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Image compression using 3D-SPIHT 2007*2008

ABSTRACT

We propose a highly scalable image compression scheme based on the set

partitioning in hierarchical trees (SPIHT) algorithm. Our algorithm called highly scalable

SPIHT (HS-SPIHT), supports spatial and SNR scalability and provides a bit stream that

can be easily adapted (reordered) to given bandwidth and resolution requirements by a

simple transcoder (parser). The HS-SPIHT algorithm adds the spatial scalability feature

without sacrificing the SNR embeddedness property as found in the original SPIHT bit

stream. HS-SPIHT finds applications in progressive Web browsing, flexible image storage

and retrieval, and image transmission over heterogeneous networks

Vivekananda Institute Of Technology - 1 -


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TABLE OF CONTENTS

CHAPTER NO. TITLE PAGE NO.

LIST OF TABLES

LIST OF FIGURES



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1 INTRODUCTION

1.1 MOTIVATION

1.2 PRINCIPLE OF COMPRESSION

1.3 COMPRESSION ALGORITHMS

1.4 IMAGE COMPRESSION PROCESS

1.5 ADAPTIVE MULTIWAVELET IMAGE

COMPRESSION

1.6 ORGANIZATION OF THE THESIS

2 LITERATURE SURVEY

3 OVERVIEW OF WAVELET CODER

3.1 WAVELET IMAGE COMPRESSION

3.2 COMPRESSION STEPS

3.3 TYPICAL IMAGE CODER

3.4 WAVELET TRANSFORM OVERVIEW

3.4.1 WAVELET TRANSFORM

3.4.2 SCALING

3.4.3 SHIFTING

3.4.4 SCALE AND FREQUENCY

3.5 DISCRETE WAVELET TRANSFORM

3.5.1 ONE STAGE FILTERING

3.5.2 MULTIPLE LEVEL DECOMPOSITION

3.5.3 WAVELET RECONSTRUCTION

3.5.4 RECONSTRUCTING APPROXIMATION

AND DETAILS

3.6 1 D WAVELET TRANSFORM

3.7 2 D TRANSFORM HIERARCHY



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3.8 AWIC FILTER CHOICE

3.9 WAVELET COMPUTATION

3.10 QUANTIZATION

3.11 ENTROPY ENCODING

4 PROPOSED SYSTEM

5ADAPTIVE MULTIWAVELET IMAGE

COMPRESSION

5.1 WAVELET IMAGE COMPRESSION PARAMETERS

5.1.1 VARYING WAVELET TRANSFORM LEVEL

5.1.2 VARYING QUANTIZATION LEVEL

5.1.3 VARYING ELIMINATION LEVEL

5.2 ADAPTIVE IMAGE COMMUNICATION

6 SIMULATION RESULTS

7 CONCLUSION

APPENDICES

1 ANALYSIS OF TRANSFORM CONSTANTS

2 PERFORMANCE METRICS

2.1 DEGREE OF COMPRESSION

2.2 IMAGE QUALITY

REFERENCES



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CHAPTER 1

INTRODUCTION

1.1 MOTIVATION

One of the major challenges in enabling mobile multimedia data services will

be the need to process and wirelessly transmit a very large volume of data. While

significant improvements in achievable bandwidth are expected with future wireless

access technologies, improvements in battery technology will lag the rapidly growing

energy requirements of future wireless data services. One approach to mitigate to this

problem is to reduce the volume of multimedia data transmitted over the wireless channel

via data compression techniques.

This has motivated active research on multimedia data compression techniques

such as JPEG [1, 2], JPEG 2000 [3, 4] and MPEG [5]. These approaches concentrate on

achieving higher compression ratio without sacrificing the quality of the image. However

these efforts ignore the energy consumption during compression and RF transmission.



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Today a lot of hospitals handle their medical image data with computers. The

use of computers and a network makes it possible to distribute the image data among the

staff efficiently. As the health care is computerized new techniques and applications are

developed, among them the MR and CT techniques. MR and CT produce sequences of

images (image stacks) each a cross-section of an object. The amount of data produced by

these techniques is vast and this might be a problem when sending the data over a

network. To overcome this image data can be compressed. For two-dimensional data

there exist many compression techniques such as JPEG, GIF and the new wavelet based

JPEG2000 standard [7]. All schemes above are used or two-dimensional data (images)

and while they are excellent for images they might not be that well suited for

compression of three-dimensional data such as image stacks.

1.2 THESIS GOAL

The purpose of this thesis is to look at coding schemes based on wavelets for medical

volumetric data. The thesis should discuss theoretical issues as well as suggest a

practically feasible implementation of a coding scheme. A short comparison between

two- and three-dimensional coding is also included.

Another goal is to implement highly scalable image compression based on SPIHT.

.

