Image Compression Black Book3

8/8/2019 Image Compression Black Book3

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Selective image compression

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INTRODUCTION

Advances over the past in many aspects of digital technology especially devices for

image acquisition data storage and bitmapped printing and display have brought

about many applications of digital imaging. With the possible exception of

facsimile, digital image are not commonplace in general purpose computing system

the way the text and geometric graphics are. The key obstacle for many applications

is the vast amount of data required to represent a digital image directly. Use of

digital images often is not viable due to high storage or transmission costs, even

when image capture and display devices are quite affordable.

In today’s digital world, when we see digital movie, listen digital music, read digital

mail, store documents digital making conversation digitally, we have to deal with

huge amount of digital data. So, data compression plays a very significant role to

keep the digital world realistic. If there were no data compression techniques, we

would have not been able to listen songs over the Internet, see digital pictures or

movies, or we would have not heard about video conferencing or telemedicine.

How data compression made it possible? What are the main advantages of data

compression in digital world? There may be many answers but the three obvious

reasons are the saving of memory space for storage, channel bandwidth and the

processing time for transmission. Every one of us might have experienced that

before the advent MP3, hardly 4 or 5 songs of wav file could be accommodated.

And it was not possible to send a wav file through mail because of its tremendous

file size. Also, it took 5 to 10 minutes or even more to download a song from theInternet.

Now, we can easily accommodate 50 to 60 songs of MP3 in a music CD of same

capacity. Because, the uncompressed audio files can be compressed 10 to 15 times

using MP3 format. And we have no problem in sending any of our favorite music to

our distant friends in any corner of the world. Also, we can download a song in

MP3 in a matter of seconds. This is a simple example of significance of data

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compression. Similar compression schemes were developed for other digital data

like images and videos. Videos are nothings but the animations of frames of images

in a proper sequence at a rate of 30 frames per second or higher. A huge amount of

memory is required for storing video files. The possibility of storing 4/5 movies inDVD CD now rather than we used 2/3 CDs for a movie file is because compression.

We will consider here mainly the image compression techniques.

Images require substantial storage and transmission resources, thus image

compression is advantageous to reduce these requirements. The report covers some

background of wavelet analysis, data compression and how wavelets have been and

can be used for image compression. An investigation into the process and problemsinvolved with image compression was made and the results of this investigation are

discussed.

This rapid development of imaging technologies has generated great interest in

efficient compression methods. Image compression can significantly help by

reducing these data storage requirements. This in turn, enables faster transmission

of image and it can also provide data security. Moreover, it makes possible the

development of efficient image processing algorithm that needs less processing time

by working directly with the compressed data.

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IMAGE COMPRESSION

Advance medical imaging requires storage of large quantities of digitized clinical

data. Due to the bandwidth and storage limitations, medical images must he

compressed before transmission and storage ,. However, the compression will

reduce the image fidelity, especially when the images are compressed at lower bit

rates. The reconstructed images suffer from blocking artifacts and the quality of the

image will be severely degraded under the circumstances of high compression ratio

which shown by JPEG standard.

In recent years, much of the research activities in image coding have been focused

on the discrete wavelet transform (DWT) as the overlapping nature of the transform

alleviates blocking artifacts, while the multiresolution character of the wavelet

decomposition leads to superior energy compaction and perceptual quality of the

decompressed image. Furthermore, the multiresolution transform domain means

that wavelet compression methods degrade much more gracefully than block-DCT

methods as the compression ratio increase.

This introduction has meant that for the first time the discrete wavelet transform is

to be used for the decomposition and reconstruction of images together with an

efficient coding scheme. The aim of multiresolution analysis is simultaneous image

representation on different resolution level. This kind of representation is well

suited to the properties of Human Visual System (HVS) . Over the last decade or so,

wavelets have had a growing impact on signal processing theory and practice, both

because of their unifying role and their successes in various applications. Filter banks, which lie at the heart of wavelet-based algorithms, have become standard

signal processing operators, used routinely in applications ranging from

compression to modems. The contributions of wavelets have often been in the

subtle interplay between discrete-time and continuous-time signal processing.

Wavelet transform represents an image as a sum of wavelet functions (wavelets)

with different locations and scales. Basis for wavelet transform can be composed of

any function that satisfies requirements of multiresolution analysis, it means that

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there exits a large selection of wavelet families depending on the choice of wavelet

function. The topic of multiresolution and wavelet transforms has received a great

deal of attention in recent years. Wavelet transforms are important because they

provide a means for localization of signals in both frequency and time . This leads tomany applications such as signal analys is image compression and applied

mathematics0. Among the most popular wavelets are Haar, Daubechies, coiflet and

biorthogonal, etc. The main purpose of this paper is to investigate the impact and

quality of orthogonal wavelet filter for SPIHT. Meanwhile, we also look into the

effect of the level of wavelet decomposition towards compression efficiency as in .

The compression simulations are done on some images. The qualitative andquantitative results of these simulations are presented. Wavelet methods involve

overlapping transforms with varying-length basis functions. This overlapping nature

of the transform alleviates blocking artifacts, while the multiresolution character of

the wavelet decomposition leads to superior energy compaction and perceptual

quality of the decompressed image. The compression simulations are done on few

images. The qualitative and quantitative results of these simulations are presented .

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LITERATURE REVIEW

Often signals we wish to process are in the time-domain, but in order to process

them more easily other information, such as frequency, is required. Mathematical

transforms translate the information of signals into different representations. For

example, the Fourier transform converts a signal between the time and frequency

domains, such that the frequencies of a signal can be seen. However the Fourier

transform cannot provide information on which frequencies occur at specific times

in the signal as time and frequency are viewed independently. To solve this problem

the Short Term Fourier Transform (STFT) introduced the idea of windows through

which different parts of a signal are viewed. For a given window in time the

frequencies can be viewed.

However Heisenburg.s Uncertainty Principle states that as the resolution of the

signal improves in the time domain, by zooming on different sections, the frequency

resolution gets worse. Ideally, a method of multiresolution is needed, which allows

certain parts of the signal to be resolved well in time, and other parts to be resolved

well in frequency.The topic of multiresolution and wavelet transforms has receiveda great deal of attention in recent years. Wavelet transforms are important because

they provide a means for localization of signals in both frequency and time . This

leads to many applications such as signal analysis, image compression and applied

mathematics.

The power and magic of wavelet analysis is exactly this multiresolution. Images

contain large amounts of information that requires much storage space, largetransmission bandwidths and long transmission times. Therefore it is advantageous

to compress the image by storing only the essential information needed to

reconstruct the image. An image can be thought of as a matrix of pixel (or intensity)

values. In order to compress the image, redundancies must be exploited, for

example, areas where there is little or no change between pixel values. Therefore

images having large areas of uniform colour will have large redundancies, and

conversely images that have frequent and large changes in colour will be less

redundant and harder to compress.

