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CHAPTER 1
Introduction
A majority of todays Internet bandwidth is estimated to be used for images and
video. Recent multimedia applications for handheld and portable devices place a limit on
the available wireless bandwidth. The bandwidth is limited even with new connection
standards. JPEG image compression that is in widespread use today took several years
for it to be perfected. Wavelet based techniques such as JPEG2000 for image
compression has a lot more to offer than conventional methods in terms of compression
ratio. Currently wavelet implementations are still under development lifecycle and are
being perfected. Flexible energy-efficient hardware implementations that can handlemultimedia functions such as image processing, coding and decoding are critical,
especially in hand-held portable multimedia wireless devices.
1.1Background
Data compression is, of course, a powerful, enabling technology that plays a vital
role in the information age. Among the various types of data commonly transferred over
networks, image and video data comprises the bulk of the bit traffic. For example,
current estimates indicate that image data take up over 40% of the volume on the
Internet. The explosive growth in demand for image and video data, coupled with
delivery bottlenecks has kept compression technology at a premium. Among the several
compression standards available, the JPEG image compression standard is in wide
spread use today. JPEG uses the Discrete Cosine Transform (DCT) as the transform,
applied to 8-by-8 blocks of image data. The newer standard JPEG2000 is based on the
Wavelet Transform (WT). Wavelet Transform offers multi-resolution image analysis,
which appears to be well matched to the low level characteristic of human vision. The
DCT is essentially unique but WT has many possible realizations. Wavelets provide us
with a basis more suitable for representing images.
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This is because it cans represent information at a variety of scales, with local
contrast changes, as well as larger scale structures and thus is a better fit for image data.
1.2 Aim of the project
The main aim of the project is to implement and verify the image compression
technique and to investigate the possibility of hardware acceleration of DWT for signal
processing applications. A hardware design has to be provided to achieve highperformance, in comparison to the software implementation of DWT. The goal of the
project is to
Implement this in a Hardware description language (Here VHDL).
Perform simulation using tools such as Xilinx ISE 8.1i.
Check the correctness and to synthesize for a Spartan 3E FPGA Kit.
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1.3 Block Diagram
Fig 1.1: Image Compression Model
Fig 1.2: Image Decompression Model
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CHAPTER 2
Description
Fig 2.1: Block Diagram of Lifting based DWT
The block diagram consists of 4 blocks:
1) DWT
2) Compression block
3) Decompression block
4) IDWT
The input image is given to DWT which consists of lifting scheme where theimage is splitted into sequence of even and odd series coefficients. These splitted series
are passed to compression block where the image is compressed using SPIHT algorithm .
Compressed image is converted into bit streams using Entropy encoder. The
reconstructed image is obtained by passing the compressed image through decompression
block and IDWT.
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INTRODUCTION TO WAVELETS AND WAVELET
TRANSFORMS
2.1 Fourier Analysis
Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps
the most well-known of these is Fourier analysis, which breaks down a signal into
constituent sinusoids of different frequencies. Another way to think of Fourier analysis is
as a mathematical technique for transformingour view of the signal from time-based to
frequency-based.
Fig 2.2Fourier analysis
For many signals, Fourier analysis is extremely useful because the signals
frequency content is of great importance
2.2 Short-Time Fourier analysis
In an effort to correct this deficiency, Dennis Gabor (1946) adapted the Fourier
transform to analyze only a small section of the signal at a timea technique calledwindowing the signal. Gabors adaptation, called the Short-Time Fourier Transform
(STFT), maps a signal into a two-dimensional function of time and frequency.
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Fig 2.3Short-Time Fourier analysis
The STFT represents a sort of compromise between the time- and frequency-
based views of a signal. It provides some information about both when and at what
frequencies a signal event occurs. However, you can only obtain this information with
limited precision, and that precision is determined by the size of the window.
While the STFT compromise between time and frequency information can be
useful, the drawback is that once you choose a particular size for the time window, that
window is the same for all frequencies. Many signals require a more flexible approach
one where we can vary the window size to determine more accurately either time or
frequency.
2.3 Problem Present in Fourier Transform2
The Fundamental idea behind wavelets is to analyze according to scale. Indeed,
some researchers feel that using wavelets means adopting a whole new mind-set or
perspective in processing data. Wavelets are functions that satisfy certain mathematical
requirements and are used in representing data or other functions. This idea is not new.
Approximation using superposition of functions has existed since the early 18OOs, when
Joseph Fourier discovered that he could superpose sines and cosines to represent other
functions. However, in wavelet analysis, the scale used to look at data plays a special
role. Wavelet algorithms process data at different scales or resolutions. Looking at a
signal (or a function) through a large window, gross features could be noticed.
Similarly, looking at a signal through a small window, small features could be noticed.
The result in wavelet analysis is to see both the forest and the trees, so to speak.
This makes wavelets interesting and useful. For many decades scientists have
wanted more appropriate functions than the sines and cosines, which are the basis of
Fourier analysis, to approximate choppy signals. By their definition, these functions are
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non-local (and stretch out to infinity). They therefore do a very poor job in approximating
sharp spikes. But with wavelet analysis, we can use approximating functions that are
contained neatly in finite domains. Wavelets are well-suited for approximating data with
sharp discontinuities.
The wavelet analysis procedure is to adopt a wavelet prototype function, called an
analyzingwavelet or mother wavelet. Temporal analysis is performed with a contracted,
high-frequency version of the prototype wavelet, while frequency analysis is performed
with a dilated, low-frequency version of the same wavelet. Because the original signal or
function can be represented in terms of a wavelet expansion (using coefficients in a linear
combination of the wavelet functions), data operations can be performed using just the
corresponding wavelet coefficients. And if wavelets best adapted to data are selected, the
coefficients below a threshold is truncated, resultant data are sparsely represented. This
sparse coding makes wavelets an excellent tool in the field of data compression. Other
applied fields that are using wavelets include astronomy, acoustics, nuclear engineering,
sub-band coding, signal and image processing, neurophysiology, music, magnetic
resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake
prediction, radar, human vision, and pure mathematics applications such as solving
partial differential equations.
