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CHAPTER 1

INTRODUCTION

Digital images are widely used in computer applications. Uncompressed

digital images require considerable storage capacity and transmission bandwidth.

Efficient image compression solutions are becoming more critical with the recent

growth of data intensive, multimedia based web applications. Data compression is

the process of converting data files into smaller files for efficiency of storage and

transmission. As one of the enabling technologies of the multimedia revolution,

data compression is a key to rapid progress being made in information technology.

It would not be practical to put images, audio, and video alone on websites without

compression. Data compression algorithms are used in those standards to reduce

the number of bits required to represent an image or a video sequence.

Compression is the process of representing information in a compact form.

Data compression treats information in digital form as binary numbers represented

by bytes of data with very large data sets. Compression is a necessary and

essential method for creating image files with manageable and transmittable sizes.

In order to be useful, a compression algorithm has a corresponding

decompression algorithm that, given the compressed file, reproduces the original

file. There have been many types of compression algorithms developed. These

algorithms fall into two broad types, lossless algorithms and lossy algorithms. A

lossless algorithm reproduces the original exactly. A lossy algorithm, as its name

implies, loses some data. Data loss may be unacceptable in many applications.

Depending on the quality required of the reconstructed image, varying amounts of

loss of information can be accepted. Video and Audio files are very large. There is

need to compress an image to utilize the bandwidth effectively. Compression

becomes part of the representation or coding scheme which have become popular

audio, image and video formats. Compression basically employs redundancy in

the data. Video and audio compression techniques are most suited to this form of

lossy compression. Lossy compression use source encoding techniques that

includes Huffman encoding.

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1.1 BACKGROUND WORK

In certain applications like video conferencing and television, large amount

of images are transmitted through internet. It necessitates an image compression

technique to efficiently utilize the bandwidth. Various lossy compression

techniques using Discrete Wavelet Transform are listed. Ability of these

techniques is dependent on the selection of transforms. The performance is

measured in terms of compression ratio. An extensive survey on various

techniques used for Image compression using DWT are listed.

1.1.1 Review of DWT based image compression techniques

Henry Y.H Chuang et al (1995) implemented VLSI architecture for fast 2D

discrete orthonormal wavelet transform. They presented a parallel pipelined VLSI

array architecture for 2D dyadic separable DWT. The 2D data array is partitioned

into non-overlapping groups of rows. All rows in a partition are processed in

parallel, and consecutive partitions are pipelined. Moreover, multiple wavelet

levels are computed in the same pipeline, and multiple DWT problems can be

pipelined also. The whole computation requires a single scan of the image data

array. Thus, it is suitable for on-line real-time applications. For an N×N image, an

m-level DWT can be computed in N22q+2m−1(2p+3) time units on a processor

costing no more than 83(Cm+Cu)p(p+2q) , where q is the partition size, p is the

length of corresponding 1D DWT filters, C m and C a are the costs of a parallel

multiplier and a parallel adder respectively, and a time unit is the time for a

multiplication and an addition. For q=N ≫ m, the computing time reduces to N2.

They found that when a large number of DWT problems are pipelined, the

computing time is about N22q per problem.

Chu Yu et al (1999) implemented a design of an efficient VLSI architecture

for 2-D Discrete Wavelet Transforms. They presented VLSI architecture for the

separable two-dimensional Discrete Wavelet Transform (DWT) decomposition.

Using a computation-schedule table, they showed how the proposed separable

architecture uses only a minimal number of filters to generate all levels of DWT

computations in real time. For the computation of an N x N 2-D DWT with a filter

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length L, this architecture spends around N2 clock cycles, and requires 2NL-2N

storage unit, 3L multipliers, as well as 3(L-1) adders. This architecture utilized a

computation-schedule table to accomplish all the resolution levels of

computations. Since this architecture has a low latency, a low hardware cost, and

ability to process 2-D digital signals in real-time, it can be applied very well to the

codec implementation for various video image processing standards, such as

MPEG-4 and JPEG-2000. They found that For a video signal processing, the

computation time of the architecture per picture spends about N 2 only, which

meets the real-time requirement of many 2-D signal-processing applications.

Ching-Hsien Chang et al (2000) presented an efficient VLSI architectures

for fast computation of the Discrete Fourier Transform and its inverse. They have

developed two new VLSI architectures for computing the point discrete Fourier

transform (DFT) and its inverse (IDFT) based on a radix-2 fast algorithm, where is

a power of two. First they have presented a linear systolic array that requires log2

complex multipliers and is able to provide a throughput of one transform sample

per clock cycle. Compared with other related systolic designs based on direct

computation or a radix-2 fast algorithm, the proposed one has the same

throughput performance but involves less hardware complexity. This design is

suitable for high-speed real-time applications, but it would not be easily realized in

a single chip when gets large. It has been verified that the proposed memory-

based architecture that can be realized in a single chip with throughput

performance up to several Mega Transform samples per second for a transform

length as high as several thousand. They found that Such area-time performance

makes this design very competitive for use in long-length DFT applications, such

as asymmetric digital subscriber lines (ADSL) and orthogonal frequency division

multiplexing (OFDM) systems.

Francescomaria Marino et al (2001) implemented two fast architectures for

the direct 2-D Discrete Wavelet Transform. They have presented two architectures

for the direct two–dimensional (2-D) discrete wavelet transform (DWT). The first

architecture is based on a modified recursive pyramid algorithm (MRPA) and

performs a “nonstandard” decomposition (i.e., Mallat’s tree) of an image in

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approximately 2 2 3 clock cycles (ccs). They found that this consistently speeds up

other known architectures that commonly need approximately 2 ccs.

Furthermore, the proposed architecture is simpler than others in terms of

hardware complexity. Subsequently, they showed how “symmetric”/“anti-

symmetric” properties of linear-phase wavelet filter bases can be exploited in order

to further reduce the VLSI area. This is used to design a second architecture that

provides one processing unit for each level of decomposition (pipelined approach)

and performs a decomposition in approximately 2 2 ccs.

Jörg Ritter et al (2001) implemented a pipelined architecture for partitioned

DWT based lossy image compression using FPGA’s. They have presented a

promising FPGA based hardware architecture for DWT in case of lossy and

lossless image compression. The FPGA implementation proposed achieves its

computational power by using the partitioned approach and taking advantage of

pipelining. Currently, an EZT algorithm based on this 2D-DWT architecture is

implemented within the scope of a master’s thesis at our institute. They

investigated the applicability of the approach presented for video compression, in

particular to motion estimation. The partitioned approach is suitable for being

applied to this area because motion seems to be usually localized through the sub

images used. Furthermore, they have been looking for methods allowing the

partitioned approach to result in higher compaction ratios than the non-partitioned

algorithms. One point of departure is to classify the sub images with respect to

similarity, to compress only one representative JC of each class C, and to

represent the remaining sub images I of class C by the image I − JC which

specifies the differences between sub image I and its representative JC. Another

point of departure is to exploit similarities between the trees of the multi-scale

representation.

