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3-D

DISCRETE

WAVELETTRANSFORM

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_______ Abstract

The digital data can be transformed using Discrete Wavelet Transform (DWT).

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

provides efficient multi-resolution. The lifting based scheme (9, 7) (Here 9 Low Pass

filter coefficients and the 7 High Pass filter coefficients) filter give lossy mode of

information. The lifting based DWT are lower computational complexity and reduced

memory requirements. Since Conventional convolution based DWT is area and power

hungry which can be overcome by using the lifting based scheme.

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 transformed 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.

In this work, the design of Lossy 3-D DWT (Discrete Wavelet Transform) using

Lifting Scheme Architecture will be modeled using the Verilog HDL and its functionality

were verified using the Modelsim tool and can be synthesized using the Xilinx tool.

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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 basis is a more effective representation of signals for

analysis as well as characterization. Continuous wavelet transform is powerful insingularity detection. A discrete and fast implementation of CWT (generally with real

valued 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.

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1.2 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 Why wavelet?

Fourier transform based spectral analysis is the dominant analytical tool for frequency

domain analysis. However, Fourier transform cannot provide any information of the

spectrum changes with respect to time. Fourier transform assumes the signal is stationary,

but PD signal is always non-stationary. To overcome this deficiency, a modified method-

short time Fourier transform allows representing the signal in both time and frequency

domain through time windowing functions. The window length determines a constant

time and frequency resolution. Thus, a shorter time windowing is used in order to capture

the transient behavior of a signal; we sacrifice the frequency resolution. The nature of the

real PD signals is non-periodic and transient as shown in such signals cannot easily be

analyzed by conventional transforms. So, an alternative mathematical tool- wavelet

transform must be selected to extract the relevant time-amplitude information from a

signal. In the meantime, we can improve the signal to noise ratio based on prior

knowledge of the signal characteristics. A continuous-time wavelet transform off(t) is

defined.

Here a, b R, a 0 and they are dilating and translating coefficients,

respectively. The asterisk denotes a complex conjugate. This multiplication of [ a ]^1/2is

for energy normalization purposes so that the transformed signal will have the same

energy at every scale. The analysis function (t) , the so-called mother wavelet, is scaled

by a, so a wavelet analysis is often called a time-scale analysis rather than a time-

frequency analysis. The wavelet transform decomposes the signal into different scales

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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.

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

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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

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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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.

1.5 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 thetransformed data, resulting in effective compression. For example, JPEG 2000 is an

image compression standard that uses biorthogonal 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 resonance imaging, speech discrimination, optics,

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

applications such as solving partial differential equations.

1.6 Area of Application

Medical application

Signal denoising

Data compression

Image processing

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CHAPTER 2CHAPTER 2

Literature Review

2.1Introduction

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 a

powerful 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.

2.2Types of compressions

There are two types of compressions

1. Lossless compression

Digitally identical to the original image. Only achieve a modest amount of

compression

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.

2. Lossy compression

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

changed from input

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Figure 2.1 Different Types of Lossy Compression Techniques

2.3 The Discrete Wavelet Transform

The Wavelet Series is just a sampled version of CWT and its computation may consume

significant amount of time and resources, depending on the resolution required. The Discrete

Wavelet Transform (DWT), which is based on sub-band coding is found to yield a fast

computation of Wavelet Transform. It is easy to implement and reduces the computation time

and resources required.

Similar work was done in speech signal coding which was named as sub-band coding. In

1983, a technique similar to sub-band coding was developed which was named pyramidal

coding. Later many improvements were made to these coding schemes which resulted in

efficient multi-resolution analysis schemes.

In CWT, the signals are analyzed using a set of basis functions which relate to each other by

simple scaling and translation. In the case of DWT, a time-scale representation of the digital

signal is obtained using digital filtering techniques. The signal to be analyzed is passed

through filters with different cutoff frequencies at different scales.

