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CHAPTER 1

Introduction

A majority of todays Internet bandwidth is estimated to be used for images and

video. Recent multimedia applications for handheld and portable devices place a limit on

the available wireless bandwidth. The bandwidth is limited even with new connection

standards. JPEG image compression that is in widespread use today took several years

for it to be perfected. Wavelet based techniques such as JPEG2000 for image

compression has a lot more to offer than conventional methods in terms of compression

ratio. Currently wavelet implementations are still under development lifecycle and are

being perfected. Flexible energy-efficient hardware implementations that can handlemultimedia functions such as image processing, coding and decoding are critical,

especially in hand-held portable multimedia wireless devices.

1.1Background

Data compression is, of course, a powerful, enabling technology that plays a vital

role in the information age. Among the various types of data commonly transferred over

networks, image and video data comprises the bulk of the bit traffic. For example,

current estimates indicate that image data take up over 40% of the volume on the

Internet. The explosive growth in demand for image and video data, coupled with

delivery bottlenecks has kept compression technology at a premium. Among the several

compression standards available, the JPEG image compression standard is in wide

spread use today. JPEG uses the Discrete Cosine Transform (DCT) as the transform,

applied to 8-by-8 blocks of image data. The newer standard JPEG2000 is based on the

Wavelet Transform (WT). Wavelet Transform offers multi-resolution image analysis,

which appears to be well matched to the low level characteristic of human vision. The

DCT is essentially unique but WT has many possible realizations. Wavelets provide us

with a basis more suitable for representing images.

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This is because it cans represent information at a variety of scales, with local

contrast changes, as well as larger scale structures and thus is a better fit for image data.

1.2 Aim of the project

The main aim of the project is to implement and verify the image compression

technique and to investigate the possibility of hardware acceleration of DWT for signal

processing applications. A hardware design has to be provided to achieve highperformance, in comparison to the software implementation of DWT. The goal of the

project is to

Implement this in a Hardware description language (Here VHDL).

Perform simulation using tools such as Xilinx ISE 8.1i.

Check the correctness and to synthesize for a Spartan 3E FPGA Kit.



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1.3 Block Diagram

Fig 1.1: Image Compression Model

Fig 1.2: Image Decompression Model



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CHAPTER 2

Description

Fig 2.1: Block Diagram of Lifting based DWT

The block diagram consists of 4 blocks:

1) DWT

2) Compression block

3) Decompression block

4) IDWT

The input image is given to DWT which consists of lifting scheme where theimage is splitted into sequence of even and odd series coefficients. These splitted series

are passed to compression block where the image is compressed using SPIHT algorithm .

Compressed image is converted into bit streams using Entropy encoder. The

reconstructed image is obtained by passing the compressed image through decompression

block and IDWT.



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INTRODUCTION TO WAVELETS AND WAVELET

TRANSFORMS

2.1 Fourier Analysis

Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps

the most well-known of these is Fourier analysis, which breaks down a signal into

constituent sinusoids of different frequencies. Another way to think of Fourier analysis is

as a mathematical technique for transformingour view of the signal from time-based to

frequency-based.

Fig 2.2Fourier analysis

For many signals, Fourier analysis is extremely useful because the signals

frequency content is of great importance

2.2 Short-Time Fourier analysis

In an effort to correct this deficiency, Dennis Gabor (1946) adapted the Fourier

transform to analyze only a small section of the signal at a timea technique calledwindowing the signal. Gabors adaptation, called the Short-Time Fourier Transform

(STFT), maps a signal into a two-dimensional function of time and frequency.



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Fig 2.3Short-Time Fourier analysis

The STFT represents a sort of compromise between the time- and frequency-

based views of a signal. It provides some information about both when and at what

frequencies a signal event occurs. However, you can only obtain this information with

limited precision, and that precision is determined by the size of the window.

While the STFT compromise between time and frequency information can be

useful, the drawback is that once you choose a particular size for the time window, that

window is the same for all frequencies. Many signals require a more flexible approach

one where we can vary the window size to determine more accurately either time or

frequency.

2.3 Problem Present in Fourier Transform2

The Fundamental idea behind wavelets is to analyze according to scale. Indeed,

some researchers feel that using wavelets means adopting a whole new mind-set 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 18OOs, when

Joseph Fourier discovered that he could superpose sines and cosines to represent other

functions. However, in wavelet analysis, the scale used to look at data plays a special

role. Wavelet algorithms process data at different scales or resolutions. Looking at a

signal (or a function) through a large window, gross features could be noticed.

Similarly, looking at a signal through a small window, small features could be noticed.

The result in wavelet analysis is to see both the forest and the 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 are the basis of

Fourier analysis, to approximate choppy signals. By their definition, these functions are



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non-local (and stretch 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

analyzingwavelet or mother wavelet. Temporal analysis is performed with a contracted,

high-frequency version of the 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 wavelets best adapted to data are selected, the

coefficients below a threshold is truncated, resultant data are sparsely represented. This

sparse coding makes wavelets an excellent tool in the field of data compression. Other

applied fields that are using wavelets include astronomy, acoustics, nuclear engineering,

sub-band coding, signal and image processing, neurophysiology, music, magnetic

resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake

prediction, radar, human vision, and pure mathematics applications such as solving

partial differential equations.

Basically wavelet transform (WT) is used to analyze non-stationary signals, i.e.,

signals whose frequency response varies in time, as Fourier transform (FT) is not suitable

for such signals. To overcome the limitation of FT, short time Fourier transform (STFT)

was proposed. There is only a minor difference between STFT and FT. In STFT, the

signal is divided into small segments, where these segments (portions) of the signal can

be assumed to be stationary. For this purpose, a window function "w" is chosen. The

width of this window in time must be equal to the segment of the signal where its still be

considered stationary. By STFT, one can get time-frequency response of a signal

simultaneously, which cant be obtained by FT.



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The short time Fourier transform for a real continuous signal is defined as:

( )

= dtetwtxtfx ftj 2* ])()([,

Where the length of the window is (t-) in time such that we can shift the window

by changing value oft and by varying the value we get different frequency response ofthe signal segments.