Abdullah AlMuhit et al (2004) implemented VLSI implementation of Discrete

Wavelet Transform (DWT) for image compression. They presented an approach

towards VLSI implementation of the Discrete Wavelet Transform (DWT) for image

compression. The design follows the JPEG2000 standard and can be used for

both lossy and lossless compression. In order to reduce complexities of the 4

design, linear algebra view of DWT and IDWT has been used. This design can be

used for image compression in a robotic system. They developed an improved and

efficient Discrete Wavelet Transform algorithm and tested in MATLAB. Then the

algorithm has been modified for implementation in VHDL.

Jose Oliver et al (2006) implemented a Huffman coding of wavelet lower

trees for very fast image compression. They have presented a very fast variation

of the Lower-Tree Wavelet (LTW) image encoder. LTW is a fast non-embedded

encoder with state-of-the-art compression efficiency, which employs a tree

structure as a fast method of coding coefficients, being faster than other encoders

like SPIHT or JPEG 2000.They have described the coding algorithm that all the

wavelet subbands are scanned in 2x2 blocks of coefficients, from the first level to

the Nth. They showed that this encoder is more efficient than other very fast

wavelet encoders, like the recently proposed PROGRESS (which is surpassed in

up to 0.5 dB), and faster than them (from 4 to 9 times in coding). Compared with

the JPEG 2000 reference software, the encoder is from 18 to 38 times faster,

while PSNR is similar at low bit-rates, and about 0.5 lower at high bit-rates. A very

fast version of the lower-tree wavelet encoder using Huffman coding and other

strategies have been presented to reduce execution time. The loss of coding

efficiency is compensated by the reduction of execution time. They found that the

encoder can be considered one of the fastest wavelet-based image encoders, and

therefore it is good for real-time interactive multimedia communications, allowing

simple implementations both in hardware and software.

Barbara Penna et al (2007) implemented transform coding techniques for

lossy hyper spectral data compression. Transform-based lossy compression has a

huge potential for hyperspectral data reduction. Hyperspectral data are 3-D, and

the nature of their correlation is different in each dimension. These calls for a

careful design of the 3-D transform to be used for compression. They have

investigated the transform design and rate allocation stage for lossy compression

of hyper spectral data. First, they selected a set of 3-D transforms, obtained by

combining in various ways wavelets, wavelet packets, the discrete cosine

transform, and the Karhunen–Loève transform (KLT), and evaluate the coding

efficiency of these combinations. Second, they proposed a low-complexity version 5

of the KLT, in which complexity and performance can be balanced in a scalable

way, allowing one to design the transform that better matches a specific

application. Third, they integrated this, as well as other existing transforms, in the

framework of Part 2 of the Joint Photographic Experts Group (JPEG) 2000

standard, taking advantage of the high coding efficiency of JPEG 2000, and

exploiting the interoperability of an international standard. An evaluation

framework based on both reconstruction fidelity and impact on image exploitation

has been introduced, and evaluate the proposed algorithm by applying this

framework to AVIRIS scenes. They found that the scheme based on the proposed

low-complexity KLT significantly outperforms previous schemes as to rate-

distortion performance.

Mukul Shirvaikar et al (2008) implemented VHDL implementation of wavelet

packet transforms using SIMULINK tools. The real-time implementation of the

Discrete Wavelet Transform (DWT) is a current area of research as it is one of the

most time consuming steps in the JPEG2000 standard. The standard implements

two different wavelet transforms: irreversible and reversible Daubechies. The

former is a lossy transform, whereas the latter is a lossless transform. Many

current JPEG2000 implementations are software based and not efficient enough

to meet real-time deadlines. Field Programmable Gate Arrays (FPGAs) are

revolutionizing image and signal processing. Many major FPGA vendors like

Altera and Xilinx have recently developed SIMULINK tools to support their FPGAs.

These tools are intended to provide a seamless path from system-level algorithm

design to FPGA implementation. They investigated FPGA implementation of 2-D

lifting-based Daubechies 9/7 and Daubechies 5/3 transforms using a

Matlab/Simulink tool that generates synthesizable VHSIC Hardware Description

Language (VHDL) code. They found that the feasibility of this approach for real

time image processing by comparing the performance of the high-level toolbox

with a handwritten VHDL implementation.

Chao Cheng et al (2008) implemented high-speed VLSI implementation of

2-D Discrete Wavelet Transform. They presented a systematic high-speed VLSI

implementation of the discrete wavelet transform (DWT) based on hardware-

efficient parallel FIR filter structures. High-speed 2-D DWT with computation time 6

as low as 2 12 can be easily achieved for an image with controlled increase of

hardware cost. Hardware efficient parallel FIR filter structures are developed to

speed up the processing speed of 2-D DWT and to control the increase of

hardware cost at the same time. They found that this design can be easily

extended to achieve higher processing speed than the given highest processing

speed with computing time and it is suitable for high-speed VLSI implementation of

2-D DWT because of its regular structure, simple control and 100% hardware

utilization for continuous images.

A. Mansouri et al (2009) implemented an efficient VLSI architecture and

FPGA implementation of high-speed and low power 2-D DWT for (9, 7) wavelet

Filter. They presented an efficient VLSI architecture of a high speed, low power 2-

D Discrete Wavelet Transform computing. This architecture, based on new and

fast lifting scheme approach for (9, 7) filter in DWT, reduces the hardware

complexity and memory accesses and this verified by simulation and successfully

implemented in a Cyclone II and Stratix III FPGAs, and the estimated frequency of

operation is 350 MHz. They found that the resulting computing rate is up to 48

frames (4096x2160) per second with 24 bpp. The architecture has regular

structure, simple control flow, small embedded buffers and low power

consumption. Thus, it is very suitable for new generation image compression

systems, such as JPEG2000. Also they found that the advantages of the proposed

architecture are saving embedded memories, fast computing time, low power

consumption, and low control complexity.