Basically wavelet transform (WT) is used to analyze non-stationary signals, i.e.,
signals whose frequency response varies in time, as Fourier transform (FT) is not suitable
for such signals. To overcome the limitation of FT, short time Fourier transform (STFT)
was proposed. There is only a minor difference between STFT and FT. In STFT, the
signal is divided into small segments, where these segments (portions) of the signal can
be assumed to be stationary. For this purpose, a window function "w" is chosen. The
width of this window in time must be equal to the segment of the signal where its still be
considered stationary. By STFT, one can get time-frequency response of a signal
simultaneously, which cant be obtained by FT.
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The short time Fourier transform for a real continuous signal is defined as:
( )
= dtetwtxtfx ftj 2* ])()([,
Where the length of the window is (t-) in time such that we can shift the window
by changing value oft and by varying the value we get different frequency response ofthe signal segments.
The Heisenberg uncertainty principle explains the problem with STFT. This
principle states that one cannot know the exact time-frequency representation of a signal,
i.e., onecannotknow what spectral components exist at what instances of times. What
one canknow are the time intervals in which certain band of frequencies exists and is
called resolution problem. This problem has to do with the widthof the window function
that is used, known as thesupportof the window. If the window function is narrow, then
it is known as compactly supported. The narrower we make the window, the better the
time resolution, and better the assumption of the signal to be stationary, but poorer the
frequency resolution:
Narrow window ===> good time resolution, poor frequency resolution.
Wide window ===> good frequency resolution, poor time resolution.
The wavelet transform (WT) has been developed as an alternate approach to
STFT to overcome the resolution problem. The wavelet analysis is done such that the
signal is multiplied with the wavelet function, similar to the window function in the
STFT, and the transform is computed separately for different segments of the time-
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domain signal at different frequencies. This approach is called multiresolution analysis
(MRA), as it analyzes the signal at different frequencies giving different resolutions.
MRA is designed to give good time resolution and poor frequency resolution at
high frequencies and good frequency resolution and poor time resolution at low
frequencies. This approach is good especially when the signal has high frequency
components for short durations and low frequency components for long durations, e.g.,
images and video frames.
So why do we need other techniques, like wavelet analysis?
Fourier analysis has a serious drawback. In transforming to the frequency domain,
time information is lost. When looking at a Fourier transform of a signal, it is impossible
to tell whena particular event took place. If the signal properties do not change much
over time that is, if it is what is called a stationarysignalthis drawback isnt very
important. However, most interesting signals contain numerous non stationary or
transitory characteristics: drift, trends, abrupt changes, and beginnings and ends of
events. These characteristics are often the most important part of the signal, and Fourier
analysis is not suited to detecting them.
2.4 Wavelet Analysis
Wavelet analysis [1] represents the next logical step: a windowing technique with
variable-sized regions. Wavelet analysis allows the use of long time intervals where we
want more precise low-frequency information, and shorter regions where we want high-
frequency information
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A wavelet is a waveform of effectively limited duration that has an average value
of zero.
Compare wavelets with sine waves, which are the basis of Fourier analysis.
Sinusoids do not have limited duration they extend from minus to plus infinity. And
where sinusoids are smooth and predictable, wavelets tend to be irregular and symmetric.
Fig 2.5 sine wave
Fourier analysis consists of breaking up a signal into sine waves of various
frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and
scaled versions of the original (or mother) wavelet. Just looking at pictures of wavelets
and sine waves, you can see intuitively that signals with sharp changes might be better
analyzed with an irregular wavelet than with a smooth sinusoid, just as some foods are
better handled with a fork than a spoon. It also makes sense that local features can be
described better with wavelets that have local extent.
2.5 What Can Wavelet Analysis Do?
One major advantage afforded by wavelets is the ability to perform local analysis,
that is, to analyze a localized area of a larger signal. Consider a sinusoidal signal with a
small discontinuity one so tiny as to be barely visible. Such a signal easily could be
generated in the real world, perhaps by a power fluctuation or a noisy switch.
Sinusoid with a small discontinuity
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Fig 2.6Sinusoidal Signal
A plot of the Fourier coefficients (as provided by the fft command) of this signal
shows nothing particularly interesting: a flat spectrum with two peaks representing a
single frequency. However, a plot of wavelet coefficients clearly shows the exact location
in time of the discontinuity.
Fig 2.7 Fourier coefficients and wavelet coefficients
Wavelet analysis is capable of revealing aspects of data that other signal analysis
techniques miss aspects like trends, breakdown points, discontinuities in higher
derivatives, and self-similarity. Furthermore, because it affords a different view of data
than those presented by traditional techniques, wavelet analysis can often compress or de-
noise a signal without appreciable degradation. Indeed, in their brief history within the
signal processing field, wavelets have already proven themselves to be an indispensableaddition to the analysts collection of tools and continue to enjoy a burgeoning popularity
today.
Thus far, weve discussed only one-dimensional data, which encompasses most
ordinary signals. However, wavelet analysis can be applied to two-dimensional data
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(images) and, in principle, to higher dimensional data. This toolbox uses only one and
two-dimensional analysis techniques.
2.6 Wavelet Transform
When we analyze our signal in time for its frequency content, Unlike Fourier
analysis, in which we analyze signals using sines and cosines, now we use wavelet
functions.
2.6.1 The Continuous Wavelet Transform
Mathematically, the process of Fourier analysis is represented by the Fourier
transform:
( )
= dtetfF tj )(
Which is the sum over all time of the signal f(t) multiplied by a complex
exponential. (Recall that a complex exponential can be broken down into real and
imaginary sinusoidal components.) The results of the transform are the Fourier
coefficients F(w), which when multiplied by a sinusoid of frequency w yields the
constituent sinusoidal components of the original signal. Graphically, the process looks
like:
Fig 2.8Continuous Wavelets of different frequencies
Similarly, the continuous wavelet transform (CWT) is defined as the sum over all
time of signal multiplied by scaled, shifted versions of the wavelet function.
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= dttpositionscaletfpositionScaleC ),,()(),(
The result of the CWT is a series many wavelet coefficients C, which are a
function of scale and position.
