr (rk) versus rk is called a histogram. The transformation function given
by the above equation is the called histogram equalization or linearization. Given
an image the process of histogram equalization consists of implementing the
transformation function which is based on information that can be extracted
directly from the given image, wit hout the need for further parameter specification.
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Fig 2.11 Left column - original images, center column - histogram equalized images
and last column - histogram of histogram equalized images.
Equalization automatically determines a transformation function that seeks to
produce an output image that has a uniform histogram. It is a good approach when
automatic enhancement is needed.
2.3.2 Histogram Matching (Specification)
In some cases, it may be desirable to specify the shape of the histogram that we
wish the processed image to have. Histogram equalization does not allow interactive image enhancement and generates only one result: an approximation to a uniform histogram. Sometimes we
need to be able to specify particular histogram shapes capable of highlighting
certain gray -level ranges. The method used to generate a processed image that has
a specified histogram is called histogram matching or histogram specification.
Algorithm
1. Compute sk=Pf (k), k = 0, , L -1, the cumulative normalized histogram of f .
2. Compute G(k), k = 0, …, L -1, the transformation function, from the given
histogram hz
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27Chapter 2: Intensity Transformations and Spatial Filtering
3. Compute G -1(sk) for each k = 0, …, L -1 using an iterative method (iterate
on z), or in effect, directly compute G -1(Pf (k))
4. Transform f using G -1(Pf (k))
Global and Local enhancement
In earlier methods pixels were modified by a transformation function based on the
gray level of an entire image. It is not suitable when enhancement is to be done in
some small areas of the image. This problem can be solved by local enhan cement
where a transformation function is applied only in the neighborhood of pixels in
the interested region.
Define square or rectangular neighborhood (mask) and move the center from pixel
to pixel. For each neighborhood
1) Calculate histogram of the points in the neighbourhood
2) Obtain histogram equalization/specification function
3) Map gray level of pixel centered in neighbourhood
4) The center of the neighbourhood region is then moved to an adjacent pixel
location and the procedure is repeated.

Fig 2.12a) Original Image b) Image obtained after global histogram equalization
and c) after local histogram equalization.

d) Result of local enhancement based on local histogram
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2.4 Fundamentals of Spatial filtering
1. Spatial filtering modifies an image by rep lacing the value of each pixel by a
function of the values of the pixel and its neighbours.
2. If the operation performed on the image pixels is linear, then the filter is
called linear spatial filter. Otherwise, the filter is a non -linear filter.
Spatial f iltering is an example of neighborhood operations, in this the operations
are done on the values of the image pixels in the neighborhood and the
corresponding value of a sub image that has the same dimensions as of the
neighborhood.
This sub image is called a filter, mask, kernel, template or window ; the values in
the filter sub image are referred to as coefficients rather than pixel. Spatial filtering
operations are performed directly on the pixel values (amplitude/gray scale) of the
image. The process consists of moving the filter mask from point to point in the
image. At each point (x, y) the response is calculated using a predefined
relationship.

Fig 2.13 The mechanics of spatial filtering using a 3 x 3 kernel
For linear spatial filtering the response is given by a sum of products of the filter
coefficient and the corresponding image pixels in the area spanned by the filter
mask.
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29Chapter 2: Intensity Transformations and Spatial Filtering
The results F (x, y) of linear filtering with the filter mask at point (x, y) in th e image
is given by
w (-1, -1) f (x -1, y-1) + w ( -1,0) f (x -1, y) + ………...+ w (1, 1) f (x + 1, y +1)
The coefficient w (0,0) coincides with image value f (x, y) indicating that mask it
centered at (x, y) when the computation of sum of products takes place.
For a mask of size M x N, we assume m=2a+1 and n=2b+1 , where a and b are
nonnegative integers. It shows that all the masks are of odd size .
In the general linear filtering of an image of size f of size M*N with a filter mask
of size m*m is given by the expression.

Where a= (m -1)/2 and b = (n -1)/2
To generate a complete filtered image this equation must be applied for x=0, 1, 2,
-----M-1 and y=0,1,2 ---, N-1. Thus, the mask processes all the pixels in the image.
The process of linear filtering is similar to frequency domain concept called
convolution. For this reason, linear spatial filtering often is referred to as convolving a mask with an image. Filter mask are sometimes called convolution
mask.
Padding the image by adding rows and columns of o’s & of padding by replicating
rows and columns when the centre of the filter approaches the border of the
image.
2.4.1 Smoothing Spatial Filters(lowpass) These filters are used for blurring and noise reduction blurring is used in preprocessing steps such as removal of small details from an image prior to object
extraction and bridging of small gaps in lines or curves.
Smoothing Linear Filters
The output of a smoothing liner spatial filter is simply the average of the pixel
contained in the neighborhood of the filter mask. These filters are also called
averaging filters or low pass filters. Smoothing filters are used in combination with
other image enhancemen t techniques.
The operation is performed by replacing the value of every pixel in the image by
the average of the gray levels in the neighborhood defined by the filter mask . This
process reduces sharp transitions in gray levels in the image.
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A major application of smoothing is noise reduction but because edges are also
provided using sharp transitions so smoothing filters have the undesirable side
effect that they blur edges. It also removes an effect false contouring which is
caused by using insufficient number of gray levels in the image.
Irrelevant details can also be removed by filters, irrelevant means which are not of
interest.
A spatial averaging filter in which all coefficients are equal is sometimes referred
to as a “box filter ”. 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 A weighted average filter is the one in which pixel are multiplied by different
coefficients.
Order Statistics Filter
These are nonlinear spatial filter whose response is based on ordering of the pixels
contained in the image area compressed by the filter and the replacing the value of
the center pixel with value determined by the ranking result.
The best example of this category is median filter. In this filter the values of the
center pix el are replaced by median of gray levels in the neighborhood of that pixel.
Median filters are popular because, for certain types of random noise, they provide
excellent noise -reduction capabilities, with considerably less blurring than linear
smoothing filters.
These filters are particularly effective in the case of impulse or salt and pepper
noise. It is called so because of its appearance as white and black dots superimposed
on an image.
In order to perform median filtering at a point in an image, we first sort the values
of the pixel in the question and its neighbors, determine their median and assign
this value to that pixel.
Low pass filtering is used for shadow correction, region extraction if used with
thresholding.
2.4.2 Sharpening Spatial Filter s (High pass)
The principal objective of sharpening is to highlight fine details in an image or to
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The applications of image sharpening range from electronic printing and medical
imaging to industrial inspection and autonomous guidance in military systems.
As smoothing can be achieved by integration, sharpening can be achieved by spatial
differentiation. The strength of respon se of derivative operator is proportional to
the degree of discontinuity of the image at that point at which the operator
is applied. Thus, image differentiation enhances edges and other discontinuities and
deemphasizes the areas with slow varying grey levels.
It is a common practice to approximate the magnitude of the gradient by using
absolute values instead of square and square roots.
A basic def inition of a first order derivative of a one -dimensional function f(x) is
the difference

A second order derivative as the difference

Fig 2.14 A scan line from an image showing ramp, step and constant segments
with its first and secon d order derivative
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2.4.3.Using LAPLACIAN - Second derivative for image sharpening
The second order derivative is calculated using Laplacian. It is simplest isotropic
filter. Isotropic filters are the ones whose response is independent of the direction
of the image to which the operator is applied.
The Laplacian for a two -dimensional function f (x, y) is defined as

In the x -direction

In the y -direction

From the above equation, the discrete La placian of two variable is

The equation can be represented using any one of the following masks

Fig 2.15 Laplacian kernels
Laplacian highlights gray -level discontinuities in an image and deemphasizes the
regions of slow varying gray levels. This makes the background a black image.
The background texture can be recovered by adding the original and Laplacian
images.