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Wavelet analysis can be used to divide the information of an image into

approximation and detail subsignals. The approximation subsignal shows the

general trend of pixel values, and three detail subsignals show the vertical,horizontal and diagonal details or changes in the image. If these details are very

small then they can be set to zero without significantly changing the image. The

value below which details are considered small enough to be set to zero is known as

the threshold. The greater the number of zeros the greater the compression that can

be achieved.

The amount of information retained by an image after compression anddecompression is known as the energy retained and this is proportional to the sum

of the squares of the pixel values. If the energy retained is 100% then the

compression is known as .lossless., as the image can be reconstructed exactly. This

occurs when the threshold value is set to zero, meaning that the detail has not been

changed. If any values are changed then energy will be lost and this is known as

lossy compression. Ideally, during compression the number of zeros and the energy

retention will be as high as possible.

However, as more zeros are obtained more energy is lost, so a balance between the

two needs to be found. The first part of the report introduces the background of

wavelets and compression in more detail. This is followed by a review of a practical

investigation into how compression can be achieved with wavelets and the results

obtained. The purpose of the investigation was to find the effect of the

decomposition level, wavelet and image on the number of zeros and energy

retention that could be achieved.

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PROBLEM DEFINITION

In certain applications, a large portion of the data traffic over the network comprises

of the transmission of digital images for various commercial purpose taking up a

considerable amount of time. Thus we decided to generate a new compression

algorithm, which focuses on a simple concept of “Wavelet Based Technique which

first decomposes an image hierarchically into oriented sub bands and then encodes

the wavelet coefficients using a zero-tree data.” This compressed data would also

reduce the amount of data traffic over the network.

Hence, our problem can be defined as follows:-

“To develop a software for compressing the image files by using the wavelet

decomposition method for image compression.”

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OBJECTIVES

To develop an algorithm that would compress the image file with a high

compression ratio.

It should support more number of image formats.

It should provide considerable savings in storage space.

It should keep the selected portion of image as it is, while compressing the

rest of the image.

Use of different wavelets for compression

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WAVELET TRANSFORM

Wavelet transform (WT) represents image as a sum of wavelets on different

resolution levels. A wavelet is a (ideally) compact function, i.e., outside a certain

interval it vanishes. Implementations are based on the fast wavelet transform, where

a given wavelet (mother wavelet) is shifted and dilated so as to provide a base in the

function space. In other words, a one-dimensional function is transformed into a

twodimensional space, where it is approximated by coefficients that depend on time

(determined by the translation parameter) and on scale, (determined by the dilation

parameter). The zoom phenomena of the WT offer high temporal localization for

high frequencies while offering good frequency resolution for low frequencies.

Consequently, the WT is especially well suited to analyze local variations such as

those in still images. Multiresolution analysis is implemented via high-pass filters

(wavelets) and low-pass filters (scaling functions).

In this context, the wavelet transform of a signal or image can be realized by means

of a filter bank via successive application of a 2-channel filter bank consisting of

high-pass and low-pass filters: the detail coefficients (from High Pass Filter) of

every iteration step are kept apart, and the iteration starts again with the remaining

approximation coefficients (from Low Pass Filter) of the transform as in .

One of the first efficient wavelet image coders reported in the literature is the

EZW . It is based on the construction of coefficient-trees and successive-

approximations, that can be implemented as bit-plane processing. Due to its

successive-approximation nature, it is SNR scalable, although at the expensive of

sacrificing spatial scalability. SPIHT is an advanced version of this algorithm,

where coefficient-trees are processed in a more efficient way. Both algorithms need

the computation of coefficient-trees and perform different iterations focusing on a

different bit plane in each iteration, what usually requires high computational

complexity .

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The DWT provides sufficient information for the analysis and synthesis of a signal,

but is advantageously, much more efficient. Discrete Wavelet analysis is computed

using the concept of filter banks. Filters of different cut-off frequencies analyse the

signal at different scales. Resolution is changed by the filtering; the scale is changed by upsampling and downsampling. If a signal is put through two filters:

(i) a high-pass filter, high frequency information is kept, low frequency information

is lost.

(ii) a low pass filter, low frequency information is kept, high frequency information

is lost.

Wavelet trnsform in a mutiresolution framework, is one of the efficient ways to

study the information content in a signal. There are two basic approaches to theconcept of multiresolution: the Filter bank approach developed by the sigrial

processing community giving rise to subband coding and the Vector Space

approach developed by the applied rnathematicians, giving rise to the Wavelet

transform.

Then the signal is effectively decomposed into two parts, a detailed part (high

frequency), and an approximation part (low frequency). The subsignal produced

from the low filter will have a highest frequency equal to half that of the original.

According to Nyquist sampling this change in frequency range means that only half

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of the original samples need to be kept in order to perfectly reconstruct the signal.

More specifically this means that upsampling can be used to remove every second

sample. The scale has now been doubled. The resolution has also been changed, the

filtering made the frequency resolution better, but reduced the time resolution.

The approximation subsignal can then be put through a filter bank, and this is

repeated until the required level of decomposition has been reached as in . Features

are extracted from the subbands generated by the wavelet transform. The optimal

number of level in decomposition is application dependent. By selecting an optimal

number of levels and features in the signal that have maximum classification

characteristics, we want to show that increases in classification accuracy andreduction of computation complexity may be achieved. Suppose we use the wavelet

packet transform as a preprocessing tool. For an n-level decomposition, 2n+1-2

subbands are created,0,.

The general idea behind image compression is to remove the redundancy in an

image so as to find a more compact representation. A popular method for image

compression is the so-called transform coding, which represents the image in a

different space than the original, such that the coefficients of the analysis in the

basis of the new space are decorrelated.

It has been known that the multiresolution wavelet decomposition is the projections

onto subspaces spanned by the scaling function basis and the wavelet basis. These

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projections on the scaling functions basis yield approximations of the signal and the

projections on the wavelet basis yield the differences between the approximations of

two adjacent levels of resolution.

Therefore, the wavelet detail images are decorrelated and can be used for image

compression. Indeed, the detail images obtained from the wavelet transform consist

of edges in the image. Since there is little correlation among the values on pixels in

the edge images, it is easily understood why the wavelet transform is useful in

image compression applications. Indeed, image compression is one of the most

popular applications of the wavelet transform as in ,.

Important properties of wavelet functions in image compression applications are

compact support (lead to efficient implementation), symmetry (useful in avoiding

depbasing in image processing), orthogonal (allow fast algorithm), regularity, and

degree of smoothness (related to filter order). The different wavelet functions make

different trade-off between how compactly the basis functions are localized in space

and how smooth they are as in . Some of the wavelet bases have fractal stlucture.