Multiplying each coefficient by the appropriately scaled and shifted wavelet
yields the constituent wavelets of the original signal:
Fig 2.9 Continuous Wavelets of different scales and positions
The wavelet transform involves projecting a signal onto a complete set of
translated and dilated versions of a mother wavelet (t). The strict definition of a mother
wavelet will be dealt with later so that the form of the wavelet transform can be examined
first. For now, assume the loose requirement that (t) has compact temporal and spectral
support (limited by the uncertainty principle of course), upon which set of basis functions
can be defined. The basis set of wavelets is generated from the mother or basic wavelet is
defined as
=
a
bt
at
ba
1)(
,; a, b R1 and a>0 (2.2)
The variable a (inverse of frequency) reflects the scale (width) of a particular
basis function such that its large value gives low frequencies and small value gives high
frequencies. The variable b specifies its translation along x-axis in time. The term 1/
a is used for normalization. The 1-D wavelet transform is given by:
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= dtttxba bafw )()(),( , (2.3)
The inverse 1-D wavelet transform is given by:
( )
=0
2,)(),(
1
a
dadbtbaW
Ctx baf
(2.4)
Where
= dC
2)
<
(2.5)
X (t) is the Fourier transform of the mother wavelet (t). Cis required to be
finite, which leads to one of the required properties of a mother wavelet. Since Cmust be
finite, then x (t) =0 to avoid a singularity in the integral, and thus the x(t) must have zero
mean. This condition can be stated as
dtt)( = 0 (2.6)
and known as the admissibility condition. The other main requirement is that the mother
wavelet must have finite energy:
dtt2
)( < (2.7)
A mother wavelet and its scaled versions are depicted in figure 2.10 indicating the
effect of scaling.
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Fig 2.10 Mother wavelet and its scaled versions
Unlike the STFT which has a constant resolution at all times and frequencies, the
WT has a good time and poor frequency resolution at high frequencies, and good
frequency and poor time resolution at low frequencies.
2.6.2 The Discrete Wavelet Transform
Calculating wavelet coefficients at every possible scale is a fair amount of work,
and it generates an awful lot of data. What if we choose only a subset of scales and
positions at which to make our calculations? It turns out rather remarkably that if we
choose scales and positions based on powers of twoso-called dyadic scales and
positionsthen our analysis will be much more efficient and just as accurate. We obtain
such an analysis from the discrete wavelet transform(DWT)[12].
An efficient way to implement this scheme using filters was developed in 1988 by
Mallat. The Mallat algorithm is in fact a classical scheme known in the signal processing
community as a two-channel sub band coder. This very practical filtering algorithm
yields a fast wavelet transform a box into which a signal passes, and out of which
wavelet coefficients quickly emerge. Lets examine this in more depth.
Now consider, discrete wavelet transform (DWT), which transforms a discrete
time signal to a discrete wavelet representation. The first step is to discretize the wavelet
parameters, which reduce the previously continuous basis set of wavelets to a discrete
and orthogonal / orthonormal set of basis wavelets.
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If were talking about sinusoids, for example the effect of the scale factor is very easy to
see:
Fig 2.11 Scaling
The scale factor works exactly the same with wavelets. The smaller the scale factor, the
more compressed the wavelet.
Fig 2.12 Scaling
It is clear from the diagrams that for a sinusoid sin (wt) the scale factor a is
related (inversely) to the radian frequency w. Similarly, with wavelet analysis the scale
is related to the frequency of the signal.
2.7.2 Shifting
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Shifting a wavelet simply means delaying (or hastening) its onset.
Mathematically, delaying a function (t) by kis represented by (t-k).
Fig 2.13 Shifting
2.8 Decomposition of Wavelets
2.8.1 One-Stage Decomposition
For many signals, the low-frequency content is the most important part. It is what
gives the signal its identity. The high-frequency content on the other hand imparts flavor
or nuance. Consider the human voice. If you remove the high-frequency components, thevoice sounds different but you can still tell whats being said. However, if you remove
enough of the low-frequency components, you hear gibberish. In wavelet analysis, we
often speak of approximations and details. The approximations are the high-scale, low-
frequency components of the signal. The details are the low-scale, high-frequency
components. The filtering process at its most basic level looks like this:
The original signal S passes through two complementary filters and emerges as
two signals. Unfortunately, if we actually perform this operation on a real digital signal,
we wind up with twice as much data as we started with. Suppose, for instance that the
original signal S consists of 1000 samples of data. Then the resulting signals will each
have 1000 samples, for a total of 2000. These signals A and D are interesting, but we get
2000 values instead of the 1000 we had. There exists a more subtle way to perform the
decomposition using wavelets.
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Fig 2.14One-Stage Decomposition
By looking carefully at the computation, we may keep only one point out of two
in each of the two 2000-length samples to get the complete information. This is the
notion of down sampling. We produce two sequences called cA and cD.
Fig 2.15 Two-Stage Decomposition
The process on the right which includes down sampling produces DWT
Coefficients. To gain a better appreciation of this process lets perform a one-stage
discrete wavelet transform of a signal. Our signal will be a pure sinusoid with high-
frequency noise added to it.
Here is our schematic diagram with real signals inserted into it:
Notice that the detail coefficients cD is small and consist mainly of a high-
frequency noise, while the approximation coefficients cA contains much less noise than
does the original signal.
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You may observe that the actual lengths of the detail and approximation
coefficient vectors are slightly morethan half the length of the original signal.
Fig 2.16
2.8.2 Multi-step Decomposition and Reconstruction
A multi step analysis-synthesis process can be represented as: filters, and thus is
associated with the approximations of the wavelet decomposition.
Fig 2.17Decomposition and Reconstruction
In the same way this process involves two aspects: breaking up a signal to obtain
the wavelet coefficients, and reassembling the signal from the coefficients. Weve
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already discussed decomposition and reconstruction at some length. Of course, there isno point breaking up a signal merely to have the satisfaction of immediately
reconstructing it. We may modify the wavelet coefficients before performing the
reconstruction step. We perform wavelet analysis because the coefficients thus obtained
have many known uses, de-noising and compression being foremost among them. But
wavelet analysis is still a new and emerging field. No doubt, many uncharted uses of the
wavelet coefficients lie in wait. The Wavelet Toolbox can be a means of exploring
possible uses and hitherto unknown applications of wavelet analysis. Explore the toolbox
functions and see what you discover.