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33Chapter 2: Intensity Transformations and Spatial Filtering

The stre ngth of the response of a derivative operator is propositional to the degree
of discontinuity of the image at that point at which the operator is applied. Thus,
image differentiation enhances eddies and other discontinuities and deemphasizes
areas with slowly varying gray levels.
The derivative of a digital function is defined in terms of differences. Any first
derivative definition
(1) Must be zero in flat segments (areas of constant gray level values)
(2) Must be nonzero at the onset of a gray level step or ramp
(3) Must be nonzero along ramps.
Any second derivative definition
;ϭͿ Must be zero in flat areas
;ϮͿ Must be nonzero at the onset and end of a gray level step or ramp
;ϯͿ Must be zero along ramps of constant slope.

Fig 2.16 Example of sharpening from blurred image to sharpened image
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It is common practice to approximate the magnitude of the gradient by using
absolute values instead of squares and square roots.
Roberts Goss gradient operators
For implementing the gradient operators Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9
The smallest filter mask is size 3x3 mask

The difference between third and first row a 3x3 mask approximates the derivate
in the x-direction and difference between the third and first column approximates
the derivative in y-direction. These masks are called sobel operators.
2.4.4 Unsharp masking and highboost filtering
Unsharp masking means subtracting a blurred version of an image form the image
itself.
It consists of following steps
1. Blur the original image
2. Subtract the blurred image from the original
3. Add the mask to the original
ZKHUHZHLQFOXGHGDZHLJKWNN•IRUJHQHUDOLW\:KHQN ZHKDYHXQVKDUS
masking, as defined above. When k > 1, the process is referred to as highboost
filtering. Choosing k < 1 reduces the contribution of the unsha rp mask
-1 0 0 1
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35Chapter 2: Intensity Transformations and Spatial Filtering
2.5 Highpass, Band reject and Band pass filters from Lowpass
filters


Fig 2.17 Low pass, Highpass, Bandreject , and Bandpass filters
Spatial and frequency -domain linear filters are classified into four broad categories:
lowpass and highpass filters, bandpass , and bandreject filters.
Lowpass filter - A lowpass filters attenuate or delete high frequencies while passing
low frequencies.
Highpass filter - A highpass filter deletes or attenuates all frequencies below a
cut-off value, u0, and passes all frequencies above this value. A highpass filter
transfer function is obtained by subtracting a lowpass function from 1. This
operation is in the frequency domain.
Bandreject filter - These filters can be constructed from the sum of a low pass and a
highpass function with different cut -off frequencies. They are known as notch
filters.
Bandpass filter - The bandpass filter transfer function can be obtained by
subtracting the bandreject function from 1 (a unit impulse in the spatia l domain).

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2.6 Spatial Intensity transformation and filtering methods using
fuzzy techniques
Fuzzy sets and its applications are very useful when it comes to image processing
specially to spatial intensity transformation and filtering. Fuzzy logic proves more
beneficial and flexible than classical set theory.
Fuzzy logic deals with degree of membership in a set and never probabilistic. They
find application in situation characterized by vagueness and imprecision.

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37Chapter 2: Intensity Transformations and Spatial Filtering

Fig: 2.18 Membership funct ions with respect to their equations
The principal steps followed in the application of rule -based fuzzy logic
1. Fuzzify the inputs
2. Perform any required fuzzy logical operations
3. Apply an implication method
4. Apply an aggregation method to the fuzzy sets from step 3
5. Defuzzify the final output fuzzy set
The problem specific knowledge can be formalized with the following IF -THEN
fuzzy rules. Consider the following rules for a fruit.
R1: IF the color is green, THEN the fruit is verdant
OR
R2: IF the color is yellow, THEN the fruit is half -mature
OR
R3: IF the color is red, THEN the fruit is mature
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Fig 2.19 Five basic steps used to implement the fuzzy rule -based system
Using Fuzzy sets for Intensity Tr ansformation
Consider a general problem of contrast enhancement. The following rules can be
defined for enhancing the contrast in a gray -scale image.
R1: IF a pixel is dark, THEN make it darker
R2: IF a pixel is gray, THEN make it gray
R3: IF a pixel is bright, THEN make it brighter
The concept of dark, gray , and bright can be expressed and evaluated using
membership functions. Fuzzy image processing is computationally intensive
because the entire process of fuzzification, processing the antecedents of all rules,
implication, aggregation , and defuzzification m ust be applied to every pixel in the
input image.
The output of these fuzzy rules whose output is interpreted as constant intensities
is considered as singletons which means their membership functions are constant.
Singletons greatly reduce the computational requirements in fuzzy image processing. Similarly , fuzzy rules can be applied for spatial filtering. When
applying fuzzy sets to spatial filtering, the basic approach is to define the
neighbourhood properties, that capture the essence of the filter are to detect.
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39Chapter 2: Intensity Transformations and Spatial Filtering
2.7 Summary
In this chapter, we learned about the various spatial domain transformations for
enhancing the image , histogram of the image, spatial filters and their application
on images. We also learned about fuzzy filtering in images.
2.8 Review Que stions
1. Explain spatial domain processing.
2. Describe point processing, contrast enhancement.
3. What are the basic intensity transformations?
4. What is filtering? What do you understand by the term spatial filtering?
5. Explain histogram equalization.
6. Write a short note on smoothing filters.
7. Write a short note on sharpening filters.
8. Explain the use of fuzzy techniques in image processing.
2.9 References
1. Digital Image Processing, third edition by Gonzalez and Woods.
2. www.Imageprocessingplace.com
3. Refer to nptel courses on digital image processing for further detailed study.
4. All Images courtesy from Digital Image Processing by Gonzalez and Woods,
4th Edition
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40IMAGE PROCESSING
Unit 2
3 FILTERING IN THE FREQUENCY DOMAIN
Unit Structure
3.0 Objectives
3.1 Introduction
3.2 Sampling and Fourier Transform of sampled functions
3.3 Discrete Fourier transform of one variable
3.4 Discrete Fourier transform of two variables
3.5 Properties of 2D - DFT and IDFT
3.6 Basics of filtering in the frequency domain
3.7 Smoothing in Frequency Domain
3.8 Sharpening in Frequency Domain
3.9 Selective filtering
3.10 Fast Fourier Transform
3.11 Summary
3.12 Review questions
3.13 References
3.0 Objective s
1. To understand the frequency domain , the difference , and advantage of
frequency domain from the spatial domain.
2. Basic steps of filtering in the frequency domain.
3. Familiarize with smoothing and sharpening filters in the frequency
domain.
4. Understand the working of Fast Fourier Transform
3.1 Introduction
Fourier transform named after the French mathematician Jean Baptiste Joseph
Fourier is one of the most prominent transforms used in Image processing.
Any function that periodically repeats itself can be expressed as a sum of sines and
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41Chapter 3: Filtering in the Frequency Domain
is called Fourier series . Even the functions whi ch are non -periodic but whose area
under the curve if finite can also be represented in such form; this is now called
Fourier transform.
A function represented in either of these forms and can be completely reconstructed
via an inverse process with no loss of information.
3.2 Sampling and Fourier transform of sampled functions
A continuous function is converted into a sequence of discrete values before they
can be processed in a computer, which requires sampling and quantization.
&RQVLGHUDFRQWLQXRXVIXQFWLRQIWWREHVDPSOHGDWXQLIRUPLQWHUYDOVǻ7RIWKH
independent variable t.
The si mplest way to sample is to multiply f(t) by a sampling function equal to the
WUDLQRILPSXOVHVǻ7XQLWVDSDUW7KHHTXDWLRQRIWKHVDPSOHGIXQFWLRQLVJLYHQE\