1.6 ORGANIZATION OF THE THESIS

CHAPTER 2: Overview of the compression schemes

CHAPTER 3: Design Specifications of wavelet coder

CHAPTER 4 Proposed system of SPIHT



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CHAPTER5: Simulation Results

CHAPTER6: Conclusion

REFERENCES

CHAPTER 2

OVERVIEW OF THE IMAGE COMPRESSION TECHNIQUES

PRINCIPLE OF COMPRESSION

Image compression addresses the problem of reducing the amount of data

required to represent a digital image. The underlying basis of the reduction process is the

removal of redundant data. From a mathematical viewpoint, this amounts to transforming

a 2-D pixel array into a statistically uncorrelated data set. The transformation is applied

prior to storage and transmission of the image. The compressed image is decompressed at

some later time, to reconstruct the original image or an approximation to it.

Different types of data redundancies:



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Interpixel redundancy: Neighboring pixels have similar values. This property is

exploited in the wavelet transform stage.

Psychovisual redundancy: Human visual system cannot simultaneously

distinguish all colors. This property is exploited in the lossy quantization stage.

Coding redundancy: Fewer bits represent frequent symbols.

1.3 COMPRESSION ALGORITHMS

There are various algorithms for image transformation:

Discrete cosine transform (DCT)

JPEG

Sub-band coding

Embedded zero wavelet transform (EZW)

SPIHT

1.4 IMAGE COMPRESSION PROCESS

Fig.1

illustrates the main block diagram of the image compression process.



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The image sample first goes through a transform, which generates a set of

frequency coefficients. The transformed coefficients are then quantized to reduce the

volume of data. The output of this step is a stream of integers, each of which corresponds

to an index of particular quantized binary. Encoding is the final step, where the stream of

quantized data is converted into a sequence of binary symbols in which shorter binary

symbols are used to encode integers that occur with relatively high probability. This

reduces the number of bits transmitted.

CHAPTER 3

OVERVIEW OF WAVELET CODER

3.1 WAVELET IMAGE COMPRESSION

The foremost goal is to attain the best compression performance possible

for a wide range of image classes while minimizing the computational and

implementation complexity of the algorithm. For a compression algorithm to be widely

useful, it must perform well on a wide



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variety of image content while maintaining a practical compression/ decompression time

on modest computers. In order to allow a broad range of implementation, an algorithm

must be amenable to both software and hardware implementation.

3.2 COMPRESSION STEPS

The steps needed to compress an image are as follows:

1. Digitize the source image into a signal, which is a string of numbers.

2. Decompose the signal into a sequence of wavelet coefficients.

3. Use quantization to convert coefficients to a sequence of binary symbols.

4. Apply entropy coding to compress it into binary strings.

The first step in the wavelet compression process is to digitize the image.

The digitized image can be characterized by its intensity levels, or scales of gray that

range from 0 (black) to 255 (white), and its resolution, or how many pixels per square

inch. The wavelets process the signal, but upto this point, compression has not yet

occurred.

The next step is quantization which converts a sequence of floating

numbers to a sequence of integers. The simplest form is to round to the nearest integer.

Another option is to multiply each number by a constant and then round to the nearest

integer. Quantization is called lossy because it introduces error into the process, since the

conversion is not a one-to-one function.

The last step is encoding that is responsible for the actual compression.

One method to compress the data is Huffman entropy coding. With this method, an

integer sequence is changed into a shorter sequence, with the numbers being 8 bit

integers. The conversion is made by an entropy coding table.



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3.3 TYPICAL IMAGE CODER

(a)

(b)

Fig. 3.1 (a) Wavelet Coder, (b) Wavelet Decoder

A typical image compression system consisting of three closely connected

components namely (a) Source Encoder (b) Quantizer, and (c) Entropy Encoder are

shown in Fig. 3.1. Compression is accomplished by applying a linear transform to

decorrelate the image data, quantizing the resulting transform coefficients, and entropy

coding the quantized values.

The source coder decorrelates the pixels. A variety of linear transforms

have been developed which include Discrete Fourier Transform (DFT), Discrete Cosine

Transform (DCT), Discrete Wavelet Transform (DWT) and many more, each with its

own advantages and disadvantages.


Source

Image

data

Compressed

Image dataMWT Quantizer

Entropy

Encoder

Compressed

Image data

Reconstructed

Image dataEntropy

Decoder

Inverse

Quantizer

Inverse

MWT


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A quantizer simply reduces the number of bits needed to store the

transformed coefficients by reducing the precision of those values. Since this is a many-

to-one mapping, it is a lossy process. Quantization can be performed on each individual

coefficient, which is known as Scalar Quantization (SQ). Quantization can also be

performed on a group of coefficients together, and this is known as Vector Quantization

(VQ).