Chengjun Zhang et al (2010) presented a pipeline VLSI architecture for

high-speed computation of the 1-D Discrete Wavelet Transform. They developed a

scheme for the design of a high-speed pipeline VLSI architecture for the

computation of the 1-D discrete wavelet transform (DWT). The main focus of the

scheme is on reducing the number and period of clock cycles for the DWT

computation with little or no overhead on the hardware resources by maximizing

the inter- and intra stage parallelisms of the pipeline. The inter stage parallelism is

enhanced by optimally mapping the computational load associated with the

various DWT decomposition levels to the stages of the pipeline and by

synchronizing their operations. The intra stage parallelism is enhanced by 7

decomposing the filtering operation equally into two subtasks that can be

performed independently in parallel and by optimally organizing the bitwise

operations for performing each subtask so that the delay of the critical data path

from a partial-product bit to a bit of the output sample for the filtering operation is

minimized. They showed that an architecture designed based on the proposed

scheme requires a smaller number of clock cycles compared to that of the

architectures employing comparable hardware resources. In fact, the requirement

on the hardware resources of the architecture designed by using the proposed

scheme also gets improved due to a smaller number of registers that need to be

employed. Based on the proposed scheme, a specific example of designing an

architecture for the DWT computation is considered. Finally, they found that the

principle of maximizing the inter- and intra stage parallelisms for the design of

architecture for the 1-D DWT computation is extendable to that for the 2-D DWT

computation.

Rohit Arora et al (2011) implemented an algorithm for image compression

using 2D wavelet transform. The standard image MASK is analyzed using wavelet

techniques for image compression. The transformations used for wavelet

transformations are Haar, Symlet, Coiflet and db4. The results in terms of

compression ratio and MSE show that the Haar transformation has the best

performance. The Haar transform is then used to compress the image up to four

levels. They sown that the reconstructed image after every level of compression.

They found that DWT has potential application in compression problem and the

use of Haar transformation for Wavelet Transformation provides the best results.

Chengjun Zhang et al (2011) implemented a pipeline VLSI architecture for

fast computation of the 2-D Discrete Wavelet Transform. They developed, a

scheme for the design of a high-speed 3-stage pipeline VLSI architecture for the

computation of the 2-D discrete wavelet transform (DWT). The main focus in the

development of the architecture is on providing a high operating frequency and a

small number of clock cycles along with an efficient hardware utilization by

maximizing the inter-stage and intra-stage computational parallelism for the

pipeline. The inter-stage parallelism is enhanced by optimally mapping the

computational task of multi decomposition levels to the stages of the pipeline and 8

synchronizing their operations. The intra-stage parallelism is enhanced by dividing

the 2-D filtering operation into four subtasks that can be performed independently

in parallel and minimizing the delay of the critical path of bit-wise adder networks

for performing the filtering operation. To validate this scheme, a circuit is designed,

simulated, and implemented in FPGA for the 2-D DWT computation. They showed

that the circuit is capable of operating with a maximum clock frequency of 134MHz

and processing 1022 frames of size 512 512 per second with this operating

frequency. They showed that the performance in terms of the processing speed of

the architecture designed based on the proposed scheme is superior to those of

the architectures designed using other existing schemes, and it has similar or

lower hardware consumption. Finally, they found that the architecture designed is

scalable in that its processing speed can be adjusted upward or downward by

changing the number of MAC cells in each of the processing units by a factor

equal to that of the reduction required in the processing speed.

V. V. Sunil Kumar et al (2012) have developed image compression

techniques by using wavelet transform. This is associated with a certain type of

compression techniques by using wavelet transforms. Wavelets are used to

characterize a complex pattern as a series of simple patterns and coefficients that,

when multiplied and summed, reproduce the original pattern. The data

compression schemes can be divided into lossless and lossy compression. Lossy

compression generally provides much higher compression than lossless

compression. Wavelets are a class of functions used to localize a given signal in

both space and scaling domains.A Minimage was originally created to test one

type of wavelet and the additional functionality was added to image to support

other wavelet types and the EZW(Embedded Zerotree Wavelet) coding algorithm

was implemented to achieve better compression results.

1.2 MOTIVATION

Various image compression techniques were developed and stopped in

simulation level itself. In order to implement an image compression technique on

FPGA, it is necessary to develop VHDL code. This motivates to propose an

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efficient image compression technique based on Discrete Wavelet Transform and

Huffman encoding using VHDL.

1.3 PROBLEM DEFINITION

In this proposed work, aims at compressing an image using 2-d Discrete

Wavelet Transform and Huffman encoding to increase the compression ratio and

reduce the Data Redundancy. Discrete wavelet transform and Huffman encoding

will be implemented using VHDL. The performance will be evaluated in terms of

compression ratio for five different images.

1.4 OBJECTIVES

To acquire the images.

To perform Discrete Wavelet Transform on the images.

To perform Huffman Encoding for the approximation coefficients.

To measure the compression ratio for different images.

1.5 SCOPE OF THE WORK

The main scope of the project is to compress an image to utilize the

bandwidth efficiently for certain applications like Television and Video

Conferencing. Video and Audio files are very large. There is a need to compress

an image to utilize the bandwidth effectively. Compression becomes part of the

representation or coding scheme which have become popular audio, image and

video formats. An efficient image compression technique that uses Huffman

encoding for the approximation coefficients obtained from Discrete Wavelet

Transform is used during the transmission of images through an internet. This

compression technique based on Discrete Wavelet Transform and Huffman

encoding is proposed to increase compression ratio and reduce data redundancy.

This proposed work is highly suitable for an applications like video conferencing.

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CHAPTER 2

OVERVIEW OF DWT AND HUFFMAN ENCODING

The wavelet transform (WT) provides an alternative approach to signal

processing, especially suited for the analysis of spatial and spectral locality. In

recent years, there has been a number of studies on wavelet transforms for signal

analysis and synthesis. Generally, the 2- D DWT is frequently applied in image

and video processing. The discrete wavelet transform (DWT) has been widely

used in audio and image processing, digital communications and other application

fields. This computation transform has been widely implemented in very-large-

scale integration (VLSI) because of the real-time requirement. Wavelets are

functions defined over a finite interval. The basic idea of the wavelet transform is

to represent an arbitrary function as a linear combination of a set of such wavelets

or basis functions. These basis functions are obtained from a single prototype

wavelet called the mother wavelet by dilations (scaling) and translations (shifts).

The purpose of wavelet transform is to change the data from time-space domain to

time-frequency domain which makes better compression results. The fundamental

idea behind wavelets is to analyze the signal at different scales or resolutions,

which is called multiresolution.

Huffman coding is based on the frequency of occurrence of a data item

(pixel in images). The principle is to use a lower number of bits to encode the data

that occurs more frequently. Codes are stored in a code Book which may be

constructed for each image or a set of images. In all cases the code book plus

encoded data must be transmitted to enable decoding.

2.1 OVERVIEW OF DWT

Wavelets are a class of functions used to localize a given signal in both

space and scaling domains. A family of wavelets can be constructed from a

mother wavelet. Compared to Windowed Fourier analysis, a mother wavelet is

stretched or compressed to change the size of the window. In this way, big

wavelets give an approximate image of the signal, while smaller and smaller

wavelets zoom in on details. Therefore, wavelets automatically adapt to both the

high-frequency and the low-frequency components of a signal by different sizes of

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windows. Any small change in the wavelet representation produces a

correspondingly small change in the original signal, which means local mistakes

will not influence the entire transform. The wavelet transform is suited for non

stationary signals, such as very brief signals and signals with interesting

components at different scales.