The Fourier Transform of a sampled function
Let F(u) denote the Fourier transform of a contin uous function f(t). The sampled
function is a product of f(t) and an impulse train.
The Fourier transform of the product of the two functions in the spatial domain is
the convolution of the transforms of the two functions in the frequency domain.
The Fouri er transform of the sampled function is given by

where the given equation below is the Fourier transform of the impulse train.

3.3 Discrete Fourier Transform of one variable
1-D Fourier Transformation and its Inverse
Let f(x) be a continuous functio n of real variable x. The Fourier transform of f(x)
is

^`³f
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42IMAGE PROCESSING
Where j = ξെͳ
F(u) is composed of an infinite sum of sine and cosine terms and each value of u
determines the frequency of its corresponding sine -cosine pair where u is a
frequency variable.
Given F(u), f(x) can be obtained by the inverse Fourier transform )( )}({1xf uF ‚

3.4 Discrete Fourier Transform of two variables
2-Dimensional Fourier transform
Images being 2 -dimensional functions, we need to define a 2 -D Fourier transform.
Fourier and Inverse Fourier transform of a two variable continuous function is
given by dxdyvy ux j yxf vuF yxf )] (2 exp[),( ),( )},({  ‚ ³³f
ff
fS dudvvy ux j vuF yxf vuF ]) (2 exp[),( ),( )},({1³³f
ff
f ‚ S
Where u, v are frequency variables.
Since the work is on digital images, rather than continuous variables discrete
Fourier transform is more appropriate
1-D Discrete Fourier Transform
The discrete Fourier transform of one variable f(x) is given by )X 
0I[H[S>MSX[0@
[ 0
¦
where u = 0,1, 2……, M -1 and I[ IXH[S>MSX[0@
X 0
¦
Where x = 0,1, 2……, M – 1 ³f
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43Chapter 3: Filtering in the Frequency Domain
To compute F(u) we substitute u=0 in the exponential term and sum for all values
of x.
Repeating for all M values of u, it takes M*M summations and multiplications to
compute discrete Fourier transform )X 
0I[H[S>MSX[0@
[ 0¦
Where u = = 0,1, 2……, M -1
For discrete functions, the Fourier transform and its inverse exist always.
F(u) is complex, F(u) = R(u) + j I (u)
Where R is real and I is imaginary
The Fourier transform of a real function is generally complex and polar coordinates
are used.
Magnitude or spectrum of the Fourier transform is given by

Magnitude specifies how much of a certain frequency component is present
Phase angle or phase spectrum of the Fourier transform is given by

The phase specifies where the frequency component is in the images
The square of the spectrum is referred to as the power spectrum of f(x) (spectral
density).
2-D Discrete Fourier transform
The Fourier transform of a 2D discrete function (image) f (x, y) of size Mx N is
given by:

where u = 0,1, 2……, M -1 and v = 0,1, 2……, N -1
The inverse 2 -D Discret e Fourier transform is given by
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where x = = 0,1, 2……, M – 1 and y = 0,1, 2……, N – 1 The 2D Fourier Spectrum, Phase Spectrum and Power Spectrum can be respectively denoted by




Properties of 2D Discrete Fourier Transform
The Periodicity property: F (u, v) in 2D DFT has a period of N in horizontal and M
in vertical directions.
The Symmetry property: The magnitude of the transform is centered on the origin

Fig 3.1 M x N Fourier Spectrum and Spectrum obtained
by multiplying f(x,y) by ( -1)x+ y
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45Chapter 3: Filtering in the Frequency Domain

Fig 3.2 a) Image b) Spectrum across four corners
c) Centred Spectrum d) Log Transformed image
The center value (at the origin) of the Frequency Spectrum corresponds to the
ZERO frequency component which also referred to as the DC component in an
image: Substituting 0,0 to the origin, the Fourier transform function yields to the
average/DC component value as follows:

F (0,0) is called “dc” component of the spectrum (current of zero frequency)
The above equation implies that the value of the Fourier transform at the origin is
equal to the average gray level of the image.
3.5 Properties of 2D DFT
1) Fourier transform is conjugate symmetric.
F (u, v) =F*( -u, -v)
so |F (u, v) |=|F ( -u, -v) |
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46IMAGE PROCESSING
which means that the Fourier spectrum is symmetric.
2) The relationship between the samples in the spatial and frequency domain
ǻX ଵ
ெο௫ DQGǻY ଵ
ேο௬
1) Translation and Rotation
Fourier transform pairs are given by

Using the polar coordinates
x= r cos ߠ ,y = r sin ߠ u = ߱ cos߮ v = ߱ sin߮
It results in the following transform pair

which indicates that rotating f(x, y) by an angle ߠ0 rotates F(u, v) by the same
angle.
2) Periodicity
The 2D Fourier tr ansform and its inverse are infinitely periodic in the u and
v directions

*Summary of the DFT pairs
Name DFT pairs
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47Chapter 3: Filtering in the Frequency Domain

3.6 Basics of filtering in the frequency domain
Filtering techniques in the frequency domain are based on modifying the Fourier
transform to achieve a specific objective, and then computing the inverse DFT to
get us back to the spatial domain.
Basic Steps for Filtering in the Frequency Domain:
1. Multi ply the input image by ( -1) x+y to center the transform.
2. Compute F (u, v), the DFT of the image from (1).
3. Multiply F (u, v) by a filter function H (u, v).
4. Compute the inverse DFT of the result in (3).
5. Obtain the real part of the result in (4).
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6. Multiply the result in (5) by ( -1) x+y
Given the filter H (u, v) (filter transfer function) in the frequency domain, the
Fourier
transform of the output image (filtered image) is given by:
G (u, v) = H (u, v) F (u, v)
The filtered image g (x, y) is simply the inverse Fourier transform of G (u, v).
g (x, y) =
3.7 Smoothing in the Frequency Domain
Smoothing is a low pass operation in the frequency domain. Smoothing is achieved
in the frequency domain by high frequency attenuation of a specified range of high
frequency components in the transform of a given image.
There are three types of lowpass filters: Ideal, Butterworth, and Gaussian Low pass
filters
1. Ideal Low Pass filter
A lowpass filter is a filter that “cuts off” all high frequency components of
the Fourier transform that are at a distance greater than a specified distance
D0 from the origin of the transform. However, ILPF is not used practically.
The transfer function of an Ideal Low pass filter is given by

Where D (u , v) is the distance from point (u, v) to the center of frequency
rectangle and D 0 is the nonnegative constant.