An entropy encoder further compresses the quantized values losslessly to

give better overall compression. It uses a model to accurately determine the probabilities

for each quantized value and produces an appropriate code based on these probabilities so

that the resultant output code stream will be smaller than the input stream.

The most commonly used entropy encoders are the Huffman encoder and

the arithmetic encoder, although for applications requiring fast execution, simple run-

length encoding (RLE) has proven very effective. It is important to note that a properly

designed quantizer and entropy encoder are absolutely necessary along with optimum

signal transformation to get the best possible compression.

3.4WAVELET TRANSFORM OVERVIEW

3.4.1 Wavelet Transform



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Wavelets are mathematical functions defined over a finite interval and

having an average value of zero that transform data into different frequency components,

representing each component with a resolution matched to its scale.

The basic idea of the wavelet transform is to represent any arbitrary

function as a superposition of a set of such wavelets or basis functions. These basis

functions or baby wavelets are obtained from a single prototype wavelet called the

mother wavelet, by dilations or contractions (scaling) and translations (shifts). They have

advantages over traditional Fourier methods in analyzing physical situations where the

signal contains discontinuities and sharp spikes. Many new wavelet applications such as

image compression, turbulence, human vision, radar, and earthquake prediction are

developed in recent years. In wavelet transform the basis functions are wavelets.Wavelets tend to be irregular and symmetric. All wavelet functions, w (2kt - m), are

derived from a single mother wavelet, w (t). This wavelet is a small wave or pulse like

the one shown in Fig. 3.2.

Fig. 3.2 Mother wavelet w (t)

Normally it starts at time t= 0 and ends at t= T. The shifted wavelet w (t-

m) starts at t= m and ends at t= m + T. The scaled wavelets w (2kt) start at t= 0 and end

at t= T/2k. Their graphs are w (t) compressed by the factor of 2kas shown in Fig. 3.3.

For example, when k= 1, the wavelet is shown in Fig 3.3 (a). If k= 2 and 3, they are

shown in (b) and (c), respectively.



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(A)w(2t) (b)w(4t) (c)w(8t)

Fig. 3.3 Scaled wavelets

The wavelets are called orthogonal when their inner products are zero. The

smaller the scaling factor is, the wider the wavelet is. Wide wavelets are comparable to

low-frequency sinusoids and narrow wavelets are comparable to high-frequency

sinusoids.

3.4.2 Scaling

Wavelet analysis produces a time-scale view of a signal. Scaling a wavelet

simply means stretching (or compressing) it. The scale factor is used to express the

compression of wavelets and often denoted by the letter a. The smaller the scale factor,

the more compressed the wavelet. The scale is inversely related to the frequency of the

signal in wavelet analysis.

3.4.3 Shifting

Shifting a wavelet simply means delaying (or hastening) its onset.

Mathematically, delaying a function f(t) by kis represented by: f(t-k) and the schematic

is shown in fig. 3.4.

(a) Wavelet function (t) (b) Shifted wavelet function (t-k)



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3.4.4 Scale and Frequency

The higher scales correspond to the most stretched wavelets. The more

stretched the wavelet, the longer the portion of the signal with which it is being

compared, and thus the coarser the signal features being measured by the wavelet

coefficients. The relation between the scale and the frequency is shown in Fig. 3.5.

Low scale High scale

Fig. 3.5 Scale and frequency

Thus, there is a correspondence between wavelet scales and frequency as

revealed by wavelet analysis:Low scale a Compressed wavelet Rapidly changing details High frequency.

High scale a Stretched wavelet Slowly changing, coarse features Low frequency .

3.5 DISCRETE WAVELET TRANSFORM

Calculating wavelet coefficients at every possible scale is a fair amount of

work, and it generates an awful lot of data. If the scales and positions are chosen based on

powers of two, the so-called dyadic scales and positions, then calculating wavelet

coefficients are efficient and just as accurate. This is obtained from discrete wavelet

transform(DWT).

3.5.1 One-Stage Filtering



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For many signals, the low-frequency content is the most important part. It

is the identity of the signal. The high-frequency content, on the other hand, imparts

details to the signal. In wavelet analysis, the approximations and details are obtained after

filtering. The approximations are the high-scale, low frequency components of the signal.

The details are the low-scale, high frequency components. The filtering process is

schematically represented as in Fig. 3.6.

Fig. 3.6 Single stage filtering

The original signal, S, passes through two complementary filters and

emerges as two signals. Unfortunately, it may result in doubling of samples and hence to

avoid this, downsampling is introduced. The process on the right, which includes

downsampling, produces DWT coefficients. The schematic diagram with real signals

inserted is as shown in Fig. 3.7.