2.1.1 Wavelet Decomposition

There are several ways wavelet transforms can decompose a signal into

various sub bands. These include uniform decomposition, octave-band

decomposition, and adaptive or wavelet-packet decomposition. Out of these,

octave-band decomposition is the most widely used.

The procedure is as follows: wavelet has two functions “wavelet “and

“scaling function”. They are such that there are half the frequencies between them.

They act like a low pass filter and a high pass filter. The decomposition of the

signal into different frequency bands is simply obtained by successive high pass

and low pass filtering of the time domain signal. This filter pair is called the

analysis filter pair. First, the low pass filter is applied for each row of data, thereby

getting the low frequency components of the row. But since the low pass filter is a

half band filter, the output data contains frequencies only in the first half of the

original frequency range. By Shannon's Sampling Theorem, they can be sub-

sampled by two, so that the output data now contains only half the original number

of samples. Now, the high pass filter is applied for the same row of data, and

similarly the high pass components are separated. In DWT, an image can be

analysed by passing it through an analysis filter bank followed by decimation

operation. An analysis filter bank consists of a low-pass and high-pass filter at

each decomposition stage. When the signal passes through these filters, it splits

into two bands. the low-pass filter, which corresponds to an averaging operation,

extracts the coarse information of the signal. The high-pass filter which

corresponds to a differencing operation, extracts the detail information of the

signal. The output of the filtering operation is then decimated by two. A two-

dimensional transform is accomplished by performing two separate one

dimensional transforms. This is shown in Fig 2.1.

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Figure 2.1 Decomposition of an image

This is a non-uniform band splitting method that decomposes the lower

frequency part into narrower bands and the high-pass output at each level is left

without any further decomposition. This procedure is done for all rows. Next, the

filtering is done for each column of the intermediate data. The resulting two-

dimensional array of coefficients contains four bands of data, each labelled as LL

(low-low), HL (high-low), LH (low-high) and HH (high-high). The LL band can be

decomposed once again in the same manner, thereby producing even more sub

bands.

2.1.2 Wavelet Transform using Haar Wavelets

Image transforms are very important in digital processing they allow to

accomplish less with more. For example the Fourier Transform may be used to

effectively compute convolutions of images1 or the Discrete Cosine Transform

may be used to significantly decrease space occupied by images without

noticeable quality loss. Wavelet Transform (WT) is a relatively new concept as a

whole, even it though it incorporates some of the transforms, which have been

know for long time. Haar wavelet is a simplest wavelet and the wavelet coefficients

are used with the scaling function.

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2.1.2.1 Algorithm of Discrete Wavelet Transform

Discrete wavelet Transform is given by

The following algorithm is also known as subband coding. Assume the

highest frequency component that exists in a signal will be π radians. To provide

sufficient information about the signal without its loss due to sampling, the signal is

sampled at Nyquist’s rate (which is twice the maximum frequency that exists in the

signal). Next, the signal is passed through two filters – one half band low pass and

one half band high band filter. The output of the high pass filter is preserved.

After passing the signal through a half band low pass filter, half of the

samples can be eliminated according to the Nyquist’s rule, since the signal now

has a highest frequency of π/2 radians instead of π radians Simply discarding

every other sample will subsample the signal by two, and the signal will then have

half the number of points. The scale of the signal is now doubled. Note that the low

pass filtering removes the high frequency information, but leaves the scale

unchanged. Only the sub sampling process changes the scale. Resolution, on the

other hand, is related to the amount of information in the signal, and therefore, it is

affected by the filtering operations. Half band low pass filtering removes half of the

frequencies, which can be interpreted as losing half of the information. Therefore,

the resolution is halved after the filtering operation. Note, however, the sub

sampling operation after filtering does not affect the resolution, since removing half

of the spectral components from the signal makes half the number of samples

redundant anyway. Half the samples can be discarded without any loss of

information. In summary, the low pass filtering halves the resolution, but leaves the

scale unchanged. The signal is then sub sampled by 2 since half of the number of

samples is redundant. This doubles the scale.

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Figure 2.2 Subband coding

The transformation

x = (x1, . . . , xN) (s | d) = (s1, . . . , sN/2 | d1, . . . , dN/2) is called the Discrete

Haar Wavelet Transformation.

2.1.2.2 Formulation for Haar wavelet Transform

Digital images require large amounts of memory to store and, when

retrieved from the internet, can take a considerable amount of time to download.

The Haar wavelet transform provides a method of compressing image data so that

it takes up less memory. A 256 x 256 pixel gray scale image is stored as a 256 x

256 matrix, with each element of the matrix being a number ranging from zero (for

black) to some positive whole number (for white). A 256 x 256 color image is

stored as three 256 x 256 matrices (One each for the colors red, green, and blue).

This matrix can be used and some linear algebra to maximize compression.

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The Haar wavelet is the simplest wavelet.

1 0 < x <1/2

wH(x) = -1 ½ < x < 1

0 otherwise

The mother wavelet obviously satisfies the two wavelet requirements, as it

is both local and oscillatory. The picture below shows the shapes of Haar wavelets

for various scales and translations.

Figure 2.3 Shape of the Haar Wavelets

Wavelet Transform is a type of signal representation that can give the

frequency content of the signal at a particular instant of time or spatial location.

The Haar wavelet transform decomposes the image into different sub-band

images, It splits component into numerous frequency bands called sub-bands.

They are LL, LH, HL, and HH sub-bands. A high-frequency sub-band contains the

edge information of input image and LL sub-band contains the clear information

about the image.

An outstanding property of the Haar functions is that except function haar(0,

t), the i–th Haar function can be generated by the restriction of the (j − 1)–th

function to be half of the interval where it is different from zero, by multiplication

with p2 and scaling over the interval [0, 1]. These properties give considerable

interest of the Haar function, since they closely relate them to the wavelet theory.