There are many types of wavelets some wavelets have symmetry (valuable in

human vision perception) such as the Biorthogonal Wavelet pairs. Shannon or

“Sinc” Wavelets can find events with specific frequencies (these are similar to the

Sinc Function filters found in traditional DSP). Haar Wavelets (the shortest) are

good for edge detection and reconstructing binary pulses. Coiflets Wavelets are

good for data with self-similarities (fractals) .

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The smoothness of wavelet functions is essential to WAVELETet-based image

compression, but unforlunately, by increasing the smoothness (higher filter order)

will also increase the complexity of calculating DWT. Therefore, in image

compression applications, we have to find a balance between filter order, degree of

smoothness and computational complexity. Inside each wavelet family, we can find

a wavelet function that represents optimal solution related to filter length and degree

of smoothness, hut this solution depends mostly on image content as in.

WHY WAVELET IMAGE COMPRESSION?

We know that lossy JPEG compression introduces blocky artifacts in the

decompressed image, which are not desirable and pleasing to the eyes. Lapped

Orthogonal Transforms (LOT) was proposed to solve this problem by using

smoothing the overlapping blocks. LOT could reduce the blocking effects but its

computational complexity is very high and hence LOT is not preferred to use over

JPEG. On the other hand, wavelet based image compression introduces no blocky

artifacts in the decompressed image. The decompressed image is much smoother

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and pleasant to eyes. Also, we can achieve much higher compression ratios much

regardless of the amount of compression achieved. Another interesting feature of

wavelet is that we can improve the quality of the image more and more and by

adding more and more detail information. This feature is attractive for what isknown as progressive transmission of images.

Another lossy compression scheme developed for image compression is the fractal

base image compression scheme. However the fractal based image compression

beginning to loss ground because it is very complex and time consuming.

Moreover, the achievable compression ratio is not high enough as can be achieved

by using wavelet. Wavelet can also be used for lossless image compression as well.

This is possible by the use of what is known as integer wavelet transforms.

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TIME FREQUENCY PLOT

Wavelet transform provides ‘TIME-FREQUENCY’ representation of a signal, it

uses MULTI-RESOLUTION technique by which different frequencies are analyzed

with different resolutions and it is mostly used to analyze NON-STATIONARY

SIGNALS.

We know that most signals in practice are TIME-DOMAIN signals and when you

plot it you get TIME-AMPLITUDE representation. But for some applications the

actual information is hidden in the frequency content of the signal. To find the

frequency content we use FOURIER TRANSFORM. So, FT gives us

FREQUENCY-AMPLITUDE representation of the time-domain signal.

So, we know that no frequency information is present in time-domain signal and no

time related information is present in frequency-domain signal. But what if we

require both time and frequency information at the same time???

FT tells us how much of each frequency exists in a signal, BUT it does not tell us

when in time these contents exist. Now if the signal is STATIONARY i.e. its

frequency content do not change in time then we do not need to know when these

contents occur because they are present at ALL TIMES!

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But this is not the case for NON STATIONARY signals i.e. signal whose frequency

response changes with time. e.g. Biomedical signals like ECG, EEG,EMG. .

Here if you are just concerned with what frequency contents are present then youcan use FT but what if you want to simultaneously know ‘When’ these frequency

contents are present. Hence, there is a need for TIME-FREQUENCY representation

of the signal i.e. to get time and frequency information simultaneously.

Need for MULTI-RESOLUTION technique:

So, many transforms were developed for time frequency representation. One of

them was Short time Fourier Transform. In STFT signal is divided into short

segments and stationary condition for non-stationary signal is assumed. Then awindow function equal to the length of the segment is selected. It is multiplied with

the

signal. Then its Fourier transform is taken. The window is then shifted and the

process is repeated. One can say STFT is windowed FT. This gives time-freq

representation of the signal. I wont go into detail it will take lot of time. So, what

was the problem with STFT...RESOLUTION!!!

STFT analyses signals using windows of finite length, which covers only a portion

of the signal. So you just know “a band of frequency” that exist in a signal not the

exact frequency components that exists in the signal. So frequency resolution is

poor. Narrower this window, poorer frequency resolution but if the window is

shorter you will be able to resolve your signal better in time i.e. good time

resolution. Conversely, wider the window, good freq resolution but poor time

resolution.

To solve this resolution problem wavelet transform :

The wavelet analysis is done similar to the STFT analysis. The signal to be analyzed

is multiplied with a wavelet function just as it is multiplied with a window function

in STFT, and then the transform is computed for each segment generated. But here

the width of the window is changed for each single frequency component. This is

called MULTI-RESOLUTION analysis i.e. it analyses the signals at different freq

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with different resolution.

The Wavelet Transform, at high frequencies, gives good time resolution and poor

frequency resolution, while at low frequencies; the Wavelet Transform gives good

frequency resolution and poor time resolution

Wavelet Time/Frequency Analysis:The power of the wavelet transform is that it

allows signal variation through time to be examined. Frequently the first example

used for wavelet packet time/frequency analysis is the so called linear chirp, which

exponentially increases in frequency over time. Rather than jumping into the linear

chirp, let us first look at a simple sine function. Figure (1) shows the function

sin(4 Pix), in the range {0..8 Pi}, sampled at 1024 evenly spaced points.

Figure (1)

A magnitude plot of the result of a Fourier transform of the sampled signal is shown

in Figure (2).(here only the relevant part of the plot is shown). This shows a signal

of about 51 cycles, which is what I get when I count the cycles by hand.

Figure (2)

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The Fourier transform plots on this web page do not show adjusted magnitude

(where adjMag = 2Mag /N), so the magnitudes do not properly represent the signal

magnitude.

Figure (3)a shows a frequency/time plot using wavelet packet frequency analysis.

As with the examples above, this plot samples the signal sin(4 Pix) in the range

{0..8 Pi} at 1024 equally spaced points. A square 32 x32 matrix is constructed from

32 elements arrays from the fifth level of the modified wavelet packet transformtree (where we count from zero, starting at the top level of the tree, which is the

original signal). Frequency is plotted on the x-axis and time on the y-axis. The z-

axis plots the founctionlog(1+s[i]2). A gradient map is also shown on the x-y plane.

Figure 3(a)

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Why are there peaks in this plot? The peaks are formed by the filtered signal at the

resolution of the level basis. Since the z-axis plots log(1+s[i] 2), the part of the sine

wave that would be below the plane is flipped up above the plane.

The wavelet frequency/time plot in 3(a) is not as easy to interpret as the Fourier

transform magnitude plot. The signal region is shown in Figure 3(b), scaled to show

the signal region in more detail.

Figure 3(b)

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The wavelet signal is spread out through a range of about 80 cycles, centered at

slightly over 100. The Fourier transform of sin(4 x) shows that there are 51 cycles in

the sample. Is the wavelet packet transform reporting a value that is double the

value reported by the Fourier transform? I don't know the answer. The wavelet

packet transform has been developed in the last decade. Where books like Richard

Lyons' Understanding Digital Signal Processing cover Fourier based frequency

analysis in detail, this depth is lacking the the literature I've seen on the wavelet

packet transform.

