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A NOVEL ARCHITECTURE OF DISCRETE WAVELET TRANSFORM USING LIFTING SCHEME ALGORITHM

CHAPTER 1CHAPTER 1

INTRODUCTION

1.1 Introduction

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

some researchers in the wavelet field feel that, by using wavelets, one is adopting a

perspective in processing data. Wavelets are functions that satisfy certain mathematical

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

Approximation using superposition of functions has existed since the early 1800's, when

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

functions. However, in wavelet analysis, the scale that we use to look at data plays a

special role. Wavelet algorithms process data at different scales or resolutions.

Fourier Transform (FT) with its fast algorithms (FFT) is an important tool for

analysis and processing of many natural signals. FT has certain limitations to characterize

many natural signals, which are non-stationary (e.g. speech). Though a time varying,

overlapping window based FT namely STFT (Short Time FT) is well known for speech

processing applications, a time-scale based Wavelet Transform is a powerful

mathematical tool for non-stationary signals.

Wavelet Transform uses a set of damped oscillating functions known as wavelet

basis. WT in its continuous (analog) form is represented as CWT. CWT with various

deterministic or non-deterministic bases is a more effective representation of signals for

analysis as well as characterization. Continuous wavelet transform is powerful in

singularity detection. A discrete and fast implementation of CWT (generally with realvalued basis) is known as the standard DWT (Discrete Wavelet Transform).With standard

DWT, signal has a same data size in transform domain and therefore it is a non-redundant

transform. A very important property was Multi-resolution Analysis (MRA) allows DWT

to view and process.

For many natural signals, the wavelet transform is a more effective tool than the

Fourier transform. The wavelet transform provides a multiresolution representation using

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a set of analysing functions that are dilations and translations of a few functions

(wavelets).

These web pages describe an implementation in Mat lab of the discrete wavelet

transforms (DWT). The programs for 1D, 2D, and 3D signals are described separately,

but they all follow the same structure. Examples of how to use the programs for 1D

signals, 2D images and 3D video clips are also described. As discrete wavelet transform

are based on perfect reconstruction two-channel filter banks, the programs below for the

(forward/inverse) DWT call programs for (analysis/synthesis) filter banks. The DWT

consists of recursively applying a 2-channel filter bank - the successive decomposition is

performed only on the low pass output. In each section below, the 2-channel filter banks

are described first.

The wavelet transform comes in several forms. The critically-sampled form of the

wavelet transform provides the most compact representation; however, it has several

limitations. For example, it lacks the shift-invariance property, and in multiple

dimensions it does a poor job of distinguishing orientations, which is important in image

processing. For these reasons, it turns out that for some applications improvements can beobtained by using an expansive wavelet transform in place of a critically-sampled one.

(An expansive transform is one that converts an N-point signal into M coefficients with

M > N.) There are several kinds of expansive DWTs; here we describe and provide an

implementation of the dual-tree complex discrete wavelet transform.

The dual-tree complex wavelet transform overcomes these limitations - it is nearly

shift-invariant and is oriented in 2D [Kin-2002]. The 2D dual-tree wavelet transform

produces six sub bands at each scale, each of which is strongly oriented at distinct angles.In addition to being spatially oriented, the 3D dual-tree wavelet transform is also motion

selective - each sub band is associated with motion in a specific direction. The 3D dual-

tree isolates in its sub bands motion in distinct directions.

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Introduction to Wavelet Transform

The wavelet transform is computed separately for different segments of the time-

domain signal at different frequencies. Multi-resolution analysis: analyzes the signal at

different frequencies giving different resolutions. Multi-resolution analysis 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. Good for signal having

high frequency components for short durations and low frequency components for long

duration, e.g. Images and video frames.

1.2.1 Wavelet Definition

A wavelet is a small wave which has its energy concentrated in time. It has an

oscillating wavelike characteristic but also has the ability to allow simultaneous time and

frequency analysis and it is a suitable tool for transient, non-stationary or time-varying

phenomena.

(a) (b)

Figure1.1 Representation of a (a) wave (b) wavelet

1.2.2 Wavelet Characteristics

The difference between wave (sinusoids) and wavelet is shown in figure 1.1.

Waves are smooth, predictable and everlasting, whereas wavelets are of limited duration,

irregular and may be asymmetric. Waves are used as deterministic basis functions in

Fourier analysis for the expansion of functions (signals), which are time-invariant, or

stationary. The important characteristic of wavelets is that they can serve as deterministic

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or non-deterministic basis for generation and analysis of the most natural signals to

provide better time-frequency representation, which is not possible with waves using

conventional Fourier analysis.

1.2.3 Wavelet Analysis

The wavelet analysis procedure is to adopt a wavelet prototype function, called an

analyzing wavelet 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. Mathematical

formulation of signal expansion using wavelets gives Wavelet Transform (WT) pair,

which is analogous to the Fourier Transform (FT) pair. Discrete-time and discrete-

parameter version of WT is termed as Discrete Wavelet Transform (DWT).

1.3 Types of Transforms

1.3.1 Fourier Transform (FT)

Fourier transform is a well-known mathematical tool to transform time-domain

signal to frequency-domain for efficient extraction of information and it is reversible also.

For a signal x(t), the FT is given by

Though FT has a great ability to capture signals frequency content as long as x(t)

is composed of few stationary components (e.g. sine waves). However, any abrupt change

in time for non-stationary signal x(t) is spread out over the whole frequency axis in X(f).

Hence the time-domain signal sampled with Dirac-delta function is highly localized in

time but spills over entire frequency band and vice versa. The limitation of FT is that it

cannot offer both time and frequency localization of a signal at the same time.

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1.3.2 Short Time Fourier Transform (STFT)

To overcome the limitations of the standard FT, Gabor introduced the initial

concept of Short Time Fourier Transform (STFT). The advantage of STFT is that it uses

an arbitrary but fixed-length window g(t) for analysis, over which the actual non-

stationary signal is assumed to be approximately stationary. The STFT decomposes such

a pseudo-stationary signal x(t) into a two dimensional time-frequency representation S( ,

f) using that sliding window g(t) at different times . Thus the FT of windowed signal x

(t) g*(t-) yields STFT as

The time-frequency resolution is fixed over the entire time-frequency plane

because the same window is used at all frequencies. There is always a trade off between

time resolution and frequency resolution in STFT.