In this setting, the first two Haar functions are called the global functions, while all

the others are denoted as the local functions. Hence, the Haar function, which is

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an odd rectangular pulse pair, is the simplest and oldest wavelet. The motivation

for using the discrete wavelet transform is to obtain information that is more

discriminating by providing a different resolution at different parts of the time–

frequency plane. The wavelet transforms allow the partitioning of the time-

frequency domain into non uniform tiles in connection with the time–spectral

contents of the signal. The wavelet methods are strongly connected with classical

basis of the Haar functions; scaling and dilation of a basic wavelet can generate

the basis Haar functions. The Haar wavelet transform decomposes the image into

different sub-band images, It splits component into numerous frequency bands

called sub-bands. They are LL, LH, HL, and HH sub-bands. A high-frequency sub-

band contains the edge information of input image and LL sub-band contains the

clear information about the image.

consider the Filter h=(h0,h1)

y=h*x

yn=1/2 xn + 1/2 xn-1

2.1.2.3. Point Haar Wavelet(oldest & simplest)

g[n] = 1/2 for n = −1, 0

g[n] = 0 otherwise

h[0] = 1/2, h[−1] = −1/2,

h[n] = 0 otherwise

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2.1.2.4 2-Dimension(analysis) of Discrete wavelet transform

In DWT, an image can be analysed by passing it through an analysis filter

bank followed by decimation operation. An analysis filter bank consists of a low-

pass and high-pass filter at each decomposition stage. When the signal passes

through these filters, it splits into two bands. the low-pass filter, which corresponds

to an averaging operation, extracts the coarse information of the signal. The high-

pass filter which corresponds to a differencing operation, extracts the detail

information of the signal. The output of the filtering operation is then decimated by

two. A two-dimensional transform is accomplished by performing two separate one

dimensional transforms. First, an image is filtered along the row and decimated by

two. It is then followed by filtering the sub image along the column and decimated

by two. This operation splits an image into four bands namely LL,LH,HL and HH

respectively. Further decomposition can be achieved by acting upon the LL sub-

band successfully and the resultant image is split into multiple bands. This process

is shown in Fig 2.4.

.

Figure 2.4 Two Dimension analysis of Discrete wavelet transform

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2.1.2.5 Working of Haar Wavelet Transform

The Haar wavelet uses a method of manipulating matrices called

averaging and differencing.

If y is a row of an image matrix, when it is averaged and differenced

will get the following results.

y 448 768 704 640 1280 1408 1600 1600

y1 608 672 1344 1600 −160 32 −64 0

y2 640 1472 −32 −128 −160 32 −64 0

y3 1056 −416 −32 −128 −160 32 −64 0

Table 2.1 Average and Difference

Blue = Approximation Coefficients

Red = Detail Coefficients

Using Matrix multiplication, but Y1=yA1

where A1 is the Matrix

1/2 0 0 0 1/2 0 0 0

1/2 0 0 0 -1/2 0 0 0

A1= 0 1/2 0 0 0 1/2 0 0

0 1/2 0 0 0 -1/2 0 0

0 0 1/2 0 0 0 1/2 0

0 0 1/2 0 0 0 -1/2 0

0 0 0 1/2 0 0 0 1/2

0 0 0 1/2 0 0 0 -1/2

and Y2=y1A2

19

where A2 is the Matrix

1/2 0 1/2 0 0 0 0 0

1/2 0 -1/2 0 0 0 0 0

0 1/2 0 1/2 0 0 0 0

0 1/2 0 -1/2 0 0 0 0

A2= 0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

and y3=y2A3

where A3 is the matrix

1/2 1/2 0 0 0 0 0 0

1/2 -1/2 0 0 0 0 0 0

0 0 1 0 0 0 0 0

A3= 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

20

This set of operations can all be done in one step:

y3 = yW

Where W is is the transformation matrix:

W = A1 A2 A3

Also note that since each column of the Ai matrices that comprise

W is orthogonal to every other, the matrices are invertible. Thus:

W^-1=A3^-1 A2^-1 A1^-1

This means that original can get data back using the equation.

y = y3W^-1

In general, it says that Q = I W, where Q is the row transformed matrix and I

is the original image matrix. But, the Haar wavelet transformation does these

transformations to each column of the image matrix, and then repeats them on

each column of the matrix. This is done by multiplying I on the left by the

transpose of W. This gives us our final equation for the row-and-column

transformed matrix T.

T = WT I W

start with an 8 x 8 image represented by matrix P

576 704 1152 1280 1344 1472 1536 1536

P= 704 640 1156 1088 1344 1408 1536 1600

768 832 1216 1472 1472 1536 1600 1600

832 832 960 1344 1536 1536 1600 1536

832 832 960 1216 1536 1600 1536 1536

960 896 896 1088 1600 1600 1600 1536

768 768 832 832 1280 1472 1600 1600

448 768 704 640 1280 1408 1600 1600

21

Last row is our vector y.

First there is a need to average and difference the rows of matrix P. In order

to get the row averaged matrix Q we simply multiply P on the right by matrix W,

our transformation matrix. This yields:

1200 −272 −288 −64 −64 −64 −64 0

1185 −288 −225 −96 32 34 −32 −32

1312 −240 −272 −48 −32 −128 −32 0

1272 −280 −160 −16 0 −192 0 32

Q= 1256 −296 −128 16 0 −128 −32 0

1272 −312 −32 16 32 −96 0 32

1144 −344 −32 −112 0 0 −96 0

1056 −416 −32 −128 −160 32 −64

0

Now average and difference the columns of Q to get row and column

transformed matrix T by multiplying Q on the left by WT .

This yields:

1212 −306 −146 −54 −24 −68 −40 4

30 36 −90 −2 8 −20 8 −4

-50 −10 -20 −24 0 72 −16 −16

T= 82 38 −24 68 48 −64 32 8

8 8 −32 16 −48 −48 −16 16

20 20 −56 −16 −16 32 −16 −16

−8 8 −48 0 −16 −16 −16 −16

44 36 0 8 80 −16 −16 0

Now matrix T is ready to be compressed.

22

2.1.2.6 Approximation and Detail Coefficients

If there is a need to average and difference each row and column of matrix

P only once (T1 = AT1 P A1), we would essentially divide our 8 x 8 matrix into four

4 x 4 sub-matrices. The resulting filtered outputs are approximation and detail

coefficients. That are approximation, vertical detail, horizontal detail and diagonal

detail subbands of the input image. These contains the clear and edge information

about an input image.

Approximation coefficients contains the information of an image. The detail

coefficients left out during the Huffman encoding to efficiently compress an image

and also to increase the quality of an image. The Haar wavelet transform

decomposes the image into different sub-band images, It splits component into

numerous frequency bands called sub-bands. They are LL, LH, HL, and HH sub-

bands. A high-frequency sub-band contains the edge information of input image

and approximation sub-band contains the clear information about the image.

These coefficients are shown in Fig 2.5.