The Heisenberg uncertainty principle explains the problem with STFT. This

principle states that one cannot know the exact time-frequency representation of a signal,

i.e., onecannotknow what spectral components exist at what instances of times. What

one canknow are the time intervals in which certain band of frequencies exists and is

called resolution problem. This problem has to do with the widthof the window function

that is used, known as thesupportof the window. If the window function is narrow, then

it is known as compactly supported. The narrower we make the window, the better the

time resolution, and better the assumption of the signal to be stationary, but poorer the

frequency resolution:

Narrow window ===> good time resolution, poor frequency resolution.

Wide window ===> good frequency resolution, poor time resolution.

The wavelet transform (WT) has been developed as an alternate approach to

STFT to overcome the resolution problem. The wavelet analysis is done such that the

signal is multiplied with the wavelet function, similar to the window function in the

STFT, and the transform is computed separately for different segments of the time-



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domain signal at different frequencies. This approach is called multiresolution analysis

(MRA), as it analyzes the signal at different frequencies giving different resolutions.

MRA is designed to give good time resolution and poor frequency resolution at

high frequencies and good frequency resolution and poor time resolution at low

frequencies. This approach is good especially when the signal has high frequency

components for short durations and low frequency components for long durations, e.g.,

images and video frames.

So why do we need other techniques, like wavelet analysis?

Fourier analysis has a serious drawback. In transforming to the frequency domain,

time information is lost. When looking at a Fourier transform of a signal, it is impossible

to tell whena particular event took place. If the signal properties do not change much

over time that is, if it is what is called a stationarysignalthis drawback isnt very

important. However, most interesting signals contain numerous non stationary or

transitory characteristics: drift, trends, abrupt changes, and beginnings and ends of

events. These characteristics are often the most important part of the signal, and Fourier

analysis is not suited to detecting them.

2.4 Wavelet Analysis

Wavelet analysis [1] represents the next logical step: a windowing technique with

variable-sized regions. Wavelet analysis allows the use of long time intervals where we

want more precise low-frequency information, and shorter regions where we want high-

frequency information



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A wavelet is a waveform of effectively limited duration that has an average value

of zero.

Compare wavelets with sine waves, which are the basis of Fourier analysis.

Sinusoids do not have limited duration they extend from minus to plus infinity. And

where sinusoids are smooth and predictable, wavelets tend to be irregular and symmetric.

Fig 2.5 sine wave

Fourier analysis consists of breaking up a signal into sine waves of various

frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and

scaled versions of the original (or mother) wavelet. Just looking at pictures of wavelets

and sine waves, you can see intuitively that signals with sharp changes might be better

analyzed with an irregular wavelet than with a smooth sinusoid, just as some foods are

better handled with a fork than a spoon. It also makes sense that local features can be

described better with wavelets that have local extent.

2.5 What Can Wavelet Analysis Do?

One major advantage afforded by wavelets is the ability to perform local analysis,

that is, to analyze a localized area of a larger signal. Consider a sinusoidal signal with a

small discontinuity one so tiny as to be barely visible. Such a signal easily could be

generated in the real world, perhaps by a power fluctuation or a noisy switch.

Sinusoid with a small discontinuity


0.5

0

-0.5


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Fig 2.6Sinusoidal Signal

A plot of the Fourier coefficients (as provided by the fft command) of this signal

shows nothing particularly interesting: a flat spectrum with two peaks representing a

single frequency. However, a plot of wavelet coefficients clearly shows the exact location

in time of the discontinuity.

Fig 2.7 Fourier coefficients and wavelet coefficients

Wavelet analysis is capable of revealing aspects of data that other signal analysis

techniques miss aspects like trends, breakdown points, discontinuities in higher

derivatives, and self-similarity. Furthermore, because it affords a different view of data

than those presented by traditional techniques, wavelet analysis can often compress or de-

noise a signal without appreciable degradation. Indeed, in their brief history within the

signal processing field, wavelets have already proven themselves to be an indispensableaddition to the analysts collection of tools and continue to enjoy a burgeoning popularity

today.

Thus far, weve discussed only one-dimensional data, which encompasses most

ordinary signals. However, wavelet analysis can be applied to two-dimensional data



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(images) and, in principle, to higher dimensional data. This toolbox uses only one and

two-dimensional analysis techniques.

2.6 Wavelet Transform

When we analyze our signal in time for its frequency content, Unlike Fourier

analysis, in which we analyze signals using sines and cosines, now we use wavelet

functions.

2.6.1 The Continuous Wavelet Transform

Mathematically, the process of Fourier analysis is represented by the Fourier

transform:

( )

= dtetfF tj )(

Which is the sum over all time of the signal f(t) multiplied by a complex

exponential. (Recall that a complex exponential can be broken down into real and

imaginary sinusoidal components.) The results of the transform are the Fourier

coefficients F(w), which when multiplied by a sinusoid of frequency w yields the

constituent sinusoidal components of the original signal. Graphically, the process looks

like:

Fig 2.8Continuous Wavelets of different frequencies

Similarly, the continuous wavelet transform (CWT) is defined as the sum over all

time of signal multiplied by scaled, shifted versions of the wavelet function.



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= dttpositionscaletfpositionScaleC ),,()(),(

The result of the CWT is a series many wavelet coefficients C, which are a

function of scale and position.