HybridPredictive Frequencyoriented

Importanc

e oriented

DCT DWT

Transfor

m

Fracta

l

Mallat Transversal

filter

CoedicLifting

Scheme

LOSSY

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2.4 Reviewed Architectures of Discrete wavelet transforms

2. 4.1 DWT and Filter Banks

Multi-Resolution Analysis using Filter BanksFilters are one of the most widely used signal processing functions. Wavelets can be realized

by iteration of filters with rescaling. The resolution of the signal, which is a measure of the

amount of detail information in the signal, is determined by the filtering operations, and the

scale is determined by up sampling and down sampling (sub sampling) operations. The DWT

is computed by successive low pass and high pass filtering of the discrete time-domain

signal. This is called the Mallat algorithm or Mallat-tree decomposition. Its significance is in

the manner it connects the continuous-time mutiresolution to discrete-time filters. In the

figure, the signal is denoted by the sequence x[n], where n is an integer. The low pass filter is

denoted by G0 while the high pass filter is denoted by H0. At each level, the high pass filter

produces detail information, d[n], while the low pass filter associated with scaling function

produces coarse approximations, a[n].

Figure2.2 Three-level wavelet decomposition tree

At each decomposition level, the half band filters produce signals spanning only half the

frequency band. This doubles the frequency resolution as the uncertainty in frequency is

reduced by half. In accordance with Nyquists rule if the original signal has a highest

frequency of , which requires a sampling frequency of 2 radians, then it now has a highest

frequency of /2 radians. It can now be sampled at a frequency of radians thus discarding

half the samples with no loss of information. This decimation by 2 halves the time resolution

as the entire signal is now represented by only half the number of samples. Thus, while the

half band low pass filtering removes half of the frequencies and thus halves the resolution,

the decimation by 2 doubles the scale.

With this approach, the time resolution becomes arbitrarily good at high frequencies, while

the frequency resolution becomes arbitrarily good at low frequencies. The time-frequency

plane is thus resolved. The filtering and decimation process is continued until the desiredlevel is reached. The maximum number of levels depends on the length of the signal. The

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DWT of the original signal is then obtained by concatenating all the coefficients, a[n] and

d[n], starting from the last level of decomposition.

Figure:2.3 Three-level wavelet reconstruction tree.

Figure 4.5 shows the reconstruction of the original signal from the wavelet coefficients.

Basically, the reconstruction is the reverse process of decomposition. The approximation and

detail coefficients at every level are up sampled by two, passed through the low pass and high

pass synthesis filters and then added. This process is continued through the same number of

levels as in the decomposition process to obtain and Hthe original signal. The Mallat

algorithm works equally well if the analysis filters, G0 and H0, are exchanged with the

synthesis filters, G1 H1.

2.4.2 Conditions for Perfect Reconstruction

In most Wavelet Transform applications, it is required that the original signal be synthesized

from the wavelet coefficients. To achieve perfect reconstruction the analysis and synthesis

filters have to satisfy certain conditions. Let G0(z) and G1(z) be the low pass analysis and

synthesis filters, respectively and H0(z) and H1(z) the high pass analysis and synthesis filters

respectively. Then the filters have to satisfy the following two conditions as given in :

G0 (-z) G1 (z) + H0 (-z). H1 (z) = 0

G0 (z) G1 (z) + H0 (z). H1 (z) = 2z-d

The first condition implies that the reconstruction is aliasing-free and the second condition

implies that the amplitude distortion has amplitude of one. It can be observed that the perfect

reconstruction condition does not change if we switch the analysis and synthesis filters.

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There are a number of filters which satisfy these conditions. But not all of them give accurate

Wavelet Transforms, especially when the filter coefficients are quantized. The accuracy of

the Wavelet Transform can be determined after reconstruction by calculating the Signal to

Noise Ratio (SNR) of the signal. Some applications like pattern recognition do not need

reconstruction, and in such applications, the above conditions need not apply.

2.4.3 Classification of wavelets

We can classify wavelets into two classes: (a) orthogonal and (b) biorthogonal. Based on the

application, either of them can be used.

(a)Features of orthogonal wavelet filter banks

The coefficients of orthogonal filters are real numbers. The filters are of the same length and

are not symmetric. The low pass filter, G0 and the high pass filter, H0 are related to each other

by

H0 (z) = z-N G0 (-z

-1)

The two filters are alternated flip of each other. The alternating flip automatically gives

double-shift orthogonality between the low pass and high pass filters ,i.e., the scalar product

of the filters, for a shift by two is zero. i.e., G[k] H[k-2l] = 0. Perfect reconstruction is

possible with alternating flip.

