32

IMAGE COMPRESSION METHODOLOGIES

The term compression refers to the process of reducing the amount of data, required to

represent given quantity of information. The interest in image compression dates back

more than 35 years. The initial focus of research efforts in this field was on the

development of analogue methods for reducing video transmission bandwidth, a process

called bandwidth compression. The advent of the digital computer and subsequent

development of advanced integrated circuits, however, caused interest to shift from

analog to digital compression approaches. With the relatively recent adoption of several

key international are image compression standards. The field has undergone significant

growth through the practical application of the theoretic work that began in the 1940s,

when C. E. Shannon and others first formulated the probabilistic view of information and

its representation, transmission and compression [1]-[3].

Image compression addresses the problem of reducing the amount of data is 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 information is applied prior to storage

Chapter 2

33

or transmission of the image. After some time, the compressed image is decompressed to

reconstruct the original image or an approximation of it [1].

The term data compression refers to the process of reducing the amount of data required

to represent a given quantity of information. They are not synonymous. In fact data are

the means by which information is conveyed. Various amounts of data may be used to

represent the same information such might be the case. For example, if the two

individuals used a different number of words to tell the same basic story, two different

versions of the story are created, and at least one includes nonessential data. That is, it

contains data that either provides no relevant information or simply restates that which is

already known. It is thus said to contain data redundancy [1]-[4].

2.1 Principles behind Compression:

A Common characteristic of most images is that neighboring pixels are correlated and

there exists redundant information. Two fundamental components of compression are

redundancy and irrelevancy reduction. Redundancy reduction aims at removing

duplication from the signal source (Image and Video). Irrelevancy reduction omits parts

of the signal that will not be noticed by the signal receiver, namely the Human Visual

System (HVS). In general, three types of redundancy are identified.

Spatial redundancy or Correlation between neighboring pixel values.

Spectral redundancy or Correlation between different color planes or

spectral bands.

Temporal redundancy or Correlation between adjacent frames in sequence

of images i.e. in Video applications.

Image compression research aims at reducing the numbers of bits needed to represent an

image by removing the spatial and spectral redundancies as much as possible [2][4]. Data

redundancy is a central issue in digital image compression. It is not an abstract concept

but a mathematically quantifiable entity.

In digital image compression three basic types of data redundancies can be identified:

coding redundancy, interpixel redundancy and psychovisual redundancy. Data

34

compression is achieved when one or more of these redundancies are reduced or

eliminated.

2.1.1 Coding redundancy:

A code is a system of symbols consists of letters, numbers, and bits etc., which are used

to represent a body of information or set of events. Each peace of information or event is

assigned a sequence of code symbols called code word. The number of symbols in each

Codeword is its length. Paul Revre used one of the most famous codes in 18th April 1775.

In this code the phrase �one if by land, two if by sea� is often used to describe that code,

in which one or two lights were used to indicate whether the British were traveling by

land or sea. This story indicates that the abbreviations have been used for the last two

centuries indicating long phrases and secrete words. The same concept is used to give

code words for the grayscale values of the image to reduce the amount of data used to

represent it.

A discrete random variable rk in the interval [0,1] represents the gray levels of an image

and that each rk occurs with probability pr(rk). The value of pr(rk) can be determined by

following equation 2.1

pr(rk)= nk/n���������������������������.(2.1)

where,

k=0,1,2,3�.L-1,

L is the number of gray levels

nk is the number of times the kth gray level appears in the image and n is the total number

of pixel in the image. If the number of bits used to represent each value of rk is l(rk), then

the average number of bits, required to represent each pixel is

1

0

)()(L

kkrkavg rprlL ������������������������.(2.2)

Equation 2.2 shows that the average length of the code words assigned to various gray

level values is found by summing the product of the number of bits used to represent each

gray level and the probability of occurrence of that gray level. Thus the total number of

