Image Compression Using Wavelet Transform

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A

Project Report on

Image Compression Using Wavelet Transform

In partial fulfillment of the requirements of

Bachelor of Technology (Computer Science &

Engineering)

Submitted By

Saurabh Sharma (Roll No.10506058)

Dewakar Prasad(Roll No.10506060)

Manish Tripathy(Roll No 10406023)

Session: 2008-09

Department of Computer Science &Engineering

National Institute of Technology


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Rourkela-769008

Orissa

AProject Report on

Image Compression Using Wavelet Transform

In partial fulfillment of the requirements ofBachelor of Technology (Computer Science &

Engineering)

Submitted By

Saurabh Sharma (Roll No.10506058)

Dewakar Prasad(Roll no.10506060)

Manish tripathy(Roll No.10406023)

Session: 2008-09

Under the guidance ofProf. Baliar Singh


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Department of Computer Science & Engineering

National Institute of Technology

Rourkela-769008

Orissa

National Institute of TechnologyRourkela

CERTIFICATE

This is to certify that that the work in this thesis report entitled Image compression using

wavelet transform submitted by Saurabh Sharma ,Dewakar Prasad and Manish tripathy in

partial fulfillment of the requirements for the degree of Bachelor of Technology in Computer

Science & Engineering Session 2005-2009 in the department of Computer Science &

Engineering, National Institute of Technology Rourkela, This is an authentic work carried out by

them under my supervision and guidance.

To the best of my knowledge the matter embodied in the thesis has not been submitted to any

other University /Institute for the award of any degree.

Date: Proff. Baliar Singh

Department of computer science & engineering

National Institute of Technology, Rourkela


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ACKNOWLEDGEMENT

I owe a debt of deepest gratitude to my thesis supervisor, Dr.Baliar Singh, Professor,

Department of Computer Science & Engineering, for his guidance, support, motivation and

encouragement through out the period this work was carried out. His readiness for consultation

at all times, his educative comments, his concern and assistance even with practical things have

been invaluable.

I am grateful to Prof. B Majhi, Head of the Department, Computer Science of

Engineering for providing us the necessary opportunities for the completion of our project. I also

thank the other staff members of my department for their invaluable help and guidance.

Saurabh Sharma(10506058)

Dewakar Prasad(10506060)

Manish tripathy(10406023)

B.Tech. Final Yr. CSE.


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CONTENTS

Chapter Page No.

Certificate 3

Acknowledgement 4

Abstract 7

Chapter 1 Literature Review 8-12

1.1 Introduction 9

1.2 Why compression is needed 10

1.3 Fundamental of image compression technique 11

1.4 Objective 12

1.5 Organization of Report 12

Chapter 2 Image compression methodology 13-25

2.1 Overview 14

2.2 Different types of transform used for coding 15

2.3 Entropy coding 24

Chapter 3 Wavelet Transform 23-47

3.1 Overview 23

3.2 What are basis function 26

3.3 Fourier Analysis 30

3.4 Similarities between Fourier and wavelet transform 32

3.5 Dissimilarities between Fourier and wavelet transform 33

3.6 List of Wavelet transform 34

Chapter 4 Results and Dicussion 48-58

4.2 Results 494.3 Conclusion 58

NOMENCLATURE 59

REFERENCES 60-61


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ABSTRACT

Abstract: - Data compression which can be lossy or lossless is requiredto decrease the storage requirement and better data transfer rate. One of

the best image compression techniques is using wavelet transform. It is

comparatively new and has many advantages over others.

Wavelet transform uses a large variety of wavelets for decomposition of

images. The state of the art coding techniques like EZW, SPIHT (set

partitioning in hierarchical trees) and EBCOT(embedded block coding

with optimized truncation)use the wavelet transform as basic and

common step for their own further technical advantages. The wavelet

transform results therefore have the importance which is dependent on

the type of wavelet used .In our project we have used different wavelets

to perform the transform of a test image and the results have been

discussed and analyzed. The analysis has been carried out in terms of

PSNR (peak signal to noise ratio) obtained and time taken for

decomposition and reconstruction.