Multiplying each coefficient by the appropriately scaled and shifted wavelet

yields the constituent wavelets of the original signal:

Fig 2.9 Continuous Wavelets of different scales and positions

The wavelet transform involves projecting a signal onto a complete set of

translated and dilated versions of a mother wavelet (t). The strict definition of a mother

wavelet will be dealt with later so that the form of the wavelet transform can be examined

first. For now, assume the loose requirement that (t) has compact temporal and spectral

support (limited by the uncertainty principle of course), upon which set of basis functions

can be defined. The basis set of wavelets is generated from the mother or basic wavelet is

defined as

=

a

bt

at

ba

1)(

,; a, b R1 and a>0 (2.2)

The variable a (inverse of frequency) reflects the scale (width) of a particular

basis function such that its large value gives low frequencies and small value gives high

frequencies. The variable b specifies its translation along x-axis in time. The term 1/

a is used for normalization. The 1-D wavelet transform is given by:



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= dtttxba bafw )()(),( , (2.3)

The inverse 1-D wavelet transform is given by:

( )

=0

2,)(),(

1

a

dadbtbaW

Ctx baf

(2.4)

Where

= dC

2)

<

(2.5)

X (t) is the Fourier transform of the mother wavelet (t). Cis required to be

finite, which leads to one of the required properties of a mother wavelet. Since Cmust be

finite, then x (t) =0 to avoid a singularity in the integral, and thus the x(t) must have zero

mean. This condition can be stated as

dtt)( = 0 (2.6)

and known as the admissibility condition. The other main requirement is that the mother

wavelet must have finite energy:

dtt2

)( < (2.7)

A mother wavelet and its scaled versions are depicted in figure 2.10 indicating the

effect of scaling.



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Fig 2.10 Mother wavelet and its scaled versions

Unlike the STFT which has a constant resolution at all times and frequencies, the

WT has a good time and poor frequency resolution at high frequencies, and good

frequency and poor time resolution at low frequencies.

2.6.2 The Discrete Wavelet Transform

Calculating wavelet coefficients at every possible scale is a fair amount of work,

and it generates an awful lot of data. What if we choose only a subset of scales and

positions at which to make our calculations? It turns out rather remarkably that if we

choose scales and positions based on powers of twoso-called dyadic scales and

positionsthen our analysis will be much more efficient and just as accurate. We obtain

such an analysis from the discrete wavelet transform(DWT)[12].

An efficient way to implement this scheme using filters was developed in 1988 by

Mallat. The Mallat algorithm is in fact a classical scheme known in the signal processing

community as a two-channel sub band coder. This very practical filtering algorithm

yields a fast wavelet transform a box into which a signal passes, and out of which

wavelet coefficients quickly emerge. Lets examine this in more depth.

Now consider, discrete wavelet transform (DWT), which transforms a discrete

time signal to a discrete wavelet representation. The first step is to discretize the wavelet

parameters, which reduce the previously continuous basis set of wavelets to a discrete

and orthogonal / orthonormal set of basis wavelets.



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If were talking about sinusoids, for example the effect of the scale factor is very easy to

see:

Fig 2.11 Scaling

The scale factor works exactly the same with wavelets. The smaller the scale factor, the

more compressed the wavelet.

Fig 2.12 Scaling

It is clear from the diagrams that for a sinusoid sin (wt) the scale factor a is

related (inversely) to the radian frequency w. Similarly, with wavelet analysis the scale

is related to the frequency of the signal.

2.7.2 Shifting



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Shifting a wavelet simply means delaying (or hastening) its onset.

Mathematically, delaying a function (t) by kis represented by (t-k).

Fig 2.13 Shifting

2.8 Decomposition of Wavelets

2.8.1 One-Stage Decomposition

For many signals, the low-frequency content is the most important part. It is what

gives the signal its identity. The high-frequency content on the other hand imparts flavor

or nuance. Consider the human voice. If you remove the high-frequency components, thevoice sounds different but you can still tell whats being said. However, if you remove

enough of the low-frequency components, you hear gibberish. In wavelet analysis, we

often speak of approximations and details. The approximations are the high-scale, low-

frequency components of the signal. The details are the low-scale, high-frequency

components. The filtering process at its most basic level looks like this:

The original signal S passes through two complementary filters and emerges as

two signals. Unfortunately, if we actually perform this operation on a real digital signal,

we wind up with twice as much data as we started with. Suppose, for instance that the

original signal S consists of 1000 samples of data. Then the resulting signals will each

have 1000 samples, for a total of 2000. These signals A and D are interesting, but we get

2000 values instead of the 1000 we had. There exists a more subtle way to perform the

decomposition using wavelets.



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Fig 2.14One-Stage Decomposition

By looking carefully at the computation, we may keep only one point out of two

in each of the two 2000-length samples to get the complete information. This is the

notion of down sampling. We produce two sequences called cA and cD.

Fig 2.15 Two-Stage Decomposition

The process on the right which includes down sampling produces DWT

Coefficients. To gain a better appreciation of this process lets perform a one-stage

discrete wavelet transform of a signal. Our signal will be a pure sinusoid with high-

frequency noise added to it.

Here is our schematic diagram with real signals inserted into it:

Notice that the detail coefficients cD is small and consist mainly of a high-

frequency noise, while the approximation coefficients cA contains much less noise than

does the original signal.



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You may observe that the actual lengths of the detail and approximation

coefficient vectors are slightly morethan half the length of the original signal.

Fig 2.16

2.8.2 Multi-step Decomposition and Reconstruction

A multi step analysis-synthesis process can be represented as: filters, and thus is

associated with the approximations of the wavelet decomposition.

Fig 2.17Decomposition and Reconstruction

In the same way this process involves two aspects: breaking up a signal to obtain

the wavelet coefficients, and reassembling the signal from the coefficients. Weve



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already discussed decomposition and reconstruction at some length. Of course, there isno point breaking up a signal merely to have the satisfaction of immediately

reconstructing it. We may modify the wavelet coefficients before performing the

reconstruction step. We perform wavelet analysis because the coefficients thus obtained

have many known uses, de-noising and compression being foremost among them. But

wavelet analysis is still a new and emerging field. No doubt, many uncharted uses of the

wavelet coefficients lie in wait. The Wavelet Toolbox can be a means of exploring

possible uses and hitherto unknown applications of wavelet analysis. Explore the toolbox

functions and see what you discover.

2.9 Wavelet Reconstruction

Weve learned how the discrete wavelet transform can be used to analyze or

decompose signals and images. This process is called decomposition or analysis. The

other half of the story is how those components can be assembled back into the original

signal without loss of information. This process is called reconstruction, or synthesis.