35

bits required to code an M x N image is MNLavg. Representing the gray levels of an

image with a natural m-bit binary code reduces the right hand side of equations 2.2 to m-

bits. That is Lavg=m when m is substituted for l(rk) then the constant m may be taken

outside the summation, leaving only the sum of the pr(rk) for 10 Lk , which is equal

to 1. This concept of coding redundancy is implemented in following example of variable

length coding:

rk pr(rk) Code 1 l1(rk) Code 2 l2(rk) r0=0 0.15 000 3 11 2 r1=1/7 0.23 001 3 01 2 r2=1/7 0.18 010 3 10 2 r 3=3/7 0.2 011 3 001 3 r 4=4/7 0.08 100 3 0001 4 r 5=5/7 0.06 101 3 00001 5 r 6=6/7 0.02 110 3 000001 6 r 7=1 0.01 111 3 000000 6

Table 2.1.1 Example of variable length coding

Variable Length Coding

0

0.05

0.1

0.15

0.2

0.25

r 0=0 r 1=1/7 r 2=2/7 r 3=3/7 r 4=4/7 r 5=5/7 r 6=6/7 r 7=1

rk

Pr(

rk)

0

1

2

3

4

5

6

7

l2(r

k)

pr(rk) l2(rk)

Figure 2.1.1 Data compression through variable length coding

In table 2.1.1 the gray level distribution of 8-level image is shown. If a natural 3-bit

binary code is used to represent the 8 possible gray levels, Lavg is three bits, because

36

l(rk)=3 bits for all rk. If code 2 in table 2.1 is used, however, the average number of bits

required to code the image is reduced to

bits 2.52

6(0.08)(0.05) 6 (0.06) 5 (0.08) 4 3(0.2) 2(0.21) 2(0.27) 2(0.17)

)()(7

02

L

kkrkavg rprlL

�(2.3)

The resulting compression ratio is equal to 52.2/3RC or 1.1905. Thus approximately

10% of the data in code 1 of table 2.1.1 is redundant. Therefore the exact value of

redundancy can be determined as

16.0

1905.1/11

DR������������������.�����.(2.4)

Figure 2.1.1 illustrates the underlying basis for the compression achieved by code 2. The

l2(rk) and pr(rk) are inversely proportional to each other. Therefore the shortest code

words in code 2 are assigned to the gray levels that occur most frequently in an image.

Assigning fewer bits to the more probable gray levels than to the less probable ones

achieves data compression. This process is commonly referred as variable length coding.

If the gray levels of an image are coded in a way that uses more code symbols than

absolutely necessary to represent each gray level, the resulting image is said to contain

coding redundancy. In general, coding redundancy is present when the codes assigned to

a set of gray level values have not been selected to take full advantage of the probabilities

of the events [3]-[7].

2.1.2 Interpixel Redundancy:

Interpixel redundancy can present in between images if the co-relation result of them

belongs to a structural or geometric relationship between the objects in the image. In this

case the histogram of such images virtually looks identical. But in reality objects in the

image are of different structure and geometry. This happens because the gray levels in

these images are not equally probable. In this case we can apply variable length coding to

reduce the coding redundancy because it would result from a straight or natural binary

encoding of their pixels. The coding process cannot alter the level of co-relation between

the pixels within the images. The codes used to represent the gray levels of each image

37

do not have the co-relation between the pixels. These co-relations are related with the

structural or geometric relationship between the objects in the image. In this case the

respective auto co-relation coefficient can be computed using equation 2.5:

1

0

1

0

),(),(*1

),(),(M

m

N

n

nymxhnmfMN

yxhyxf �����.����..(2.5)

where,

f(x,y) and h(x,y) are the two functions whose auto co-relation coefficients to be

compute.

f* denotes the complex conjugate of f

The normalize form of equation 2.5 is equation 2.6 which is also used for the same

purpose.

)(

)()(

OA

nAn

�.(2.6)

Where,

nN

y

nyxfyxfnN

nA1

0

),(),(1

)(

The scaling factor of equation 2.6 accounts for the varying number of sum terms that

arise for each integer value of n . The value of n must be strictly less than N. The

variable x is the coordinate of the line used in the computation. The dramatic difference

between the shapes of the functions can easily be identified in their respective plots of

n versus .

