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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
e oriented
DCT DWT
Transfor
m
Fracta
l
MallatTransversal
filter CoedicLifting
Scheme
LOSSY
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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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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.