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Fig. 3.7 Decomposition and decimation

3.5.2 Multiple-Level Decomposition

The decomposition process can be iterated, with successive approximations

being decomposed in turn, so that one signal is broken down into many lower resolution

components. This is called the wavelet decomposition tree and is depicted as in Fig. 3.8.

Fig. 3.8 Multilevel decomposition

3.5.3 Wavelet Reconstruction



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The reconstruction of the image is achieved by the inverse discrete wavelet

transform (IDWT). The values are first upsampled and then passed to the filters. This is

represented as shown in Fig. 3.9.

Fig. 3.9 Wavelet Reconstruction

The wavelet analysis involves filtering and downsampling, whereas the

wavelet reconstruction process consists of upsampling and filtering. Upsampling is the

process of lengthening a signal component by inserting zeros between samples as shown

in Fig. 3.10.

Fig. 3.10 Reconstruction using upsampling

3.5.4 Reconstructing Approximations and Details

It is possible to reconstruct the original signal from the coefficients of the

approximations and details. The process yields a reconstructed approximation which has

the same length as the original signal and which is a real approximation of it.



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The reconstructed details and approximations are true constituents of the

original signal. Since details and approximations are produced by downsampling and are

only half the length of the original signal they cannot be directly combined to reproduce

the signal. It is necessary to reconstruct the approximations and details before combining

them. The reconstructed signal is schematically represented as in Fig. 3.11.

Fig. 3.11 Reconstructed signal components

3.6 1-D WAVELET TRANSFORM

The generic form for a one-dimensional (1-D) wavelet transform is shown

in Fig. 3.12. Here a signal is passed through a low pass and high pass filter, hand g,

respectively, then down sampled by a factor of two, constituting one level of transform.

Fig. 3.12 1D Wavelet Decomposition.

Repeating the filtering and decimation process on the lowpass branch

outputs make multiple levels or scales of the wavelet transform only. The process is

typically carried out for a finite number of levels K, and the resulting coefficients are

called wavelet coefficients.



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The one-dimensional forward wavelet transform is defined by a pair of

filters sand t that are convolved with the data at either the even or odd locations. The

filters s and t used for the forward transform are called analysis filters.

nL nH

li = sjx2i+j and hi = tjx2i+1+j

j=-nl j=-nH

Although land hare two separate output streams, together they have the

same total number of coefficients as the original data. The output stream l, which is

commonly referred to as the low-pass data may then have the identical process applied

again repeatedly. The other output stream, h (or high-pass data), generally remainsuntouched. The inverse process expands the two separate low- and high-pass data streams

by inserting zeros between every other sample, convolves the resulting data streams with

two new synthesis filters s and t, and adds them together to regenerate the original

double size data stream.

nH nl

yi = tjl

i+j + s

j h

i+j where l

2i = li, l

2i+1 = 0

j= -nH j= -nH h2i+1 = hi, h

2i = 0

To meet the definition of a wavelet transform, the analysis and synthesis

filters s, t, s andt must be chosen so that the inverse transform perfectly reconstructs the

original data. Since the wavelet transform maintains the same number of coefficients as

the original data, the transform itself does not provide any compression. However, the

structure provided by the transform and the expected values of the coefficients give a

form that is much more amenable to compression than the original data. Since the filters

s, t, s and t are chosen to be perfectly invertible, the wavelet transform itself is lossless.

Later application of the quantization step will cause some data loss and can be used to

control the degree of compression. The forward wavelet-based transform uses a 1-D

subband decomposition process; here a 1-D set of samples is converted into the low-pass

subband (Li) and high-pass subband (Hi). The low-pass subband represents a down



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sampled low-resolution version of the original image. The high-pass subband represents

residual information of the original image, needed for the perfect reconstruction of the

original image from the low-pass subband

3.7 2-D TRANSFORM HEIRARCHY

The 1-D wavelet transform can be extended to a two-dimensional (2-D)

wavelet transform using separable wavelet filters. With separable filters the 2-D

transform can be computed by applying a 1-D transform to all the rows of the input, and

then repeating on all of the columns.

LL1 HL1

LH1 HH1

Fig. 3.13 Subband Labeling Scheme for a one level, 2-D Wavelet Transform

The original image of a one-level (K=1), 2-D wavelet transform, with

corresponding notation is shown in Fig. 3.13. The example is repeated for a three-level

(K=3) wavelet expansion in Fig. 3.14. In all of the discussion Krepresents the highest

level of the decomposition of the wavelet transform.



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LL1 HL1HL2

HL3

LH1 HH1

LH2 HH2

LH3 HH3

Fig. 3.14 Subband labeling Scheme for a Three Level, 2-D Wavelet Transform

The 2-D subband decomposition is just an extension of 1-D subband

decomposition. The entire process is carried out by executing 1-D subband

decomposition twice, first in one direction (horizontal), then in the orthogonal (vertical)

direction. For example, the low-pass subbands (Li) resulting from the horizontal direction

is further decomposed in the vertical direction, leading to LLi and LHi subbands.