Time Frequency Analysis of a Signal Composed of the Sum of Two Sine Waves

The Fourier transform is a powerful tool for decomposing a signal that is composedof the sum of sine (or cosine) waves. The plot below super imposes two sine waves,

sin(16 Pix) and sin(4 Pix).

Figure 4

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When these two signals are added together we get the signal show below in Figure

(5) (shown in detail)

Figure 5

The same signal, plotted through a range of {0..32 Pi} and sampled at 1024 equally

spaced points is shown below.

Figure 6

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The Fourier transform result in Figure 10 shows that this signal is composed the 51

cycle sin(4 Pix) signal and another signal of about 200 cycles (sin(16 Pix)). This is a

case where the Fourier transform really shines as a signal analysis tool. The two

signal components are widely spaced, allowing clear resolution.

Figure 7

The wavelet packet transform plotted in Figure (8). shows two signal components,

the sin(4 Pix) component we saw in Figures 3(a) and 3(b) and the higher frequency

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component from sin(16 Pix). Again, the wavelet packet transform result is not

entirely clear. As the Fourier transform result shows, the higher frequency signal

component is about four times the frequency of the lower frequency component.

This is not quite the case with the wavelet packet transform, where the secondfrequency component appears to be slightly less than four times the frequency of the

sin(4 Pix) component. The surface plot also shows two echo artifacts at higher

frequencies.

Figure 8

For stationary signals that repeat through infinity, where the signal components are

sufficiently separated, the Fourier transform can clearly separate the signal

components. However, the Fourier transform result is only in the frequency domain.

The time component ceases to exist. Also, the basic Fourier tranform does not

provide very useful answers for signals that vary through time. Figure 12 shows a

plot of a sine wave where the frequency increases by Pi/2 in each of eight steps.

Figure 9

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Figure 10(a) shows a surface plot of the modified wavelet packet transform applied

to this signal (using the Haar wavelet). The surface ridge shows the increasing

frequency, although the steps cannot be clearly isolated, perhaps because the

frequency difference between the steps is not sufficiently large. The ridges above

512 on the frequency spectrum are artifacts.

Figure 10 (a)

Figure 10(b) shows a gradient plot, using the same data as Figure 13 (a). As with

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the surface plot representation, we can see the frequency increase, but the step wise

nature of this increase cannot be seen.

Figure 10 (b)

The modified wavelet packet transform is frequently demonstrated using a "linear

chirp" signal. This is a signal with an exponentially increasing frequency, calculated

from the equation:

Figure 11 shows a plot of the linear chirp signal in the region {0..2}, sampled at

1024 points. As the linear chirp frequency increases, the signal becomes

undersampled, which accounts for the jagged arrowhead shape of the signal around

0 as xi gets closer to 2.

Figure 11

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Figure (12) shows the result of the modified wavelet packet transform, using the

Haar wavelet, applied to the linear chirp. The peaks exist because the signal is

sampled at a particular resolution and the absolute value of the signal is plotted.

Note that as the frequency increases the peaks seem to disappear as the signal

cycles get close together. The ridge along the diagonal shows that the signal

frequency increases through time. In theory the linear chirp frequency increases

exponentially, not linearly as this plot suggests. However, the signal is sampled at a

finite number of points, so the exponential nature of the signal disappears as the

signal becomes under sampled. The ridges that are perpendicular to the main

diagonal line are artifacts.

Figure 12

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In theory the Daubechies D4 wavelet transform (e.g., four scaling (H) and four

wavelet coefficients (G)) is closer that the Haar wavelet transform to a perfect filter

that exactly divides the frequency spectrum. The closer the (H, G) filters are to an

ideal filter, the fewer the artifacts in the wavelet packet transform result. The result

of applying the modified wavelet packet transform, using Daubechies D4 filters, to

the linear chirp is shown in figure (13 )

Figure 13

The result in Figure 12 is certainly not better than that obtained using the Haar

transform and, in fact, may be worse. In Ripples in Mathematics, the authors give an

example of wavelet packet transform results using Daubechies D12 filters. There

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are notable fewer artifacts in this case. Jensen and la Cour-Harbo mention that as

the filter length approaches the signal size, the filter approaches an ideal filter.

The plots in Figures 11 and 12 come from 32x32 matrices (where the original

sample consisted of 1024 points). Time is divided up into 32 regions (as is

frequency). Can we get better time/frequency resolution by decreasing the range of

the time regions and increasing the number of frequency regions?

Figure 13 shows a surface plot of a 16x64 matrix generated from the next "linear

basis" (e.g., a horizontal slice through the wavelet packet tree at the next level). As

the gradient plot on the x-y plane shows, the time frequency localization is not

improved.

Figure 13

The plot in Figure 14 is generated from an 8x128 matrix. By further reducing the

time regions, all the frequency bands become compressed into a smaller time

region. Multiple frequency bands become associated with a given time region.

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Figure 14

Discrete Fourier transform:The sequence of N complex numbers x0, ..., x N −1 is

transformed into the sequence of N complex numbers X 0, ..., X N −1 by the DFT

according to the formula:

where i is the imaginary unit and is a primitive N'th root of unity.

(This expression can also be written in terms of a DFT matrix; when scaled

appropriately it becomes a unitary matrix and the X k can thus be viewed as

coefficients of x in an orthonormal basis.)

The transform is sometimes denoted by the symbol , as in or

or .

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

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

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

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

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

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

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

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

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

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


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




The inverse discrete Fourier transform (IDFT) is given by

A simple description of these equations is that the complex numbers X k represent

the amplitude and phase of the different sinusoidal components of the input "signal"

xn. The DFT computes the X k from the xn, while the IDFT shows how to compute

the xn as a sum of sinusoidal components with frequency k / N

cycles per sample. By writing the equations in this form, we are making extensive

use of Euler's formula to express sinusoids in terms of complex exponentials, which

are much easier to manipulate. In the same way, by writing X k in polar form, we

obtain the sinusoid amplitude Ak / N and phase φk from the complex modulus and

argument of X k , respectively:

Note that the normalization factor multiplying the DFT and IDFT (here 1 and 1/ N )

and the signs of the exponents are merely conventions, and differ in some

treatments. The only requirements of these conventions are that the DFT and IDFT

have opposite-sign exponents and that the product of their normalization factors be

1/ N . A normalization of for both the DFT and IDFT makes the transforms

unitary, which has some theoretical advantages, but it is often more practical in

numerical computation to perform the scaling all at once as above (and a unit

scaling can be convenient in other ways).