1.3.3 Wavelet Transform (WT)Fixed resolution limitation of STFT can be resolved by letting the resolution in

time-frequency plane in order to obtain Multi resolution analysis. The Wavelet Transform

(WT) in its continuous (CWT) form provides a flexible time-frequency, which narrows

when observing high frequency phenomena and widens when analyzing low frequency

behaviour. Thus time resolution becomes arbitrarily good at high frequencies, while the

frequency resolution becomes arbitrarily good at low frequencies. This kind of analysis is

suitable for signals composed of high frequency components with short duration and low

frequency components with long duration, which is often the case in practical situations.

1.3.4 Comparative Visualisation

The time-frequency representation problem is illustrated in figure1.2. A

comprehensive visualization of various time-frequency representations shown in figure

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1.2, demonstrates the time-frequency resolution for a given signal in various transform

domains with their corresponding basis functions.

Figure1.2 Comparative visualizations of time-frequency representation of an

arbitrary non-stationary signal in various transform domains

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Difference between Continuous Wavelet Transform and Discrete

Wavelet Transform

Wavelet transforms are classified into discrete wavelet transforms (DWTs) and

continuous wavelet transforms (CWTs). Note that both DWT and CWT are continuous-

time (analog) transforms. They can be used to represent continuous-time (analog) signals.

CWTs operate over every possible scale and translation whereas DWTs use a specific

subset of scale and translation values or representation grid.

The word wavelet is due to Morlet and Grossmann in the early 1980s. They used

the French word ondelette, meaning "small wave". Soon it was transferred to English by

translating "onde" into "wave", giving "wavelet".

The Wavelet transform is in fact an infinite set of various transforms, depending

on the merit function used for its computation. This is the main reason, why we can hear

the term "wavelet transforms" in very different situations and applications.

Orthogonal wavelets are used to develop the discrete wavelet transform

Non-orthogonal wavelets are used to develop the continuous wavelet transform

There are more wavelet types and transforms, but those two are most widely used and can

serve as examples of two main types of the wavelet transform: redundant and non-

redundant ones.

The discrete wavelet transform returns a data vector of the same length as the

input is. Usually, even in this vector many data are almost zero. This corresponds

to the fact that it decomposes into a set of wavelets (functions) that are orthogonal

to its translations and scaling. Therefore we decompose such a signal to a same or

lower number of the wavelet coefficient spectrum as is the number of signal data

points. Such a wavelet spectrum is very good for signal processing and

compression, for example, as we get no redundant information here.

The continuous wavelet transform in contrary returns an array one dimension

larger than the input data. For a 1D data we obtain an image of the time-frequency

plane. We can easily see the signal frequencies evolution during the duration of

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the signal and compare the spectrum with other signals spectra. As here is used

the non-orthogonal set of wavelets, data are correlated highly, so big redundancy

is seen here. This helps to see the results in a more humane form.

Applications of Discrete Wavelet Transform

Generally, an approximation to DWT is used for data compression if signal is

already sampled, and the CWT for signal analysis. Thus, DWT approximation is

commonly used in engineering and computer science, and the CWT in scientific research.

One use of wavelet approximation is in data compression. Like some other transforms,

wavelet transforms can be used to transform data and then encode the transformed data,resulting in effective compression. For example, JPEG 2000 is an image compression

standard that uses orthogonal wavelets. A related use is that of smoothing/denoising data

based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively

thresholding the wavelet coefficients that correspond to undesired frequency components

smoothing and/or denoising operations can be performed. Other applied fields that are

making use of wavelets include astronomy, acoustics, nuclear engineering, sub-band

coding, signal and image processing, neurophysiology, music, magnetic resonanceimaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar,

human vision, and pure mathematics applications such as solving partial differential

equations.

Area of Application

Medical application

Signal denoising Data compression

Image processing

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CHAPTER 2CHAPTER 2

Literature Review

2.1Introduction

VLSI Architecture to deign the Discrete Wavelet Transform for medical images

storage and retrieval is carried out. Lossless is usually required in the medical image field.

The word length required for lossless makes too expensive Thus, there is a clear need for

designing architecture to implement the lossless DWT for medical images. The data path

word-length has been selected to ensure the lossless accuracy criteria leading a high speed

implementation with small chip area.The DWT represents the signal in dynamic sub-band decomposition. Generation of

the DWT in a wavelet packet allows sub-band analysis without the constraint of dynamic

decomposition. The discrete wavelet packet transform (DWPT) performs an adaptive

decomposition of frequency axis. The specific decomposition will be selected according to an

optimization criterion

The Discrete Wavelet Transform (DWT), based on time-scale representation,

provides efficient multi-resolution sub-band decomposition of signals. It has become apowerful tool for signal processing and finds numerous applications in various fields such as

audio compression, pattern recognition, texture discrimination, computer graphics etc.

Specifically the 3-D DWT play a significant role in many image/video coding applications.

2.1Types of compressions

There are two types of compressions

1. Lossless compressionDigitally identical to the original image. Only achieve a modest amount of

compression

2. Lossy compression

Discards components of the signal that are known to be redundant. Signal is therefore

changed from input

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Lossless compression involves with compressing data, when decompressed data will

be an exact replica of the original data. This is the case when binary data such as

executable are compressed.

Figure 2.1 Different Types of Lossy Compression Techniques

2.2 Reviewed Architectures of Discrete wavelet transforms and inverse

discrete wavelet transforms

2.2.1 Discrete Wavelet Transform

The Discrete Wavelet Transform (DWT) is a popular signal processing technique best

known for its results in data compression. As hardware designers, we are concerned more

with the algorithmic details of the DWT, rather than the mathematical details discussed in the

many papers which provide the foundations for wavelets. Algorithmically, the DWT is a

recursive filtering process. At each level, the input data is filtered by two related filters to

produce two result data-streams. These data-streams are then sub samples by two (or

decimated) to reduce the output to the same number of data-words as the original signal.