Similarly, the high pass subband (Hi) is further decomposed into HLi and

HHi. After one level of transform, the image can be further decomposed by applying the

2-D subband decomposition to the existing LLi subband. This iterative process results in

multiple transform levels. In Fig. 3.14 the first level of transform results in LH1, HL1,

and HH1, in addition to LL1, which is further decomposed into LH2, HL2, HH2, LL2 at

the second level, and the information of LL2 is used for the third level transform. The

subband LLi is a low-resolution subband and high-pass subbands LHi, HLi, HHi are

horizontal, vertical, and diagonal subband respectively since they represent the

horizontal, vertical, and diagonal residual information of the original image. An example



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of three-level decomposition into subbands of the image CASTLE is illustrated in Fig.

3.15.

H2H1HH

Fig. 3.15 The process of 2-D wavelet transform applied through three

transform levels

To obtain a two-dimensional wavelet transform, the one-dimensional

transform is applied first along the rows and then along the columns to produce four

subbands: low-resolution, horizontal, vertical, and diagonal. (The vertical subband is

created by applying a horizontal high-pass, which yields vertical edges.) At each level,

the wavelet transform can be reapplied to the low-resolution subband to further

decorrelate the image. Fig. 3.16 illustrates the image decomposition, defining level and

subband conventions used in the AWIC algorithm. The final configuration contains a

small low-resolution subband. In addition to the various transform levels, the phrase level

0 is used to refer to the original image data. When the user requests zero levels of

transform, the original image data (level 0) is treated as a low-pass band and processing

follows its natural flow.

Low Resolution Subband

43

Level 2

Level 1

Vertical subband

HL

4 43 3

Level 2 Level 2



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Level 1

Horizontal Subband

LH

Level 1

Diagonal Subband

HH

Fig. 3.16 Image Decomposition Using Wavelets

3.8 AWIC FILTER CHOICE

The main difference between subband and wavelet coding is the choice of

filters to be used in the transform. The filters used in wavelet coding systems were

typically designed to satisfy certain smoothness constraints. In contrast, subband filters

were designed to approximately satisfy the criteria of non-overlapping frequency

responses.

There are two types of filter choices, orthogonal and biorthogonal. The

biorthogonal wavelet transform has the advantage that it can use linear phase filters, but

the disadvantage is that it is not energy preserving. The fact that biorthogonal wavelets

are not energy preserving does not turn out to be a big problem, since there are linear

phase biorthogonal filter coefficients, which are close to being orthogonal.

The 9-7 and 5-3 Daubechies biorthogonal filters, were chosen as ideal for

image compression. Both the biorthogonal 9-7 and 5-3 filters were used in AWIC

(Rushanan 1997). The detracting visual artifact at high compression rates using the

longer filter is ringing, while the shorter filter produces staircasing. The design trade-off

between these two filters is speed performance and quality performance. Visually, the

level of ringing artifacts and staircasing artifacts appears to be nearly equivalent,



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dependent on subjective opinion. The peak-signal-to-noise ratio favors the ringing artifact

over the staircasing.

The Daubechies 5-tap/3-tap filter is used in forward wavelet transform and

it has good localization and symmetric properties, which allows simple edge treatment,

high-speed computation, and high quality compressed image. The following equation

represents the Daubechies 5-tap/3-tap filter.

3.9 WAVELET COMPUTATION

In order to obtain an efficient wavelet computation, it is important to

eliminate as many unnecessary computations as possible. A careful examination of the

forward and reverse transforms shows that about half the operations either lead to data

which are destroyed or are null operations (as in multiplication by 0).

The one-dimensional wavelet transform is computed by separately

applying two analysis filters at alternating even and odd locations. The inverse process

first doubles the length of each signal by inserting zeros in every other position, then

applies the appropriate synthesis filter to each signal and adds the filtered signals to get

the final reverse transform.

3.10 QUANTIZATION

Quantization refers to the process of approximating the continuous set of

values in the image data with a finite set of values. The input to a quantizer is the original

data, and the output is always one among a finite number of levels. The quantizer is a

function whose set of output values is discrete, and usually finite. Obviously, this is a



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process of approximation, and a good quantizer is one, which represents the original

signal with minimum loss or distortion.

There are two types of quantization - Scalar Quantization and Vector

Quantization. In scalar quantization, each input symbol is treated separately in producing

the output. If the input range is divided into levels of equal spacing, then the quantizer is

termed as a Uniform Quantizer, and if not, it is termed as a Non-Uniform Quantizer. A

uniform quantizer can be easily specified by its lower bound and the step size. Also,

implementing a uniform quantizer is easier than a non-uniform quantizer.