656 1169 1392 1552 −16 −15 −48 −16

816 1248 1520 1584 −16 −160 −16 16

880 1040 1584 1552 16 −112 −16 16

688 752 1360 1600 −80 16 −80 0

T1= −16 47 16 −16 −48 −49 −16 16

−16 96 −16 16 −16 32 −16 −16

−48 48 −16 −16 −16 −16 −16 −16

80 80 16 0 80 −16 −16 0

23

Figure 2.5 Coefficients of DWT

2.1.2.7 Applications of Haar Transform

Due to its low computing requirements, the Haar transform has been mainly

used for pattern recognition and image processing . Hence, two dimensional signal

and image processing is an area of efficient applications of Haar transforms due to

their wavelet-like structure. In this area, it is usually reported that the simplest

possible orthogonal wavelet system is generated from the Haar scaling function

and wavelet. Moreover, wavelets are considered as a generalization of the Haar

functions and transforms. Such a transform is also well suited in communication

technology for data coding, multiplexing and digital filtering. For example,

application of non-normalized Haar transform in a sequence division multiplex

system is described. Bandwidth economy for multiplexed digital channels based

on Haar transform is presented in. For real time applications, hardware-based fast

Haar chips have been developed. Different generalizations of Haar functions and

24

transform are used in digital speech processing with applications in voice

controlled computing devices and robotics. The advantages of computational and

memory requirements of the Haar transform make it of a considerable interest to

VLSI designers as well. Digital MOS ICs as a tool external to the circuit, but also

as a possibility for a self-test strategy. The use of this set of CAD tools allowed the

derivation of strategies for testing MOS circuits when memory states were

encountered as a consequence of some type of faults. The advantage of using

Haar functions instead of Walsh functions in CAD system based on spectra

methods for some classes of Boolean.

2.2 OVERVIEW HUFFMAN ENCODING

Huffman code is a Optimum prefix code developed by D. Huffman in a class

assignment. Construction of Huffman codes is based on two ideas:

In an optimum code, symbols with higher probability should have shorter

code words

In an optimum prefix code, the two symbols that occur least frequently will

have the same length (otherwise, the truncation of the longer codeword to

the same length still produce a decodable code)

Huffman coding is a popular method for compressing data with variable-

length codes. Given a set of data symbols (an alphabet) and their frequencies of

occurrence (or, equivalently, their probabilities), the method constructs a set of

variable-length code words with the shortest average length and assigns them to

the symbols. Huffman coding serves as the basis for several applications

implemented on popular platforms. Some programs use just the Huffman method,

while others use it as one step in a multistep compression process. The Huffman

method [Huffman 52] is somewhat similar to the Shannon–Fano method, proposed

independently by Claude Shannon and Robert Fano in the late 1940s ([Shannon

48] and [Fano 49]). It generally produces better codes, and like the Shannon–Fano

method, it produces the best variable-length codes when the probabilities of the

symbols are negative powers of 2. The main difference between the two methods

is that Shannon–Fano constructs its codes from top to bottom (and the bits of each

codeword are constructed from left to right), while Huffman constructs a code tree 25

from the bottom up (and the bits of each codeword are constructed from right to

left).

Since its inception in 1952 by D. Huffman, the method has been the subject

of intensive research in data compression. The long discussion in [Gilbert and

Moore 59] proves that the Huffman code is a minimum-length code in the sense

that no other encoding has a shorter average length. A much shorter proof of the

same fact was discovered by Huffman himself [Motil 07]. An algebraic approach to

constructing the Huffman code is introduced in [Karp 61]. In [Gallager 78], Robert

Gallager shows that the redundancy of Huffman coding is at most p1 + 0.086

where p1 is the probability of the most-common symbol in the alphabet. The

redundancy is the difference between the average Huffman codeword length and

the entropy. Given a large alphabet, such as the set of letters, digits and

punctuation marks used by a natural language, the largest symbol probability is

typically around 15–20%, bringing the value of the quantity p1 + 0.086 to around

0.1. This means that Huffman codes are at most 0.1 bit longer (per symbol) than

an ideal entropy encoder, such as arithmetic coding. The Huffman encoding

algorithm starts by constructing a list of all the alphabet symbols in descending

order of their probabilities. It then constructs, from the bottom up, a binary tree with

a symbol at every leaf. This is done in steps, where at each step two symbols with

the smallest probabilities are selected, added to the top of the partial tree, deleted

from the list, and replaced with an auxiliary symbol representing the two original

symbols. When the list is reduced to just one auxiliary symbol (representing the

entire alphabet), the tree is complete. The tree is then traversed to determine the

codewords of the symbols.



that occurs more frequently. Codes are stored in a Code Book which may be


encoded data must be transmitted to enable decoding.

26

2.2.1 Principle of Huffman codes

Starting with two least probable symbols, g and d, of an alphabet A, if the

codeword for g is [m]0, the codeword for d would be [m]1, where [m] is a string of

1s and 0s. Now, the two symbols can be combined into a group, which represents

a new symbol y in the alphabet set.

Five steps:

Find the gray-level probabilities for the image by finding the histogram

Order the input probabilities (histogram magnitudes)from smallest to largest

Combine the smallest two by addition

Go to step2,until only two probabilities are left

By working backward along the tree, generate code by alternating

assignment of 0 and 1.

The symbol y has the probability P(g) + P(d). Recursively determine the bit

pattern [m] using the new alphabet set.

Let A = {a1, …, a5}, P(ai) = {0.2, 0.4, 0.2, 0.1, 0.1}.

Table 2.2 Principle of Huffman codes

27

2.2.2 Efficiency of Huffman codes

Redundancy – the difference between the entropy and the average length

of a code.

Table 2.3 Efficiency of Huffman codes

The average codeword length for this code is

l = 0.4 x 1 + 0.2 x 2 + 0.2 x 3 + 0.1 x 4 + 0.1 x 4 = 2.2 bits/symbol.

The redundancy is 0.078 bits/symbol.

For Huffman code, the redundancy is zero when the probabilities are negative

powers of two.

2.2.3 Length of Huffman codes

Given a sequence of positive integers {l1, l2, …, lk} satisfies

there exists a uniquely decodable code whose codeword lengths are given by { l1,

l2, …, lk}.

The optimal code for a source S has an average code length lavg with the

following bounds:

where H(S) is the entropy of the source.

28

The lower-bound can be obtained by showing that:

For the upper-bound, notice that given an alphabet {a1, a2, …, ak}, and a set of

codeword lengths.

the code satisfies the Kraft-McMillan inequality and

has lavg < H(S) + 1.

2.2.4 Encoding Procedure



that occurs more frequently. Codes are stored in a Code Book which may be


encoded data must be transmitted to enable decoding. Huffman coding is a

popular method for compressing data with variable-length codes. Given a set of

data symbols (an alphabet) and their frequencies of occurrence (or, equivalently,

their probabilities), the method constructs a set of variable-length code words with

the shortest average length and assigns them to the symbols.