Fig 3.3 a) Perspective plot of ILPF b) Function as an image
c) Radial cross section ‚*XY>@
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49Chapter 3: Filtering in the Frequency Domain

Fig 3.4 a) Original image b - f) Results of filtering using ILPFs with cut -off
frequencies set at radii values 10, 30, 60, 160, and 460
2. Butterworth Low pass filter
It has a parameter called the filter order. For high values of filter order, it
approaches the form of the ideal filter whereas for low filter order values it
acts as a Gaussian filter. It may be viewed as a transition between two
extremes. The transfer function of a Butterworth low pass filter (BLPF) of
order n with cut off frequency at distance Do from the origin is defined as

A BLPF approaches the sharpness of an ILPF function with considerably
less ringing.
The kernel corresponding to the BLPF of order 1 has neither ringing nor
negative values. The kernel corre sponding to a BLPF of order 2 shows mild
ringing and small negative values. As the remaining images show, ringing
becomes significant for higher order filters. A BLPF of order 20 has a spatial
kernel that exhibits ringing characteristics similar to those o f the ILPF.
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Fig 3.5 a) Perspective plot of BLPF b) Function as an image
c) Radial cross section of order n=1 through 4
3. Gaussian Low pass filter
The transfer function of GLPF is of the form

Where D (u, v) - the distance of point (u, v) from the center of the transform
ı '0- specified cut off frequency
The inverse Fourier transform of a frequency domain Gaussian function is
also Gaussian. This means that a spatial Gaussian filter kern el, obtained by
computing the IDFT, will have no ringing effect.

Fig 3.6 a) Perspective plot of GLPF b) Function as an image
c) Radial cross section for D 0 values.
3.8 Sharpening in the Frequency Domain
Image sharpening can be achieved by a high pass filtering process, which attenuates
the low -frequency components without disturbing high -frequency information.
These are radially symmetric and completely specified by a cross section.
If we have the transfer function of a low pass filter the correspondin g high pass
filter can be obtained using the equation
HHP (u, v) = 1 - HLP (u, v)
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51Chapter 3: Filtering in the Frequency Domain
Where H LP (u, v) is the transfer function of the low pass filter.
1) Ideal High Pass filter
An ideal highpass filter (IHPF) transfer function is given by

where, D( u, v) is the distance from the center of the frequency rectangle

Fig 3.7 Perspective plot , image and radial cross section of ILPF, GLPF and BLPF
2) Butterworth High pass filter
The transfer function of Butterworth High Pass filter of order n is given by
the equa tion

3) Gaussian High Pass filter
The transfer function of a Gaussian High Pass Filter is given by the
equation

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Homomorphic Filtering
Homomorphic filters are widely used in image processing for compensating
the effect of no uniform illumination in an image. Pixel intensities in an
image represent the light reflected from the corresponding points in the
objects. As per as image model, image f (x, y) may be characterized by two
components: (1) the amount of source light incident on the scene being
viewed, and (2) the amount of light reflected by the objects in the scene.
These portions of light are called the illumination and reflectance components, and are denoted i (x, y) and r (x , y) respectively. The functions
i (x , y) and r ( x , y) combine multiplicatively to give the image function f (
x , y):
f (x, y) = i (x, y). r (x, y) (1)
where 0 < i (x, y) < a and 0 < r (x, y) < 1. Homomorphic filters are used in
such situations where the image is subjected to the multiplicative interference or noise as depicted in Eq. 1. We cannot easily use the above
product to operate separately on the frequency components of illumination
and reflection because the Fourier transform of f (x, y) is not separable; that
is
F [f (x, y)) not equal to F [i (x, y)]. F [r (x, y)].
We can separate the two components by taking the logarithm of the two sides
ln f (x, y) = ln i (x, y) + ln r (x, y).
Taking Fourier transforms on both sides we get,
F [ln f (x, y)} = F [ln i (x, y)} + F[ln r(x, y)].
that is, F (x, y) = I (x, y) + R (x, y), where F, I and R are the Fourier transforms
ln f (x, y), ln i (x, y ), and ln r (x, y). respectively. The function F represents
the Fourier transform of the sum of two images: a low -frequency illumination
image and a high -frequency reflectance image. If we now apply a filter with
a transfer function that suppresses low- frequency components and enhances
high-frequency components, then we can suppress the illumination component and enhance the reflectance component. Taking the inverse
transform of F (x, y) and then ant i-logarithm, we get
f’ (x, y) = i’

Fig 3.8 Steps in Homomorphic filtering
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53Chapter 3: Filtering in the Frequency Domain
3.9 Selective Filtering
Selective filtering is to process specific bands of frequencies or small regions of
the frequency rectangle. Filters in the first category are called band filters. If
frequencies in the band are filtered out, the band filter is called a bandreject filter;
similarly, if the frequencies are passed, the filter is called a bandpass filter. Filters
in the second category are called notch filters. They are further classified as whether
the frequencies in the notch areas are rejected or passed.
BANDREJECT AND BANDPASS FILTERS
Bandpass and Bandreject filter transfer functions in the frequency domain can be
constructed by combining lowpa ss and highpass filter transfer functions. Lowpass
filter transfer functions are the basis for forming highpass, bandreject, and bandpass
filter functions. A bandpass filter transfer function is obtained from a bandreject
function.
HBP (u, v) = 1 – HBR (u, v)