The mathematical manipulation that effects synthesis is called the inverse

discrete wavelet transforms(IDWT). To synthesize a signal in the Wavelet Toolbox, we

reconstruct it from the wavelet coefficients:



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Fig 2.18Wavelet Reconstruction

Up sampling is the process of lengthening a signal component by inserting zeros

between samples:

Signal component Upsampled signal component



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The Wavelet Toolbox includes commands like IDWT and wavered that perform

single-level or multilevel reconstruction respectively on the components of one-

dimensional signals. These commands have their two-dimensional analogs, idwt2 and

waverec2.

2.10 Dissimilarities between Fourier and Wavelet Transforms

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 usingwavelets\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. One way to see the time-frequency

resolution differences between the Fourier transform and the wavelet transform is to look

at the basis function coverage of the time-frequency plane. Figure 1.1 shows a windowed

Fourier transform, where the window is simply a square wave. The square wave window

truncates the sine or cosine function to a window of a particular width. Because a single

window is used for all frequencies in the WFT, the resolution of the analysis is the same

at all locations in the time-frequency plane.

An advantage of wavelet transforms is that the windows vary. In order to isolate

signal discontinuities, one would like to have some very short basis functions. At the

same time, in order to obtain detailed frequency analysis, one would like to have some

very long basis functions.



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Fig 2.19 Fourier basis functions, time-frequency tiles

and coverage of the time-frequency plane.

A way to achieve this is to have short high-frequency basis functions and long

low-frequency ones. This happy medium is exactly what you get with wavelet

transforms. Figure 1.2 shows the coverage in the time-frequency plane with one wavelet

function, the Daubechies [15] wavelet.

One thing to remember is that wavelet transforms do not have a single set of basis

functions like the Fourier transform, which utilizes just the sine and cosine functions.

Instead, wavelet transforms have an infinite set of possible basis functions. Thus wavelet

analysis provides immediate access to information that can be obscured by other time-

frequency methods such as Fourier analysis.



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Fig 2.20 Daubechies wavelet basis functions, time-frequency tiles

and coverage of the time-frequency plane.

2.11 WAVELET APPLICATIONS

The following applications show just a small sample of what researchers can do

with wavelets.

Computer and Human Vision

FBI Fingerprint compression

De-noising Noisy Data

Detecting self similar behavior in a time series

Musical tone generation



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CHAPTER 3

DWT Architecture

3.1 Discrete Wavelet Transform

The next step toward developing a DWT is to be able to transform a discrete time

signal. The wavelet transform can be interpreted as a applying a set of filters. Digital

filters are very efficient to implement and thus provide us with the needed tool for

performing the DWT, and are usually applied as equivalent low and high-pass filters. The

design of these filters is similar to subband coding, i.e., only the low pass filter has to be

designed such that the high pass filter has additional phase shift of 180 degree as

compared to the low pass filter. Unlike subband coding, these filters are designed to give

flat or smooth spectral response and are bi-orthogonal.

=

>== XMD debug Options. The XMD Debug Options dialog box

allows the user to specify the connections type and JTAG chain Definition . The

connection types are available for MicroBlaze:

Simulator enables XMD to connect to the MicroBlaze ISS

Hardware enables XMD to connect to the MDM peripheral in the hardware

Stub enables XMD to connect to the JTAG UART or UART via XMDSTUB

Virtual platform enables a virtual (c model) to be used

Verify that Hardware is selected



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Select Save

Select Debug -> Launch XMD

CHAPTER 5

Implementation

5.1 Fundamentals of Digital Image

Digital image is defined as a two dimensional function f(x, y), where x and y are

spatial (plane) coordinates, and the amplitude of f at any pair of coordinates (x, y) iscalled intensity or grey level of the image at that point. The field of digital image

processing refers to processing digital images by means of a digital computer. The digital

image is composed of a finite number of elements, each of which has a particular location

and value. The elements are referred to as picture elements, image elements, pels, and

pixels. Pixel is the term most widely used.

5.1.1 Image Compression

Digital Image compression addresses the problem of reducing the amount of data

required to represent a digital image. The underlying basis of the reduction process is

removal of redundant data. From the mathematical viewpoint, this amounts to

transforming a 2D pixel array into a statically uncorrelated data set. The data redundancy

is not an abstract concept but a mathematically quantifiable entity. If n1 and n2 denote

the number of information-carrying units in two data sets that represent the same

information, the relative data redundancy DR [2] of the first data set (the one

characterized by n1) can be defined as,

R

DC

R1

1=

Where RC called as compression ratio [2]. It is defined as

RC =2

1

n

n



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In image compression, three basic data redundancies can be identified and

exploited: Coding redundancy, interpixel redundancy, and phychovisal redundancy.

Image compression is achieved when one or more of these redundancies are reduced or

eliminated.

The image compression is mainly used for image transmission and storage. Image

transmission applications are in broadcast television; remote sensing via satellite, aircraft,

radar, or sonar; teleconferencing; computer communications; and facsimile transmission.

Image storage is required most commonly for educational and business documents,

medical images that arise in computer tomography (CT), magnetic resonance imaging

(MRI) and digital radiology, motion pictures, satellite images, weather maps, geological

surveys, and so on.

Fig 5.1 Block Diagram

5.1.2 Image Compression Types

There are two types image compression techniques.

1. Lossy Image compression

2. Lossless Image compression

5.1.2.1 Lossy Image compression

Lossy compression provides higher levels of data reduction but result in a less

than perfect reproduction of the original image. It provides high compression ratio.



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Lossy image compression is useful in applications such as broadcast television,

videoconferencing, and facsimile transmission, in which a certain amount of error is an

acceptable trade-off for increased compression performance.

5.1.2.2 Lossless Image compression

Lossless Image compression is the only acceptable amount of data reduction. It

provides low compression ratio while compared to lossy. In Lossless Image compression

techniques are composed of two relatively independent operations: (1) devising an

alternative representation of the image in which its interpixel redundancies are reduced

and (2) coding the representation to eliminate coding redundancies. Lossless Image

compression is useful in applications such as medical imaginary, business documents and

satellite images.

