Anusha Final Tech Doc

8/2/2019 Anusha Final Tech Doc

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Chapter 1

INTRODUCTION TO IMAGE TRANSFORMATION AND COMPRESSION

USING WAVELETS1. Introduction

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

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

large amounts of information that requires much storage space, large transmission 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. 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. 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.

Chapter 2

LITERATURE SURVEY


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2.1 The need for Wavelets

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. A good analogy for this idea is given by

Hubbard. In order to do this calculation we would find it easier to first translate the numerals in toour number system, and then translate the answer back into a roman numeral. The result is the

same, but taking the detour into an alternative number system made the process easier and quicker.

Similarly we can take a detour into frequency space to analysis or process a signal.

2.1.1 Multiresolution and Wavelets

The power of Wavelets comes from the use of multiresolution. High frequency parts of the

signal use a small window to give good time resolution, low frequency parts use a big window to

get good frequency information. As frequency resolution gets bigger the time resolution must get

smaller. .

In Fourier analysis a signal is broken up into sine and cosine waves of different

frequencies, and it effectively re-writes a signal in terms of different sine and cosine waves.

Wavelet analysis does a similar thing, it takes a mother wavelet, then the signal is translated into

shifted and scale versions of this mother wavelet.


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Figure 2.1 The different transforms provided different resolutions of time and frequency.

2.1.2 The Continuous Wavelet Transform (CWT)

The continuous wavelet transform is the sum over all time of scaled and shifted versions of

the mother wavelet . Calculating the CWT results in many coefficients C, which are functions of

scale and translation.

The translation, , is proportional to time information and the scale, s, is proportional to the

inverse of the frequency information. To find the constituent wavelets of the signal, the coefficients

should be multiplied by the relevant version of the mother wavelet. The scale of a wavelet simply

means how stretched it is along the x-axis, larger scales are more stretched:


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Figure 2.2 The db8 wavelet shown at two different scales

The translation is how far it has been shifted along the x-axis. Figure 2.3 shows a wavelet,

figure 2.4 shows the same mother wavelet translated by k:

Figure 2.3 The translated wavelet.


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Figure 2.4 The wavelet translated by k

The Wavelet Toolbox Users Guide [7] suggests five easy steps to compute the CWT

coefficients for a signal.

1. Choose a wavelet, and compare this to the section at the start of the signal.

2. Calculate C, which should measure how similar the wavelet and the section of the signal are.

3. Shift the wavelet to the right by translation , and repeat steps 1 and 2, calculating values of C

for all translations.

4. Scale the wavelet, and repeat steps 1-3.

5. Repeat 4 for all scales.

Chapter 3

WAVELET TRANSFORMS

Wavelet transform is capable of providing the time and frequency information

simultaneously, hence it gives a time-frequency representation of the signal. When we are

interested in knowing what spectral component exists at any given instant of time, we want to

know the particular spectral component at that instant. In these cases it may be very beneficial to

know the time intervals these particular spectral components occur. For example, in EEGs, the

latency of an event-related potential is of particular interest (Event-related potential is the response

of the brain to a specific stimulus like flash-light, the latency of this response is the amount of time

elapsed between the onset of the stimulus and the response)examplecomes from a reference.


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Wavelets (small waves) are functions defined over a finite interval and having an average

value of zero. The basic idea of the wavelet transform is to represent any arbitrary function (t) as

a superposition of a set of such wavelets or basis functions. These basis functions are obtained

from a single wave, by dilations or contractions (scaling) and translations (shifts). For pattern

recognition continuous wavelet transform is better. The Discrete Wavelet Transform of a finite

length signal x (n) having N components, for example, is expressed by an N x N matrix similar to

the Discrete cosine transform (DCT).

3.1 WAVELET-BASED COMPRESSION

Digital image is represented as a two-dimensional array of coefficients, each coefficient

representing the brightness level in that point. We can differentiate between coefficients as more

important ones, and lesser important ones. Most natural images have smooth color variations, with

the fine details being represented as sharp edges in between the smooth variations. Technically, the

smooth variations in color can be termed as low frequency variations and the sharp variations as

high frequency variations.