The low-pass filter output of this result is then further processed by the same two filters, and

this continues recursively for the desired depth oruntil no further filtering can occur. This

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HybridPredictive Frequencyoriented

Importanc

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DCT DWT

Transfor

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Fracta

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MallatTransversal

filter CoedicLifting

Scheme

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recursive filtering process of the one-dimensional DWT is shown in Figure 1, where z is the

input data-stream, a and dare approximation (low-pass filter output) and difference (high-

pass filter output) data-streams respectively. The subscript values show the level of output.

Figure 2.2 The DWT filtering process.

The filtering steps are multiply and accumulate operations. A filter in the algorithmic,

discrete sense is a number of coefficient values. The number of these values is referred to

as the filter width and these coefficients are also referred to as taps. At each data-word of

the input, the filter spans across that data-word and its neighboring data-words as a

window. The values within this window are multiplied by their corresponding filter

coefficient and all the results are added together to give the filtered result for this data-word.

The filtering operation extracts certain frequency information from the data depending on the

characteristics of the filter. This filtering operation can be done with a systolic array. It is

simple to implement a systolic array for each level of the DWT, but the arrays are poorly

utilized due to the decreasing data-rates of the levels. It is possible, through some complex

timing, to use a single array to perform all levels of the DWT.

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2.2.2 Inverse discrete wavelet transforms

The inverse DWT (IDWT) is the computational reverse. The lowest low-pass and

high pass data-streams are up-sampled (i.e. a zero is placed between each data-word) and

then filtered using filters related to the decomposition filters. The two resulting streams are

simply added together to form the low-pass result of the previous level of processing. This

can be combined with the high-pass result in a similar fashion to produce further levels, the

process continuing until the original data-stream is reconstructed. This process is shown in

figure

Figure 2.3 The Inverse DWT filtering process.

We have previously developed an array for the DWT and are now designing an arrayfor the IDWT. We present a simple discussion of the array with no detailed implementation

specifics to allow the reader to understand the issues we are dealing with by the input

buffering approach.

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Discrete Wavelet Transform Architecture

The discrete wavelet transform (DWT) is being increasingly used for image coding.

This is due to the fact that DWT supports features like progressive image transmission (by

quality, by resolution), ease of compressed image manipulation, region of interest coding, etc.

DWT has traditionally been implemented by convolution. Such an implementation demands

both a large number of computations and a large storage features that are not desirable for

either high-speed or low-power applications. Recently, a lifting-based scheme that often

requires far fewer computations has been proposed for the DWT. The main feature of the

lifting based DWT scheme is to break up the high pass and low pass filters into a sequence of

upper and lower triangular matrices and convert the filter implementation into banded matrix

multiplications. Such a scheme has several advantages, including in-place computation of

the DWT, integer-to-integer wavelet transform (IWT), symmetric forward and inverse

transform, etc. Therefore, it comes as no surprise that lifting has been chosen in the

upcoming.

The proposed architecture computes multilevel DWT for both the forward and the

inverse transforms one level at a time, in a row-column fashion. There are two row

processors to compute along the rows and two column processors to compute along the

columns. While this arrangement is suitable or filters that require two banded-matrix

multiplications filters that require four banded-matrix multiplications require all four

processors to compute along the rows or along the columns. The outputs generated by the

row and column processors (that are used for further computations) are stored in memory

modules.

The memory modules are divided into multiple banks to accommodate high

computational bandwidth requirements. The proposed architecture is an extension of the

architecture for the forward transform that was presented. A number of architectures havebeen proposed for calculation of the convolution-based DWT. The architectures are mostly

folded and can be broadly classified into serial architectures (where the inputs are supplied to

the filters in a serial manner) and parallel architectures (where the inputs are supplied to the

filters in a parallel manner).

Recently, a methodology for implementing lifting-base DWT That redues the

Memory requirements and communication between the processors, when the image is broken

up info blocks. For a system that consists of the lifting-based DWT transform followed by an

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embedded zero-tree algorithm, a new interleaving scheme that reduces the number of

memory accesses has been proposed. Finally, a lifting-based DWT architecture capable of

performing filters with one lifting step, i.e., one predict and one update step. The outputs are

generated in an interleaved fashion.

Figure 2.4 Lifting Schemes. (a) Scheme 1. (b) Scheme 2.

The basic principle of the lifting scheme is to factorize the poly phase matrix of a

wavelet filter into a sequence of alternating upper and lower triangular matrices and a

diagonal matrix. This leads to the wavelet implementation by means of banded-matrix

multiplications.

Let and be the low pass and high pass analysis filters, and let and

be the low pass and high pass synthesis filters. The corresponding poly-phase matrices

are defined as

If is a complementary filter pair, then can always be factored into lifting steps as

Where K is a constant. The two types of lifting schemes are shown in Figure

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Scheme 1 which corresponds to the factorization consists of three steps:

Predictstep, where the even samples are multiplied by the time domain equivalent of

and are added to the odd samples. Update step, where updated odd samples are multiplied by the time domain

equivalent of and are added to the even samples.

Scalingstep, where the even samples are multiplied by 1/k and odd samples by k.

The inverse DWT is obtained by traversing in the reverse direction, changing the

factor K to 1/K, K factor to i/K, and reversing the signs of coefficients in and . In

Scheme 2 which corresponds to the factorization, the odd samples are calculated inthe first step, and the even samples are calculated in the second step. The inverse is obtained

by traversing in the reverse direction.

The lifting scheme is a technique for both designing wavelets and performing the

discrete wavelet transform. Actually it is worthwhile to merge these steps and design the

wavelet filters while performing the wavelet transform. This is then called the second

generation wavelet transform. The technique was introduced by Wim Sweldens.

The discrete wavelet transform applies several filters separately to the same signal. In

contrast to that, for the lifting scheme the signal is divided like a zipper. Then a series of

convolution-accumulate operations across the divided signals is applied.

The basic idea of lifting is the following: If a pair of filters (h,g) is complementary,

that is it allows forperfect reconstruction, then for every filters the pair (h',g) with

allows for perfect reconstruction, too. Of course, this is

also true for every pair (h,g') of the form . The converse is

also true: If the filter banks (h,g) and (h',g) allow for perfect reconstruction, then there is a

unique filters with .

Each such transform of the filter bank (or the respective operation in a wavelet

transform) is called a lifting step. A sequence of lifting steps consists of alternating lifts, that

is, once the low pass is fixed and the high pass is changed and in the next step the high pass is

fixed and the low pass is changed. Successive steps of the same direction can be merged.