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CHAPTER1

LITURATURE REVIEW


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CHAPTER 1:LITURATURE REVIEW

1.1 INTRODUCTION

Uncompressed multimedia (graphics, audio and video) data

requires considerable storage capacity and transmission bandwidth.

Despite rapid progress in mass-storage density, processor speeds, and

digital communication system performance, demand for data storage

capacity and data-transmission bandwidth continues to outstrip the

capabilities of available technologies. The recent growth of data

intensive multimedia-based web applications have not only sustained theneed for more efficient ways to encode signals and images but have

made compression of such signals central to storage and communication

technology.

To enable Modern High Bandwidth required in wireless data services

such as mobile multimedia, email, mobile, internet access, mobile

commerce, mobile data sensing in sensor networks, Home and Medical

Monitoring Services and Mobile Conferencing, there is a growingdemand for rich Content Cellular Data Communication, including

Voice, Text, Image and Video.

One of the major challenges in enabling mobile multimedia data services

will be the need to process and wirelessly transmit very large volume of

this rich content data. This will impose severe demands on the battery

resources of multimedia mobile appliances as well as the bandwidth of

the wireless network. While significant improvements in achievable

bandwidth are expected with future wireless access technology,

improvements in battery technology will lag the rapidly growing energy

requirements of the future wireless data services. One approach to

mitigate this problem is to reduce the volume of multimedia data

transmitted over the wireless channel via data compression technique

such as JPEG, JPEG2000 and MPEG . These approaches concentrate on

achieving higher compression ratio without sacrificing the quality of the


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Image. However these Multimedia data Compression Technique ignore

the energy consumption during the compression and RF transmission.

Here one more factor, which is not considered, is the processing power

requirement at both the ends i.e. at the Server/Mobile to Mobile/Server.

Thus in this paper we have considered all of these parameters like theprocessing power required in the mobile handset which is limited and

also the processing time considerations at the server/mobile ends which

will handle all the loads.

Since images will constitute a large part of future wireless data, we focus

in this paper on developing energy efficient, computing efficient and

adaptive image compression and communication techniques. Based on a

popular image compression algorithm, namely, wavelet image

compression, we present an Implementation of Advanced ImageCompression Algorithm Using Wavelet Transform.

1.2 Why Compression is needed?

In the last decade, there has been a lot of technological transformation in

the way we communicate. This transformation includes the ever present,

ever growing internet, the explosive development in mobilecommunication and ever increasing importance of video

communication.

Data Compression is one of the technologies for each of the aspect of

this multimedia revolution. Cellular phones would not be able to provide

communication with increasing clarity without data compression. Data

compression is art and science of representing information in compact

form.

Despite rapid progress in mass-storage density, processor speeds, and

digital communication system performance, demand for data storage

capacity and data-transmission bandwidth continues to outstrip the

capabilities of available technologies. In a distributed environment large

image files remain a major bottleneck within systems.

Image Compression is an important component of the solutions available

for creating image file sizes of manageable and transmittable


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dimensions. Platform portability and performance are important in the

selection of the compression/decompression technique to be employed.

Four Stage model of Data Compression

Almost all data compression systems can be viewed as comprising four

successive stages of data processing arranged as a processing pipeline

(though some stages will often be combined with a neighboring stage,

performed "off-line," or otherwise made rudimentary).

The four stages are

(A) Preliminary pre-processing steps.

(B) Organization by context.(C) Probability estimation.

(D) Length-reducing code.

The ubiquitous compression pipeline (A-B-C-D) is what is of interest.

With (A) we mean various pre-processing steps that may be appropriate

before the final compression engine


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Lossy compression often follows the same pattern as lossless, but with

one or more quantization steps somewhere in (A). Sometimes clever

designers may defer the loss until suggested by statistics detected in (C);

an example of this would be modern zero tree image coding.

(B) Organization by context often means data reordering, for

which a simple but good example is JPEG's "Zigzag" ordering.

The purpose of this step is to improve the estimates found by the

next step.

(C) A probability estimate (or its heuristic equivalent) is formed

for each token to be encoded. Often the estimation formula will

depend on context found by (B) with separate 'bins' of state

variables maintained for each conditioned class.