5.1.3 Image Compression Standards

There are many methods available for lossy and lossless, image compression. The

efficiency of these coding standardized by some Organizations. The International

Standardization Organization (ISO) and Consultative Committee of the International

Telephone and Telegraph (CCITT) are defined the image compression standards for both

binary and continuous tone (monochrome and Colour) images. Some of the Image

Compression Standards are

1. JBIG1

2. JBIG2

3. JPEG-LS

4. DCT based JPEG

5. Wavelet based JPEG2000

Currently, JPEG2000 [4] [5] is widely used because; the JPEG-2000 standard

supports lossy and lossless compression of single-component (e.g., grayscale) and

multicomponent (e.g., color) imagery. In addition to this basic compression functionality,

however, numerous other features are provided, including: 1) progressive recovery of an

image by fidelity or resolution; 2) region of interest coding, whereby different parts of an

image can be coded with differing fidelity; 3) random access to particular regions of an



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image without the needed to decode the entire code stream; 4) a flexible file format with

provisions for specifying opacity information and image sequences; and 5) good error

resilience. Due to its excellent coding performance and many attractive features, JPEG

2000 has a very large potential application base.

Some possible application areas include: image archiving, Internet, web browsing,

document imaging, digital photography, medical imaging, remote sensing, and desktop

publishing.

The main advantage of JPEG2000 over other standards, First, it would addresses a

number of weaknesses in the existing JPEG standard. Second, it would provide a number

of new features not available in the JPEG standard.

5.2 Lifting Scheme

The wavelet transform of image is implemented using the lifting scheme [3]. The

lifting operation consists of three steps. First, the input signal x[n] is down sampled into

the even position signal xe (n) and the odd position signal xo(n) , then modifying these

values using alternating prediction and updating steps.

]2[)( nxnxe = and ]12[)( += nxnxo

A prediction step consists of predicting each odd sample as a linear combination

of the even samples and subtracting it from the odd sample to form the prediction error.

An update step consists of updating the even samples by adding them to a linear

combination of the prediction error to form the updated sequence. The prediction and

update may be evaluated in several steps until the forward transform is completed. The

block diagram of forward lifting and inverse lifting is shown in figure 5.2

xe (n) s

x (n)

xo(n) d

Fig (a)

s


Split UP

MergeU P
http://www.final-yearprojects.co.cc/http://www.troubleshoot4free.com/fyp/http://www.final-yearprojects.co.cc/http://www.troubleshoot4free.com/fyp/http://www.troubleshoot4free.com/fyp/http://www.final-yearprojects.co.cc/http://www.troubleshoot4free.com/fyp/http://www.final-yearprojects.co.cc/http://www.troubleshoot4free.com/fyp/http://www.final-yearprojects.co.cc/http://www.troubleshoot4free.com/fyp/


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x (n)

d

Fig (b)

Fig 5.2 The Lifting Scheme. (a) Forward Transform (b) Inverse Transform

The inverse transform is similar to forward. It is based on the three operations undo

update, undo prediction, and merge. The simple lifting technique using Haar wavelet is

explained in next section.

5.2.1 Lifting using Harr

The lifting scheme is a useful way of looking at discrete wavelet transform. It is

easy to understand, since it performs all operations in the time domain, rather than in the

frequency domain, and has other advantages as well. This section illustrates the lifting

approach using the Haar Transform [6].

The Haar transform is based on the calculations of the averages (approximation

co-efficient) and differences (detail co-efficient). Given two adjacent pixels a and b, the

principle is to calculate the average2

)( bas

+= and the difference bad = . If a and b

are similar, s will be similar to both and d will be small, i.e., require few bits to represent.

This transform is reversible, since2

dsa = and

2

dsb += and it can be written using

matrix notation as

=

2/1

2/1),(),( bads

1

1= (a,b)A,

=2/1

1),(),( dsba

2/1

1= 1),(

Ads

Consider a row of n2 pixels values lnS , fornl 20


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Wavelet decompositions are widely used in signal and image processing

applications. Classical linear wavelet transforms perform homogeneous smoothing of

signal contents. In a number of cases, in particular in applications in image and video

processing, such homogeneous smoothing is undesirable. This has led to a growing

interest in nonlinear wavelet representations called as adaptive wavelet decomposition

that can preserve discontinuities such as transitions in signals and edges in images.

Adaptive wavelet decomposition is very useful in various applications, such as

image analysis, compression, and feature extraction and denoising. The adaptive

multiresolution representations take into account the characteristics of the underlying

signal and do leave intact important signal characteristics, such as sharp transitions,

edges, singularities, and other region of interests.

In the adaptive update lifting framework, the update lifting step, which yields a

low pass filter or a moving average process, is performed first on the input polyphase

components (the output from the splitting process, according to the lifting terminology),

followed by a fixed prediction step yielding the wavelet coefficients.

5.2.2 General Adaptive Update Lifting

We consider a (K + 1) band Filter bank decomposition with inputs x, y (1), y (2),

y (3).y (k), with 1K , which represent the polyphase components of the analyzed

signal. The first polyphase component, x, is updated using the neighboring signal

elements from the other polyphase components, thus yielding an approximation signal.

Subsequently, the signal elements in the polyphase components y (1), y (2)y (K) are

predicted using the neighboring signal elements from the approximated polyphase

component and the other polyphase components. The prediction steps, which are non-

adaptive, result in detail coefficients. The adaptive update step is illustrated in Figure 5.3.

x xx

y(1) .. .

y(2)


D Ud

+


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y(k)

Fig 5.3 Adaptive update lifting scheme

Here, x and y (1), y (2)y (K) are the input for a decision map D, whose output

at location n is binary decision

dn = D {y (1), y (2)y (K)} {0,1}

Which triggers the update filter Ud and the addition d. More precisely, if dn is the

binary decision at location n, then the updated value x1(n) is given by

))(()()(1

nyUnxnx idndn= ------------ (5.1)

We assume that the addition d is of the form xdu = d(x+u) with d #0, so that

the operation is invertible. The update filter is taken to be of the form

)())((2

1

, nynyU j

L

Lj

jdd =

= -------------

(5.2)

Where yj(n) = y(n+j) and L1 and L2 are nonnegative integers. The filter

coefficients d,j depend on the decision d at location n. Henceforth, we will usej to

denote the summation from L1 to L2.