Fig 3.9 Transfer functions of Bandreject filters
NOTCH FILTERS
Notch filters are the most useful of the selective filters. A notch filter rejects (or
passes) frequencies in a predefined neighbourhood of the frequency rectangle.
Zerophase -shift filters must be symmetric about the origin (center of the frequency
rectang le), so a notch filter transfer function with center at (u 0, v0) must have a
corresponding notch at location ( -u0, - v0).Notch reject filter transfer functions are
constructed as products of highpass filter transfer functions whose centres have
been transl ated to the centres of the notches.
3.10 Fast Fourier Transform Algorithm
Discrete Fourier transform and Inverse Discrete Fourier transform takes a lot of
computations in real-time as learned theoretically. To implement the transform
practically, the Fast Fouri er Transform algorithm is used. Fast Fourier transform
(FFT), reduces computations to the order of MN log2 MN multiplications and
additions.
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3.11 Summary
In this chapter, we learned about the frequency domain, filtering in the frequency
domain , and its eff ective usage in image processing. The use of the fast Fourier
algorithm and its purpose practically.
3.12 Review questions
1. Compare and contrast spatial domain and frequency domain
2. How does filtering in the frequency domain works?
3. What are the basic steps of filtering in the frequency domain?
4. What are smoothing filters? Explain the various types of low pass filters.
5. Explain sharpening in the frequency domain.
6. Write a short note on selective filtering.
7. Explain homomorphic filtering b riefly.
8. What is FFT? Why FFT is used practically and not DFT?
9. Mention the properties of DFT.
3.13 References
1. Digital Image Processing, fourth edition by Gonzalez and Woods.
2. www.Imageprocessingplace.com
3. Refer to nptel courses on digital image processing for further detailed study
on FFT and other related concepts.
4. All Images courtesy from Digital Image Processing by Gonzalez an d Woods,
4th Edition
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55Chapter 4: Image Restoration and Reconstruction
Unit 2
4 IMAGE RESTORATION AND
RECONSTRUCTION
Unit Structure
4.0 Objectives
4.1 Introduction
4.2 Model of the Image degradation/restoration process
4.3 Noise models
4.4 Restoration in Spatial and Noise Reduction in Frequency Domain
4.5 Inverse Filtering
4.6 Minimum Mean Square Error (Wiener) Filtering
4.7 Geometric mean filter
4.8 Image Reconstruction from projections
4.9 Summary
4.10 Review questions
4.11 References
Objectives
1. Understand the characteristics of various noise models used in image
processing.
2. Understand the reduction of noise in the frequency domain.
3. Familiarize with Inverse, Geometric mean filters and understand
reconstruction of images using projections
4.1 Introduction
Restoration attempts to recover an image that has been degraded by using a priori
knowledge of the degradation phenomenon. Thus, r estoration techniques are
oriented toward modelling the degradation and applying the inverse process in
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It deals with the characteristics of various noise models used in image processing,
and how to estimate from image data the parameters that define those models.
The chapter gives an overview of minimum mean -square -error (Wiener) filtering
and its advantages over direct inverse filtering.
Image Restoration refers to a class of methods that aim to remove or reduce the
degradations that have occurred while the digital image was being obtained. All-
natural images when displayed have gone through some sort of degradation:
a) During display mode
b) Acquisition mode.
c) Processing mode
The degradations may be due to
a) Sensor noise
b) Blur due to camera mis focus
c) Relative object -camera motion
d) Random atmospheric turbulence
4.2 A model of the Image Degradation/Restoration process

Fig 4.1 A model of the Image Degradation/Restoration process
The degradation process operates on a degradation function that operates on an
input image with an additive noise term. The input image is represented by using
WKHQRWDWLRQI[\QRLVHWHUPFDQEHUHSUHVHQWHGDVȘ[\ These two terms when
combined give the result as g(x,y). If we are given g(x,y), some knowledge about
the degradation function H or J and some knowledge about the additive noise
WHHPȘ[\WKH objective of restoration is to obtain an estimate f'(x,y) of the
original image. We want the estimate to be as close as possible to the original image.
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57Chapter 4: Image Restoration and Reconstruction
7KHPRUHZHNQRZDERXWKDQGȘWKH closer f(x,y) will be to f'(x,y).
If it is a linear position invariant process, then degraded image is given in the spatial
domai n by
J[\ I[\ K[\Ș[\
h(x,y) is spatial representation of degradation function and symbol * represents
convolution. In frequency domain we may write this equation as
G (u, v) =F (u, v) H (u, v) +N (u, v)
The terms in the capital letters are the Fourier Transform of the corresponding
terms in the spatial domain.
4.3 Noise models
The principal sources of noise in digital images arise during image acquisition
and/or transmission.
For example, in acquiring images with a CCD camera, light levels and sensor
temperature are major factors affecting the amount of noise in the resulting image. Images are corrupted during transmission principally by interference in the transmission channel. For example, an image transmitted using a wireless netwo rk
might be corrupted by lightning or other atmospheric disturbance.
Spatial and Frequency properties of Noise
1. When the Fourier spectrum of noise is constant, the noise is called white
noise.
2. Noise is independent of spatial coordinates. Noise is uncorrela ted with
respect to the image itself
Common Noise found in Image Processing applications
1. Gaussian Noise
These noise models are used frequently in practices because of its tractability
in both spatial and frequency domain.
The PDF of Gaussian random variable, z, is given by

where, z represents intensity, z(bar) is the mean (average) value of z and ߪ is
the standard deviation.


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2. Rayleigh Noise
The PDF of Rayleigh noise is given by

The mean and variance of z when this random variable is characterized by a
Rayleigh PDF

3. Erlang (gamma) noise
The PDF of Erlang noise is given by

where the parameters are such that a > b, b is a positive integer, and “!”
indicates factorial. The mean and variance of z are

4. Exponential Noise
The PDF of exponential noise is given by

where a > 0. The mean and variance of z are


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59Chapter 4: Image Restoration and Reconstruction
5. Uniform Noise
The PDF of uniform noise is

The mean and variance of z are

6. Salt-and-Pepper Noise
The PDF of salt -and-pepper noise is given by

where V is any integer value in the range 0 < V < 2k – 1
Fig 4.2 Noise Models with their probability density functions
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4.4 Restoration in Spatial and Noise Reduction in Frequency
Domain
When an image is degraded only by additive noise, the equation becomes
J[\ I[\Ș[\
or
G (u, v) = F (u, v) + N (u, v)
The noise terms are unknown so subtracting them from g (x, y) or G (u, v) is not a
realistic approach. In the case of periodic noise it is possible to estimate N(u,v) from
the spectrum G(u, v). So, N (u, v) can be subtracted from G (u, v) to obtain an
estimate of original image. Spatial filtering can be done when only additive noise
is present.
The following techniques can be used to reduce the noise effect
Mean Filters
1) Arithmetic Mean Filters
It is the simplest mean filter. Let S xy represents the set of coordinates in the
sub image of size m*n centered at point (x,y). The arithmetic mean filter
computes the average value of the corrupted image g(x,y ) in the area defined
by S xy. The value of the restored image f at any point (x,y) is the arithmetic
mean computed using the pixels in the region defined by Sxy

This operation can be using a convolution mask in which all coefficients
have value 1/mn
A mean filter smoothes local variations in image Noise is reduced as a result
of blurring.
2) Geometric mean filters
An image restored using a geometric mean filter is given by the expression

where indicates multiplication. Here, each restored pixel is given by the
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61Chapter 4: Image Restoration and Reconstruction
product of all the pixels in the subimage area, raised to the power 1/ mn
A geometric mean filter achieves smoothing comparable to an arithmetic
mean filter, but it tends to lose le ss image detail in the process.
3) Harmonic mean filter
The harmonic mean filter works well for salt noise, but fails for pepper noise.
It does well also with other types of noise like Gaussian noise.
The harmonic mean filtering operation is give n by the expression

4) Contraharmonic mean filter
The Contraharmonic mean filter yields a restored image based on the
expression

where Q is called the order of the filter. This filter is well suited for reducing
or virtually eliminating the effects of salt -and-pepper noise. For positive
values of Q, the filter eliminates pepper noise. For negative values of Q, it
eliminates salt noise. It cannot do both simultaneously.
The Contraharmonic filter reduces to the arithmetic mean filter if Q = 0, and
WRWKHKDUPRQLFPHDQILOWHULI4 í
ORDER STATISTICS FILTERS
Order statistics filters are spatial filters whose response is based on ordering the
pixel contained in the image area encompassed by the filter. The response of the
filter at any point is determined by the ranking result.
1) Median Filters
These filters replace the value of a pixel by the median of the intensity levels
in a predefined neighbourhood of that pixel