2.9 Wavelet Reconstruction
Weve learned how the discrete wavelet transform can be used to analyze or
decompose signals and images. This process is called decomposition or analysis. The
other half of the story is how those components can be assembled back into the original
signal without loss of information. This process is called reconstruction, or synthesis.
The mathematical manipulation that effects synthesis is called the inverse
discrete wavelet transforms(IDWT). To synthesize a signal in the Wavelet Toolbox, we
reconstruct it from the wavelet coefficients:
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Fig 2.18Wavelet Reconstruction
Up sampling is the process of lengthening a signal component by inserting zeros
between samples:
Signal component Upsampled signal component
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The Wavelet Toolbox includes commands like IDWT and wavered that perform
single-level or multilevel reconstruction respectively on the components of one-
dimensional signals. These commands have their two-dimensional analogs, idwt2 and
waverec2.
2.10 Dissimilarities between Fourier and Wavelet Transforms
The most interesting dissimilarity between these two kinds of transforms is that
individual wavelet functions are localized in space.
Fourier sine and cosine functions are not. This localization feature, along with
wavelets' localization of frequency, makes many functions and operators usingwavelets\sparse. When transformed into the wavelet domain. This sparseness, in turn,
results in a number of useful applications such as data compression, detecting features in
images, and removing noise from time series. One way to see the time-frequency
resolution differences between the Fourier transform and the wavelet transform is to look
at the basis function coverage of the time-frequency plane. Figure 1.1 shows a windowed
Fourier transform, where the window is simply a square wave. The square wave window
truncates the sine or cosine function to a window of a particular width. Because a single
window is used for all frequencies in the WFT, the resolution of the analysis is the same
at all locations in the time-frequency plane.
An advantage of wavelet transforms is that the windows vary. In order to isolate
signal discontinuities, one would like to have some very short basis functions. At the
same time, in order to obtain detailed frequency analysis, one would like to have some
very long basis functions.
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Fig 2.19 Fourier basis functions, time-frequency tiles
and coverage of the time-frequency plane.
A way to achieve this is to have short high-frequency basis functions and long
low-frequency ones. This happy medium is exactly what you get with wavelet
transforms. Figure 1.2 shows the coverage in the time-frequency plane with one wavelet
function, the Daubechies [15] wavelet.
One thing to remember is that wavelet transforms do not have a single set of basis
functions like the Fourier transform, which utilizes just the sine and cosine functions.
Instead, wavelet transforms have an infinite set of possible basis functions. Thus wavelet
analysis provides immediate access to information that can be obscured by other time-
frequency methods such as Fourier analysis.
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Fig 2.20 Daubechies wavelet basis functions, time-frequency tiles
and coverage of the time-frequency plane.
2.11 WAVELET APPLICATIONS
The following applications show just a small sample of what researchers can do
with wavelets.
Computer and Human Vision
FBI Fingerprint compression
De-noising Noisy Data
Detecting self similar behavior in a time series
Musical tone generation
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CHAPTER 3
DWT Architecture
3.1 Discrete Wavelet Transform
The next step toward developing a DWT is to be able to transform a discrete time
signal. The wavelet transform can be interpreted as a applying a set of filters. Digital
filters are very efficient to implement and thus provide us with the needed tool for
performing the DWT, and are usually applied as equivalent low and high-pass filters. The
design of these filters is similar to subband coding, i.e., only the low pass filter has to be
designed such that the high pass filter has additional phase shift of 180 degree as
compared to the low pass filter. Unlike subband coding, these filters are designed to give
flat or smooth spectral response and are bi-orthogonal.
=
>== XMD debug Options. The XMD Debug Options dialog box
allows the user to specify the connections type and JTAG chain Definition . The
connection types are available for MicroBlaze:
Simulator enables XMD to connect to the MicroBlaze ISS
Hardware enables XMD to connect to the MDM peripheral in the hardware
Stub enables XMD to connect to the JTAG UART or UART via XMDSTUB
Virtual platform enables a virtual (c model) to be used
Verify that Hardware is selected
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Select Save
Select Debug -> Launch XMD
CHAPTER 5
Implementation
5.1 Fundamentals of Digital Image
Digital image is defined as a two dimensional function f(x, y), where x and y are
spatial (plane) coordinates, and the amplitude of f at any pair of coordinates (x, y) iscalled intensity or grey level of the image at that point. The field of digital image
processing refers to processing digital images by means of a digital computer. The digital
image is composed of a finite number of elements, each of which has a particular location
and value. The elements are referred to as picture elements, image elements, pels, and
pixels. Pixel is the term most widely used.
5.1.1 Image Compression
Digital Image compression addresses the problem of reducing the amount of data
required to represent a digital image. The underlying basis of the reduction process is
removal of redundant data. From the mathematical viewpoint, this amounts to
transforming a 2D pixel array into a statically uncorrelated data set. The data redundancy
is not an abstract concept but a mathematically quantifiable entity. If n1 and n2 denote
the number of information-carrying units in two data sets that represent the same
information, the relative data redundancy DR [2] of the first data set (the one
characterized by n1) can be defined as,
R
DC
R1
1=
Where RC called as compression ratio [2]. It is defined as
RC =2
1
n
n
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In image compression, three basic data redundancies can be identified and
exploited: Coding redundancy, interpixel redundancy, and phychovisal redundancy.
Image compression is achieved when one or more of these redundancies are reduced or
eliminated.
The image compression is mainly used for image transmission and storage. Image
transmission applications are in broadcast television; remote sensing via satellite, aircraft,
radar, or sonar; teleconferencing; computer communications; and facsimile transmission.
Image storage is required most commonly for educational and business documents,
medical images that arise in computer tomography (CT), magnetic resonance imaging
(MRI) and digital radiology, motion pictures, satellite images, weather maps, geological
surveys, and so on.
Fig 5.1 Block Diagram
5.1.2 Image Compression Types
There are two types image compression techniques.