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2.3.1 Advantages

1. Lifting schema of DWT has been recognized as a faster approach.

2. No need to divide the input coding into non-overlapping 3-D blocks. It has higher

compression ratios avoid blocking artefacts.

3. Allows good localization both in time and spatial frequency domain

4. Better identification of which data is relevant to human perception higher

compression ratio

2.3.2 Disadvantages

1. The cost of computing DWT as compared to DCT is higher because the number of

logic gates used for DWT is more compared to DCT.

2. The use of larger DWT basis functions or wavelet filters produces blurring and

ringing noise near edge regions in the images or video frames.

2.4 Lifting Implementation of the Discrete Wavelet Transform

The DWT has been traditionally implemented by convolution or FIR filter bank

structures. The DWT implementation is basically a frame-based as opposed /to the block-

based implementation ofdiscrete cosine transforms (DCT) /or similar transformations. Such

an implementation requires both a large number of arithmetic computations and a large

memory for storage features /that are not desirable for either high-speed or low-power

image and /video processing applications. Recently, a new mathematical formulation for

/wavelet transformation has been proposed by Swelden based on spatial construction of the

wavelets and a very versatile scheme for its factorization /has been suggested in. This new

approach is called the lifting-based /wavelet transform, or simply lifting. The main feature of

the lifting-based DWT scheme is to break up the high-pass and low-pass wavelet filters into

a sequence of smaller filters that in turn can be converted into a sequence of upper and lower

triangular matrices, which will be discussed in the subsequent section.

This scheme often requires far fewer computations compared to the convolution-

based DWT, and its computational complexity can be reduced up to 50%. It has several other

advantages, including in-place computation of the DWT, integer-to-integer wavelet

transform (IWT), symmetric forward and inverse transform, requiring no signal boundary

extension, etc. Asa result, lifting-based hardware implementations provide an efficient way

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to compute wavelet transforms compared to traditional approaches. So it comes as no surprise

that lifting has been suggested for implementation of the DWT in the upcoming JPEG2000

standard [8]. In a traditional forward DWT using a filter bank, the input signal (x) is filtered

separately by a low-pass filter( h ) and a high-pass filter ( g ) at each/transform level. The

two output streams are then subsampled by simply dropping the alternate output samples in

each stream to produce the lowpass ( y ~a)nd high-pass ( y ~su)b bands as shown in Figure

4.6. These two filters (k , i j ) form the analysis filter bank. The original signal can be

reconstructed by asynthesis filter bank(h,g) starting fromy~ and Y Has shown in Figure 4.6.

We have adopted the discussion on lifting from the celebrated paper by Daubechies and

Sweldens (141. It should also be noted that we adopted the notation (h, g ) for the analysis

filter and (h, g ) as the synthes is filter in this section and onward in this chapter. Given a

discrete signalx ( n ) , arithmetic computation of above can be expressed as follows:

Where TL arid TH are the lengths of the low-pass (K) and high-pass ( 3 ) filters

respectively. During the inverse transform to reconstruct the signal, both y~ and Y Hare first

up-sampled by inserting zeros between two samples and then they are filtered by low-pass(h) and high-pass (9) filters respectively. These two filtered output streams are added together

to obtain the reconstructed signal (2') as shown in Figure 4.6.

Figure 2.5 Signal Analysis and Reconstruction in DWT

There are two types of lifting. One is called primal liftingand the other is called dual

lifting. We define these two types of lifting based on the mathematical formulations shown in

the previous section.

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2.5 Summary of Literature Review

1. Lifting scheme of discrete wavelet transform and Inverse discrete wavelet transform

has been recognized as a faster approach

2. Allows good localization both in time and spatial frequency domain

3. Transformation of the whole image introduces inherent scaling

4. Better identification of which data is relevant to human perception, higher

compression ratio

5. It is perceived that the wavelet transform is an important tool for analysis and

processing of signals. The wavelet transform in its continuous form can accurately

represent minor variations in signal characteristics. Critically sampled version of

continuous wavelet transform, known as standard DWT

6. DWT is very popular for de-noising and compression in a number of applications by

the virtue of its computational simplicity through fast algorithms, and non-redundant.

There are certain signal processing applications (e.g. Time-division multiplexing in

Telecommunication)

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Chapter 3 .

WAVELET TRANSFORM DESIGN

3.1 1-D Discrete Wavelet Transform

3.1.1 2-Channel Perfect Reconstruction Filter Bank

The analysis filter bank decomposes the input signal x (n) into two sub band

signals, c (n) and d (n). The signal c (n) represents the low frequency (or coarse) part

of x (n), while the signal d (n) represents the high frequency (or detail) part of x(n).The analysis filters bank first filters x (n) using a low pass and a high pass filter. We

denote the low pass filter by af1 (analysis filter 1) and the high pass filter by af2

(analysis filter 2). As shown in the figure, the output of each filter is then down-

sampled by 2 to obtain the two sub band signals, c(n) and d(n).

Figure 3.1 sub band signal model

The Matlab program below, afb.m, implements the analysis filter bank. The program

uses the Matlab function upfirdn (in the Signal Processing Toolbox) to implement the

filtering and downsampling.

The synthesis filter bank combines the two subband signals c(n) and d(n) to

obtain a single signal y(n). The synthesis filter bank first up-samples each of the two

subband signals. The signals are then filtered using a lowpass and a highpass filter.

We denote the lowpass filter by sf1 (synthesis filter 1) and the highpass filter by sf2

(synthesis filter 2). The signals are then added together to obtain the signal y(n). If the

four filters are designed so as to guarantee that the output signal y(n) equals the input

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signal x(n), then the filters are said to satisfy the perfect reconstruction condition. The

Matlab program below, sfb.m, implements the synthesis filter bank.

There are many sets of filters that satisfy the perfect reconstruction conditions.

One set of filters, from [AS-2001], is shown in the Table below. These filters are

approximately symmetric.

The following code fragment shows an example of how to use the Matlab

functions, afb.m and sfb.m. This example verifies the perfect reconstruction property.