An Encoding procedure for the Huffman encoding are as follows:


Order the input probabilities (histogram magnitudes)from smallest to largest


Go to step2,until only two probabilities are left

By working backward along the tree, generate code by alternating

assignment of 0 and 1.This is shown in Fig 2.6.

29

Figure 2.6 Encoding Procedure

2.2.5 Example of Huffman coding

An example of Huffman encoding is shown if Fig 2.7. In such an alphabet,

one symbol can be assigned the code 0 and the other code 1. The Huffman

method cannot assign to any symbol a code shorter than one bit, so it cannot

improve on this simple code. If the original data (the source) consists of individual

bits, such as in the case of a bi-level (monochromatic) image, it is possible to

combine several bits (perhaps four or eight) into a new symbol and pretend that

the alphabet consists of these (16 or 256) symbols. The problem with this

approach is that the original binary data may have certain statistical correlations

between the bits, and some of these correlations would be lost when the bits are

combined into symbols. When a typical bi-level image is digitized by scan lines, a

pixel is more likely to be followed by an identical pixel than by the opposite one.

30

Figure 2.7 Huffman code for the 26-letter Alphabet

Therefore have a file that can start with either a 0 or a 1 (each has 0.5

probability of being the first bit). A zero is more likely to be followed by another 0

and a 1 by another 1. Figure 2.7 is a finite-state machine illustrating this situation.

If these bits are combined into, say, groups of eight, correlated by the original pixel

probabilities. If the input data contains, e.g., the two adjacent groups 00011100

and 00001110, they will be encoded independently, ignoring the correlation

between the last 0 of the first group and the first 0 of the next group. Selecting

larger groups improves this situation but increases the number of groups, which

implies more storage for the code table and longer time to calculate the table.

31

CHAPTER 3

DWT BASED IMAGE COMPRESSION TECHNIQUE FOR VIDEO

CONFERENCING

A scheme of compression technique is used based on the combination of

2-D DWT/Huffman encoding is proposed to compress an input image by reducing

the coding redundancy. Here, the Discrete wavelet transform and Huffman coding

are implemented using VHDL. Input image size of 256x256 is read using MATLAB

R2012 and image pixels are stored in Memory block module to perform Discrete

Wavelet Transform. Here, the image compression technique is based on Discrete

wavelet transform and Huffman coding. Approximation coefficients are obtained

from DWT using VHDL source code. Huffman coding of approximation coefficients

are obtained using VHDL and Compression ratio is measured. Simulation of DWT

and Huffman Coding using Modelsim is performed. The block diagram of

proposed work is shown in Fig 3.1.

Input image

256x256

compressed

image

Figure 3.1 Block Diagram of proposed system

32

Image acquisition

Memory block module

Discrete Wavelet Transform with Haar module

Huffman Encoding module

Compressed image in terms of compression ratio

3.1 INPUT IMAGE AND MEMORY BLOCK

An input image is red and it is converted into gray scale image. An image

pixel values are created and stored in memory. This is show in figure 4.1.

Figure 3.2 Flowchart to read an input image and Memory block module

3.2 DISCRETE WAVELET TRANSFORM MODULE

This module computes the wavelet transform coefficients of the input image

pixels obtained from the memory module. In this band splitting method that

decomposes the lower frequency part into narrower bands and the high-pass

output at each level is left without any further decomposition. This procedure is

done for all rows. Next, the filtering is done for each column of the intermediate

data. The resulting two-dimensional array of coefficients contains four bands of

data, each labelled as LL (low-low), HL (high-low), LH (low-high) and HH (high-

high). The Low pass and High pass coefficients are computed. This is shown in

figure 3.2.33

Image acquisition

Convert an input image into gray scale image

Create text file containing pixel values

Store the image pixel values in memory block module

Figure 3.3 Flowchart of DWT

The Approximation coefficients are applied as an input to Huffman

encoding module to code the data. The Huffman data for those coefficients are

determined. Further decomposition can be achieved by acting upon the LL sub-

band successfully and the resultant image is split into multiple bands. An input

sequence is decomposed into low-pass and high-pass sub bands in this method of

DWT with Haar wavelet. This Haar wavelet transform decomposes the image into

different sub-band images, It splits component into numerous frequency bands

called sub-bands. They are LL, LH, HL, and HH sub-bands. A high-frequency sub-

band contains the edge information of input image and LL sub-band contains the

clear information about the image.

34

Start

Image acquisition

Apply horizontal decomposition

Apply vertical decomposition

DWT Coefficients

Stop

3.3 HUFFMAN ENCODING MODULE

The Huffman encoding of DWT output has been computed. The compressed

values are viewed serially.

No

Yes

Figure 3.4 Flowchart of Huffman encoding module

35


Order the input probabilities (histogram magnitudes) from smallest to largest


Are only two

probabilities are left?

By working backward along the tree, generate code by alternating assignment of 0 and 1

3.3.1 Huffman Code

The reference codeword for the approximation coefficients along with

codeword length are shown in Table 2.4.

Table 2.4 Huffman Code

3.4 COMPRESSED IMAGE

The compressed image is displayed in terms of compression ratio. The

compressed values are viewed serially. This s shown if figure 3.4.

Figure 3.5 Flowchart for compressed image in terms of compression ratio

36

View the compressed values serially

Obtain the compression ratio

Measure the performance by obtaining compression ratio for different images

3.5 OVERALL FLOWCHART

An image is red and the pixels are stored as a text file. The approximation

coefficients are obtained by implementing the DWT with haar. Then, the Huffman

encoding of Approximation coefficients are described to find the compressed

values. Compressed image is obtained in terms of Compression ratio.

Figure 3.6 Flowchart of proposed system

CHAPTER 4

37

Start

Image acquisition

Perform DWT with Haar Wavelet and obtain the approximation ad detail

coefficients

Perform Huffman Encoding of

Approximation coefficients

Compressed imageStop

RESULTS AND DISCUSSION

4.1 INPUT IMAGES

The Different colour images are given as an input for the DWT based

imaged compression. The input images are shown below.

Figure 4.1. Input Image 1 Figure 4.2. Input Image 2

Figure 4.3. Input Image 3 Figure 4.4. Input Image 4

38

Figure 4.5. Input Image 5

4.2 GRAY SCALE IMAGES

The gray scale images for the given input images are shown below.

Figure 4.6. Gray scale Image 1 Figure 4.7. Gray scale Image 2

Figure 4.8. Gray scale Image 3 Figure 4.9. Gray scale Image 4

Figure 4.10 Gray scale Image 5

4.3 SIMULATION OUTPUT FOR DISCRETE WAVELET TRANSFORM

39

Discrete Wavelet Transform with Haar wavelet is simulated using VHDL.

Memory bock module is used to store the image pixel values, then the pixels are

read from memory through the data lines and wavelet coefficients are obtained.