1. Lossy Image compression
2. Lossless Image compression
5.1.2.1 Lossy Image compression
Lossy compression provides higher levels of data reduction but result in a less
than perfect reproduction of the original image. It provides high compression ratio.
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Lossy image compression is useful in applications such as broadcast television,
videoconferencing, and facsimile transmission, in which a certain amount of error is an
acceptable trade-off for increased compression performance.
5.1.2.2 Lossless Image compression
Lossless Image compression is the only acceptable amount of data reduction. It
provides low compression ratio while compared to lossy. In Lossless Image compression
techniques are composed of two relatively independent operations: (1) devising an
alternative representation of the image in which its interpixel redundancies are reduced
and (2) coding the representation to eliminate coding redundancies. Lossless Image
compression is useful in applications such as medical imaginary, business documents and
satellite images.
5.1.3 Image Compression Standards
There are many methods available for lossy and lossless, image compression. The
efficiency of these coding standardized by some Organizations. The International
Standardization Organization (ISO) and Consultative Committee of the International
Telephone and Telegraph (CCITT) are defined the image compression standards for both
binary and continuous tone (monochrome and Colour) images. Some of the Image
Compression Standards are
1. JBIG1
2. JBIG2
3. JPEG-LS
4. DCT based JPEG
5. Wavelet based JPEG2000
Currently, JPEG2000 [4] [5] is widely used because; the JPEG-2000 standard
supports lossy and lossless compression of single-component (e.g., grayscale) and
multicomponent (e.g., color) imagery. In addition to this basic compression functionality,
however, numerous other features are provided, including: 1) progressive recovery of an
image by fidelity or resolution; 2) region of interest coding, whereby different parts of an
image can be coded with differing fidelity; 3) random access to particular regions of an
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image without the needed to decode the entire code stream; 4) a flexible file format with
provisions for specifying opacity information and image sequences; and 5) good error
resilience. Due to its excellent coding performance and many attractive features, JPEG
2000 has a very large potential application base.
Some possible application areas include: image archiving, Internet, web browsing,
document imaging, digital photography, medical imaging, remote sensing, and desktop
publishing.
The main advantage of JPEG2000 over other standards, First, it would addresses a
number of weaknesses in the existing JPEG standard. Second, it would provide a number
of new features not available in the JPEG standard.
5.2 Lifting Scheme
The wavelet transform of image is implemented using the lifting scheme [3]. The
lifting operation consists of three steps. First, the input signal x[n] is down sampled into
the even position signal xe (n) and the odd position signal xo(n) , then modifying these
values using alternating prediction and updating steps.
]2[)( nxnxe = and ]12[)( += nxnxo
A prediction step consists of predicting each odd sample as a linear combination
of the even samples and subtracting it from the odd sample to form the prediction error.
An update step consists of updating the even samples by adding them to a linear
combination of the prediction error to form the updated sequence. The prediction and
update may be evaluated in several steps until the forward transform is completed. The
block diagram of forward lifting and inverse lifting is shown in figure 5.2
xe (n) s
x (n)
xo(n) d
Fig (a)
s
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Split UP
MergeU P
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x (n)
d
Fig (b)
Fig 5.2 The Lifting Scheme. (a) Forward Transform (b) Inverse Transform
The inverse transform is similar to forward. It is based on the three operations undo
update, undo prediction, and merge. The simple lifting technique using Haar wavelet is
explained in next section.
5.2.1 Lifting using Harr
The lifting scheme is a useful way of looking at discrete wavelet transform. It is
easy to understand, since it performs all operations in the time domain, rather than in the
frequency domain, and has other advantages as well. This section illustrates the lifting
approach using the Haar Transform [6].
The Haar transform is based on the calculations of the averages (approximation
co-efficient) and differences (detail co-efficient). Given two adjacent pixels a and b, the
principle is to calculate the average2
)( bas
+= and the difference bad = . If a and b
are similar, s will be similar to both and d will be small, i.e., require few bits to represent.
This transform is reversible, since2
dsa = and
2
dsb += and it can be written using
matrix notation as
=
2/1
2/1),(),( bads
1
1= (a,b)A,
=2/1
1),(),( dsba
2/1
1= 1),(
Ads
Consider a row of n2 pixels values lnS , fornl 20
-
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Wavelet decompositions are widely used in signal and image processing
applications. Classical linear wavelet transforms perform homogeneous smoothing of
signal contents. In a number of cases, in particular in applications in image and video
processing, such homogeneous smoothing is undesirable. This has led to a growing
interest in nonlinear wavelet representations called as adaptive wavelet decomposition
that can preserve discontinuities such as transitions in signals and edges in images.
Adaptive wavelet decomposition is very useful in various applications, such as
image analysis, compression, and feature extraction and denoising. The adaptive
multiresolution representations take into account the characteristics of the underlying
signal and do leave intact important signal characteristics, such as sharp transitions,
edges, singularities, and other region of interests.
In the adaptive update lifting framework, the update lifting step, which yields a
low pass filter or a moving average process, is performed first on the input polyphase
components (the output from the splitting process, according to the lifting terminology),
followed by a fixed prediction step yielding the wavelet coefficients.
5.2.2 General Adaptive Update Lifting
We consider a (K + 1) band Filter bank decomposition with inputs x, y (1), y (2),
y (3).y (k), with 1K , which represent the polyphase components of the analyzed
signal. The first polyphase component, x, is updated using the neighboring signal
elements from the other polyphase components, thus yielding an approximation signal.
Subsequently, the signal elements in the polyphase components y (1), y (2)y (K) are
predicted using the neighboring signal elements from the approximated polyphase
component and the other polyphase components. The prediction steps, which are non-
adaptive, result in detail coefficients. The adaptive update step is illustrated in Figure 5.3.
x xx
y(1) .. .
y(2)
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D Ud
+
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y(k)
Fig 5.3 Adaptive update lifting scheme
Here, x and y (1), y (2)y (K) are the input for a decision map D, whose output
at location n is binary decision
dn = D {y (1), y (2)y (K)} {0,1}
Which triggers the update filter Ud and the addition d. More precisely, if dn is the
binary decision at location n, then the updated value x1(n) is given by
))(()()(1
nyUnxnx idndn= ------------ (5.1)
We assume that the addition d is of the form xdu = d(x+u) with d #0, so that
the operation is invertible. The update filter is taken to be of the form
)())((2
1
, nynyU j
L
Lj
jdd =
= -------------
(5.2)
Where yj(n) = y(n+j) and L1 and L2 are nonnegative integers. The filter
coefficients d,j depend on the decision d at location n. Henceforth, we will usej to
denote the summation from L1 to L2.