First, we create a random input signal x of length 64. Then the analysis and synthesis

filters are obtained with the Matlab functionfarras.m. The two subband signals (here

called lo and hi) are computed with the function afb.m. The output signal y is then

computed using the Matlab function sfb.m. The maximum value of the error x - y is

computed, and it is equal to zero (within computer precision). This verifies the perfect

reconstruction property.

A couple of remarks about the programs afb.m and sfb.m. Suppose the input

signal x(n) is of length N. For convenience, we will like the subband signals c(n) and

d(n) to each be of length N/2. However, these subband signals will exceed this length

by L/2, where L is the length of the analysis filters.

To avoid this excessive length, the last L/2 samples of each subband signal isadded to the first L/2 samples. This procedure (periodic extension) can create

undesirable artifacts at the beginning and end of the subband signals, however, it is

the most convenient solution. When the analysis and synthesis filters are exactly

symmetric, a different procedure (symmetric extension) can be used, that avoids the

artifacts associated with periodic extension.

A second detail also arises in the implementation of the perfect reconstruction

filter bank. If all four filters are causal, then the output signal y(n) will be a translated(or circularly shifted) version of x(n). To avoid this, we perform a circular shift in

both the analysis and synthesis filter banks. The circular shift is implemented with the

Matlab function cshift.m.

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3.2 Discrete Wavelet Transform (Iterated Filter Banks)

The discrete wavelet transform (DWT) gives a multiscale representation of a

signal x(n). The DWT is implemented by iterating the 2-channel analysis filter bank

described above. Specifically, the DWT of a signal is obtained by recursively

applying the lowpass/highpass frequency decomposition to the lowpass output as

illustrated in the diagram. The diagram illustrates a 3-scale DWT. The DWT of the

signal x is the collection of subband signals. The inverse DWT is obtained by

iteratively applying the synthesis filter bank.

Figure 3.2 Signal Analysis and Reconstruction in DWT

The Matlab function dwt.m below computes the J-scale discrete wavelettransform w of the signal x. We use the cell array data structure of Matlab to store the

subband signals. For j = 1..J, w{j} is the high frequency subband signal produced at

stage j. w{J+1} is the low frequency subband signal produced at stage J.

The inverse DWT is computed with the Matlab functionidwt.m. The perfect

reconstruction of the DWT is verified in the following example. First we create a

random input signal x of length 64. Then the analysis and synthesis filters are

obtained with the Matlab function farras.m. The 3-scale DWT is computed with thefunction dwt.m. The inverse DWT is then computed to get the signal y. As verified

below, y = x within computer precision.

The wavelet associated with a set of synthesis filters can be computed using

the following Matlab code fragment. In this example, we set all of the wavelet

coefficients to zero, for the exception of one wavelet coefficient which is set to one.

We then take the inverse wavelet transform.

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Figure 3.3 standard 1-D wavelet

3.3 2-D Discrete Wavelet Transform

3.3.1 2-D Filter Banks

To use the wavelet transform for image processing we must implement a 2D

version of the analysis and synthesis filter banks. In the 2D case, the 1D analysis filter

bank is first applied to the columns of the image and then applied to the rows. If the

image has N1 rows and N2 columns, then after applying the 1D analysis filter bank to

each column we have two sub band images, each having N1/2 rows and N2 columns;

after applying the 1D analysis filter bank to each row of both of the two sub band

images, we have four sub band images, each having N1/2 rows and N2/2 columns.

This is illustrated in the diagram below. The 2D synthesis filter bank combines the

four sub band images to obtain the original image of size N1 by N2.

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Figure 3.4 One stage in multi-resolution wavelet decomposition of an image

The 2D analysis filter bank is implemented with the Mat lab function afb2D.m.

This function calls a sub-function,afb2D_A.m, which applies the 1D analysis filter

bank along one dimension only (either along the columns or along the rows). The

function afb2D.m returns two variables: lo is the low pass sub band image; hi is a cell

array containing the 3 other sub band images.

The 2D synthesis filter bank is similarly implemented with the

commands sfb2D.m and sfb2D_A.m.

3.3.2 2D Discrete Wavelet Transform

As in the 1D case, the 2D discrete wavelet transform of a signal x is

implemented by iterating the 2D analysis filter bank on the low pass sub band image.

In this case, at each scale there are three sub bands instead of one. The

function, dwt2D.m, computes the J-scale 2D DWT w of an image x by repeatedly

calling afb2D.m.

Again, w is a Mat lab cell array; for j = 1..J, d = 1..3, w{j}{d} is one of the

three sub band images produced at stage j. w{J+1} is the low pass sub band image

produced at the last stage. The function idwt2D.m computes the inverse 2D DWT.

The perfect reconstruction of the 2D DWT is verified in the following example. We

create a random input signal x of size 128 by 64, apply the DWT and its inverse, and

show it reconstructs x from the wavelet coefficients in w.

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http://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D_A.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D_A.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/sfb2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/sfb2D_A.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/dwt2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/idwt2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/afb2D_A.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/sfb2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/sfb2D_A.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/dwt2D.mhttp://eeweb.poly.edu/iselesni/WaveletSoftware/allcode/idwt2D.m


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There are three wavelets associated with the 2D wavelet transform. The

following figure illustrates three wavelets as gravy scale images.

Figure 3.5 gravy scale images

Note that the first two wavelets are oriented in the vertical and horizontal

directions; however, the third wavelet does not have a dominant orientation. The third

wavelet mixes two diagonal orientations, which gives rise to the checkerboard

artefact. (The 2D DWT is poor at isolating the two diagonal orientations.) This figure

was produced with the following Mat lab code fragment (dwt2D_plots.m).

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CHAPTERCHAPTER44

Design of Hardware Model

4.1 Discrete Wavelet Transforms

The discrete wavelet transform (DWT) became a very versatile signal processing tool

after Mallet proposed the multi-resolution representation of signals based on wavelet

decomposition. The method of multi-resolution is to represent a function (signal) with a

collection of coefficients, each of which provides information about the position as well as

the frequency of the signal (function). The advantage of the DWT over Fourier

transformation is that it performs multi-resolution analysis of signals with localization both intime and frequency, popularly known as time-frequency localization. As a result, the DWT

decomposes a digital signal into different sub bands so that the lower frequency sub bands

have finer frequency resolution and coarser time resolution compared to the higher frequency

sub bands. The DWT is being increasingly used for image compression due to the fact that

the DWT supports features like progressive image transmission (by quality, by resolution),

ease of compressed image manipulation] region of interest coding, etc. Because of these

characteristics, the DWT is the basis of the new JPEG2000 image compression standard.