The LL, LH,HL,HH coefficients are calculated using wavelet decomposition. The

simulation output of DWT for an image 1 is shown in Fig 4.11.

Figure 4.11 Simulation output of DWT for image 1

The DWT coefficients of an image 1 are LL(Low-Low), LH(Low-High),

HL(High-Low), HH(High-High) are shown in Fig 4.12.

Figure 4.12 DWT coefficients for image 1


The DWT coefficients of an image 2 are shown in Fig 4.14.

40






pixel values and the corresponding coefficients of those pixel values are shown in

Fig 4.13. An address lines and data lines which are used in memory block are also

specified. Whenever the clock pin is high the process will be initialised and the

coefficients are determined.

41




read from memory through the data lines and Wavelet coefficients are obtained.


pixel values and the corresponding coefficients of those pixel values are shown in



coefficients are determined. The DWT coefficients for an image 3 are shown in Fig

4.16.

42





The LL, LH,HL,HH Coefficients are calculated using wavelet decomposition. The

Pixel values and the corresponding coefficients of those pixel values are shown in



coefficients are determined. The DWT coefficients for an image 4 are shown in Fig

4.18.

43



44






Pixel values and the corresponding coefficients of those pixel values are shown in



coefficients are determined. The DWT coefficients for an image 5 are shown in

Fig 4.20.

45


4.4 SIMULATION OUTPUT FOR HUFFMAN ENCODING

The approximation coefficients are given as an input to Huffman data table

to obtain the Huffman data for each coefficient. Huffman code variable has an

input of codeword length and the compressed data is viewed as serially. Huffman

code for the approximation coefficients of an image 1 is shown in Fig 4.21.

Figure 4.21 Simulation output of Huffman coding for image 1

46





Figure 4.22 Simulation output for image 2





47




input of codeword length and the compressed data is viewed as serially.

48

The Huffman codewords for the approximation coefficients of an image 4 is

shown in Fig 4.24.


49






50

4.5 COMPRESSED VALUES AND COMPRESSION RATIO

The compressed values and compression ratio of an image 1 is shown Fig

4.26.

Figure 4.26 Compressed values and compression ratio for image 1

The compressed values and compression ratio of an image 2 is shown Fig 4.27.


51





52

The compressed values and compression ratio of an image 5 is shown

Fig 4.30.The parallel compressed values are converted into serial and these are

viewed serially by column wise. All the serial values are showing the

corresponding Huffman code for the approximation coefficients.


53

The Compression ratio for five different images are compared and these

ratio are shown in Table 5.1.

Table 5.1 Comparison of Compression ratio for five different images

The compression ratio for five different images are plotted as a graph. This

is shown in Fig 4.31.

Figure 4.31 Flow graph of compression ratio for five different images

54

Images Compression ratio

Desert 0.9478

Hydrangeas 0.9389

Lighthouse 0.9532

Penguins 0.9524

Tulips 0.9536

CHAPTER 5

CONCLUSION

In this work, an efficient compression technique that uses Huffman

encoding for the approximation coefficients of the given images has been

successfully developed. In the first stage, the discrete wavelet transform is

performed using Haar wavelet filter coefficients. The DWT coefficients are

approximation and detail coefficients. These coefficients are determined using

wavelet decomposition. In the second stage, the Huffman encoding for the

approximation coefficients are performed. The compressed values are viewed

serially by converting the parallel values. The Huffman code for each coefficient is

determined.

The performance of the proposed technique is evaluated in terms of

compression ratio for five different images. It is found that the compression ratio

has decreased significantly as only approximation coefficients are considered as

the same time the quality of image has been compromised as detail coefficients

are left out. The proposed lossy compression technique is highly suitable for

applications like video conferencing during the transmission of images through an

internet but not that suitable for medical image transmission also the compression

ratio can be further reduced by incorporating quantization of the approximation

coefficients.

55

CHAPTER 6

BIBLIOGRAPHY

[1] Angelopoulou. M, Masselos. K, Cheung. P and Andreopoulos. Y (2007), “A comparison of 2-D discrete wavelet transform computation schedules on FPGAs,” in Proc. IEEE Int. Conf. Field Programmable Technology (FPT), Bangkok, Tailand, pp. 181–188.

[2] Averbuch. A, Lazar. D and Israeli. M (1996), “Image compression using wavelet transform and multiresolution decomposition. ” IEEE Transactions on Image Processing, vol. 5, no. 1, pp. 4-15.

[3] Chao Cheng (2008), Member, IEEE “High-Speed VLSI Implementation of 2-D Discrete Wavelet Transform,” IEEE Trans, Signal Processing, Vol. 56, No. 1.

[4] Chengjun Zhang, Chunyan Wang (2010), Senior Member, IEEE” “A Pipeline VLSI Architecture for High-Speed Computation of the 1-D Discrete Wavelet Transform,” IEEE Transactions On Circuits and Systems—I: Regular Papers, Vol. 57, No. 10.

[5] Ching-Hsien Chang, Chin-Liang Wang (2000), Member, IEEE “Efficient VLSI Architectures for Fast Computation of the Discrete Fourier Transform and Its Inverse,” IEEE Transactions On Signal Processing, Vol. 48, No. 11.

[6] Chrysytis. C and Ortega. A (2000), “Line-based, reduced memory, waveletimage compression,” IEEE Trans. Circuits Syst. Video Technol., vol. 9, no. 3, pp. 378–389.

[7] Grzesczak. A, Mandal. M. K and Panchanathan. S (1996), “VLSI implementation of discrete wavelet transform,” IEEE Trans. Very Large ScaleIntegr. (VLSI) Syst., vol. 4, no. 4, pp. 421–433.

[8] Mallat. S (2009), “A theory for multiresolution signal decomposition : The wavelet representation, ” IEEE Trans. Pattern Anal. and Machine Intell.

56

[9] Mansouri. A (2009) “An Efficient VLSI Architecture and FPGA Implementation of High-Speed and Low Power 2-D DWT for (9, 7) Wavelet Filter,” IJCSNS International Journal of Computer Science and Network Security, VOL.9 No.3.

[10] Raghunath. S and Aziz. S. M (2006), “High speed area efficient multi-resolution 2-D 9/7 filter DWT processor,” in Proc. Int. Conf. Very Large Scale Integration (IFIP), vol. 16–18, pp. 210–215.

[11] Rioul. O and Vetterli. M (1991), “Wavelets and signal processing,” IEEE Signal Processing Magazine, vol. 8, no.4, pp. 14-38.

[12] Sunil kumar. V. V (2012) “Image Compression Techniques by using Wavelet Transform,” Journal of International Engineering and Applications ISSN 2224-5782 (print) ISSN 2225-0506 (online) Vol 2, No.5.

.

57

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