Just the same way a quantizer partitions its input and outputs discrete

levels, a inverse quantizer is one which receives the output levels of a quantizer andconverts them into normal data, by translating each level into a 'reproduction point' in the

actual range of data. The quantization error (x - x')is used as a measure of the optimality

of the quantizer and inverse quantizer.

3.11 ENTROPY ENCODING

There are many ways of compressing images. An image is composed ofmany dots, called pixels, and each pixel has a color. Pictures contain some number of

rows (R) and columns (C) of pixels. An image is stored in the computer as a set of values,

each of which represents the color of a pixel. The total number of values in the image is

R times C. To transmit an image, all of these values could be sent. Alternatively a set of

instructions that allow the picture to be redrawn when it is decompressed can also be sent.

Encoding is done by a number of methods. The simple and basic encoding

method being Run length coding (Sayood 2000), it has a low compression performance

but is lossless. The Huffman coding is also a lossless coding method which gained

popularity for its compression. It is performed by the use of standard tables. The recent

standard methods such as arithmetic coding and LZW coding are very effective.



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CHAPTER 4

PROPOSED SPIHT

Parents and children

SPIHT was introduced in [9]. It is a refinement of the algorithm presented by Shapiro in

[10]. SPIHT assumes that the decomposition structure is the octave-band structure and

then uses the fact that sub-bands at different levels but of the same orientation display

similar characteristics. As is seen in Figure 4.1 the band LL HL has similarities with the

band HL (both have high-pass filtered rows). To utilize the above observation SPIHT

defines spatial parent-children relationships in the decomposition structure. The squares

in Figure 4.1 represent the same spatial location of the original image and the same

orientation, but at different scales. The different scales of the subbands imply that a

region in the sub-band LL HL is spatially co-located (represent the same region in the

original image) with a region 4 times larger (in the two dimensional case) in the band HL.

SPIHT describes this collocation with one to four parent-children relationships, where the



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parent is in a sub-band of the same orientation as the children but at a smaller scale .If.

this prediction is successful then SPIHT can represent the parent and all its descendants

with a single symbol called a zero-tree, introduced in [10]. To predict energy of

coefficients in lower level sub-bands (children) using coefficients in higher level sub-

bands (parents) makes sense since there should be more energy per coefficient in these

small bands, than in the bigger ones. To see how SPIHT uses zerotrees the workings of

SPIHT are briefly explained below. For more information the reader is referred to [9].

SPIHT consists of two passes, the ordering pass and the refinement pass. In the

ordering pass SPIHT attempts to order the coefficients according to their magnitude. In

the refinement pass the quantization of coefficients is refined. The ordering and refining

is made relative to a threshold. The threshold is appropriately initialised and then

continuously made smaller with each round of the algorithm. SPIHT maintains three lists

of coordinates of coefficients in the decomposition. These are the List of Insignificant

Pixels (LIP), the List of Significant Pixels (LSP) and the List of Insignificant Sets (LIS).

To decide if a coefficient is significant or not SPIHT uses the following definition. A

coefficient is deemed significant at a certain threshold if its magnitude is larger then or

equal to the threshold. Using the notion of significance the LIP, LIS and LSP can be

explained. The LIP contains coordinates of coefficients that are insignificant at the

current threshold, The LSP contains the coordinates of coefficients that are significant at

the same threshold.The LIS contains coordinates of the roots of the spatial parent-

children trees .

1. In the ordering pass of SPIHT (marked by the dotted

line in the schematic above) the LIP is first searched for coefficients that are significant at

the current threshold, if one is found 1 is output then the sign of the coefficient is marked

by outputting either 1 or 0 (positive or negative). Now the significant coefficient is

moved to the LSP. If a coefficient in LIP is insignificant a 0 is outputted.

2. Next in the ordering pass the sets in LIS are processed. For every set in the LIS it is

decided whether the set is significant or insignificant. A set is deemed significant if at

least one coefficient in the set is significant. If the set is significant the immediate



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children of the root are sorted into LIP and LSP depending on their significance and 0s

and 1s are output as when processing LIP. After sorting the children a new set (spatial

coefficient tree) for each child is formed in the LIS. If the set is deemed insignificant, that

is this set was a zero-tree, a 0 is outputted and no more processing is needed. The above

is a simplification of the LIS processing but the important thing to remember is that entire

sets of insignificant coefficients, zero-trees, are represented with a single 0. The idea

behind defining spatial parent-children relationships as in (4.1) is to increase the

possibility of finding these zero-trees.

3.The SPIHT algorithm continues with the refinement pass. In the refinement pass the

next bit in the binary representation of the coefficients in LSP is outputted. The next

bit is related to the current threshold. The processing of LSP ends one round of the

SPIHT algorithm, before the next round starts the current threshold is halved.