Also, for perfect reconstruction, the synthesis filters are identical to the analysis filters except

for a time reversal. Orthogonal filters offer a high number of

en into account and the appropriate mother wavelet should be chosen in order tvanishing

moments. This property is useful in many signal and image processing applications. They

have regular structure which leads to easy implementation and scalable architecture.

(b)Features of biorthogonal wavelet filter banks

In the case of the biorthogonal wavelet filters, the low pass and the high pass filters do not

have the same length. The low pass filter is always symmetric, while the high pass filter

could be either symmetric or anti-symmetric. The coefficients of the filters are either real

numbers or integers. For perfect reconstruction, biorthogonal filter bank has all odd length or

all even length filters. The two analysis filters can be symmetric with odd length or one

symmetric and the other antisymmetric with even length. Also, the two sets of analysis and

synthesis filters must be dual. The linear phase biorthogonal filters are the most popular

filters for data compression applications.

Wavelet Families

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There are a number of basis functions that can be used as the mother wavelet for Wavelet

Transformation. Since the mother wavelet produces all wavelet functions used in the

transformation through translation and scaling, it determines the characteristics of the

resulting Wavelet Transform. Therefore, the details of the particular application should be

take use the Wavelet Transform effectively.

Figure 2.4 illustrates some of the commonly used wavelet functions. Haar wavelet is one of

the oldest and simplest wavelet. Therefore, any discussion of wavelets starts with the Haar

wavelet. Daubechies wavelets are the most popular wavelets. They represent the foundations

of wavelet signal processing and are used in numerous applications. These are also called

Maxflat wavelets as their frequency responses have maximum flatness at frequencies 0 and .

This is a very desirable property in some applications.

The Haar, Daubechies, Symlets and Coiflets are compactly supported orthogonal wavelets.

These wavelets along with Meyer wavelets are capable of perfect reconstruction. The Meyer,

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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.

2.4.6 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 have

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been 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-based DWT that reduces the

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

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

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.

Figure2.6 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.

2.5 Lifting Implementation of the Discrete Wavelet Transform

As the DWT intrinsically constitutes a pair of filtering operations, a unified

representation of the poly-phase matrix is introduced as follows

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where h(z) andg(z) stand for the transfer functions for the low pass and high pass filter banks,

respectively, and all suffixes e and o in the literature correspond to even and odd terms,

respectively. Thus, the transform is symbolized with the equation

with(z) and (z) signifying the filtered low pass and high pass parts of the input x(z).

The lifting scheme factorizes the poly phase representation into a cascade of upper and lower

triangular matrices and a scaling matrix which subsequently return a set of linear algebraic

equations in the time domain bringing forth the possibility of a pipelined processor.

For instance, the common Daubechies (9, 7) filter bank can be factorized as

Figure 2.7 The lifting scheme :Split,Predict,Update and Scale phases

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Figure 2.8 1-D lifting Scheme of daubechies 9/7 for forward wavelet DWT

Advantages

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

2. No need to divide the input coding into non-overlapping 2-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

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.

2.6 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.

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

Telecommunication)

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CHAPTER 3CHAPTER 3

Design of Hardware Model

3.1 Discrete Wavelet Transforms

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

after Mallat 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 in

time 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.

3.2 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 components

of 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 3.1 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 3.2 One dimensional inverse wavelet transforms

3.3 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 2-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

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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 3.3 Row-column computation of 2-D DWT

Figure 3.4 Two channel filter bank at level 3

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

mainly classified as either Convolution 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.

3.4 3D DISCRETE WAVELET TRANSFORM

Figure 3.5 3D-DWT

The 3D DWT can be considered as a combination of three 1D DWT in the x,y andz

directions, as shown in Fig. 1. The preliminary work in the DWT processor design is to build

1D DWT modules, which are composed of high-pass and low-pass filters that perform a

convolution of filter coefficients and input pixels. After a one-level of 3D discrete wavelet

transform, the volume of image is decomposed into HHH, HHL, HLH, HLL, LHH, LHL,

LLH and LLL signals as shown in the Figure

Lifting Scheme

The basic idea behind the lifting scheme is very simple; try to use the correlation in

the data to remove redundancy. First split the data into two sets (split phase) i.e., odd samples

and even samples as shown in Fig. 2. Because of the assumed smoothness of the data, we

predict that the odd samples have a value that is closely related to their neighboring even

samples. We use N even samples to predict the value of a neighboring odd value (predict

phase). With a good prediction method, the chance is high that the original odd sample is in

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the same range as its prediction. We calculate the difference between the odd sample and its

prediction and replace the odd sample with this difference. As long as the signal is highly

correlated, the newly calculated odd samples will be on the average smaller than the original

one and can be represented with fewer bits. The odd half of the signal is now transformed. To

transform the other half, we will have to apply the predict step on the even half as well.