(The convention of a negative sign in the exponent is often convenient because it

means that X k is the amplitude of a "positive frequency" 2πk / N . Equivalently, the

DFT is often thought of as a matched filter : when looking for a frequency of +1, one

correlates the incoming signal with a frequency of −1.)

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

http://en.wikipedia.org/wiki/Euler's_formula

http://en.wikipedia.org/wiki/Complex_number#Polar_form

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



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

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

http://en.wikipedia.org/wiki/Euler's_formula

http://en.wikipedia.org/wiki/Complex_number#Polar_form




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




Wavelet:A wavelet is a wave-like oscillation with an amplitude that starts out at

zero, increases, and then decreases back to zero. It can typically be visualized as a

"brief oscillation" like one might see recorded by a seismograph or heart monitor.

Generally, wavelets are purposefully crafted to have specific properties that makethem useful for signal processing. Wavelets can be combined, using a "shift,

multiply and sum" technique called convolution, with portions of an unknown

signal to extract information from the unknown signal.

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".

For example, a wavelet could be created to have a frequency of Middle ‘C’ and a

short duration of roughly a 32nd note. If this wavelet were to be convolved at

periodic intervals with a signal created from the recording of a song, then the results

of these convolutions would be useful for determining when the Middle C note was

being played in the song. Mathematically, the wavelet will resonate if the unknown

signal contains information of similar frequency - just as a tuning fork physically

resonates with sound waves of its specific tuning frequency. This concept of resonance is at the core of many practical applications of wavelet theory.

As wavelets are a mathematical tool they can be used to extract information from

many different kinds of data, including - but certainly not limited to - audio signals

and images. Sets of wavelets are generally needed to analyze data fully. A set of

"complementary" wavelets will deconstruct data without gaps or overlap so that the

deconstruction process is mathematically reversible. Thus, sets of complementary

wavelets are useful in wavelet based compression/decompression algorithms where

it is desirable to recover the original information with minimal loss.

More technically, a wavelet is a mathematical function used to divide a given

function or continuous-time signal into different scale components. Usually one can

assign a frequency range to each scale component. Each scale component can then

be studied with a resolution that matches its scale. A wavelet transform is the

representation of a function by wavelets. The wavelets are scaled and translated

36




copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating

waveform (known as the "mother wavelet").

In formal terms, this representation is a wavelet series representation of a square-

integrable function with respect to either a complete, orthonormal set of basis

functions, or an over complete set or Frame of a vector space, for the Hilbert space

of square integrable functions.

Wavelet classification: 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.

CONTINUOUS WAVELET TRANSFORM: IN continuous wavelet transforms, a

given signal of finite energy is projected on a continuous family of frequency bands

(or similar subspaces of the L pfunction space ). For instance the signal may

be represented on every frequency band of the form [ f ,2 f ] for all positive

frequencies f>0. Then, the original signal can be reconstructed by a suitable

integration over all the resulting frequency components.

The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at

scale 1. This subspace in turn is in most situations generated by the shifts of one

generating function , the mother wavelet . For the example of the scale

one frequency band [1,2] this function is

with the (normalized) sinc function. Other example mother wavelets are:

37

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







Meyer Morlet Mexican Hat

The subspace of scale a or frequency band is generated by the functions

(sometimes called child wavelets)

,

Where=a is positive and defines the scale

= b is any real number and defines the shift.

The pair (a,b) defines a point in the right halfplane .

The projection of a function x onto the subspace of scale a then has the form

With wavelet coefficients

.

For the analysis of the signal x, one can assemble the wavelet coefficients into a

scaleogram of the signal.

Discrete wavelet transforms :

It is computationally impossible to analyze a signal using all wavelet coefficients,

38

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

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

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

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




so one may wonder if it is sufficient to pick a discrete subset of the upper half plane

to be able to reconstruct a signal from the corresponding wavelet coefficients. One

such system is the affine system for some real parameters a>1, b>0. The

corresponding discrete subset of the half plane consists of all the points

with integers . The corresponding baby wavelets are now

given as

ψm,n(t ) = a− m / 2ψ(a− mt − nb).

A sufficient condition for the reconstruction of any signal x of finite energy by the

formula

is that the functions form a tight frame of .

Multiresolution discrete wavelet transforms

D4 wavelet

In any discretized wavelet transform, there are only a finite number of wavelet

39




coefficients for each bounded rectangular region in the upper half plane. Still, each

coefficient requires the evaluation of an integral. To avoid this numerical

complexity, one needs one auxiliary function, the father wavelet .

Further, one has to restrict a to be an integer. A typical choice is a=2 and b=1. The

most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet.

From the mother and father wavelets one constructs the subspaces

, where φm,n(t ) = 2 − m / 2φ(2 − mt − n)

and

, where ψm,n(t ) = 2 − m / 2ψ(2 − mt − n).

From these one requires that the sequence

forms a multiresolution analysis of and that the subspacesare the orthogonal "differences" of the above

sequence, that is, W m is the orthogonal complement of V m inside the subspace V m − 1.

In analogy to the sampling theorem one may conclude that the space V m with

sampling distance 2m more or less covers the frequency baseband from 0 to 2 − m − 1.

As orthogonal complement, W m roughly covers the band [2 − m − 1,2− m].

From those inclusions and orthogonality relations follows the existence of

sequences and that satisfy the identities

and

and

and .

40




The second identity of the first pair is a refinement equation for the father wavelet

φ. Both pairs of identities form the basis for the algorithm of the fast wavelet

transform.

Mother wavelet: For practical applications, and for efficiency reasons, one prefers

continuously differentiable functions with compact support as mother (prototype)

wavelet (functions). However, to satisfy analytical requirements (in the continuous

WT) and in general for theoretical reasons, one chooses the wavelet functions from

a subspace of the space

This is the space of measurable functions that are absolutely and square integrable:

and

Being in this space ensures that one can formulate the conditions of zero mean and

square norm one:

is the condition for zero mean, and

is the condition for square norm one.

For ψ to be a wavelet for the continuous wavelet transform (see there for exact

statement), the mother wavelet must satisfy an admissibility criterion (loosely

speaking, a kind of half-differentiability) in order to get a stably invertible

transform.

For the discrete wavelet transform, one needs at least the condition that the wavelet

series is a representation of the identity in the space . Most constructions of

discrete WT make use of the multiresolution analysis, which defines the wavelet by

a scaling function. This scaling function itself is solution to a functional equation.

In most situations it is useful to restrict ψ to be a continuous function with a higher

41




number M of vanishing moments, i.e. for all integer m<M

Some example mother wavelets are:

Meyer Morlet Mexican Hat

The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted)

by a factor of b to give (under Morlet's original formulation):

For the continuous WT, the pair (a,b) varies over the full half-plane ; for

the discrete WT this pair varies over a discrete subset of it, which is also called

affine group.These functions are often incorrectly referred to as the basis functions

of the (continuous) transform. In fact, as in the continuous Fourier transform, there

is no basis in the continuous wavelet transform. Time-frequency interpretation uses

a subtly different formulation (after Delprat).