From (5.1) and (5.2), we infer the update equation used at analysis:

=

+=N

j

jjdndn nynxnx1

,

1 )()()( ---------- (5.3)

Where jddjd ,, = . Clearly, we can easily invert (3.3) through

))()((1

)( ,1

nynxnx jj

jdn

dn

=

------------

(5.4)

Presumed that the decision dn is known at every location n. Thus, in order to have perfect

reconstruction, it must be possible to recover the decision dn = D(x, y)(n) from x1 (rather

than x which is not available at synthesis) and y. This amounts to the problem of finding

another decision map D1 such that



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))(,())(,(11

nyxDnyxD jj = ------------

(5.5)

Where x1 is given by (5.1). It can be shown that a necessary, but in no way sufficient,

condition for perfect reconstruction is that the value 11

, =+ =

N

j

jdndn .

5.2.3 Threshold Technique

The input images x, y1, y2 and y3 are obtained by a polyphase decomposition of an

original image x0 is given by,

x (m, n) = x0(2m, 2n)

y1 (m, n) = x0(2m, 2n+1)

y2 (m, n) = x0(2m+1, 2n)

y3 (m, n) = x0(2m+1, 2n+1)

Where x(m,n) represents the current location pixel value. y1(m,n),y2(m,n) and

y3(m,n) are horizontal, vertical and diagonal pixel value respectively. This is obtained by

using context formation, as shown in figure 5.3.The gradient vector v (n), with

components, {v1(n),v2(n),v3(n)}T

(where T represents transposition) is given by)()()(

11nynxnv = ;)()()( 22 nynxnv = ;

)()()(33

nynxnv = ;

Then the L2 norm of v (n) is given by

2

3

2

2

2

1)( vvvsd ++=

Here, the decision map takes binary decision. i.e.) either 1 or 0.

If, Tsd >)( then dn=1 and else dn=0.

Here d(s) is called as seminorm and T denotes the given Threshold. It can take

arbitrary value (user defined). Depending on the condition, the Decision map chooses the



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different update filters, followed the fixed prediction step. The filter equation for different

regions is given by

Decision Region I

If, Tsd >)( xx = 0.4 * y +0.2*yh+0.2*yv+0.2*yd

Decision Region II

If, Tsd )( xx = 0.5 * y + 0.2*yh+0.15*yv+0.15*yd.

Moreover, we are exclusively interested in the case where the decision map D 1 at

synthesis is of the same form as D, but possibly with a different threshold. Thus we need

that

11 )()( TsdTsd >>

The d1(s) is the norm for the gradient vector v 1(n) at synthesis. The v1 (n) is given by

)()()(1

1

11

1 nynxnv =

)()()(1

2

11

2 nynxnv =

)()()(1

3

11

3 nynxnv =

Where )(),(),(1

3

1

2

1

1nynyny corresponds the horizontal, the vertical, and diagonal

detail bands respectively.

The L2 norm of v1(n) is given by

23

122

121

11)()()()( vvvsd ++=

Then decision map is given by

If, Tsd >)(1 then d1n=1 and else d1n=0.

and the filter for different region is given by

Decision Region I

If, Tsd >)(1 x= (1/0.4) * (ry - (0.2*xh+0.2*xv+0.2*xd))

Decision Region II

If, Tsd )( x = (1/0.5)*(ry - (0.2*xh+0.15*xv+0.15*xd))



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Here, Different threshold values are choose arbitrarily and reconstructed. Then the

reconstructed image is compared to original image and found the Peak Signal to Noise

Ratio.

5.3 SPIHT Algorithm

Embedded zero tree wavelet (EZW) coding, introduced by J. M. Shapiro, is a very

effective and computationally simple technique for image compression. Here we offer an

alternative explanation of the principles of its operation, so that the reasons for its

excellent performance can be better understood. These principles are partial ordering by

magnitude with a set partitioning sorting algorithm, ordered bit plane transmission, and

exploitation of self-similarity across different scales of an image wavelet transform.

Moreover, we present a new and different implementation based on set partitioning in

hierarchical trees (SPIHT), which provides even better performance than our previously

reported extension of EZW that surpassed the performance of the original EZW. The

image coding results, calculated from actual file sizes and images reconstructed by the

decoding algorithm, are either comparable to or surpass previous results obtained through

much more sophisticated and computationally complex methods. In addition, the new

coding and decoding procedures are extremely fast, and they can be made even faster,

with only small loss in performance.

5.3.1 Progressive Image Transmission

After converting the image pixels into wavelet coefficient SPIHT [10] is applied.

We assume, the original image is defined by a set of pixel values jip , , where (i, j) the

pixel coordinates. The wavelet transform is actually done to the array given by,

)},({),( jipDWTjic = . --------------- (5.6)

Where c (i, j) is the wavelet coefficients.



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In SPIHT, initially, the decoder sets the reconstruction vector c to zero and

updates its components according to the coded message. After receiving the value

(approximate or exact) of some coefficients, the decoder can obtain a reconstructed

image by taking inverse wavelet transform,

)},({),( jicIDWTjip = --------------- (5.7)

called as progressive transmission.

A major objective in a progressive transmission scheme is to select the most

important information-which yields the largest distortion reduction-to be transmitted first.

For this selection, we use the mean squared-error (MSE) distortion measure

2

,,

2)(

1

1)( jiji

i j

MSE ppN

ppN

ppD == ---------- (5.8)

Where Nis the number of image pixels. jip , is the Original pixel value and jip , is the

reconstructed pixel value. Furthermore, we use the fact that the Euclidean norm is

invariant to the unitary transformation

2

,, )(1

)()( jijii j

MSEMSE ccN

ccDppD == ----------- (5.9)

From the above the equation, it is clear that if the exact value of the transform

coefficient jic , is sent to the decoder, then the MSE decreases by Nc ji /2

, . This means

that the coefficients with larger magnitude should be transmitted first because they have a

larger content of information. This is the progressive transmission method. Extending this

approach, we can see that the information in the value of jic , can also be ranked

according to itsbinary representation, and the most significant bits should betransmitted

first. This idea is used, for example, in the bit-plane method for progressive transmission.