(D) Finally, based on its estimated probability, each compressed file

token is represented as bits in the compressed file. Ideally, a 12.5%-

probable token should be encoded with three bits, but details become

complicated

Principle behind Image Compression

Images have considerably higher storage requirement than text; Audio

and Video Data require more demanding properties for data storage. An

image stored in an uncompressed file format, such as the popular BMP

format, can be huge. An image with a pixel resolution of 640 by 480

pixels and 24-bit colour resolution will take up 640 * 480 * 24/8 =

921,600 bytes in an uncompressed format.

The huge amount of storage space is not only the consideration but also

the data transmission rates for communication of continuous media are

also significantly large. An image, 1024 pixel x 1024 pixel x 24 bit,without compression, would require 3 MB of storage and 7 minutes for

transmission, utilizing a high speed, 64 Kbits /s, ISDN line.

Image data compression becomes still more important because of the

fact that the transfer of uncompressed graphical data requires far more

bandwidth and data transfer rate. For example, throughput in a


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multimedia system can be as high as 140 Mbits/s, which must be

transferred between systems. This kind of data transfer rate is not

realizable with todays technology, or in near the future with reasonably

priced hardware.

1.3 Fundamentals of Image Compression Techniques

A digital image, or "bitmap", consists of a grid of dots, or "pixels", with

each pixel defined by a numeric value that gives its colour. The term

data compression refers to the process of reducing the amount of datarequired to represent a given quantity of information. Now, a particular

piece of information may contain some portion which is not important

and can be comfortably removed. All such data is referred as Redundant

Data. Data redundancy is a central issue in digital image compression.

Image compression research aims at reducing the number of bits needed

to represent an image by removing the spatial and spectral redundancies

as much as possible.

A common characteristic of most images is that the neighboring pixels

are correlated and therefore contain redundant information. The

foremost task then is to find less correlated representation of the image.

In general, three types of redundancy can be identified:

1. Coding Redundancy

2. Inter Pixel Redundancy

3.PsychovisualRedundancy

Coding Redundancy

If the gray levels of an image are coded in a way that uses more

code symbols than absolutely necessary to represent each gray level, the

resulting image is said to contain coding redundancy. It is almost always

present when an images gray levels are represented with a straight or

natural binary code. Let us assume that a random variable rK

lying in the


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interval [0, 1] represents the gray levels of an image and that each rK

occurs with probability Pr(r

K).

Pr(r

K) = N

k/ n where k = 0, 1, 2 L-1

L = No. of gray levels.N

k=No. of times that gray appears in that image

N = Total no. of pixels in the image

If no. of bits used to represent each value of rK

is l (rK), the

average no. of bits required to represent each pixel is

Lavg

= l (rK) P

r(r

K)

That is average length of code words assigned to the various gray

levels is found by summing the product of the no. of bits used to

represent each gray level and the probability that the gray level occurs.Thus the total no. of bits required to code an MN image is MN L

avg.

Inter Pixel Redundancy

The Information of any given pixel can be reasonably predictedfrom the value of its neighbouring pixel. The information carried by an

individual pixel is relatively small.

In order to reduce the inter pixel redundancies in an image, the 2-D

pixel array normally used for viewing and interpretation must be

transformed into a more efficient but usually non visual format. For

example, the differences between adjacent pixels can be used to

represent an image. These types of transformations are referred as

mappings. They are called reversible if the original image elements canbe reconstructed from the transformed data set.


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Psycho visual Redundancy

Certain information simply has less relative importance than otherinformation in normal visual processing. This information is said to be

Psycho visually redundant, it can be eliminated without significantly

impairing the quality of image perception.

In general, an observer searches for distinguishing features such as

edges or textual regions and mentally combines them in recognizable

groupings. The brain then correlates these groupings with prior

knowledge in order to complete the image interpretation process.

The elimination of psycho visually redundant data

results in loss of quantitative information; it is commonly referred as

quantization. As this is an irreversible process i.e. visual information is

lost, thus it results in Lossy Data Compression. An image reconstructed

following Lossy compression contains degradation relative to the

original. Often this is because the compression scheme completely

discards redundant information.