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where, as before, Sxy is a sub-image (neighb ourhood) centered on point (x,
y). The value of the pixel at (x, y) is included in the computation of the
median. Median filters provide excellent noise -reduction capabilities, with
considerably less blurring than linear smoothing filters of similar size.
Median filters are particularly effective in the presence of both bipolar and
unipolar impulse noise.
2) Max and Min filters
Using the l00th percentile of a ranked set of numbers is called the max filter
and is given by the equation

This filter is useful for finding the brightest points in an image or for eroding
dark regions adjacent to bright areas
The 0th percentile filter is min filter
This filter is useful for flinging the darkest point in the image. Also, it
reduces salt noise of the min o peration.
3. Mid-point filter
The midpoint filter simply computes the midpoint between the maximum
and minimum values in the area encompassed by the filter

It combines the order statistics and averaging filter. This filter works best
for randomly distributed noise like Gaussian or uniform noise.
4. Alpha -Trimmed mean filter
A filter formed by averaging these remaining pixels is called an alpha -
trimmed mean filter. The form of this filter is given by

ZKHUHWKHYDOXHRIGFDQUDQJHIURPWRPQí:KHQG WKHDOSKD -
trimmed filter reduces to the arithmetic mean filter. ,IZHFKRRVHGPQ í
the filter becomes a median filter.
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Periodic Noise Reduction using frequency domain filtering
The basic idea is that periodic noise appears as concentrated bursts of energy in the
Fourier transform, at locations corresponding to t he frequencies of the periodic
interference. The approach is to use a selective filter to isolate the noise. In
restoration of images corrupted by periodic interference, the tool of choice is a
notch filter.
The general form of a notch filter transfer func tion is

A notch pass filter transfer function is obtained from a notch reject function using
the expression

4.5 Inverse Filtering
It is a process of restoring an image degraded by a degradation function H. This
function can be obtained by any method. The simplest approach to restoration is
direct, inverse filtering.
Inverse filtering provides an estimate F(u,v) of the transform of the original image
simply by the transform of the degraded image G(u,v) by the degradation function.

It shows an interesting result that even if we know the depredation function we
cannot recover the underrated image exactly because N(u,v) is not known. If the
degradation value has zero or very small values then the ratio N(u,v)/H(u,v) could
easily dominat e the estimate F(u,v).
4.6 Minimum Mean Square Error (Wiener) Filtering
Wiener Filtering is an approach that incorporates both the degradation function and
statistical characteristics of noise into the restoration process. The method is
founded on considering images and noise as random variables, and the objective is
to find a n estimate ˆ f of the uncorrupted image f such that the mean square error
between them is minimized. This error measure is defined as

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The minimum error function of the above expression is given by the expression

The product of a complex quantity with its conjugate is equal to the magnitude of
the complex quantity squared. This result is known as the Wiener filter
The terms used are as follows

4.7 Geometric Mean filter
The generalized form of Wiener filter is the geometric mean filt er.

where ઺ and ߚ are nonnegative, real constants. The geometric mean filter transfer
function consists of the two expressions in brackets raised to the powers ઺ DQGí
઺, respectively.
When ઺ = 1 the geometric mean filter reduces to the inverse filter . With a = 0 the
filter becomes the so -called parametric Wiener filter, which reduces to the “standard” Wiener filter when ߚ =1
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65Chapter 4: Image Restoration and Reconstruction
4.8 Image Reconstruction from projections
The basic principles of projections apply to other CT imaging modalities, such as SPECT (single photon emission tomography), PET (positron emission tomography), MRI (magnetic resonance imaging), and some modalities of ultrasound imaging. One can use projec tions with Radon transform or fan-beam projections or slice theorem for understanding image reconstruction from projections. A filter function can be used for image reconstruction. This approach is appropriately called filtered back projection. In practice , the data are discrete, so
all frequency -domain computations are carried out using a 1 -D FFT algorithm .
4.9 Summary
In this chapter, we learned about the various noise models regarding image
degradation and reconstruction. The different sources of noise. T he purpose of
filters and the working of spatial and frequency -based filters to reconstruct images.
4.10 Review questions
1. Explain the model of image degradation and reconstruction with a diagram?
2. What are the different sources of noise?
3. Explain the various noise models.
4. Describe the various spatial filters/frequency -based filters used to reconstruct
images.
5. Write a short note on Wiener Filtering.
4.11 References
1. Digital Image Processing, fourth edition by Gonzalez and Woods.
2. www.Imageprocessingplace.com
3. Refer to nptel courses on digital image processing for a further detailed study
on FFT and other related concepts
4. All Images courtesy from Digital Image Processing by Gonzalez and Woods,
4th Edition
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Unit 3
5 WAVELET AND OTHER IMAGE TRANSFORMS
Unit Structure
5.0 Objective
5.1 Preliminaries
5.2 Matrix -based transforms
5.3 Correlation
5.4 Basis Functions in the Time -Frequency Plane
5.5 Basis Images
5.6 Fourier -Related Transforms
5.7 Walsh -Hadamard Transforms
5.8 Slant Transform
5.9 Haar Transform
5.10 Wavelet Transforms
5.11 Color Fundamentals
5.12 Color Models
5.13 Pseudocolor Image Processing
5.14 Full-Color Image Processing
5.15 Color Transformations
5.16 Color Imag e Smoothing and Sharpening
5.17 Using Color in Image Segmentation
5.18 Noise in Color Images
5.19 Color Image Compression
5.20 Summary
5.21 Unit End questions
5.22 Reference for further reading
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67Chapter 5: Wavelet and Other Image Transforms
5.0 Objective
In image compression, the wavelet transformation plays an extremely important
role. Wavelet transformation is a more appropriate technique than Fourier for
image compression applications. Fourier transform is not practical for spectral
computing information as all the previous and future i nformation on the signal are
essential throughout the entire of the time domain and frequencies cannot be
observed varying in time, as the resulting function is time -independent. The Fourier
discrete transform is member of an essential class of linear tran sformations,
including Hartley, sine, cosine, Walsh -Hadamard, Slant, Haar and the wavelet
transform. Such transformations, decompose tasks into weighted amounts of
orthogonal or biorthogonal functions that can be studied with the use of linear
algebra tool s and functional analysis tools.
5.1 Preliminaries
In Linear Algebra, A vector space (called as linear space) is a group of objects that
can be added and multiplied ('scaled') by numbers called scalars. Scalars are
sometimes taken by real numbers, but there are vector space with scalar
multiplication by complex numbers, rational numbers or usually any field also
exists. Vector addition and scalar multiplication operations must fulfil some
requirements, known as vector axioms. An inner product is a widespread of the dot
product. In a vector space, it is a method of multiplying vectors together, the result
being a scalar one.
An inner product <·,·> fulfils four characteristics for a real vector space. Let u, v
and w be scalar vectors, an GĮEHDVFDODUWKHQ
1. =+.
DOSKDYZ! ĮYZ!
3. =.
4. >=0 and equal if and only if v=0.
It's known as the positive -definite fourth condition in the above list. Notice that
certain authors define an inner product coul d be a function <·,·> that satisfies only
the third of the above terms with a (weaker) non -degenerated (i.e., when = 0 for all
w, then v=0) addition of the (weaker) condition. Functions fulfilling all four criteria
are usually referred to in such literatur e as positive -definite interior products, while
internal products that are not defined positively are often referred to as uncertain.
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68IMAGE PROCESSING
that are not positive -definite, for exampl e, can create 'standards' under which those
vectors (such vectors are called spacelike) have imaginary magnitude and cause
'metrics' that are not actual metrics. The inner product of Lorentzia is an example
of an indefinite inner product.
An inner product space is called a vector space with an inner product this
description also refers in any field to an abstract vector space.
1. The real numbers of the inner product
=xy.
Euclidean space RN is an infinite set of all real N -tuples.
eq -1
Where u and v are column vectors
2. Unit space through complex C number with inner product function. Inner
product spaces over the field of complex numbers are sometimes referred to
as unitary spaces.
eq -2
where * indicates the complex operation of the conjugate, and u and v are
complex values column vector
3. The vector space C[a; b] of all real -valued continuous functions on a closed
interval [a; b] is an inner product spa ce, whose inner product is defined by
eq -3
4. The norm or length of vector z in all three inner product spaces is as specified
eq -4