From (5.1) and (5.2), we infer the update equation used at analysis:
=
+=N
j
jjdndn nynxnx1
,
1 )()()( ---------- (5.3)
Where jddjd ,, = . Clearly, we can easily invert (3.3) through
))()((1
)( ,1
nynxnx jj
jdn
dn
=
------------
(5.4)
Presumed that the decision dn is known at every location n. Thus, in order to have perfect
reconstruction, it must be possible to recover the decision dn = D(x, y)(n) from x1 (rather
than x which is not available at synthesis) and y. This amounts to the problem of finding
another decision map D1 such that
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))(,())(,(11
nyxDnyxD jj = ------------
(5.5)
Where x1 is given by (5.1). It can be shown that a necessary, but in no way sufficient,
condition for perfect reconstruction is that the value 11
, =+ =
N
j
jdndn .
5.2.3 Threshold Technique
The input images x, y1, y2 and y3 are obtained by a polyphase decomposition of an
original image x0 is given by,
x (m, n) = x0(2m, 2n)
y1 (m, n) = x0(2m, 2n+1)
y2 (m, n) = x0(2m+1, 2n)
y3 (m, n) = x0(2m+1, 2n+1)
Where x(m,n) represents the current location pixel value. y1(m,n),y2(m,n) and
y3(m,n) are horizontal, vertical and diagonal pixel value respectively. This is obtained by
using context formation, as shown in figure 5.3.The gradient vector v (n), with
components, {v1(n),v2(n),v3(n)}T
(where T represents transposition) is given by)()()(
11nynxnv = ;)()()( 22 nynxnv = ;
)()()(33
nynxnv = ;
Then the L2 norm of v (n) is given by
2
3
2
2
2
1)( vvvsd ++=
Here, the decision map takes binary decision. i.e.) either 1 or 0.
If, Tsd >)( then dn=1 and else dn=0.
Here d(s) is called as seminorm and T denotes the given Threshold. It can take
arbitrary value (user defined). Depending on the condition, the Decision map chooses the
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different update filters, followed the fixed prediction step. The filter equation for different
regions is given by
Decision Region I
If, Tsd >)( xx = 0.4 * y +0.2*yh+0.2*yv+0.2*yd
Decision Region II
If, Tsd )( xx = 0.5 * y + 0.2*yh+0.15*yv+0.15*yd.
Moreover, we are exclusively interested in the case where the decision map D 1 at
synthesis is of the same form as D, but possibly with a different threshold. Thus we need
that
11 )()( TsdTsd >>
The d1(s) is the norm for the gradient vector v 1(n) at synthesis. The v1 (n) is given by
)()()(1
1
11
1 nynxnv =
)()()(1
2
11
2 nynxnv =
)()()(1
3
11
3 nynxnv =
Where )(),(),(1
3
1
2
1
1nynyny corresponds the horizontal, the vertical, and diagonal
detail bands respectively.
The L2 norm of v1(n) is given by
23
122
121
11)()()()( vvvsd ++=
Then decision map is given by
If, Tsd >)(1 then d1n=1 and else d1n=0.
and the filter for different region is given by
Decision Region I
If, Tsd >)(1 x= (1/0.4) * (ry - (0.2*xh+0.2*xv+0.2*xd))
Decision Region II
If, Tsd )( x = (1/0.5)*(ry - (0.2*xh+0.15*xv+0.15*xd))
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Here, Different threshold values are choose arbitrarily and reconstructed. Then the
reconstructed image is compared to original image and found the Peak Signal to Noise
Ratio.
5.3 SPIHT Algorithm
Embedded zero tree wavelet (EZW) coding, introduced by J. M. Shapiro, is a very
effective and computationally simple technique for image compression. Here we offer an
alternative explanation of the principles of its operation, so that the reasons for its
excellent performance can be better understood. These principles are partial ordering by
magnitude with a set partitioning sorting algorithm, ordered bit plane transmission, and
exploitation of self-similarity across different scales of an image wavelet transform.
Moreover, we present a new and different implementation based on set partitioning in
hierarchical trees (SPIHT), which provides even better performance than our previously
reported extension of EZW that surpassed the performance of the original EZW. The
image coding results, calculated from actual file sizes and images reconstructed by the
decoding algorithm, are either comparable to or surpass previous results obtained through
much more sophisticated and computationally complex methods. In addition, the new
coding and decoding procedures are extremely fast, and they can be made even faster,
with only small loss in performance.
5.3.1 Progressive Image Transmission
After converting the image pixels into wavelet coefficient SPIHT [10] is applied.
We assume, the original image is defined by a set of pixel values jip , , where (i, j) the
pixel coordinates. The wavelet transform is actually done to the array given by,
)},({),( jipDWTjic = . --------------- (5.6)
Where c (i, j) is the wavelet coefficients.
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In SPIHT, initially, the decoder sets the reconstruction vector c to zero and
updates its components according to the coded message. After receiving the value
(approximate or exact) of some coefficients, the decoder can obtain a reconstructed
image by taking inverse wavelet transform,
)},({),( jicIDWTjip = --------------- (5.7)
called as progressive transmission.
A major objective in a progressive transmission scheme is to select the most
important information-which yields the largest distortion reduction-to be transmitted first.
For this selection, we use the mean squared-error (MSE) distortion measure
2
,,
2)(
1
1)( jiji
i j
MSE ppN
ppN
ppD == ---------- (5.8)
Where Nis the number of image pixels. jip , is the Original pixel value and jip , is the
reconstructed pixel value. Furthermore, we use the fact that the Euclidean norm is
invariant to the unitary transformation
2
,, )(1
)()( jijii j
MSEMSE ccN
ccDppD == ----------- (5.9)
From the above the equation, it is clear that if the exact value of the transform
coefficient jic , is sent to the decoder, then the MSE decreases by Nc ji /2
, . This means
that the coefficients with larger magnitude should be transmitted first because they have a
larger content of information. This is the progressive transmission method. Extending this
approach, we can see that the information in the value of jic , can also be ranked
according to itsbinary representation, and the most significant bits should betransmitted
first. This idea is used, for example, in the bit-plane method for progressive transmission.