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Image Compression Model

As figure shows a compression system consists of two distinct structural

blocks: an encoder and a decoder. An input image f(x, y) is fed into the

encoder, which creates a set of symbols from the input data.

Image Compression Techniques

There are basically two methods of Image Compression:

1. Lossless Coding Techniques

2. Lossy Coding Techniques

Lossless Coding Techniques:In Lossless Compression schemes, the reconstructed image, after

compression, is numerically identical to the original image. However

Lossless Compression can achieve a modest amount of Compression.

Lossless coding guaranties that the decompressed image is absolutely

identical to the image before compression. This is an important

requirement for some application domains, e.g. Medical Imaging, where

not only high quality is in the demand, but unaltered archiving is a legal

requirement. Lossless techniques can also be used for the compression

of other data types where loss of information is not acceptable, e.g. text

documents and program executables. Lossless compression algorithms

can be used to squeeze down images and then restore them again for

viewing completely unchanged.

Lossless Coding Techniques are as follows: Source Encoder Input

Image F(x, y)

1. Run Length Encoding

2. Huffman Encoding3. Entropy Encoding

4. Area Encoding


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Lossy Coding Techniques:

Lossy techniques cause image quality degradation in each

Compression / De-compression step. Careful consideration of the

Human Visual perception ensures that the degradation is oftenunrecognizable, though this depends on the selected compression ratio.

An image reconstructed following Lossy compression contains

degradation relative to the original. Often this is because the

compression schemes are capable of achieving much higher

compression. Under normal viewing conditions, no visible loss is

perceived (visually Lossless).

Lossy Image Coding Techniques normally have three Components:

1. Image Modeling:It is aimed at the exploitation of statistical characteristics of the

image (i.e. high correlation, redundancy). It defines such things as

the transformation to be applied to the Image.

2. Parameter Quantization:

The aim of Quantization is to reduce the amount of data used to

represent the information within the new domain.

3. Encoding:

Here a code is generated by associating appropriate code words to

the raw produced by the Quantizer. Encoding is usually error free. It

optimizes the representation of the information and may introduce

some error detection codes.

Measurement of Image Quality

The design of an imaging system should begin with an analysis of the

physical characteristics of the originals and the means through which the

images may be generated. For example, one might examine a

representative sample of the originals and determine the level of detail


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that must be preserved, the depth of field that must be captured, whether

they can be placed on a glass platen or require a custom book-edge

scanner, whether they can tolerate exposure to high light intensity, and

whether specular reflections must be captured or minimized. A detailed

examination of some of the originals, perhaps with a magnifier ormicroscope, may be necessary to determine the level of detail within the

original that might be meaningful for a researcher or scholar. For

example, in drawings or paintings it may be important to preserve

stippling or other techniques characteristic


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CHAPTER 1:LITURATURE REVIEW

1.4 OBJECTIVE

The objective of this project is to compress an

image using haar wavelet transform.


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CHAPTER2

Image Compression Methodology


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CHAPTER 2:Image compression methodology

2.1 Overview

The storage requirements for the video of a typical

Angiogram procedure is of the order of several hundred Mbytes

*Transmission of this data over a low bandwidth network results in very

high latency* Lossless compression methods can achieve compression ratios of ~2:1

* We consider lossy techniques operating at much higher compression

ratios (~10:1)

* Key issues:

- High quality reconstruction required

- Angiogram data contains considerable high-frequency spatial texture

* Proposed method applies a texture-modelling scheme to the high-

frequency texture of some regions of the image

* This allows more bandwidth allocation to important areas of the image

2.2 Different types of Transforms used for coding are:

1. FT (Fourier Transform)


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2. DCT (Discrete Cosine Transform)

3. DWT (Discrete Wavelet Transform)

2.2.2 The Discrete Cosine Transform (DCT):

The discrete cosine transform (DCT) helps separate the image into parts

(or spectral sub-bands) of differing importance (with respect to the

image's visual quality). The DCT is similar to the discrete Fourier

transform: it transforms a signal or image from the spatial domain to the

frequency domain.