The basic difference wavelet-based and Fourier-based techniques is that short-time Fourier-

based techniques use a fixed analysis window, while wavelet-based techniques can be considered

using a short window at high spatial frequency data and a long window at low spatial frequency

data. This makes DWT more accurate in analyzing image signals at different spatial frequency, and

thus can represent more precisely both smooth and dynamic regions in image. The compressor

includes forward wavelet transform, Quantizer, and Lossless entropy encoder. The corresponding

decompressed is formed by Lossless entropy decoder, de-Quantizer, and an inverse wavelet

transform. Wavelet-based image compression has good compression results in both rate and

distortion sense.

3.2 PRINCIPLES OF IMAGE COMPRESSION

Image Compression is different from data compression (binary data). When we apply

techniques used for binary data compression to the images, the results are not optimal. In Lossless

compression the data (binary data such as executables, documents etc) are compresses such that

when decompressed, it give an exact replica of the original data. They need to be exactly

reproduced when decompressed. On the other hand, images need not be reproduced 'exactly'.


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In images the neighboring pixels are correlated and therefore contain redundant

information. Before we compress an image, we first find out the pixels, which are correlated. The

fundamental components of compression are redundancy and irrelevancy reduction. Redundancy

means duplication and Irrelevancy means the parts of signal that will not be noticed by the signal

receiver, which is the Human Visual System (HVS).

There are three types of redundancy can be identified:

1) Spatial Redundancy i.e. correlation between neighboring pixel values.

2) Spectral Redundancy i.e. correlation between different color planes or spectral bands.

3) Temporal Redundancy i.e. correlation between adjacent frames in a sequence of images (in

video applications).

Image compression focuses on reducing the number of bits needed to represent an image

by removing the spatial and spectral redundancies.

3.3 PROCEDURE FOR COMPRESSING AN IMAGE

Just as in any other image compression schemes the wavelet method for image

compression also follows the same procedure:

i) Decomposition of signal using filters banks.

ii) Down sampling of Output of filter bankiii) Quantization of the above

iv) Finally encoding

For decoding

i) Up samples

ii) Re composition of signal with filter bank.

3.4 PRINCIPLES OF USING TRANSFORM AS SOURCE ENCODER

Before going into details of wavelet theory and its application to image compression, I willgo ahead and describe few terms most commonly used in this report.

Mathematical transformations are applied to signals to obtain further information from that

signal that is not readily available in the raw signal.

Various wavelet transforms

Discrete Wavelet Transform


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Haar Wavelet Transform

3.5 DISCRETE WAVELET TRANSFORM

At the heart of the analysis (or compression) stage of the system is the forward discrete

wavelet transform (DWT). A block diagram of a wavelet based image compression system is

shown in Figure 3.1. Here, the input image is mapped from a spatial domain, to a scale-shift

domain. This transform separates the image information into octave frequency subbands. The

expectation is that certain frequency bands will have zero or negligible energy content; thus,

information in these bands can be thrown away or reduced so that the image is compressed without

much loss of information.

The DWT coefficients are then quantized to achieve compression. Information lost during

the quantization process cannot be recovered and this impacts the quality of the reconstructed

image. Due to the nature of the transform, DWT coefficients exhibit spatial correlation, that are

exploited by quantization algorithms like the embedded zero-tree wavelet (EZW) and set

partitioning in hierarchical trees (SPIHT) for efficient quantization. The quantized coefficients may

then be entropy coded; this is a reversible process that eliminates any redundancy at the output of

the quantizer.

Figure 3.1: A wavelet based image compression system.

In the synthesis (or decompression) stage, the inverse discrete wavelet transform recovers

the original image from the DWT coefficients. In the absence of any quantization the reconstructed

image will be identical to the input image. However, if any information was discarded during the


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quantization process, the reconstructed image will only be an approximation of the original image.

Hence this is called lossy compression. The more an image is compressed, the more information is

discarded by the quantizer; the result is a reconstructed image that exhibits increasingly more

artifacts. Certain integer wavelet transforms exist that result in DWT coefficients that can be

quantized without any loss of information. These result in lossless compression, where the

reconstructed image is an exact replica of the input image. However, compression ratios achieved

by these transforms are small compared to lossy transforms (e.g. 4:1 compared to 40:1).