Following, we present a progressive transmission scheme that incorporates these two
concepts: ordering the coefficients by magnitude and transmitting the most significant
bits first. To simplify the exposition, we first assume that the ordering information is
explicitly transmitted to the decoder. Later, we show a much more efficient method to
code the ordering information.
5.3.2 Set Partitioning Sorting Algorithm
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One of the main features of the proposed coding method is that the ordering data
is not explicitly transmitted. Instead, it is based on the fact that the execution path of any
algorithm is defined by the results of the comparisons on its branching points. So, if the
encoder and decoder have the same sorting algorithm, then the decoder can duplicate the
encoders execution path if it receives the results of the magnitude comparisons, and the
ordering information can be recovered from the execution path.
One important fact used in the design of the sorting algorithm is that we do not
need to sort all coefficients. Actually, we need an algorithm that simply selects the
coefficients such that1
, 22+
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=)(TSn 1;n
ji
Tji
cMaxm
2}{ ,),(
---------------- (5.11)
0; otherwise.
to indicate the significance of a set of coordinates T . To simplify the notation of single
pixel sets, we write )}),({( jiSn as ),( jiSn .
5.3.3 Spatial Orientation Trees
A tree structure, called spatial orientation tree, naturally defines the spatial
relationship on the hierarchical pyramid. Fig.5.4 shows how our spatial orientation tree is
defined in a pyramid constructed with recursive four-subband splitting. Each node of the
tree corresponds to a pixel and is identified by the pixel coordinate. Its direct descendants
(offspring) correspond to the pixels of the same spatial orientation in the next finer level
of the pyramid. The tree is defined in such a way that each node has either no offspring
(the leaves) or four offspring, which always form a group of 2 x 2adjacent pixels.
In Fig 5.4, the arrows are oriented from the parent node to its four offspring. The
pixels in the highest level of the pyramid are the tree roots and are also grouped in 2 x 2
adjacent pixels. However, their offspring branching rule is different, and in each group,
one of them (indicated by the star in Fig.5.4)has no descendants.
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Fig 5.4 Examples of parent-offspring dependencies in the
Spatial-orientation tree
The following sets of coordinates are used to present the new coding method:
O (i,j): set of coordinates of all offspring of node (i, j);
D (i, j ) : set of coordinates of all descendants of the node
H: set of coordinates of all spatial orientation tree roots (nodes in the highest
pyramid level);
L(i, j ) = D(i, j ) - O(i, j )
For instance, except at the highest and lowest pyramid levels, we have
O(i,j) = {(2i,2j),(2i,2j+1),(2i+1,2j),(2i+1,2j+1)}
We use parts of the spatial orientation trees as the partitioning subsets in the sorting
algorithm. The set partition rules are simply the following.
I)The initial partition is formed with the sets ((i, j ) ) and D(i,j ), for all (i,j)
H.2) If D(i,j) is significant, then it is partitioned into L(i,j ) plus the four single-element sets
with ( k , I ) O(i,j ) .
3)If L(i,j ) is significant, then it is partitioned into the four sets D(k,I), with (k,I) O(i,j )
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Future Scope
Future work aims at extending this frame work for color images, video
compressions, and De-noising applications
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Conclusion
The architecture has been implemented using .c; The model has been verifiedwith a set of text data. the derived architecture has many advantages especially the
reduction in the number of operations per sample, leading to a reduced chip area and
power consumption .it can be said that the original design criteria has been considered
and an effective and feasible architecture has been designed.
The major disadvantage of this design is that it cannot perform online transform
i.e. the output will not be generated in all instance of data input, this work can be further
extended by using multi processor architecture to speed up the data usage. The maximumclock speed achievable is around 131.6 MHz.
.
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BIBLIGRAPHY
1. Ajith Boopardikar, Wavelet Theory and Application,TMH
2. I.Daubechies W. Sweldens, Factoring wavelet transforms into lifting schemes,
J. Fourier Anal. Appl., vol. 4, pp. 247269, 1998.
3. W. Sweldens, The lifting scheme: A new philosophy in biorthogonal wavelet
constructions, inProc. SPIE, vol. 2569, 1995, [3]
4. JPEG2000 Committee Drafts [Online].
Available: http://www.jpeg.org/CDs15444.htm
5. JPEG2000 Verification Model 8.5 (Technical Description), Sept. 13,2000.
6. K. Andra, C. Chakrabarti, and T. Acharya, A VLSI architecture for lifting basedwavelet transform, in Proc. IEEE Workshop SignlProces. Syst., Oct. 2000,
pp7079.
7. Real-time image compression based on wavelet vector quantization, algorithm
and VLSI architecture ,Hatami, S. Sharifi, S. Ahmadi, H. Kamarei, M.
Dept. of Electr. & Comput. Eng., Univ. of Tehran, Iran; IEEE Trans,May 2005.
8. Fourier Analysis- http://www.sunlightd.com/Fourier/
9. A VLSI Architecture for Lifting-Based Forward and Inverse Wavelet Transform
Kishore Andra, Chaitali Chakrabarti, Member, IEEE, and Tinku Acharya, Senior
Member, IEEE, IEEE Trans. on Signal Processing, vol. 50,No.4, April 2002.
10. K. K. Parhi and T. Nishitani, VLSI architectures for discrete wavelet
transforms,IEEE Trans. VLSI Syst., vol. 1, pp. 191202, June 1993.
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11. M. Ferretti and D. Rizzo, A parallel architecture for the 2-D discrete wavelet
transform with integer lifting scheme, J. VLSI Signal Processing, vol. 28, pp.
165185, July 2001.