Below is a schematic of how SPIHT works.



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CHAPTER 5



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Results

FIRST LEVEL DECOMPOSITION

SECOND LEVEL DECOMPOSITION



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THIRD LEVEL DECOMPOSITION



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CHAPTER 6

IMPLEMENTATION

.

This is the 3-dimensional view of brain image which we are using for the purpose of

compression.


BRAIN IMAGE IN 3D


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The image shown above is sliced into 8 frame (bit planes) which is as shown below-



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mri1.bmp mri2.bmp

mri3.bmp mri4.bmp

mri5.bmp mri6.bmp


Brain images from frames 1-8

5 6


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mri7.bmp mri8.bmp

the above images are applied to discerte wavelet transform in order to sample and

quantizeIn image compression, such a decomposition scheme should be extended to

twodimensions.

By choosing separable filters, a two-dimensional subband coding scheme

can be implemented as a straightforward extension of the one-dimensional version. This

is performed by applying the same filters along the two dimensions iteratively. For an M

x M image, M one-dimensional transforms are carried out along the rows (one for each

row). This results in two M x M/2 subimages/subbands, one corresponding to the lowpass

filtered components and the other corresponding to the high-pass filtered rows of the

image. Next, each of these subbands is filtered further along the columns to partition the

resulting data into four M/2 x M/2 subbands (low-pass row and low-pass column,

lowpass

row and high-pass column, high-pass row and low-pass column, high-pass row and

high-pass column). This constitutes one stage of the subband decomposition of an image,

and is depicted in Figure 2.3. In this thesis, we are concerned with 3D datasets, so one

stage of decomposition results in eight sub-volumes. This is accomplished by

transforming each slice of the volumetric dataset using two-dimensional transform and

then applying one-dimensional transform along the slices.


7 8


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coding stage:

Once the image is decomposed into subbands, an encoding scheme is used to compress

the transform coefficients. Traditionally, scalar quantization is used in this stage [16].

Two state-of-the-art coding schemes based on wavelet transform, SPIHT encoding is

done.

SPIHT Coding Algorithm

Set Partitioning in Hierarchical Trees (SPIHT) algorithm is based on embedded zerotree

wavelet (EZW) coding method; it employs spatial orientation trees and uses set

partitioning sorting algorithm [24].

An octave-band decomposition structure is used in SPIHT. Coefficients

corresponding to the same spatial location in different subbands in the pyramid structure



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display self-similarity characteristics. Note that, SPIHT defines parent-children

relationships between these self-similar subbands to establish spatial orientation trees.

Such parent-children relationships in two-dimensional case can be expressed as in

Equation below.

An example showing octave-band decomposition is depicted in Figure

The arrows in the figure indicate the parent-children relationship in subband pyramid.

The start of arrow line is parent coefficient, and end of arrow indicates four children

coefficients. The coefficients in the shaded area are all the descendants of root

coefficient,

A. It is assumed that lower frequency subbands have more energy and if the magnitude of

a parent coefficient in a lower frequency subband is less than a threshold, it is highly



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likely that magnitudes of its descendants are also less than that threshold. Thus,

significance map of transformed coefficients with a preset threshold can be efficiently

coded by a zero-tree coding scheme. For most images, the assumption used for the

construction of spatial orientation tree is satisfied.

The next step is the set partitioning sorting algorithm, which consists of two passes:

sorting pass and refinement pass. In sorting pass, the ordering information of the

coefficients is found according to the results of a magnitude test. In the refinement pass,

the quantization for each significant coefficient is refined in a successive manner. A

quantization threshold is used in coefficient magnitude test. The threshold is initialized as

. . . and then successively decreased by a factor of two in each pass of the

algorithm. When the bit budget is reached, the algorithm will stop.

Three ordered lists, in which each element indicates coordinate pair (i, j) of a

coefficient, are used to implement the coding algorithm. They are named as List of

Insignificant Pixels (LIP), List of Significant Pixels (LSP) and List of Insignificant Sets

(LIS). Magnitude test is performed to decide whether a coefficient is significant or not. A

coefficient is said to be significant at a given threshold if its magnitude is larger than or

equal to that threshold. The construction of LIP, LIS and LSP takes place according to

the significance of the coefficients with respect to the threshold value.

_

LIP contains coordinates of coefficients that are insignificant with respect to the

current threshold

_

LSP contains the coordinates of coefficients that are significant at the same

threshold

_

LIS contains coordinates of the roots of the spatial orientation parent-children

trees



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The flowchart given in Figure illustrates how the SPIHT algorithm works.