4.2 One dimensional DWT

Any signal is first applied to a pair of low-pass and high-pass filters. Then down

sampling (i.e., neglecting the alternate coefficients) is applied to these filtered coefficients.

The filter pair (h, g) which is used for decomposition is called analysis filter-bank and the

filter pair which is used for reconstruction of the signal is called synthesis filter bank.(g`,

h`).The output of the low pass filter after down sampling contains low frequency componentsof the signal which is approximate part of the original signal and the output of the high pass

filter after down sampling contains the high frequency components which are called details

(i.e., highly textured parts like edges) of the original signal.

This approximate part can still be further decomposed into low frequency and high

frequency components. This process can be continued successively to the required number of

levels. This process is called multi level decomposition, shown in Figure 3.1

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Figure 4.1 One dimensional two level wavelet decomposition

In reconstruction process, these approximate and detail coefficients are first up-

sampled and then applied to low-pass and high-pass reconstruction filters. These filtered

coefficients are then added to get the reconstructed version of the original image. This

process can be extended to multi level reconstruction i.e., the approximate coefficients to this

block may have been formed from pairs of approximate and detail coefficients. Shown in

Figure 3.2

Figure 4.2 One dimensional inverse wavelet transforms

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4.3 Two-Dimensional DWT

One dimensional DWT can be easily extended to two dimensions which can be used

for the transformation of two dimensional images. A two dimensional digital image which

can be represented by a 3-D array X [m,n] with m rows and n columns, where m, n are

positive integers. First, a one dimensional DWT is performed on rows to get low frequency L

and high frequency H components of the image. Then, once again a one dimensional DWT is

performed column wise on this intermediate result to form the final DWT coefficients LL,

HL, LH, HH. These are called sub-bands.

The LL sub-band can be further decomposed into four sub-bands by following the

above procedure. This process can continue to the required number of levels. This process is

called multi level decomposition. A three level decomposition of the given digital image is as

shown. High pass and low pass filters are used to decompose the image first row-wise and

then column wise. Similarly, the inverse DWT is applied which is just opposite to the

forward DWT to get back the reconstructed image, shown in Figure 3.3

Figure 4.3 Row-column computation of 3-D DWT

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Figure 4.4 Two channel filter bank at level 3

Various architectures have been proposed for computation of the DWT. These can be

mainly classified as either Convolutional Architectures or Lifting Based Architectures. The

number of computations required to find the DWT coefficients by the filter method is large

for higher level of decomposition. This leads to the implementation of new technique called

lifting scheme for computing DWT coefficients. This scheme reduces the number of

computations and also provides in-place computation of DWT coefficients.

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4.4 GENERAL IMPLEMENTATION FLOW

The generalized implementation flow diagram of the project is represented as follows.

Figure 4.5 General Implementation Flow Diagram

Initially the market research should be carried out which covers the previous version

of the design and the current requirements on the design. Based on this survey, the

specification and the architecture must be identified. Then the RTL modelling should be

carried out in VERILOG HDL with respect to the identified architecture. Once the RTL

modelling is done, it should be simulated and verified for all the cases. The functional

verification should meet the intended architecture and should pass all the test cases.

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Once the functional verification is clear, the RTL model will be taken to the synthesis

process. Three operations will be carried out in the synthesis process such as

Translate

Map

Place and Route

The developed RTL model will be translated to the mathematical equation format

which will be in the understandable format of the tool. These translated equations will be then

mapped to the library that is, mapped to the hardware. Once the mapping is done, the gates

were placed and routed. Before these processes, the constraints can be given in order to

optimize the design. Finally the BIT MAP file will be generated that has the design

information in the binary format which will be dumped in the FPGA board.

4.5 Implementation

The 3D (5, 3) wavelet transform block and for the recovery stage 3D (5, 3) Inverse

wavelet transform were designed.

4.5.1 Integer Wavelet Transform

In conventional DWT realizations, partial transform results need to be represented

with a high precision. This raises storage and complexity problems. On the other hand, the

Integer Wavelet Transform (IWT) produces integer intermediate results. Thus, it is possible

to use integer arithmetic without encountering rounding error problems. There are different

types of integer transforms like S(sequential) transform which is popularly known as Haar

wavelet transform, S(sequential)+P(prediction) transform, CDF(4,4), CDF(2,2) also known

as (5,3) transform etc.

The two filter banks supported by JPEG2000 standard are Debauchies (9, 7) and

Debauchies (5, 3) filter banks. Since the integer to integer wavelet transform coefficients are

integers, it can be used in lossless compression. Since the aim of the thesis is to suggest a

reversible (lossless) watermarking method so we will consider only (5, 3) Integer Wavelet

Transform.

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3-D (5, 3) DWT Lossless Transformation

The analysis and the synthesis filter coefficients ( both low pass and high pass) for Le

Gall 5/3 Integer Wavelet Transform are as shown in the table 3.1.Table 4.1 Le Gall 5/3 Analysis and Synthesis Filter coefficients [17].

Analysis Filter Coefficients

N Low Pass filter h(n) High pass filter g(n)

0 6/8 1+ 2/8 -1/2+ -1/8

Equation 3.1 and Equation 3.2 shows the lifting steps for the 5/3 le Gall Integer

Wavelet Transform. The rational coefficients allow the transform to be invertible with finiteprecision analysis, hence giving a chance for performing lossless compression. The equations

show the lifting steps for (5, 3) le gall Integer Wavelet Transform. The even and odd

coefficient equations for (5, 3) Inverse Integer Wavelet Transform are

( ) ( )( ) ( )

+++=+

2

2221212

nnxnxny .. (3.1)

( ) ( ) ( ) ( )[ ]121222 +++= nynynxny .. (3.2)

4.5.2 The 3-D (5, 3) Discrete Wavelet Transform

Initially the Pixel values of any image will be taken with the help of MATLAB, which

will be used as the primary inputs to the DWT Block.