Because the even half is merely a sub-sampled version of the original signal, it has lost some

properties that we might want to preserve. In case of images we would like to keep the

intensity (mean of the samples) constant throughout different levels. The third step (update

phase) updates the even samples using the newly calculated odd samples such that the desired

property is preserved. Now the circle is round and we can move to the next level.

We apply these three steps repeatedly on the even samples and transform each time

half of the even samples, until all samples are transformed.

3.4.1 Rationalization of Filter Coefficients

As already stated lifting scheme is one of the most efficient algorithms for the

implementation of discrete wavelet transform. But one of the major shortcomings with this

scheme is that the lifting coefficients obtained for the implementation of bi-orthogonal 9/7

wavelet transformation are irrational numbers [7, 8]. Hence the direct irrational coefficient

implementation requires lot of hardware resources and the processing time at the cost of

slight improvement in the compression performance. On the other hand, lower precision in

filter coefficients results in smaller and faster hardware at the cost of compression

performance. In addition to this rationalization also determines other critical hardware

properties such as throughput and power consumption. Hence it is suggested that they should

be optimally rationalized without much affecting the compression performance.

Table 1 shows the irrational and approximated rational counterpart for 9/7 filter which

are considered as a very good alternative to irrational coefficients. When these coefficients

are applied to image coding, the compression performance is almost same as that of

irrationalized filter coefficient implementation, while the computational complexity is

reduced remarkably. The heart of 3-D DWT implementation is designing of 1-D processor

which is clearly elaborated in Fig. 4. The different lifting coefficients can be easily obtained

for Daubechies 9/7 filter by factorization of poly phase matrix. Fig. 3 shows the

implementation of 9/7 lifting scheme. This figure is direct implementation of Fig. 2 for the

required scheme. When the signal passes through various steps, it is split into three separate

one dimensional transforms, the high pass component (HHH) and a low pass component

(LLL). Because of sub sampling the total number of transformed coefficients is same as that

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of original one. These transformed coefficients are then processed by x-coordinate Processor,

which have the same architecture as that of y and z-processor, to complete 3-D

transformation.

The bi-orthogonal 9/7 wavelet can be implemented as four lifting steps followed by

scaling requires that the following equations be implemented in hardware.

The original data to be filtered is denoted by x[n]; and the 1-D DWT outputs are the

detail coefficients x5[n] and approximation coefficients x6[n]. The lifting step coefficients

scaling coefficientsare constants given by Table 1. The above equations are implemented on

Verilog to obtain the coefficients x5[n] andx6[n]. These coefficients correspond to H and L

respectively. Now these coefficients are passed through the 1-D processor 3 times. Where, z-

coordinate processor gives the final output as the eight subsets of original image as shown in

Fig. 1. These coefficients are then stored in external memory in the form of binary file. Forthe multiple level of decomposition this binary file can be invoked iteratively to obtain

further sublevels.

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. 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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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.

3.5 GENERAL IMPLEMENTATION FLOW

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

Figure 3.6 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

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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.

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 tooptimize 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.

3.6 Implementation

The 3D (9,7) wavelet transform block were designed.

3.6.1 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.

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

lossy method so we will consider only (9,7) Integer Wavelet Transform.

3-D (9,7) DWT Lossy Transformation

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

Wavelet Transform are as shown in the table 3.1.

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The rational coefficients allow the transform to be invertible with finite precision

analysis, hence giving a chance for performing lossy compression.

3.6.2 The 3-D (9, 7) 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 (9, 7) DWT block diagram is developed based on the equations. 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,b,c andd 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.

Figure 3.7 Computation of Basic (9,7) DWT Block in which coefficients a,b,c

and d are lifting coefficients

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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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CHAPTER 4CHAPTER 4

Results and Discussion

4.1 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 netlist 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 describes

the operation of the process has been achieved. This proves that the modelled design worksproperly as per its functionality.