Comparisons with Fourier Transform (Continuous-Time)

The wavelet transform is often compared with the Fourier transform, in which

signals are represented as a sum of sinusoids. The main difference is that wavelets

are localized in both time and frequency whereas the standard Fourier transform is

only localized in frequency. The Short-time Fourier transform (STFT) is more

similar to the wavelet transform, in that it is also time and frequency localized, but

there are issues with the frequency/time resolution trade-off. Wavelets often give a

better signal representation using Multiresolution analysis, with balanced resolution

42




at any time and frequency.

The discrete wavelet transform is also less computationally complex, taking O( N )

time as compared to O( N log N ) for the fast Fourier transform. This computational

advantage is not inherent to the transform, but reflects the choice of a logarithmic

division of frequency, in contrast to the equally spaced frequency divisions of the

FFT.It is also important to note that this complexity only applies when the filter size

has no relation to the signal size. A wavelet without compact support such as the

Shannon wavelet would require O( N^2). (For instance, a logarithmic Fourier

Transform also exists with O( N ) complexity, but the original signal must be

sampled logarithmically in time, which is only useful for certain types of signals .

Scaling filter:

An orthogonal wavelet is entirely defined by the scaling filter - a low-pass finite

impulse response (FIR) filter of length 2N and sum 1. In biorthogonal wavelets,

separate decomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter is calculated as the

quadrature mirror filter of the low pass, and reconstruction filters are the time

reverse of the decomposition filters.

Daubechies and Symlet wavelets can be defined by the scaling filter.

Scaling function: Wavelets are defined by the wavelet function ψ(t ) (i.e. the

mother wavelet) and scaling function φ(t ) (also called father wavelet) in the time

domain.

The wavelet function is in effect a band-pass filter and scaling it for each level

halves its bandwidth. This creates the problem that in order to cover the entire

spectrum, an infinite number of levels would be required. The scaling function

filters the lowest level of the transform and ensures all the spectrum is covered.

For a wavelet with compact support, φ(t ) can be considered finite in length and is

43




equivalent to the scaling filter.

44




BLOCK DIAGRAM

IMAGE ACQUISITION:-

Most image processing programs are designed to start by loading an image from

disk. There are some facilities for acquiring images directly from a camera or from

a video source. This capability means that you can skip the steps involved in using

two separate programs: the first to control the acquisition and the second for data

analysis.

Image acquisition is hardware dependent. The first stage of any vision system is the

image acquisition stage.

After the image has been obtained, various methods of processing can be applied to

the image to perform the many different vision tasks required today.

However, if the image has not been acquired satisfactorily then the intended tasks

may not be achievable, even with the aid of some form of image enhancement.

CompressedImage

Input

Image

Selective ImageCompression

45




SELECTIVE IMAGE COMPRESSION:-

It consists of three main blocks:-

Region of Interest:-In this block, user has to define the region of interest. Thisselected region and the remaining image are then applied to wavelet decomposition.

After this the selected portion of image is quantized and encoded with lossless

image compression techniques. The remaining portion of image is also quantized

and encoded with lossy image compression techniques.

WAVELET DECOMPOSITION:-

Wavelet signifies small wave. It was first used in approximating a function by linear

combination of various waveforms obtained by translating and scaling the wavelet

at various position and scales. It was very old from the time of Alfred Haars. But it

was not so popular then because it found no application area. It becomes popular

only when Ingrid Daubechies shows that QMF (Quadrature Mirror Filter) filters

used in filterbank for subband coding can be generated from the wavelet by using

the perfect reconstruction relation of the filter bank. So, what we obtain from the

wavelet is a set of QMF filter banks that can be used for subband coding. In a QMFfilter bank a signal is first decomposed into low pass and high pass components

using low filters.

The filter components are reduced their size by half either by rejecting the even or

odd samples thereby the total size of the original signal is preserved. The low pass

filter component retains almost all distinguishable features of the original signal.And the high pass filter component has little or no resemblance of the original

WaveletDecomposition

Quantization& Encoding

Selecting the part of Image

46




signal. The low pass component is again decomposed into two components. The

decomposition process can be continued up to the last possible level or up to a

certain desired level. As the high pass filter components have less information

discernible to the original signal, we can eliminate the information contents of thehigh pass filters partially or significantly at each level of decomposition during the

reconstruction process. It is this possibility of elimination of the information

contents of the high pass filter components that gives higher compression ratio in

the case of wavelet based image compression. Simple decomposition and

reconstruction by eliminating less informative parts of the high pass filter

components may not always lead to the compression.

For we have to use certain coding scheme to get higher compression ratios. Some of

the most cited coding algorithms for wavelet based image compression are EZW

(Embedded Zerotree Wavelet) , SPIHT (Set Partitioning in Hierarchical Tree) and

EBCOT (Embedded Block Coding with Optimal Truncation) . EZW is no longer

popular as it has been improved to SPIHT. Now SPIHT and EBCOT are the two

main contenders for wavelet based image coding. JPEG group has accepted

EBCOT as their wavelet based coding scheme to upgrade their JPEG to JPEG2000

version to achieve higher compression ratio and less distortion in the decompressed

signal. So, now the main contender is between SPIHT and JPEG2000.

In any wavelet based image compression scheme, the achievable compression ratio

is not only dependent on the efficiency of the coding scheme, it is also dependent on

the choice of appropriate wavelet filters. Different filters give different compression

ratios for the same image and coding scheme . There are numerous wavelet filters.

It is our task to choice appropriate filters for our compression scheme. For

JPEG2000, filters used 9/7 biothogonal wavelet filters for lossy image compression

and 5/3 for lossless image compression. For SPIHT coding scheme, usually 9/7

biorthogonal filters is used for lossy image compression and S+P transform filters

for lossless image compression. However, 9/7 is not the optimal filters for lossy

image compression, still lot of research is going on about the finding of optimal

filters for lossy image compression for different image types.

47




integer c. After that quantization we produce a finite set of values which can be

encoded by say binary techniques.

Quantization, involved in image processing, is a lossy compression technique

achieved by compressing a range of values to a single quantum value. When the

number of discrete symbols in a given stream is reduced, the stream becomes more

compressible. For example, reducing the number of colors required to represent a

digital image makes it possible to reduce its file size. Specific applications include

DCTdata quantization in JPEG and DWT data quantization in JPEG 2000.

There are two types of quantization:-

Scalar Quantization and

Vector Quantization

The Wavelet Scalar Quantization algorithm (WSQ) is a compression algorithm

used for gray-scale fingerprint images. It is based onwavelet theory and has become

a standard for the exchange and storage of fingerprint images. WSQ was developed

by the FBI, the Los Alamos National Laboratory, and the National Institute of

Standards and Technology (NIST).