2.2.3 Discrete Wavelet Transform (DWT):

The discrete wavelet transform (DWT) refers to wavelet transforms for

which the wavelets are discretely sampled. A transform which localizes

a function both in space and scaling and has some desirable properties

compared to the Fourier transform. The transform is based on a wavelet

matrix, which can be computed more quickly than the analogous Fourier

matrix. Most notably, the discrete wavelet transform is used for signal

coding, where the properties of the transform are exploited to represent adiscrete signal in a more redundant form, often as a preconditioning for

data compression. The discrete wavelet transform has a huge number of

applications in Science, Engineering, Mathematics and Computer

Science.


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Wavelet compression is a form of data compression well suited for

image compression (sometimes also video compression and audio

compression). The goal is to store image data in as little space as

possible in a file. A certain loss of quality is accepted (lossy

compression).Using a wavelet transform, the wavelet compression methods are better

at representing transients, such as percussion sounds in audio, or high-

frequency components in two-dimensional images, for example an

image of stars on a night sky. This means that the transient elements of a

data.

signal can be represented by a smaller amount of information than wouldbe the case if some other transform, such as the more widespread

discrete cosine transform, had been used.

First a wavelet transform is applied. This produces as many coefficients

as there are pixels in the image (i.e.: there is no compression yet since it

is only a transform). These coefficients can then be compressed more

easily because the information is statistically concentrated in just a few

coefficients. This principle is called transform coding. After that, the

coefficients are quantized and the quantized values are entropy encoded

and/or run length encoded.

Examples for Wavelet Compressions:

JPEG 2000

Ogg

Tarkin

SPIHT

MrSID


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2.3 Quantization:

Quantization involved in image processing. Quantization techniques

generally compress by compressing a range of values to a single

quantum value. By reducing the number of discrete symbols in a givenstream, the stream becomes more compressible. For example seeking to

reduce the number of colors required to represent an image. Another

widely used exampleDCT data quantization in JPEG and DWT data

quantization in JPEG 2000.

Quantization in image compression

The human eye is fairly good at seeing small differences in brightness

over a relatively large area, but not so good at distinguishing the exact

strength of a high frequency brightness variation. This fact allows one to

get away with greatly reducing the amount of information in the high

frequency components. This is done by simply dividing each component

in the frequency domain by a constant for that component, and then

rounding to the nearest integer. This is the main lossy operation in the

whole process. As a result of this, it is typically the case that many of thehigher frequency components are rounded to zero, and many of the rest

become small positive or negative numbers.

2.3 Entropy Encoding

An entropy encoding is a coding scheme that assigns codes to symbols

so as to match code lengths with the probabilities of the symbols.

Typically, entropy encoders are used to compress data by replacing

symbols represented by equal-length codes with symbols represented by

codes proportional to the negative logarithm of the probability.

Therefore, the most common symbols use the shortest codes.


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According to Shannon's source coding theorem, the optimal code length

for a symbol is logbP, where b is the number of symbols used to make

output codes and P is the probability of the input symbol.

Three of the most common entropy encoding techniques are Huffman

coding, range encoding, and arithmetic coding. If the approximateentropy characteristics of a data stream are known in advance (especially

for signal compression), a simpler static code such as unary coding,

Elias gamma coding, Fibonacci coding, Golomb coding, or Rice coding

may be useful.

There are three main techniques for achieving entropy coding:

Huffman Coding - one of the simplest variable length coding

schemes.

Run-length Coding (RLC) - very useful for binary datacontaining long runs of ones of zeros.

Arithmetic Coding - a relatively new variable length coding

scheme that can combine the best features of Huffman and run-

length coding, and also adapt to data with non-stationary statistics.

We shall concentrate on the Huffman and RLC methods for simplicity.

Interested readers may find out more about Arithmetic Coding inchapters 12 and 13 of the JPEG Book.

First we consider the change in compression performance if simple

Huffman Coding is used to code the subimages of the 4-level Haar

transform.