4.2 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.

4.2.1 DWT Block

The initial block of the design is that the Discrete Wavelet Transform (DWT) blockwhich 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 vary because of the multiplier. The DWT

consists of registers and adders. When ever 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 when ever 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 that the low pass and the high pass again divided into LL, LH and HH, HL. The 3-D

discrete wavelet transform is that the low pass and the high pass again divided into LLL,

LLH, LHL, LHH, HLL, HLH, HHL, and HHH. The output is verified in the Modelsim.

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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.

Figure 4.1 Simulation Result of DWT-1 Block with Both High and Low Pass Coefficients


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Figure 4.8 Simulation Result of 3D-DWT(TOP MODULE) Block with High and Low Pass

Coefficients LLL, LLH, LHL, LHH, HLL, HLH, HHL and HHH.

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

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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. The design of DWT is synthesized and its results

were analyzed as follows.

4.4.1 DWT Synthesis Result

This device utilization includes the following.

Logic Utilization

Logic Distribution

Total Gate count for the Design

Device utilization summary:

=================================================================

=

HDL Synthesis Report

Macro Statistics

# RAMs : 14

10x120-bit dual-port RAM : 8



# Multipliers : 42

200x10-bit multiplier : 42

# Adders/Subtractors : 56200-bit adder : 56

# Counters : 14

4-bit up counter : 14

# Registers : 132

1-bit register : 20

200-bit register : 112

# Comparators : 21

4-bit comparator greater : 7

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4-bit comparator lessequal : 14

==================================================================

Advanced HDL Synthesis Report

Macro Statistics

# RAMs : 14

10x120-bit dual-port block RAM : 8

12x60-bit dual-port block RAM : 4

8x10-bit dual-port distributed RAM : 2

# Multipliers : 42

200x10-bit registered multiplier : 42

# Adders/Subtractors : 56

200-bit adder : 56

# Counters : 14

4-bit up counter : 14# Registers : 8870

Flip-Flops : 8870

# Comparators : 21

4-bit comparator greater : 7

4-bit comparator lessequal : 14

=================================================================

=

* Final Report *

=================================================================

=

Final Results

RTL Top Level Output File Name : top_dwt_m97.ngrTop Level Output File Name : top_dwt_m97

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Output Format : NGC

Optimization Goal : Speed

Keep Hierarchy : NO

Design Statistics

# IOs : 1612

Cell Usage :

# BELS : 54288

# GND : 1

# INV : 30

# LUT1 : 1022

# LUT2 : 17056

# LUT3 : 133

# LUT4 : 38

# LUT5 : 4

# LUT6 : 8

# MUXCY : 18001

# VCC : 1

# XORCY : 17994

# FlipFlops/Latches : 14166

# FDR : 14070

# FDRE : 83

# FDSE : 7

# FDSE_1 : 6

# RAMS : 46

# RAM32M : 2

# RAM32X1D : 8

# RAMB18 : 8

# RAMB36_EXP : 24

# RAMB36SDP_EXP : 4

# Clock Buffers : 2

# BUFG : 1

# BUFGP : 1

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# IO Buffers : 1611

# IBUF : 11

# OBUF : 1600

# DSPs : 670

# DSP48E : 670

=================================================================

=

Device utilization summary:

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

Selected Device : 5vfx200tff1738-1

Slice Logic Utilization:

Number of Slice Registers: 14166 out of 122880 11%

Number of Slice LUTs: 18315 out of 122880 14%

Number used as Logic: 18291 out of 122880 14%

Number used as Memory: 24 out of 36480 0%

Number used as RAM: 24

Slice Logic Distribution:

Number of LUT Flip Flop pairs used: 27459

Number with an unused Flip Flop: 13293 out of 27459 48%

Number with an unused LUT: 9144 out of 27459 33%

Number of fully used LUT-FF pairs: 5022 out of 27459 18%

Number of unique control sets: 42

IO Utilization:

Number of IOs: 1612

Number of bonded IOBs: 1612 out of 960 167% (*)

Specific Feature Utilization:

Number of Block RAM/FIFO: 32 out of 456 7%

Number using Block RAM only: 32

Number of BUFG/BUFGCTRLs: 2 out of 32 6%

Number of DSP48Es: 670 out of 384 174% (*)

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4.5 Summary 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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