This compression method is preferred over standard compression algorithms like

JPEG because at the same compression ratios WSQ doesn't present the "blocking

artifacts" and loss of fine-scale features that are not acceptable for identification in

financial environments and criminal justice.

Vector quantization is a classical quantization technique from signal processing

which allows the modeling of probability density functions by the distribution of

prototype vectors. It was originally used for data compression. It works by dividing

a large set of points (vectors) into groups having approximately the same number of

49

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

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

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

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

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

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


http://en.wikipedia.org/wiki/JPEG_2000

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

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

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

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

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



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

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

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

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




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



http://en.wikipedia.org/wiki/JPEG_2000

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

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

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

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




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

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


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




points closest to them. Each group is represented by its centroid point, as ink-means

and some other clustering algorithms.

The density matching property of vector quantization is powerful, especially for

identifying the density of large and high-dimensioned data. Since data points are

represented by the index of their closest centroid, commonly occurring data have

low error, and rare data high error. This is why VQ is suitable for lossy data

compression. It can also be used for lossy data correction and density estimation.

Vector quantization is based on the competitive learning paradigm, so it is closely

related to the self-organizing map model.

LOSSLESS ENCODING :-

The existing systems used for lossless encoding are:-

Run length encoding:-

Run-length encoding (RLE) is a very simple form of data compression in

which runs of data (that is, sequences in which the same data value occurs in many

consecutive data elements) are stored as a single data value and count, rather than as

the original run. This is most useful on data that contains many such runs: for example, relatively simple graphic images such as icons, line drawings, and

animations. It is not recommended for use with files that don't have many runs as it

could potentially double the file size.

Simple implementation of each RLE algorithm

Compression efficiency restricted to a particular type of contents

Mainly utilized for encoding of monochrome graphic data

May cause data explosion i.e. the output may be larger than the input or

original.

Huffman coding:-

Huffman coding is an entropy encoding algorithm used for lossless data

compression. The term refers to the use of a variable-length code table for encoding

a source symbol (such as a character in a file) where the variable-length code table

has been derived in a particular way based on the estimated probability of

50

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

http://en.wikipedia.org/wiki/K-means

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


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


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

http://en.wikipedia.org/wiki/Self-organizing_map


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

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

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


http://en.wikipedia.org/wiki/Variable-length_code

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

http://en.wikipedia.org/wiki/K-means




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

http://en.wikipedia.org/wiki/Self-organizing_map










occurrence for each possible value of the source symbol.

Huffman coding uses a specific method for choosing the representation for each

symbol, resulting in a prefix code (sometimes called "prefix-free codes") (that is,

the bit string representing some particular symbol is never a prefix of the bit stringrepresenting any other symbol) that expresses the most common characters using

shorter strings of bits than are used for less common source symbols.

Works on image brightness histogram

Uses shortest codes to represent most commonly occurring brightness

patterns.

Compression rates of 1.5 - 2:1

Lossless predictive coding:-

For typical images, the values of adjacent pixels are highly correlated; that is, a

great deal of information about a pixel value can be obtained by inspecting its

neighboring pixel values. This property is exploited in predictive coding techniques

where an attempt is made to predict the value of a given pixel based on the values of

the surrounding pixels.

Stores the difference between successive pixel’s brightness in fewer bits. Relies on the image having smooth changes in brightness; at sharp changes

in the image we need overload patterns.

Gives up to 2:1 image compression rates – can be improved by iterative

application.

Arithmetic coding:-

Arithmetic coding is a method for lossless data compression. Normally, a string of

characters such as the words "hello there" is represented using a fixed number of

bits per character, as in the ASCII code. Like Huffman coding, arithmetic coding is

a form of variable-length entropy encoding that converts a string into another form

that represents frequently used characters using fewer bits and infrequently used

characters using more bits, with the goal of using fewer bits in total. As opposed to

other entropy encoding techniques that separate the input message into its

component symbols and replace each symbol with a code word, arithmetic coding

encodes the entire message into a single number, a fraction n where (0.0 ≤ n < 1.0).

51

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


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

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



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


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

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






L-Z-W Coding:-

Lempel–Ziv–Welch (LZW) is a universal lossless data compression algorithm. A

dictionary is initialized to contain the single-character strings corresponding to all

the possible input characters (and nothing else). The algorithm works by scanning

through the input string for successively longer substrings until it finds one that is

not in the dictionary. When such a string is found, it is added to the dictionary, and

the index for the string less the last character (i.e., the longest substring that is in the

dictionary) is retrieved from the dictionary and sent to output. The last input

character is then used as the next starting point to scan for substrings.

In this way successively longer strings are registered in the dictionary and made

available for subsequent encoding as single output values. The algorithm works best

on data with repeated patterns, so the initial parts of a message will see little

compression. As the message grows, however, the compression ratio tends

asymptotically to the maximum.

Lossy Encoding:-

Transform Coding:-

Transform coding is a type of data compression for "natural" data like audio signals

or photographic images. The transformation is typically lossy, resulting in a lower

quality copy of the original input. In transform coding, knowledge of the application

is used to choose information to discard, thereby lowering its bandwidth. The

remaining information can then be compressed via a variety of methods. When the

output is decoded, the result may not be identical to the original input, but is

52



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


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

http://en.wikipedia.org/wiki/Signal_(information_theory)



http://en.wikipedia.org/wiki/Bandwidth_(computing)



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


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

http://en.wikipedia.org/wiki/Signal_(information_theory)



http://en.wikipedia.org/wiki/Bandwidth_(computing)




expected to be close enough for the purpose of the application.

53




SELECTIVE IMAGE COMPRESSION

i. In most of the cases, it is required to keep some information in the imageintact as compressing the entire information in frequency domain lead to

loof data. The main aim of selective compression is to partially compress the

given image. To achieve the same, the required part of the image which

needs to be retained is cropped. The rest of the image is compressed using

selected wavelets to achieve compression. The cropped image is overlapped

on compressed image to get the output image.

Original image

ii. Select the masked image and obtain the transform coefficients by

transforming the masked image in accordance with the wavelet transforming

technique.

SELECTED REGION IN AN IMAGE

54




iii. Determining a quantizing interval which minimizes quantization errors in a

predetermined step size, wherein the quantizing interval is determined

according to statistical characteristics of the masked image.

Masked image

iv. Generating a simple tree structure based on the transform coefficients

modifying the simple tree structure by using statistical characteristics of the

transform coefficients to produce a monotonically decreasing tree structure,

wherein the monotonically decreasing tree structure comprises a parent node

and child nodes which correspond to the parent node.

v. Modifying the monotonically decreasing tree structure by limiting a

maximum height difference between the parent node and the child nodes to

produce a modified tree list.