This is an example DCT coefficient matrix:

A common quantization matrix is:

Using this quantization matrix with the DCT coefficient matrix from

above results in:

For example, using 415 (the DC coefficient) and rounding to the

nearest integer


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Chapter3WAVELET TRANSFORM


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CHAPTER 3:NUMERICAL MODELING

3.1 OVERVIEW

The fundamental idea behind wavelets is to analyze according to

scale. Indeed, some researchers in the wavelet field feel that, by using

wavelets, one is adopting a whole new mindset or perspective in

processing data.

Wavelets are functions that satisfy certain mathematical requirements

and are used in representing data or other functions. This idea is not

new. Approximation using superposition of functions has existed

since the early 1800's, when Joseph Fourier discovered that he could

superpose sines and cosines to represent other functions. However, in

wavelet analysis, the scale that we use to look at data plays a special

role. Wavelet algorithms process data at different scales or

resolutions. If we look at a signal with a large "window," we wouldnotice gross features. Similarly, if we look at a signal with a small

"window," we would notice small features. The result in wavelet

analysis is to see both the forest andthe trees, so to speak.

This makes wavelets interesting and useful. For many decades,

scientists have wanted more appropriate functions than the sines and

cosines which comprise the bases of Fourier analysis, to approximate

choppy signals . By their definition, these functions are non-local (andstretch out to infinity). They therefore do a very poor job in

approximating sharp spikes. But with wavelet analysis, we can use

approximating functions that are contained neatly in finite domains.

Wavelets are well-suited for approximating data with sharp

discontinuities.

The wavelet analysis procedure is to adopt a wavelet prototype

function, called an analyzing waveletor mother wavelet. Temporal

analysis is performed with a contracted, high-frequency version of the


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prototype wavelet, while frequency analysis is performed with a

dilated, low-frequency version of the same wavelet. Because the

original signal or function can be represented in terms of a wavelet

expansion (using coefficients in a linear combination of the wavelet

functions), data operations can be performed using just the

corresponding wavelet coefficients. And if you further choose the best

wavelets adapted to your data, or truncate the coefficients below a threshold,

your data is sparsely represented. This sparse coding makes wavelets

an excellent tool in the field of data compression.

Other applied fields that are making use of wavelets include

astronomy, acoustics, nuclear engineering, sub-band coding, signal

and image processing, neurophysiology, music, magnetic resonanceimaging, speech discrimination, optics, fractals, turbulence,

earthquake-prediction, radar, human vision, and pure mathematics

applications such as solving partial differential equations.

3.2 What are Basis Functions?

It is simpler to explain a basis function if we move out of the realm of

analog (functions) and into the realm of digital (vectors) (*). Every

two-dimensional vector (x,y) is a combination of the vector (1,0) and

(0,1). These two vectors are the basis vectors for (x,y). Why? Notice

thatx multiplied by (1,0) is the vector (x,0), andy multiplied by (0,1)

is the vector (0,y). The sum is (x,y).

The best basis vectors have the valuable extra property that the

vectors are perpendicular, or orthogonal to each other. For the basis

(1,0) and (0,1), this criteria is satisfied.

Now let's go back to the analog world, and see how to relate these

concepts to basis functions. Instead of the vector (x,y), we have a

functionf(x). Imagine thatf(x) is a musical tone, say the noteA in aparticular octave. We can constructA by adding sines and cosines


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using combinations of amplitudes and frequencies. The sines and

cosines are the basis functions in this example, and the elements of

Fourier synthesis. For the sines and cosines chosen, we can set the

additional requirement that they be orthogonal. How? By choosing

the appropriate combination of sine and cosine function terms whose

inner product add up to zero. The particular set of functions that are

orthogonal andthat constructf(x) are our orthogonal basis functions

for this problem.

What are Scale-Varying Basis Functions?

A basis function varies in scale by chopping up the same function or

data space using different scale sizes. For example, imagine we have asignal over the domain from 0 to 1. We can divide the signal with two

step functions that range from 0 to 1/2 and 1/2 to 1. Then we can

divide the original signal again using four step functions from 0 to

1/4, 1/4 to 1/2, 1/2 to 3/4, and 3/4 to 1. And so on. Each set of

representations code the original signal with a particular resolution or

scale.