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1. In the sorting pass, the LIP is scanned to determine whether an entry is significant

at the current threshold. If an entry is found to be significant, output a bit 1 and

another bit for the sign of the coefficient, which is marked by either 1 for

positive or 0 for negative. Now the significant entry is moved to the LSP. If an

entry in LIP is insignificant, a bit 0 is output.

2. Entries in LIS are processed. When an entry is the set of all descendants of a

coefficient, named type A, magnitude tests for all descendants of the current

entry are carried out to decide whether they are significant or not. If the entry is

found to be as significant, the direct offsprings of the entry are undergone

magnitude tests. If direct offspring is significant, it is moved into LIP; otherwise it

is moved into LSP. If the entry is deemed to be insignificant, this spatial

orientation tree rooted by the current entry was a zero-tree, so a bit 0 is output

and no further processing is needed. Finally, this entry is moved to the end of LIS

as type B, which stands for the set of all descendants except for the immediate

offsprings of a coefficient, if there are still descendants of its direct offsprings.

Alternatively, if the entry in LIS is type B, significance test is performed on the

descendants of its direct offspring. If significance test is true, the spatial

orientation tree with root of type B entry is split into four sub-trees that are rooted

by the direct offspring and these direct offsprings are added in the end of LIS as

type A entries. The important thing in LIS sorting is that entire sets of

insignificant coefficients, zero-trees, are represented with a single 0. The purpose

behind defining spatial parent-children relationships is to increase the possibility

of finding these zero-trees.

3. Finally, refinement pass is used to output the refinement bits (nthbit) of the

coefficients in LSP at current threshold. Before the algorithm proceeds to the next

round, the current threshold is halved.

After the encoding process the compressed data is being obtained and it is used to store or

transmit.



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At the receiver :

The inverse process is being performed.

Initially inverse SPIHT is applied.

It uses the threshold and the received bitstream in order to obtain the image matrix same

as the original image matrix.

The next step is inverse DWT

Synthesis Stage

In the previous section, we described the subband decomposition of images into different

frequent components. In order to reconstruct the input image, a synthesis scheme is

needed to restore the original input. The process of decomposition can be reversed, so it

is possible to obtain reconstructed image given that the decomposed components are

available. Such synthesis process can be realized by up-sampling followed by filtering,

and then combining the results at the output of the filters. For the analysis scheme given

in Figure corresponding synthesis scheme is shown in Figure



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thus we obtain reconstructed image which is same as original image

When lossy compression is employed, we need to measure the difference (distortion)

between the original signal and the reconstructed one. A common metric for measuring

distortion is the Mean Squared Error, or MSE. In the MSE measurement the total squared

difference between the original signal and the reconstructed one is averaged over the

entire signal and can be computed as follows:

Another distortion measure is the Peak Signal to Noise Ratio, or PSNR. Since the MSE is

sensitive to the range of signal scale, the PSNR is used to incorporate the maximum

amplitude of the original signals into the metric, making this measurement independent

of the range of the data. PSNR is usually measured in decibels as:



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For 8 bit images or image volumes for instance, peakx = 255.

The graph of these distortion values is as shown below

1 2 3 4 5 6 7 8900

950

1000

1050

1100

1150

1200

1250MSE V/S BIT PLANE

bit plane

MSE



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1 2 3 4 5 6 7 865.4

65.6

65.8

66

66.2

66.4

66.6

66.8

67PSNR V/S BIT PLANE

bit plane

PSNR

IN

DB



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The original image ,reconstructed image and their differences is obtained as shown below

Original Frame1 Reconst Frame1 DIFFERENCE1



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CHAPTER 7

CONCLUSION AND FUTURE WORK

The performance of the Wavelet -based coder could be raised if a better cost-

function was developed. If the gain in PSNR from using this new cost function is

significant maybe the Wavelet coder would be the preferred choice.

In the next phase the proposed system was simulated and implemented in VLSI.



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REFERENCES

[1] Forchheimer R. (1999), Image coding and data compression, Linkping:

Department of electrical engineering at Linkpings University.

[2] Chui C. K. (1992), An introduction to wavelets, Boston, Academic Press, ISBN

0121745848

[3] Sayood K. (2000), Introduction to data compression, Morgan Kaufmann Publrs,

US, 2 revised edition, ISBN 1558605584.

[4] Daubechies I., Barlaud M., Antonini M., Mathieu P. (1992), Image coding using

wavelet transform in IEEE Transactions on image processing, Vol. 1, No. 2, pp.

205-220.

[5] Brislawn C. M.. (1996), Classification of nonexapnsive symmetric extension

transforms for multi rate filter banks in Applied and computational harmonic

analysis, Vol. 3, pp. 337-357.

[6] Azpiroz-Leehan J., Lerallut J.-F. (2000), Selection of biorthogonal filters for

image compression of MR images using wavelet packets in Medcial engineering &

physics, Vol. 22, pp. 225-243.



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