Following, we present a progressive transmission scheme that incorporates these two

concepts: ordering the coefficients by magnitude and transmitting the most significant

bits first. To simplify the exposition, we first assume that the ordering information is

explicitly transmitted to the decoder. Later, we show a much more efficient method to

code the ordering information.

5.3.2 Set Partitioning Sorting Algorithm



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One of the main features of the proposed coding method is that the ordering data

is not explicitly transmitted. Instead, it is based on the fact that the execution path of any

algorithm is defined by the results of the comparisons on its branching points. So, if the

encoder and decoder have the same sorting algorithm, then the decoder can duplicate the

encoders execution path if it receives the results of the magnitude comparisons, and the

ordering information can be recovered from the execution path.

One important fact used in the design of the sorting algorithm is that we do not

need to sort all coefficients. Actually, we need an algorithm that simply selects the

coefficients such that1

, 22+


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=)(TSn 1;n

ji

Tji

cMaxm

2}{ ,),(

---------------- (5.11)

0; otherwise.

to indicate the significance of a set of coordinates T . To simplify the notation of single

pixel sets, we write )}),({( jiSn as ),( jiSn .

5.3.3 Spatial Orientation Trees

A tree structure, called spatial orientation tree, naturally defines the spatial

relationship on the hierarchical pyramid. Fig.5.4 shows how our spatial orientation tree is

defined in a pyramid constructed with recursive four-subband splitting. Each node of the

tree corresponds to a pixel and is identified by the pixel coordinate. Its direct descendants

(offspring) correspond to the pixels of the same spatial orientation in the next finer level

of the pyramid. The tree is defined in such a way that each node has either no offspring

(the leaves) or four offspring, which always form a group of 2 x 2adjacent pixels.

In Fig 5.4, the arrows are oriented from the parent node to its four offspring. The

pixels in the highest level of the pyramid are the tree roots and are also grouped in 2 x 2

adjacent pixels. However, their offspring branching rule is different, and in each group,

one of them (indicated by the star in Fig.5.4)has no descendants.



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Fig 5.4 Examples of parent-offspring dependencies in the

Spatial-orientation tree

The following sets of coordinates are used to present the new coding method:

O (i,j): set of coordinates of all offspring of node (i, j);

D (i, j ) : set of coordinates of all descendants of the node

H: set of coordinates of all spatial orientation tree roots (nodes in the highest

pyramid level);

L(i, j ) = D(i, j ) - O(i, j )

For instance, except at the highest and lowest pyramid levels, we have

O(i,j) = {(2i,2j),(2i,2j+1),(2i+1,2j),(2i+1,2j+1)}

We use parts of the spatial orientation trees as the partitioning subsets in the sorting

algorithm. The set partition rules are simply the following.

I)The initial partition is formed with the sets ((i, j ) ) and D(i,j ), for all (i,j)

H.2) If D(i,j) is significant, then it is partitioned into L(i,j ) plus the four single-element sets

with ( k , I ) O(i,j ) .

3)If L(i,j ) is significant, then it is partitioned into the four sets D(k,I), with (k,I) O(i,j )



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Future Scope

Future work aims at extending this frame work for color images, video

compressions, and De-noising applications



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Conclusion

The architecture has been implemented using .c; The model has been verifiedwith a set of text data. the derived architecture has many advantages especially the

reduction in the number of operations per sample, leading to a reduced chip area and

power consumption .it can be said that the original design criteria has been considered

and an effective and feasible architecture has been designed.

The major disadvantage of this design is that it cannot perform online transform

i.e. the output will not be generated in all instance of data input, this work can be further

extended by using multi processor architecture to speed up the data usage. The maximumclock speed achievable is around 131.6 MHz.

.



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BIBLIGRAPHY

1. Ajith Boopardikar, Wavelet Theory and Application,TMH

2. I.Daubechies W. Sweldens, Factoring wavelet transforms into lifting schemes,

J. Fourier Anal. Appl., vol. 4, pp. 247269, 1998.

3. W. Sweldens, The lifting scheme: A new philosophy in biorthogonal wavelet

constructions, inProc. SPIE, vol. 2569, 1995, [3]

4. JPEG2000 Committee Drafts [Online].

Available: http://www.jpeg.org/CDs15444.htm

5. JPEG2000 Verification Model 8.5 (Technical Description), Sept. 13,2000.

6. K. Andra, C. Chakrabarti, and T. Acharya, A VLSI architecture for lifting basedwavelet transform, in Proc. IEEE Workshop SignlProces. Syst., Oct. 2000,

pp7079.

7. Real-time image compression based on wavelet vector quantization, algorithm

and VLSI architecture ,Hatami, S. Sharifi, S. Ahmadi, H. Kamarei, M.

Dept. of Electr. & Comput. Eng., Univ. of Tehran, Iran; IEEE Trans,May 2005.

8. Fourier Analysis- http://www.sunlightd.com/Fourier/

9. A VLSI Architecture for Lifting-Based Forward and Inverse Wavelet Transform

Kishore Andra, Chaitali Chakrabarti, Member, IEEE, and Tinku Acharya, Senior

Member, IEEE, IEEE Trans. on Signal Processing, vol. 50,No.4, April 2002.

10. K. K. Parhi and T. Nishitani, VLSI architectures for discrete wavelet

transforms,IEEE Trans. VLSI Syst., vol. 1, pp. 191202, June 1993.



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11. M. Ferretti and D. Rizzo, A parallel architecture for the 2-D discrete wavelet

transform with integer lifting scheme, J. VLSI Signal Processing, vol. 28, pp.

165185, July 2001.