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3.3 Fourier analysis

FOURIER TRANSFORM

The Fourier transform's utility lies in its ability to analyze a signal inthe time domain for its frequency content. The transform works by

first translating a function in the time domain into a function in the

frequency domain. The signal can then be analyzed for its frequency

content because the Fourier coefficients of the transformed function

represent the contribution of each sine and cosine function at each

frequency. An inverse Fourier transform does just what you'd expect,

transform data from the frequency domain into the time domain.

DISCRETE FOURIER TRANSFORM

The discrete Fourier transform (DFT) estimates the Fourier transform

of a function from a finite number of its sampled points. The sampled

points are supposed to be typical of what the signal looks like at all

other times.

The DFT has symmetry properties almost exactly the same as thecontinuous Fourier transform. In addition, the formula for the inverse

discrete Fourier transform is easily calculated using the one for the

discrete Fourier transform because the two formulas are almost

identical.

WINDOWED FOURIER TRANSFORM

Iff(t) is a nonperiodic signal, the summation of the periodic functions,

sine and cosine, does not accurately represent the signal. You couldartificially extend the signal to make it periodic but it would require

additional continuity at the endpoints. The windowed Fourier

transform (WFT) is one solution to the problem of better representing

the non periodic signal. The WFT can be used to give information

about signals simultaneously in the time domain and in the frequency

domain.

With the WFT, the input signalf(t) is chopped up into sections, and

each section is analyzed for its frequency content separately. If the


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signal has sharp transitions, we window the input data so that the

sections converge to zero at the endpoint. This windowing is

accomplished via a weight function that places less emphasis near the

interval's endpoints than in the middle. The effect of the window is tolocalize the signal in time.

FAST FOURIER TRANSFORM

To approximate a function by samples, and to approximate the

Fourier integral by the discrete Fourier transform, requires applying a

matrix whose order is the number sample points n. Since multiplyingan matrix by a vector costs on the order of arithmetic operations,

the problem gets quickly worse as the number of sample points

increases. However, if the samples are uniformly spaced, then the

Fourier matrix can be factored into a product of just a few sparse

matrices, and the resulting factors can be applied to a vector in a total

of order arithmetic operations. This is the so-calledfast Fourier

transform or FFT.

3.4 SIMILARITIES BETWEEN FOURIER AND WAVELET

TRANSFORM

The fast Fourier transform (FFT) and the discrete wavelet transform

(DWT) are both linear operations that generate a data structure that

contains segments of various lengths, usually filling and

transforming it into a different data vector of length .

The mathematical properties of the matrices involved in the

transforms are similar as well. The inverse transform matrix for both

the FFT and the DWT is the transpose of the original. As a result,

both transforms can be viewed as a rotation in function space to a

different domain. For the FFT, this new domain contains basis

functions that are sines and cosines. For the wavelet transform, this

new domain contains more complicated basis functions called

wavelets, mother wavelets, or analyzing wavelets.


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Both transforms have another similarity. The basis functions are

localized in frequency, making mathematical tools such as power

spectra (how much power is contained in a frequency interval) and

scale grams (to be defined later) useful at picking out frequencies and

calculating power distributions.

3.5 DISSIMILARITIES BETWEEN FOURIER AND

WAVELET TRANSFORM

The most interesting dissimilarity between these two kinds of

transforms is that individual wavelet functions are localized in space.

Fourier sine and cosine functions are not. This localization feature,

along with wavelets' localization of frequency, makes many functions

and operators using wavelets "sparse" when transformed into the

wavelet domain. This sparseness, in turn, results in a number of useful

applications such as data compression, detecting features in images,

and removing noise from time series.


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3.6 LIST OF WAVELET RELATED TRANSFORM

1. Continuous wavelet transform

A continuous wavelet transform is used to divide a continuous-timefunction into wavelets. Unlike Fourier transform, the continuous

wavelet transform possesses the ability to construct a time frequency

represented of a signal that offers very good time and frequency

localization.