3.6 2-D DISCRETE WAVELET TRANSFORM

A 2-D separable discrete wavelet transform is equivalent to two consecutive 1-D

transforms. For an image, a 2-D DWT is implemented as a 1-D row transform followed by a 1-D

column transform. Transform coefficients are obtained by projecting the 2-D input image x(u, v)

onto a set of 2-D basis functions that are expressed as the product of two 1-D basis.

The 2-D DWT (analysis) can be expressed as the set of equations shown below. The scaling

function and wavelets and (and the corresponding

transform coefficients X(N, j,m), X(1)(N, j,m), X(2)(N, j,m) and X(3)(N, j,m)) correspond to different

subbands in the decomposition. X(N, j,m) are the coarse coefficients that constitute the LL

subband. The X(1)(N, j,m) coefficients contain the vertical details and correspond to the LH

subband. The X(2)(N, j,m) coefficients contain the horizontal details and correspond to the HL

subband. The X(3)(N, j,m) coefficients represent the diagonal details in the image and constitute the

HH subband.

The histograms for coefficients in all subbands shown in, illustrate the advantage of

transform coding. The original 512 512 lighthouse image had few zero pixel values. Thus a

single-level decomposition at scale (N +1) has four subbands of coefficients as shown in Figure

3.2.


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Figure 3.2: Single-level 2-D wavelet decomposition (analysis).

The synthesis bank performs the 2-D IDWT to reconstruct . 2-D IDWT is

given in equation below.

A single stage of a 2-D filter bank is shown in Figure 3.3. First, the rows of the input image

are filtered by the highpass and lowpass filters. The outputs from these filters are downsampled by


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two, and then the columns of the outputs are filtered and downsampled to decompose the image

into four subbands. The synthesis stage performs upsampling and filtering to reconstruct the

original image.

Multiple levels of decomposition are achieved by iterating the analysis stage on only the

LL band.

Figure 3.3: One level filter bank for computation of 2-D DWT and IDWT.

For l levels of decomposition, the image is decomposed into 3l + 1 subbands. Figure 3.4

shows the filter bank structure for three levels of decomposition of an input image, and Figure 3.5

shows the subbands in the transformed image.

Figure 3.4: Analysis section of a three level 2-D filter bank.


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Figure 3.5: Three level wavelet decomposition of an image.

Chapter 4

HAAR WAVELETS

4.1 THE HAAR WAVELET

The Haar wavelet algorithms published here are applied to time series where the number of

samples is a power of two (e.g., 2, 4, 8, 16, 32, 64...) The Haar wavelet uses a rectangular window

to sample the time series. The first pass over the time series uses a window width of two. The

window width is doubled at each step until the window encompasses the entire time series. Each

pass over the time series generates a new time series and a set of coefficients. The new time series

is the average of the previous time series over the sampling window. The coefficients represent the

average change in the sample window. For example, if we have a time series consisting of the

values v0, v1, ... vn, a new time series, with half as many points is calculated by averaging the pointsin the window. If it is the first pass over the time series, the window width will be two, so two

points will be averaged:

for (i = 0; i < n; i = i + 2)

si = (vi + vi+1)/2;

The 3-D surface below graphs nine wavelet spectrums generated from the 512 point AMAT

close price time series. The x-axis shows the sample number, the y-axis shows the average value at

that point and the z-axis shows log2 of the window width.


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Figure 4.1: The Wavelet Spectrum

The wavelet coefficients are calcalculated along with the new average time series values.

The coefficients represent the average change over the window. If the windows width is two this

would be:

for (i = 0; i < n; i = i + 2)

ci = (vi - vi+1)/2;

The graph below shows the coefficient spectrums. As before the z-axis represents the log2

of the window width. The y-axis represents the time series change over the window width.

Somewhat counter intutitively, the negative values mean that the time series is moving upward

vi is less than vi+1

so vi - vi+1 will be less than zero

Positive values mean the the time series is going down, since vi is greater than vi+1. Note

that the high frequency coefficient spectrum (log2(windowWidth) = 1) reflects the noisiest part of

the time series. Here the change between values fluctuates around zero.