Basically 1-D (5, 3) DWT block diagram is developed based on the equations (2) and

(3). The registers in the top half will operate in even clock where as the ones in bottom half

work in odd clock.The input pixels arrive serially row-wise at one pixel per clock cycle and it will get

split into even and odd. So after the manipulation with the lifting coefficients a and b is

done, the low pass and high pass coefficients will be given out. Hence for every pair of pixel

values, one high pass and one low pass coefficients will be given as output respectively.

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The internal operation of the DWT block has been explained above and hence the

high pass and low pass coefficients of the taken image were identified and separated. The

generated low pass and high pass coefficients are stored in buffers for further calculations.

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Figure 4.6 Computation of Basic (5, 3) DWT Block in which a and b are lifting

coefficients (a = -1/2 and b = 1)


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CHAPTER 5 .

SOFTWARE DISCRIPTION

5.1 Starting the ISE Software

To start ISE, double-click the desktop icon,

or start ISE from the Start menu by selecting:

Start All Programs -> Xilinx ISE 13.4 -> Project Navigator

5.2 Accessing Help

At any time during the tutorial, you can access online help for additional information about

the ISE software and related tools.

To open Help, do either of the following:

Press F1 to view Help for the specific tool or function that you have selected or

highlighted.

Launch the ISE Help Contents from the Help menu. It contains information about

creating and maintaining your complete design flow in ISE.

Table 5.1: ISE Help Topics

5.3 Create a New Project

Create a new ISE project which will target the FPGA device on the Spartan-3 Start-up Kit

Demo board.

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To create a new project:

1. Select File New Project The New Project Wizard appears.

2. Type tutorial in the Project Name field.

3. Enter or browse to a location (directory path) for the new project. A tutorial subdirectory is

created automatically.

4. Verify that HDL is selected from the Top-Level Source Type list.

5. ClickNext to move to the device properties page.

6. Fill in the properties in the table as shown below:

Product Category: All

Family: Spartan3E

Device: XC3S100E

Package: CP132

Speed Grade: -5

Top- Level Source Type: HDL

Synthesis Tool: XST (VHDL/Verilog)

Simulator: ISE Simulator (VHDL/Verilog)

Preferred Language: Verilog (orVHDL)

Verify that Enable Enhanced Design Summary is selected.

Leave the default values in the remaining fields.

When the table is complete, your project properties will look like the following:

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Table 5.2: ISE New project wizard

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Table 5.3: ISE project navigator

7. ClickNext to proceed to the Create New Source window in the New Project Wizard. At

the end of the next section, your new project will be complete.

5.4 Creating a Verilog Source

Create the top-level Verilog source file for the project as follows:

1. ClickNew Source in the New Project dialog box.

2. Select Verilog Module as the source type in the New Source dialog box.

3. Type in the file name test_module.

4. Verify that the Add to Project checkbox is selected.

5. ClickNext.

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6. Declare the ports for the counter design by filling in the port information as shown below:

Table 5.4: ISE new source wizard

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Table 5.5: ISE Help Topics

7. ClickNext, then Finish in the New Source Information dialog box to complete the new

source file template.

8. ClickNext, then Next, then Finish.

The source file containing the test_ module module displays in the Workspace, and the

counter displays in the Sources tab, as shown below:

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Table 5.6: Hierarchy and Process windows

In the above window

1. Hierarchy window

2. Process window

5.5 Checking the Syntax of the new test module Module

When the source files are complete, check the syntax of the design to find errors and

typos.

1. Verify that Implementation is selected from the drop-down list in the Sources

Window.

2. Select the test module design source in the Sources window to display the related

Processes in the Processes window.

3. Click the + next to the Synthesize-XST process to expand the process group.

4. Double-click the Check Syntaxprocess.

Note:

You must correct any errors found in your source files. You can check for errors in the

Console tab of the Transcript window. If you continue without valid syntax, you will not be

able to

Simulate or synthesize your design.

5. Close the HDL file.

5.6 To Synthesize the Code

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Select the test module in hierarchy window and double click on synthesize in process

window. The code will synthesize and show if any errors. We can also see rtl view and

technological view after this.

5.7 To implement the design

Select test module in hierarchy window & double click on the implement design. Then tool

will perform all operations regarding translation, mapping & place and route operations

Now you can see the design summery/reports on the window

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CHAPTER 6 .

SIMULATION RESULTS

6.1 Introduction

The DWT process and the developed architecture for the required functionality were

discussed in the previous chapters. Now this chapter deals with the simulation and synthesis

results of the DWT process. Here Modelsim tool is used in order to simulate the design and

checks the functionality of the design. Once the functional verification is done, the design

will be taken to the Xilinx tool for Synthesis process and the net list generation.

The Appropriate test cases have been identified in order to test this modelled DWT

process architecture. Based on the identified values, the simulation results which describesthe operation of the process has been achieved. This proves that the modelled design works

properly as per its functionality.

6.2 Simulation Results

The test bench is developed in order to test the modeled design. This developed test

bench will automatically force the inputs and will make the operations of algorithm to

perform.6.2.1 DWT Block

The initial block of the design is that the Discrete Wavelet Transform (DWT) block

which is mainly used for the transformation of the image. In this process, the image will be

transformed and hence the high pass coefficients and the low pass coefficients were

generated. Since the operation of this DWT block has been discussed in the previous chapter,

here the snapshots of the simulation results were directly taken in to consideration and

discussed.The input is 16 bits each input bit width is 20 bit width. The DWT consists of

registers and adders. Whenever the input is send, the data divided into even data and odd

data. The even data and odd data is stored in the temporary registers. When the reset is high

the temporary register value consists of zero whenever the reset is low the input data split into

the even data and odd data. The input data read up to sixteen clock cycles after that the data

read according to the lifting scheme. The output data consists of low pass and high pass

elements. This is the 1-D discrete wavelet transform. The 2-D discrete wavelet transform is

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that the low pass and the high pass again divided into LL, LH and HH, HL. The output is

verified in the Modelsim.

For this DWT block, the clock and reset were the primary inputs. The pixel values of

the image, that is, the input data will be given to this block and hence these values will be

split in to even and odd pixel values. In the design, this even and odd were taken as a array

which will store its pixel values in it and once all the input pixel values over, then load will

be made high which represents that the system is ready for the further process.