12. Discrete Wavelet Transform

http://en.wikipedia.org/wiki/Discrete_wavelet_transform

13. A. Rushton, "VHDL for logic synthesis," Wiley, 1998 5

14. C. H. Roth, "Digital systems design using VHDL," PWS Pub. Co., 1998 5

15. I. Daubechies, "Orthonormal Basis of Compactly Supported Wavelets," Comm. in

Pure and Applied Math Vol. 41, No. 7, pp. 909 -996, 1988 3

16. Chih-Chi Cheng; Chao-Tsung Huang; Ching-Yeh Chen; Chung-Jr Lian; Liang-

Gee Chen, "On-Chip Memory Optimization Scheme for VLSI Implementation of

Line-Based Two-Dimentional Discrete Wavelet Transform," Circuits and

Systems for Video Technology, IEEE Transactions on , vol.17, no.7, pp.814-822,

July 2007



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APPENDIX

Source Code#include

#include

#define ROW 64

#define COL 32

#define R_1 64

#define C_1 64

#define True 1

#define False 0

#define true1 1

#define false1 0

#define Len 4096

int Input[64][64];

int REDB[R_1][C_1];

int st[40000];

int Len_Array=10000;

int Ad[64][64];

int Adr[64][64];

int desc[4][3];

int xDim=64;

int yDim=64;



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int level=1;

int rowS=32;

int colS=32;

int rowL=64;

int colL=64;

int L;

int max;

int currSet[1][4];

int currset;

int D=0;

int T=256;

int Level=2;

int rows=64;

int columns=64;

int Even[64][32];

int Odd[64][32];

int Low[64][32];

int High[64][32];

int LEven[32][32];

int wavedecode[64][64];

int waveencodeImage[64][64];

int HOdd[32][32];

int LOdd[32][32];

int HEven[32][32];

int LL[32][32];

int LH[32][32];

int HL[32][32];

int HH[32][32];

int RLL[32][32];

int RHL[32][32];

int RHH[32][32];



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int RLH[32][32];

int totalbitCount;

int RL[ROW][COL];

int RH[ROW][COL];

int R[ROW][COL];

int H[ROW][COL];

int Output[ROW][ROW];

FILE *R1=0x22000000;

FILE *R2;

int aa;

int cc;

struct descArr{

int desc1[4];

int desc2[4];

int desc3[4];

};

struct descArr descArrS;

void integerdwt();

void reversedwt();

/////////////////////////////////////////////////////////////////////

void integerdwt()

{//begin of integer dwt

int i,j,k,a;

int columns1;

int rows1;

rows1=rows/2;

columns1=columns/2;

//////// one dimensional Even component////////



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for (j=0;j


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}

}

///////////////////one dimensional L even

for (j=0;j


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for (k=0;k


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for(i=0;i


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k=0;

for(i=1;i


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}

for(i=0;i


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int v,m,t;

char ch;

int manti,tempI,remind;

char charVa;

int i,j,ind;

int index2=0;

int temp1,temp2,temp3;

int min,index,k;

int val;

char *cp=0x22000000;

int matVal=0;

int intValC;

for( i=0;i


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matVal=(matVal*10)+3;

}

else if(ch=='4'){


}

else if(ch=='5'){


}

else if(ch=='6'){


}

else if(ch=='7'){


}

else if(ch=='8'){


}

else if(ch=='9'){


}

else{

//printf(" %d ",matVal);

REDB[i][j]=matVal;

matVal=0;

cp++;

ch=*cp;

while(ch=='\n'||intValC==010||intValC==020||intValC==003||intValC==032){

cp++;

ch=*cp;

}

break; //break while



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}

intValC=*cp;

if(ch=='\n'||intValC==010||intValC==020||intValC==003||intValC==032){

//printf("LB");

}

cp++;

}//while loop end

}// for j

}//for i

for( i=0;i


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}

integerdwt();

// printf("matrix all: \n");

for(i=0;i


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for( i=0;i


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}

}

else{

if(j


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return 0;

}//end of main loop

Simulation & Synthesis Result

1 Simulation Results

1.1 Matlab output



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The above figure shows the Matlab GUI interface. The input image is loaded

using image button and the image is converted into text file using convert_text

button.

1.2 Xilinx Results

Input file:



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This figure shows the text files of even, odd, LL, LH, HL, HH.

Comparing input text file with the retrieval text file (after decompression)



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Verifying input text file in Matlab GUI interface.



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Verifying retrieval text file in Matlab GUI interface.



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Synthesis Results

Input Image:



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Compressed Image:-



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Reconstructed Image:-



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3 Synthesis Report


Overview

Generated on Fri Apr 03 12:03:07 2010

Source D:/New_Folder/lifting/system.xmp

EDK Version 8.1.02

FPGA Family spartan3e

Device xc3s500efg320-4

# IP Instantiated 17

# Processors 1

# Busses 3


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Timing Information

Post Synthesis Clock Limits

These are the post synthesis clock frequencies. The critical frequencies are marked with green.The values reported here are post synthesis estimates calculated for each individual module. These values will changeafter place and route is performed on the entire system.

MODULE CLK Port MAX FREQ

microblaze_0 FSL3_S_CLK 65.595MHz

microblaze_0 DBG_CLK 65.595MHz

DDR_SDRAM_16Mx16 Device_Clk 83.914MHz

DDR_SDRAM_16Mx16 OPB_Clk 83.914MHz

DDR_SDRAM_16Mx16 DDR_Clk90_in 83.914MHz

DDR_SDRAM_16Mx16 Device_Clk90_in 83.914MHz

DDR_SDRAM_16Mx16 Device_Clk90_in_n 83.914MHz

DDR_SDRAM_16Mx16 Device_Clk_n 83.914MHz

DDR_SDRAM_16Mx16 DDR_Clk90_in_n 83.914MHz

opb_intc_0 OPB_Clk 117.233MHz

RS232_DCE OPB_Clk 138.083MHz

debug_module debug_module/drck_i 146.951MHz



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debug_module OPB_Clk 146.951MHz

debug_module bscan_update 146.951MHz

mb_opb OPB_Clk 181.719MHz

ilmb LMB_Clk 249.128MHz

dlmb LMB_Clk 249.128MHz
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