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Figure 4.2: Haar coefficient spectrum

Plot of the Haar coefficient spectrum. The surface plots the highest frequency spectrum in

the front and the lowest frequency spectrum in the back. Note that the highest frequency spectrum

contains most of the noise.

4.2 FILTERING SPECTRUM

The wavelet transform allows some or all of a given spectrum to be removed by setting the

coefficients to zero. The signal can then be rebuilt using the inverse wavelet transform. Plots of the

AMAT close price time series with various spectrums filtered out are shown here.

4.3 NOISE FILTERS

Each spectrum that makes up a time series can be examined independently. A noise filter

can be applied to each spectrum removing the coefficients that are classified as noise by setting the

coefficients to zero.

4.4 WAVELETS VS. SIMPLE FILTERS

How do Haar wavelet filters compare to simple filters, like windowed mean and median

filters. Whether a wavelet filter is better than a windowed mean filter depends on the application.


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The wavelet filter allows specific parts of the spectrum to be filtered. For example, the entire high

frequency spectrum can be removed Or selected parts of the spectrum can be removed, as is done

with the gaussian noise filter. The scaling function (which uses the hn coefficients) produces N/2

elements that are a smoother version of the original data set. In the case of the Haar transform,

these elements are a pairwise average. The last step of the Haar transform calculates one scaling

value and one wavelet value. In the case of the Haar transform, the scaling value will be the

average of the input data.

The power of Haar wavelet filters is that they can be efficiently calculated and they

provide a lot of flexibility. They can potentially leave more detail in the time series, compared to

the mean or median filter. To the extent that this detail is useful for an application, the wavelet

filter is a better choice.

4.5 LIMITATIONS OF THE HAAR WAVELET TRANSFORM

The Haar wavelet transform has a number of advantages:

It is conceptually simple.

It is fast.

It is memory efficient, since it can be calculated in place without a temporary array.

It is exactly reversible without the edge effects that are a problem with other wavelettrasforms.

The Haar transform also has limitations, which can be a problem for some applications. In

generating each set of averages for the next level and each set of coefficients, the Haar transform

performs an average and difference on a pair of values. Then the algorithm shifts over by two

values and calculates another average and difference on the next pair. The high frequency

coefficient spectrum should reflect all high frequency changes. The Haar window is only two

elements wide. If a big change takes place from an even value to an odd value, the change will notbe reflected in the high frequency coefficients.

For example, in the 64 element time series graphed below, there is a large drop between

elements 16 and 17, and elements 44 and 45.


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Figure 4.3: frequency coefficient spectrum

Chapter 5

IMAGE COMPRESSION USING WAVELETS

Images require much storage space, large transmission bandwidth and long transmission

time. The only way currently to improve on these resource requirements is to compress images,

such that they can be transmitted quicker and then decompressed by the receiver. In image

processing there are 256 intensity levels (scales) of grey. 0 is black and 255 is white. Each level isrepresented by an 8-bit binary number so black is 00000000 and white is 11111111. An image can

therefore be thought of as grid of pixels, where each pixel can be represented by the 8-bit binary

value for grey-scale.

Figure 5.1 Intensity level representation of image


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The resolution of an image is the pixels per square inch. (So 500dpi means that a pixel is

1/500th of an inch). To digitise a one-inch square image at 500 dpi requires 8 x 500 x500 = 2

million storage bits. Using this representation it is clear that image data compression is a great

advantage if many images are to be stored, transmitted or processed. According to [6] "Image

compression algorithms aim to remove redundancy in data in a way which makes image

reconstruction possible." This basically means that image compression algorithms try to exploit

redundancies in the data; they calculate which data needs to be kept in order to reconstruct the

original image and therefore which data can be thrown away. By removing the redundant data,

the image can be represented in a smaller number of bits, and hence can be compressed. During

compression however, energy is lost because thresholding changes the coefficient values and hence

the compressed version contains less energy.But what is redundant information? Redundancy reduction is aimed at removing

duplication in the image. According to Saha there are two different types of redundancy relevant to

images:

(i) Spatial Redundancy . correlation between neighbouring pixels.