12. Discrete Wavelet Transform
http://en.wikipedia.org/wiki/Discrete_wavelet_transform
13. A. Rushton, "VHDL for logic synthesis," Wiley, 1998 5
14. C. H. Roth, "Digital systems design using VHDL," PWS Pub. Co., 1998 5
15. I. Daubechies, "Orthonormal Basis of Compactly Supported Wavelets," Comm. in
Pure and Applied Math Vol. 41, No. 7, pp. 909 -996, 1988 3
16. Chih-Chi Cheng; Chao-Tsung Huang; Ching-Yeh Chen; Chung-Jr Lian; Liang-
Gee Chen, "On-Chip Memory Optimization Scheme for VLSI Implementation of
Line-Based Two-Dimentional Discrete Wavelet Transform," Circuits and
Systems for Video Technology, IEEE Transactions on , vol.17, no.7, pp.814-822,
July 2007
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APPENDIX
Source Code#include
#include
#define ROW 64
#define COL 32
#define R_1 64
#define C_1 64
#define True 1
#define False 0
#define true1 1
#define false1 0
#define Len 4096
int Input[64][64];
int REDB[R_1][C_1];
int st[40000];
int Len_Array=10000;
int Ad[64][64];
int Adr[64][64];
int desc[4][3];
int xDim=64;
int yDim=64;
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int level=1;
int rowS=32;
int colS=32;
int rowL=64;
int colL=64;
int L;
int max;
int currSet[1][4];
int currset;
int D=0;
int T=256;
int Level=2;
int rows=64;
int columns=64;
int Even[64][32];
int Odd[64][32];
int Low[64][32];
int High[64][32];
int LEven[32][32];
int wavedecode[64][64];
int waveencodeImage[64][64];
int HOdd[32][32];
int LOdd[32][32];
int HEven[32][32];
int LL[32][32];
int LH[32][32];
int HL[32][32];
int HH[32][32];
int RLL[32][32];
int RHL[32][32];
int RHH[32][32];
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int RLH[32][32];
int totalbitCount;
int RL[ROW][COL];
int RH[ROW][COL];
int R[ROW][COL];
int H[ROW][COL];
int Output[ROW][ROW];
FILE *R1=0x22000000;
FILE *R2;
int aa;
int cc;
struct descArr{
int desc1[4];
int desc2[4];
int desc3[4];
};
struct descArr descArrS;
void integerdwt();
void reversedwt();
/////////////////////////////////////////////////////////////////////
void integerdwt()
{//begin of integer dwt
int i,j,k,a;
int columns1;
int rows1;
rows1=rows/2;
columns1=columns/2;
//////// one dimensional Even component////////
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for (j=0;j
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}
}
///////////////////one dimensional L even
for (j=0;j
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for (k=0;k
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for(i=0;i
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k=0;
for(i=1;i
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}
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int v,m,t;
char ch;
int manti,tempI,remind;
char charVa;
int i,j,ind;
int index2=0;
int temp1,temp2,temp3;
int min,index,k;
int val;
char *cp=0x22000000;
int matVal=0;
int intValC;
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matVal=(matVal*10)+3;
}
else if(ch=='4'){
matVal=(matVal*10)+4;
}
else if(ch=='5'){
matVal=(matVal*10)+5;
}
else if(ch=='6'){
matVal=(matVal*10)+6;
}
else if(ch=='7'){
matVal=(matVal*10)+7;
}
else if(ch=='8'){
matVal=(matVal*10)+8;
}
else if(ch=='9'){
matVal=(matVal*10)+9;
}
else{
//printf(" %d ",matVal);
REDB[i][j]=matVal;
matVal=0;
cp++;
ch=*cp;
while(ch=='\n'||intValC==010||intValC==020||intValC==003||intValC==032){
cp++;
ch=*cp;
}
break; //break while
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}
intValC=*cp;
if(ch=='\n'||intValC==010||intValC==020||intValC==003||intValC==032){
//printf("LB");
}
cp++;
}//while loop end
}// for j
}//for i
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}
integerdwt();
// printf("matrix all: \n");
for(i=0;i
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}
}
else{
if(j
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return 0;
}//end of main loop
Simulation & Synthesis Result
1 Simulation Results
1.1 Matlab output
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The above figure shows the Matlab GUI interface. The input image is loaded
using image button and the image is converted into text file using convert_text
button.
1.2 Xilinx Results
Input file:
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This figure shows the text files of even, odd, LL, LH, HL, HH.
Comparing input text file with the retrieval text file (after decompression)
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Verifying input text file in Matlab GUI interface.
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Verifying retrieval text file in Matlab GUI interface.
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Synthesis Results
Input Image:
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Compressed Image:-
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Reconstructed Image:-
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3 Synthesis Report
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Overview
Generated on Fri Apr 03 12:03:07 2010
Source D:/New_Folder/lifting/system.xmp
EDK Version 8.1.02
FPGA Family spartan3e
Device xc3s500efg320-4
# IP Instantiated 17
# Processors 1
# Busses 3
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Timing Information
Post Synthesis Clock Limits
These are the post synthesis clock frequencies. The critical frequencies are marked with green.The values reported here are post synthesis estimates calculated for each individual module. These values will changeafter place and route is performed on the entire system.
MODULE CLK Port MAX FREQ
microblaze_0 FSL3_S_CLK 65.595MHz
microblaze_0 DBG_CLK 65.595MHz
DDR_SDRAM_16Mx16 Device_Clk 83.914MHz
DDR_SDRAM_16Mx16 OPB_Clk 83.914MHz
DDR_SDRAM_16Mx16 DDR_Clk90_in 83.914MHz
DDR_SDRAM_16Mx16 Device_Clk90_in 83.914MHz
DDR_SDRAM_16Mx16 Device_Clk90_in_n 83.914MHz
DDR_SDRAM_16Mx16 Device_Clk_n 83.914MHz
DDR_SDRAM_16Mx16 DDR_Clk90_in_n 83.914MHz
opb_intc_0 OPB_Clk 117.233MHz
RS232_DCE OPB_Clk 138.083MHz
debug_module debug_module/drck_i 146.951MHz
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debug_module OPB_Clk 146.951MHz
debug_module bscan_update 146.951MHz
mb_opb OPB_Clk 181.719MHz
ilmb LMB_Clk 249.128MHz
dlmb LMB_Clk 249.128MHz
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