Equations through (5 to 15) are valid for all inner product spaces, including
those defined by Eqs. (3) to (4) a nd the angle between two nonzero vectors z
and w is
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69Chapter 5: Wavelet and Other Image Transforms
5. Although the meaning must always be considered, the term 'vector' is usually
used for abstract vectors. A matrix (e.g., column vector) or a continuous function may be a vector .
eq-5
6. If the z norm is 1, z is considered to have been normal ized. If it is said that
Eq-5, z and w are orthogonal. A natural resu lt of these definitions is that a
number of non -zero vectors are jointly or equally orthogonal, provided it only
eq-6
7. They constitute an orthogonal basis for the inner product space .They are an
orthonormal basis, when the basic vectors are normatively
eq-7
8. Similarly, a set of w0, W1, W2, ... vectors and an additional set of
dual vectors is said to be biorthogonal and biorthogonal basis vector space
eq-8
9. They are a biorthonormal basis if and only if
eq-9
10. As a method for coherently representing an infinite set of vectors, the
foundation of inner product space is among the most important principles in
linear algebra. The subsequent derivation that depends on the orthogonally
of the basis vectors is fundamen tal to the matrix -based transformations of the
next section. Let W = {} 0 1 {}, w 2,... Be the orthogonal basis of the inner
product area V, and let z V be supported. The vector z can then be expressed
as the following linear combination of base vectors.
eq-10


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11. whose inner product with basis vector w i is
eq-11
12. Since they are mutually orthogonal the inner product s on the right side of Eq.
(11) are 0 unless the vector subscriptions for which the inner products are
computed match [see Eq. ( 7)]. Thus, the only nonzero word is to exclude zero
terms and to divide the two sides of the equation by giving
eq-12
13. Which reduces to
eq-13
14. If the basic vector norms are 1
eq-14
15. And
eq-15
5.2 Matrix -Based Transforms
The 1 -D discrete Fourier transform is one of a class of significant transformations
that can be represented in a general relation. The word transform is used in
mathematics to describe a change in form without any corresponding change in
value.
eq-16
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Where x is a spatial variable, T(u) is the transform of f(x), r(x,u) is a forward
transformation kernel, and integer u is a transform variable with values in the range
Similarly, the inverse transform of T(u) is
eq-17
If s(x,u) is a inverse transformation kernel and x is a transformation kernel for the
range r(x,u), and s(x,u) in Eqs.(16) and (17) that only depends on x and u and not
on f(x) or T(u) values, then the existence of the transform pair that is defined will
be determined.
Graphical equat ions ( 17) are shown in Fig.5. 1. It should be taken into consideration
that the f(x) is a weighted total of N inverse (e.g. s(x,u) for) kernel functions and
that the T(u) for weight . Each N s(x,u) contributes f(x) to each x. Expanding the
right side of Eq. (17)

FIGURE 5.1
(Reference from "Digital Image Processing fourth edition by
Rafael C. Gonzalez and Richard E. Woods)

A graphical illustration of Eq. (18)
eq-18
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It is readily evident which a linear expan sion is represented as Eq. (11 ), with Eq.
(18) s(x,u), T(u) in place (e.g. th e basis of vectors) and Equ. (11 ). In this case, a
linear expansion is immediately apparent. When we suppose s(x,u) is orthonormal
vectors of a product's inner space in Eq. (18), Eq. (13 ) tells us that
eq-19
And T(u) can be calculated for transform through the inner products
We frequently use subscripts to define the matrix or vector elements. This means
the first column vector f element, which is f (0), and the first c olumn of column
vector element s (0, 3).
eq-20

eq-21

And

eq-22


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73Chapter 5: Wavelet and Other Image Transforms
and using them to rewrite Eq. (19) as
eq-23

Combining the N -base vectors in the transform matrix
eq-24
Then Eq. (23) is replaced into Eq. (21) and EQ. (1) can be used to achieve
By using Eq. (1), we assume that real -value vector is the most common case. For
complex inner product space, equation (2) must be used.
eq-25
Or
eq-26

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eq-27
Where Eqs. (1) and (7) respectively follow the two las t steps. Since AAT=I it gives
Eq (26) pre -multiplication by AT and simplification gives f=ATt Eqs. (16) and ( 17)
thus become the matrix -based pair of transformations.
eq-28

And
eq-29
It is essential to understand that we assumed that the N transform basis vectors (i.e.
s u for u =0, 1… .N-1) of the Transformation matrix A were true and orthonormal in
the derivations of Eqs (28) and (29). In compliance with Eq (7)
eq-30
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75Chapter 5: Wavelet and Other Image Transforms
The assumed orthonormality makes it possible to calculate forward t ransforms
without explicit references to a kernel of forward transformation, i.e. t=Af , when
A is a function acts in the reverse transformation kernel s(x,u).
Because of the reality that the base vector A is real and orthonormal, the transform
is termed the orthogo nal transform defined in Eq. ( 28). It preservatives the inner
products is i.e., ۃۃf1,f2 ۄۄ =ۃۃt1,t2 ۄۄ =ۃۃA f1,A f 2 ۄۄ and therefore, the distances and
angles pre and post transformation among vectors. The rows and columns of A are
orthonormal AAT = ATA = I, so Aí1 = AT. The outcome in the Eqs. (28) And Eq
(29) are a reversible transfor m pair . Substituted by Eq. ( 29) with ( 28) produces t=
Af= AATt=t, while replacing Eq. (28) with ( 29), f= ATt=ATAF=f gives.
eq-31

And
eq-32
Where the inverse transformation kernels r(x,y,u,v) and s( x,y,u,v) respectively are forward . Transform T(u,v) and invert kernel transformation s(x,y,u,v), respectively, with eq. (32) way to set a linear expansions to f(x,y), are regarded as
another weighting coefficients and basis vectors . Forward transformatio n kernel
r(x,y,u,v) is separable if
eq-33
and symmetric if r 1is functionally equal to r 2so
eq-34
When the transformation kernels are real and orthonormal; both r and s are
distinguishable and symmetrical, then the matrix equivalents for Eqs. (6 -31) and
(6-32) shall be substitute r in Eqs. (33) and r in Eqs. (34), respectively, for
separable, separable symmetrical inverse kernels.
T=AFAT eq-35
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And
F=ATTA eq-36
When F is a matrix of f( x, y), T is a transform and A is as defined in the Equation
(24) Equation. The column and row transformations of F are measured according
to the F pre and post -multiplication by A and Eq. (35). This actually breaks the 2 -
D transformation into two 1 -D trans formations, which mirror the 2 -D DFT process.
EXAMPLE 1 A simple orthogonal transformation.
Consider the 2 -element basis vectors