55




Transformed image

vi. Quantizing the transform coefficients based on the quantizing interval to

produce quantized transform coefficients.

vii. Arithmetically coding the modified tree list and the quantized transform

coefficients to produce the coded data, wherein the coded data corresponds

to the output compressed image. Reconstruct the compressed image and

place the cropped image at its original position.

Output image

56




57




RESULTS

WAVEL

ET

INPUT IMAGE_1

CAMERAMAN.TIF

INPUT IMAGE_2

LIFTINGBODY.PN

G

ORIGINA

L IMAGE

(KB)

OUTPUT

IMAGE

(KB)

ORIGINA

L IMAGE

(KB)

OUTPUT

IMAGE

(KB)

BIORTH

ONORM

AL-1.3

63.9 11.0 122 15.4

COIFLET

-1

63.9 10.9 122 15.2

HAAR 63.9 9.71 122 12.4

DAUBEC

HIES-2

63.9 10.8 122 15.1

SYMLET-1

63.9 9.71 122 12.4

58




In this paper, we found that wavelet-based image compression prefers smooth

functions of relatively short length.We have applied the input image to various

wavelets such as Daubuchies, Haar, Coiflets, Symlets and Biorthogonal for image

compression and then reconstructed the image. Our results show that differentwavelet filters performed differently for different images, but generally the

difference between each other was not great. As the results shown, HAAR and

SYMLET-1 are the best-suited wavelet filters for an in all compression bit rate and

it also aligns the cropped image more perfectly than others. But BIORTHOGONAL

wavelet shows worst result as compared to others. As conclusion, the choice of the

best performing wavelet filter in medical image compression is mostly depends on

the image content.

60




The many benefits of image compression include less required storage

space, quicker sending and receiving of images, and less time lost on image

viewing and loading. But where and how is image compression used today?

The digital form of image compression is also being put to work in

industries such as fax transmission, satellite remote sensing, and high

definition television, etc.

In certain industries, the archiving of large numbers of images is required. A

good example is the health industry, where the constant scanning and/or

storage of medical images and documents take place. Image compression

offers many benefits here, as information can be stored without placing largeloads on system servers. Depending on the type of compression applied,

images can be compressed to save storage space, or to send to multiple

physicians for examination. And conveniently, these images can uncompress

when they are ready to be viewed, retaining the original high quality and

detail that medical imagery demands.

Image compression is also useful to any organization that requires the

viewing and storing of images to be standardized, such as a chain of retail

stores or a federal government agency. In the retail store example, the

introduction and placement of new products or the removal of discontinued

items can be much more easily completed when all employees receive, view

and process images in the same way. Federal government agencies that

standardize their image viewing, storage and transmitting processes can

eliminate large amounts of time spent in explanation and problem solving.

The time they save can then be applied to issues within the organization,

such as the improvement of government and employee programs.

In the security industry, image compression can greatly increase the

efficiency of recording, processing and storage. However, in this application

it is imperative to determine whether one compression standard will benefit

all areas. For example, in a video networking or closed-circuit television

62

http://www.lockergnome.com/windows/2006/12/25/real-world-applications-of-image-compression/#%23












application, several images at different frame rates may be required. Time is

also a consideration, as different areas may need to be recorded for various

lengths of time. Image resolution and quality also become considerations, as

does network bandwidth, and the overall security of the system.

Museums and galleries consider the quality of reproductions to be of the

utmost importance. Image compression, therefore, can be very effectively

applied in cases where accurate representations of museum or gallery items

are required, such as on a Web site. Detailed images that offer short

download times and easy viewing benefit all types of visitors, from the

student to the discriminating collector. Compressed images can also be used

in museum or gallery kiosks for the education of that establishment’svisitors. In a library scenario, students and enthusiasts from around the

world can view and enjoy a multitude of documents and texts without

having to incur traveling or lodging costs to do so.

Regardless of industry, image compression has virtually endless benefits

wherever improved storage, viewing and transmission of images are

required. And with the many image compression programs available today,

there is sure to be more than one that fits your requirements best.

63










FUTURE SCOPE

64




Our project basically aims at reducing storage space requirement and

transmission time during image data transfer so that cost effective

communication can be achieved.

Further developments can be made in order to achieve still better

compression ratio since use of the self similarity of images under different

scale is possible.

Further efforts can be made to develop a good Graphic User Interface so as

to make the software more user friendly.

65




Selective image compression is a technique where explicitly defined

regions of interest are kept as it is, to maintain the important data and

compressing the remaining image.

Images require substantial storage and transmission resources, thus image

compression is advantageous to reduce these requirements.The main

advantages of the image compression are as follows: it saves memory space

for storage, channel bandwidth and the processing time for transmission.

Multiresolution is method, which allows certain parts of the signal to beresolved well in time, and other parts to be resolved well in frequency.

Multiresolution character of the wavelet decomposition leads to superior

energy compaction and perceptual quality of the decompressed image.

There are three methods which are used for image compression: Fourier

Transform, Cosine Transform and Wavelet Transform.

Important properties of wavelet functions in image compression applications

are compact support (lead to efficient implementation), symmetry (useful in

avoiding depbasing in image processing), orthogonal (allow fast algorithm),

regularity, and degree of smoothness (related to filter order).

There are many types of wavelets some wavelets have symmetry (valuable

in human vision perception) such as the Biorthogonal Wavelet pairs.

Shannon or “Sinc” Wavelets can find events with specific frequencies (these

are similar to the Sinc Function filters found in traditional DSP). Haar

Wavelets (the shortest) are good for edge detection and reconstructing

binary pulses. Coiflets Wavelets are good for data with self-similarities

(fractals).

Wavelet signifies small wave. It was first used in approximating a function

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by linear combination of various waveforms obtained by translating and

scaling the wavelet at various position and scales. It was very old from the

time of Alfred Haars.

Wavelet uses the concept of Subband coding which uses two filters: low

pass filter and high pass filter. The low pass filter component retains almost

all distinguishable features of the original signal. And the high pass filter

component has little or no resemblance of the original signal. The low pass

component is again decomposed into two components.

The decomposition process can be continued up to the last possible level or up to a certain desired level.

As the high pass filter components have less information discernible to the

original signal, we can eliminate the information contents of the high pass

filters partially or significantly at each level of decomposition during the

reconstruction process.

It is this possibility of elimination of the information contents of the high

pass filter components that gives higher compression ratio in the case of

wavelet based image compression.

Quantization, involved in image processing, i s a lossy compression

technique achieved by compressing a range of values to a single quantum

value.

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REFERENCES

[l] N. Jayant. J. Johntson, and R. Safranek, "Signal compression based on human

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IEEE

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[11] Michael E. Zervakis, Taek Mu Kwon and Andreas E. Savakis, “Operator

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June 1996.

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