Once the load signal is set to high, then the each value from the even and odd array

will be taken and used for the Low Pass Coefficients generation process. Hence each value

will be given to the adder and in turn given to the multiplication process with the filter

coefficients. Finally the Low Pass Coefficients will be achieved from the addition process of

multiplied output and the odd pixel value.

Again this Low Pass Coefficient will be taken and it will be multiplied with the filter

coefficients. The resultant will be added with the even pixel value which gives the High Pass

Coefficient. Hence all the values from even and odd array will be taken and then above said

process will be carried out in order to achieve the High and Low Pass Coefficients of the

image.

Now these low pass coefficients and the high pass coefficients were taken as the inputfor the further process. Hence for the DWT-2 process, low pass coefficients will be taken as

the inputs and will do the process in order to calculate the low pass and high pass coefficients

from the transformed coefficients of DWT-1. In DWT-2, the same process as in DWT-1 will

be carried out. Hence the simulated waveform is shown in the figure 4.2.

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Fig 6.1 Simulation Result of DWT with Both High and Low Pass Coefficients

Similarly the high pass coefficients from the DWT-1 block were taken as input to the

DWT-3 block and hence further transformed low pass and high pass coefficients will beobtained.

6.3 Introduction to FPGA

FPGA stands for Field Programmable Gate Array which has the array of logic

module, I /O module and routing tracks (programmable interconnect). FPGA can be

configured by end user to implement specific circuitry. Speed is up to 100 MHz but at present

speed is in GHz.Main applications are DSP, FPGA based computers, logic emulation, ASIC and

ASSP. FPGA can be programmed mainly on SRAM (Static Random Access Memory). It is

Volatile and main advantage of using SRAM programming technology is re-configurability.

Issues in FPGA technology are complexity of logic element, clock support, IO support and

interconnections (Routing).

In this work, design of a DWT and IDWT is made using Verilog HDL and is

synthesized on FPGA family of Spartan 3E through XILINX ISE Tool. This process includes

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Translate

Map

Place and Route

6.3.1 FPGA Flow

The basic implementation of design on FPGA has the following steps.

Design Entry

Logic Optimization

Technology Mapping

Placement

Routing Programming Unit

Configured FPGA

Above shows the basic steps involved in implementation. The initial design entry of

may be Verilog HDL, schematic or Boolean expression. The optimization of the Boolean

expression will be carried out by considering area or speed.

Figure 6.2 Logic Block

In technology mapping, the transformation of optimized Boolean expression to FPGA

logic blocks, that is said to be as Slices. Here area and delay optimization will be taken place.

During placement the algorithms are used to place each block in FPGA array. Assigning the

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FPGA wire segments, which are programmable, to establish connections among FPGA

blocks through routing. The configuration of final chip is made in programming unit.

6.4 Synthesis Result

The developed DWT is simulated and verified their functionality. Once the functional

verification is done, the RTL model is taken to the synthesis process using the Xilinx ISE

tool. In synthesis process, the RTL model will be converted to the gate level net list mapped

to a specific technology library. Here in this Spartan 3E family, many different devices were

available in the Xilinx ISE tool. In order to synthesis this DWT and IDWT design the device

named as XC3S500E has been chosen and the package as FG320 with the device speed

such as -4.

The design of DWT is synthesized and its results were analysed as follows.

6.4.1 DWT Synthesis Result

This device utilization includes the following.

Logic Utilization

Logic Distribution

Total Gate count for the Design

Device utilization summary:

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The device utilization summery is shown above in which its gives the details of

number of devices used from the available devices and also represented in %. Hence as the

result of the synthesis process, the device utilization in the used device and package is shown

above.

Table 6.1 device utilization summary of 3D DWT

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Table 6.2 device utilization summary of 2D DWT

Timing Summary

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

Speed Grade: -5

Minimum period: 8.123ns (Maximum Frequency: 123.113MHz)

Minimum input arrival time before clock: 3.932ns

Maximum output required time after clock: 12.204ns

Maximum combinational path delay: No path found

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RTL Schematic

The RTL (Register Transfer Logic) can be viewed as black box after synthesize of

design is made. It shows the inputs and outputs of the system. By double-clicking on the

diagram we can see gates, flip-flops and MUX.

Figure 6.3 DWT Schematic with Basic Inputs and Output

Here in the above schematic, that is, in the top level schematic shows all the inputs

and final output of DWT design.

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Figure 6.4 DWT Schematic with top module design

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Figure 6.5 Blocks inside the Developed Top Level DWT Design

The internal blocks available inside the design includes DWT-1, DWT-2 and DWT-3

which were clearly shown in the above schematic level diagram. Inside each block the gate

level circuit will be generated with respect to the modelled HDL code.

6.5 Summary

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The developed DWT are modelled and are simulated using the Modelsim tool.

The simulation results are discussed by considering different cases.

The RTL model is synthesized using the Xilinx tool in Spartan 3E and their synthesis

results were discussed with the help of generated reports.

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CHAPTER 7CHAPTER 7

CONCLUSION AND FUTURE WORK

7.1 Conclusion

Basically the medical images need more accuracy without loosing of information. The

Discrete Wavelet Transform (DWT) was based on time-scale representation, which provides

efficient multi-resolution. The lifting based scheme(5, 3) (The high pass filter has five taps

and the low pass filter has three taps) filter give lossless mode of information. A more

efficient approach to lossless whose coefficients are exactly represented by finite precision

numbers allows for truly lossless encoding.This work ensures that the image pixel values given to the DWT process which gives

the high pass and low pass coefficients of the input image. The simulation results of DWT

were verified with the appropriate test cases. Once the functional verification is done, discrete

wavelet transform is synthesized by using Xilinx tool in Spartan 3E FPGA family. Hence it

has been analyzed that the discrete wavelet transform (DWT) operates at a maximum clock

frequency of 99.197 MHz respectively.

7.2 Future scope of the Work

As future work,

This work can be extended in order to increase the accuracy by increasing the level of

transformations.

This can be used as a part of the block in the full-fledged application, i.e., by using

these DWT, the applications can be developed such as compression, watermarking,

etc.

Documents

Documentation 03