And note they are orthonormal in accordance with Eq. (30) :

Substitute and transformation matrix in or out of Eq. (6-24).
eq-37
and the transform of matrix

follows from Eq. (35)
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In accordance with Eq. (36) , the inverse of transform T is

In conclusion, we note that A is an orthogonal transformation matrix for which

Rectangular Arrays
When rectangular arrays are transformed, Eqs. (35) and (36) are changed. In
comparison to square .
eq-38
And
eq-39
where F, AM and AN are of size MXN ,MXM,NXN and respectively. Both and are
defined in accordance with Eq. ( 24)
EXAMPLE 2: Computing the transform of a rectangular array.
A simple transformation in which M and N are 2 and 3, respectively, is

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Where matrices F, A2 and A3 in the first step of the computation are defined. As
expected, 2X3 the transforming output T is the same as F
To demonstrate that it is an orthogonal matrix of transformation, and that Eq
reverses transformation (39)
Complex Orthonormal Basis Vectors
An orthonormal basis is the basis of which vectors are orthogonal to each other and
have a unit norm.
Orthonormal bases are essential in applications because a vector is especially
simple to represent on an orthonormal basis, called Fourier expansi on.
We have to be familiar with the concepts of internal product and standard to
understand this concept.
Orthonormal sets
Remember that if the inner product equals zero, two vectors are orthogonal.
Definition Let S have an inner product vector space <...>. A set of K vectors
S1…Sk ȯS, if and only if are an orthonormal set.
eq-40
Where * is the complex operation of the conjugate. If vectors of basis (35) and (36)
are problematic as opposed to real -value,
T=AFAT eq-41
And
F=A*T TA* eq-42
Transformation matrix A is then considered the unitary matrix and the unitary
transformation pair Eq (41) and Eq(42) is equal. The 1 -D counterpart of Eq. (41)
and (42) are thus an essential and useful property of A:
Orthogonal transformations are a unique case wherein the functions of expansion
are real-valued.
Both transforms maintain inner products.
t=Af eq-43
f=A*T t eq-44

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EXAMPLE 3: A transform with complex -valued basis vectors.
The inverse of a unitary transformation matrix, which is the inverse of the
transformation matrix, is the inverse.
eq-45
is its conjugate transpose. Thus,

Where the matrix and where A is a single matrix, which in Eqs (41)
can be used through eq(44).. It can be shown easily if the base vectors in A satisfy
Eq. (40).
Biorthonormal Basis Vectors
Expansion functions S 0,S1…,S N-1 are biorthonormal in Eq. ( 24) if there are dual
expansion functions ~S0, ~S 1…,~S N-1 that
eq-46
Even the expansion functions and dual functions do not need to be orthonormal.
With a set of biorthonormal expansion features, Eqs. (35) and (36) are
transformed
eq-47
And
eq-48
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80IMAGE PROCESSING
Transformation matrix A remains as described by Eq. ( 24); dual transformation
matrix is an NX N matrix .The dual expansion functions
of its rows are transposed. When the function of expansion and its duals are
identical, i.e. , reduces the Equal Eq(47) and Eq(48) respectively to Eq.
(35) and Eq(36) are
eq-49
eq-50
EXAMPLE 4 : A biorthonormal transform.
Consider the real biorthonormal transformation matrices

A and are biorthonormal .The transform of 1 -D column vector f=[30 11 210
6] T is


And =tTt =65,084 which is unequal to the transformation have not
preserve inner products. but, reversible :

The forward and inverse transforms are calculated by using Eqs. (49) and (50) ,
respectively.
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81Chapter 5: Wavelet and Other Image Transforms
Finally, we mention that the majority of the concepts in this area can be used to
extend the form continuously.

The coefficients and base vectors for the expansion of the internal product space
&>DE@E\Į u and S u for u= for given f(x) and basis s u(x) for u=
.The appropriate coefficients of expansion may be calculated by
defining of Integral Internal Product of the C([a, b]) -i.e., Eq's -3 and all internal
product spaces' gener al properties, that is, Eqs. (10) by ( 15). Thus, for e.g. su(x) for
u= . are orthonormal basis vectors of C([a, b])
eq-52
EXAMPLE 5 : The Fourier series and discrete Fourier transform
Consider the representation in linear expanding of orthonormal base vectors of
the form of the continuous periodic function T
eq-53
In accordance with Eqs. (51) and (52)
eq-54

And
eq-55
yields the following discrete counterparts of Eqs. (53) through (55)
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82IMAGE PROCESSING
eq-56
eq-57
And
eq-58
The discrete complex basis vectors of Eq. (56) are an orthonormal basis of inner
product space CN Equations (58) and (59)
Consider, Eqs (55) and (58) , in the calculation of Fourier and the Fourier discrete
transfor m of I[ VLQʌ[ of period of T=1 are now used . In accordance with
Eq. (55)
DQGLQWKHVDPHZD\Į -1= j0.5 hence entire another coefficients are zero, the
resultant Fourier series is
eq-59
5.3 Correlation
The correlation of f and g indicated by two continual functions f(x) and g(x) are
denoted by , is defined as
,QIDFWWKHWHUPFURVVFRUUHODWLRQVKRXOGEHXVHGZKHQI[J[DQGDXWR -
correlation when f(x)=g(x) Equation ( 61) is valid for both cases
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83Chapter 5: Wavelet and Other Image Transforms
eq-61
Here the final step are follows from Eq. (3) ZLWKĮ -’DQGĮ ’6RPHWLPHVUHODWHG
to as the sliding inner product f and g, the correlation tests the similarity of f(x) and
g(x) according to their relative displacement ᶭx . if ᶭx =0.
As the name suggests, sliding inner product, display s lide between functions,
multiply them and compute the area. The functions become increasingly identical
as the area increases.
f յ g(0) = < f(x) ,g(x) > eq-62
And Eq. ( 52) defining continuous orthonormal expa nsion coefficients in Eq. ( 51)
can be written as alternatively
Įu = =f յ su (0) Thus, the expansion coefficients are correlations of one point, with zero displacement ᶭ. The similarity of f(x) and su(x) is measured E\Įx
The discrete equivalents of Eqs. (61) Through (63) are Integers n and m, fn
indicates the nth element in f, and g n+m implies the element (n+m) in g. Equation
(66) is the result of Equation (65) and (23)
eq-64
f յ g(0) =< f,g> eq -65
and
T(u)= =S u յ f (0)
Similar comments on Eq (63) and continuous series expansions and continuing
transforms in Eq (66) may be made. Every orthogonal transformation element and
orthogonal transformations are discrete. Each orthogonal transformation element.
5.4 Basis Functions in the Time -Frequency Plane
As Fig.5.2, which shows the basis vectors of certain commonly encountered
transformations, most orthogonal basis sets of sinusoids, square waves, ramps and
other small waves called wavelets are associated with mathemat ical activity. If h(t)
is a vector base and g(t) is the function transformed, the transform coefficient is a
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