(ii) Spectral Redundancy - correlation between different colour planes and spectral bands.

Where there is high correlation, there is also high redundancy, so it may not be necessary to record

the data for every pixel.

There are two parts to the compression:

1. Find image data properties; grey-level histogram, image entropy, correlation functions etc..

2. Find an appropriate compression technique for an image of those properties.

5.1 Image Data Properties relavent to wavelets

In order to make meaningful comparisons of different image compression techniques it is

necessary to know the properties of the image. One property is the image entropy; a highly

correlated picture will have a low entropy. For example a very low frequency, highly correlated

image will be compressed well by many different techniques. A compression algorithm that is good

for some images will not necessarily be good for all images, it would be better if we could say

what the best compression technique would be given the type of image we have. One way of

calculating entropy is suggested by [6] Information redundancy, r, is r = b - He [6], where b is the

smallest number of bits for which the image quantisation levels can be represented.


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Information redundancy can only be evaluated if a good estimate of image entropy is

available, but this is not usually the case because some statistical information is not known. 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. An estimate of He can be obtained from a grey-

level histogram. If h(k) is the frequency of grey-level k in an image f, and image size is MxN then

an estimate of P(k) can be made:

The Compression ratio K = b / He.

5.2 WAVELETS AND COMPRESSION

Wavelets are useful for compressing signals but they also have far more extensive uses.They can be used to process and improve signals, in fields such as medical imaging where image

degradation is not tolerated they are of particular use. They can be used to remove noise in an

image, for example if it is of very fine scales, wavelets can be used to cut out this fine scale,

effectively removing the noise.

5.2.1 The Fingerprint example

The FBI have been using wavelet techniques in order to store and process fingerprint

images more efficiently. The problem that the FBI were faced with was that they had over 200

Million sets of fingerprints, with up to 30,0000 new ones arriving each day, so searching through

them was taking too long. The FBI thought that computerizing the fingerprint images would be a

better solution, however it was estimated that checking each fingerprint would use 600Kbytes of

memory and even worse 2000 terabytes of storage space would be required to hold all the image

data. The FBI then turned to wavelets for help, adapting a technique to compress each image into

just 7% of the original space. Even more amazingly, according to Kiernan[8], when the images are

decompressed they show "little distortion". However this was unsatisfactory, trying to compressimages this way into less than 10% caused "tiling artefacts" to occur, leaving marked boundaries in

the image. The basic steps used in the fingerprint compression were:

(1) Digitise the source image into a signals

(2) Decompose the signal s into wavelet coefficients


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(3) Modify the coefficients from w, using thresholding to a sequence w.

(4) Use quantisation to convert w to q.

(5) Apply entropy encoding to compress q to e.

5.3 WAVELETS AND DECOMPRESSION

Whenever we compress an image, at the receiver end to get the original form of the image

we have to decompress it. This can be done by image decompression models. The main blocks in

this decompression models are channel decoder and source decoder. A source decoder consists of

two stages, namely symbol decoder and inverse mapper. These blocks do the exact opposite

operation of channel as well as source encoder blocks.

The source decoder contains only two components: a symbol decoder and an inverse

mapper. These blocks perform, in reverse order, the inverse of the source encoders symbol

encoder and mapper blocks. Because quantization results in irreversible information loss, an

inverse quantizer block is not included in the general decoder model.

Chapter 6

SUMMARY


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The importance of the threshold value on the energy level was something of which we did not have

an appreciation before collecting the results. To be more specific we understood that hresholding

had an effect but didn.t realise the extent to which thresholding could change the energy retained

and compression rates. Therefore when the investigation was carried out more attention was paid

to choosing the wavelets, images and decomposition levels than the thresholding strategy.

ADVANTAGES:

We can avoid creating of blocking artefacts from an transformed image.

Multi-Resolution is possible with Wavelet transformation.

Image compression.

DISADVANTAGES:

When compared with DCT(Discrete Cosine Transformation) algorithm, Performance is

low.

Blockiness can be occurred after this transformation.