2 .Multiresolution analysis

A multiresolution analysis (MRA) or multiscale approximation(MSA) is the design methods of most of the practically relevant

discrete wavelet transform (DWT) and the justification for the

algorithm of the fast Fourier wavelet transform (FWT)

3. Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelettransform (DWT) is any wavelet transform for which the wavelets are

discretely sampled. As with other wavelet transforms, a key

advantage it has over Fourier transforms is temporal resolution: it

captures both frequency andlocation information.

4. Fast wavelet transform

The Fast Wavelet Transform is a mathematical algorithm designed toturn a waveform or signal in the time domain into a sequence of

coefficients based on an orthogonal basis of small finite waves, or

wavelets. The transform can be easily extended to multidimensional

signals, such as images, where the time domain is replaced with the

space domain
http://en.wikipedia.org/wiki/Time-frequency_representationhttp://en.wikipedia.org/wiki/Time-frequency_representationhttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Wavelet_transformhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Orthogonal_basishttp://en.wikipedia.org/wiki/Waveletshttp://en.wikipedia.org/wiki/Waveletshttp://en.wikipedia.org/wiki/Orthogonal_basishttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Wavelet_transformhttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Time-frequency_representation


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3.2 HAAR WAVELET

In mathematics, the Haar wavelet is a certain sequence of functions.

It is now recognised as the first known wavelet.

This sequence was proposed in 1909 by Alfred Haar. Haar used these

functions to give an example of a countable orthonormal system for

the space of square integrable functions on the real line. The study of

wavelets, and even the term "wavelet", did not come until much later.

The Haar wavelet is also the simplest possible wavelet. The technical

disadvantage of the Haar wavelet is that it is not continous , and

therefore not differentiable.

The Haar wavelet's mother wavelet function (t) can be described as

and its scaling function (t) can be described as
http://en.wikipedia.org/wiki/File:Haar_wavelet.svghttp://en.wikipedia.org/wiki/File:Haar_wavelet.svghttp://en.wikipedia.org/wiki/File:Haar_wavelet.svg


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Wavelets are mathematical functions that were developed by

scientists working in several different fields for the purpose of sorting

data by frequency. Translated data can then be sorted at a resolution

which matches its scale. Studying data at different levels allows forthe development of a more complete picture. Both small features and

large features are discernable because they are studied separately.

Unlike the discrete cosine transform, the wavelet transform is not

Fourier-based and therefore wavelets do a better job of handling

discontinuities in data.

The Haar wavelet operates on data by calculating the sums and

differences of adjacent elements. The Haar wavelet operates first on

adjacent horizontal elements and then on adjacent vertical elements.

The Haar transform is computed using:


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CHAPTER4

RESULTS AND DISCUSSION


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CHAPTER 4 : RESULTS AND DISCUSSION

4.1 RESULTS

The image on the left is the original image and the image on

the right is the compressed one

(The point is that the image on the left you are right now

viewing is compressed using Haar wavelet method and the

loss of quality is not visible. Of course, image compression

using Haar Wavelet is one of the simplest ways.)

Original image compressed image


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4.2 CONCLUSION

Haar wavelet transform for image compression is simple and

crudest algorithm.as compared to other algorithms it is more

effective.The quality of compressed image is also maintained


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BIBILOGRAPHY:- [1

[1] Aldroubi, Akram and Unser, Michael (editors), Wavelets inMedicine and Biology, CRC Press, Boca Raton FL, 1996.[2] Benedetto, John J. and Frazier, Michael (editors), Wavelets;Mathematics and Applications, CRC Press, Boca RatonFL, 1996.[3] Brislawn, Christopher M., \Fingerprints go digital," AMS

Notices 42(1995), 1278{1283.[4] Chui, Charles, An Introduction to Wavelets, AcademicPress, San Diego CA, 1992.[5] Daubechies, Ingrid, Ten Lectures on Wavelets, CBMS 61,SIAM Press, Philadelphia PA, 1992.[6] Glassner, Andrew S., Principles of Digital Image Synthesis,

Morgan Kaufmann, San Francisco CA, 1995.


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Image Compression Using Wavelet Transform