Chapter 7

CONCLUSION

The general aim of the project was to investigate how wavelets could be used for image

compression. We believe that we have achieved this. We have gained an understanding of whatwavelets are, why they are required and how they can be used to compress images. We understand

the problems involved with choosing threshold values. While the idea of thresholding is simple and

effective, finding a good threshold is not an easy task. We have also gained a general

understanding of how decomposition levels, wavelets and images change the division of energy


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between the approximation and detail subsignals. So we did achieve an investigation into

compression with wavelets.

Using global thresholding is not incorrect, it is a perfectly valid solution to threshold, the

problem is that using global thresholding masked the true effect of the decomposition levels in

particular on the results. This meant that the true potential of a wavelet to compress an image

whilst retaining energy was not shown. The investigation then moved to local thresholding which

was better than global thresholding because each detail subsignal had its own threshold based on

the coefficients that it contained. This meant it was easier to retain energy during compression.

However even better local thresholding techniques could be used. These techniques would be

based on the energy contained within each subsignal rather than the range of coefficient values and

use cumulative energy profiles to find the required threshold values. If the actual energy retained

value is not important, rather a near optimal trade off is required then a method called BayesShrink

could be used. This method performs a denoising of the image which thresholds the insignificant

details and hence produces Zeros while retaining the significant energy.

We were perhaps too keen to start collecting results in order to analyse them when we

should have spent more time considering the best way to go about the investigation. Having

analysed the results it is clear that the number of thresholds analysed (only 10 for each

combination of wavelet, image and level) was not adequate to conclude which is the best wavelet

and decomposition level to use for an image. There is likely to be an optimal value that the

investigation did not find. So it was difficult to make quantative predictions for the behaviour of

wavelets with images, only the general trends could be investigated. We feel however that the

reason behind my problems with thresholding was that thresholding is a complex problem in

general. Perhaps it would have been better to do more research into thresholding strategies and

images prior to collecting results. This would not have removed the problem of thresholding but

allowed us to make more informed choices and obtain more conclusive results. At the start of the

project our aim was to discover the effect of decomposition levels, wavelets and images on the

compression. We believe that we have discovered the effect each have.

How well a wavelet can compact the energy of a subsignal into the approximation

subsignal depends on the spread of energy in the image. An attempt was made to study image

properties but it is still unclear as to how to link image properties to a best wavelet basis function.

Therefore an investigation into the best basis to use for a given image could be another extension.


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Only one family of wavelets was used in the investigation, the Daubechies wavelets. However

there are many other wavelets that could be used such as Meyer, Morlet and Coiflet. Wavelets can

also be used for more than just images, they can be used for other signals such as audio signals.

They can also be used for processing signals not just compressing them. Although compression

and denoising is done in similar ways, so compressing signals also performs a denoising of the

signal. Overall we feel that we have achieved quite a lot given the time constraints considering that

before we could start investigating wavelet compression we had to first learn about wavelets and

how to use VC++. We feel that we have learned a great deal about wavelets, compression and how

to analyse image.

REFERENCES

1. D. Donoho and I. Johnstone, Adapting to unknown smoothness via wavelet shrinkage,

JASA, v.19, pp. 1200-1224, 1995

2. A. Chambolle, R. DeVore, N. Lee and B. Lucier, Nonlinear Wavelet Image Processing, IEE

Transactions on Image Processing, v.7, pp319-335, 1998

3. K. Castleman, Digital Image Processing', Prentice Hall 1996

4. V. Cherkassky and F. Mulier, Learning from Data, Wiley, N.Y. 1998

5. V. Cherkassky and X. Shao, Model Selection for Wavelet-based Signal Estimation, Proc.

International Joint Conference on Neural Networks, Anchorage, Alaska, 1998

6. Amir Said and William A. Pearlman, A New Fast and Efficient Image Code Based on Set

Partitioning in Hierarchical Trees, IEEE Transactions on Circuits and Systems for Video

Technology, vol. 6, pp. 243-250, June 1996


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7. Amir Said and William A. Pearlman, An Image Multi-resolution Representation for

Lossless and Lossy Image Compression, first submitted July 1994, and published in IEEE

Transactions on Image Processing, vol. 5, pp. 1303-1310, Sept. 1996

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