(a) (b)

38

(c) (d)

Figure 2.1.2: Two images and respective gray level histograms

In figure 2.1.2 (a) and (b) are the images having virtually identical histograms as shown

in figure c and d. In these histograms two dominant ranges of gray level values indicate

the presence of number of gray levels values. The codes used to represent the gray levels

of such images do not have the correlation between their pixels. These correlations

depend on structural or geometric relationship between the objects in the image. Such

correlation can be calculated using equation 2.6. This example shows that figure 2.3 (a)

and (b) contains interpixel redundancies. In order to reduce the interpixel redundancies in

an image, the two dimensional pixel arrays are normally used for human viewing and

interpretation must be transformed into a more efficient non-visual format. It means that

the differences between adjacent pixels can be used to represent an image.

Transformation of this type is called as mappings. If the original image elements can be

reconstructed from the transformed data set then it will be called as reversible mappings

[6],[10],[11],[13], [19],[20].

2.1.3 Psychovisual Redundancy:

The brightness of region of image perceived by human eyes depends upon factors other

than light reflected by the region. For example intensity variation can be perceived in an

area of constant intensity, such phenomena results in the fact that the eye does not

respond with equal sensitivity to all visual information. Certain information simply has

less relative importance than other information in normal visual processing. This

information is said to be Psychovisual redundant. This type of redundancy can be

39

eliminated without significantly impairing the quality of image perception. This type of

redundancy can present in image as human perception of the information in an image

normally does not involve quantitative analysis of every pixel values in the image. In

general an observer searches for distinguishing features such as edges or texture region

and mentally combines them into recognizable groupings. Afterwards the brain then

correlates these grouping with prior knowledge in order to complete the image

interpretation process. Psychovisual redundancy is associated with real or quantifiable

visual information. The elimination of such information is possible only because of such

information are not essential for normal visual processing. Since the elimination of

Psychovisually redundant data results in a loss of quantitative information, this

commonly refers to as quantization. This is irreversible operation hence it is called as

lossy data compression.

(a)

(b)

Figure 2.1.3 Compression by quantization.

40

In figure 2.1.3 a the gray scale image of 256 possible gray levels of 8-bit image is shown,

whereas in figure b in the same image after uniform quantization of 1-bit or 128 possible

gray levels are shown. By visual interpretation both of these images look identical. This

happens because the peculiarities of human visual system. This process of allocating

number varying bits is done at quantization level. The same can also be possible with the

help of bit plane slicing in which, it highlights the contribution made in total image

appearance by specific bits. Suppose, each pixel in an image is represented by 8-bits, this

image is compost of eight 1-bit planes ranging from bit plane 0 to bit plane 7. In 8-bit

image plane 0 contains all the lowest order bits in the bytes comprising the pixels in the

image and plane 7 contains all the high order bits. This high order information carries

more details therefore it looks like identical with original image. This type of redundancy

can also be called as decomposition [3]-[8].

2.1.4 Types of Compression

Many methods have been presented over the past years to perform image compression

having one common goal: to alter the representation of information contained in an

image, so that it can be represented sufficiently well with less information, regardless of

the details of each compression method. These methods fall into two broad categories

1. Lossless algorithms.

2. Lossy algorithms.

In Lossless data compression, the original data can be recovered exactly from the

compressed data. It is generally used for application where any difference between the

compressed and the reconstructed data cannot be allowed. Lossless compression

techniques generally are composed of relatively two independent operations 1. A

representation in which its interpixel redundancies are reduced. 2. Codification of the

representation to eliminate coding redundancies. Variable length coding, Huffman

coding, Arithmetic coding, LZW coding, Bit plane coding, Lossless Predictive coding are

the most commonly used coding techniques for Lossless data compression and normally

41

providing a compression ratio of 2 to 10 and they are equally applicable to both binary to

gray scale images [4][5].

Lossy compression techniques involve some loss of information and data cannot

be recovered from the same. These methods are used where the some loss of data can be

acceptable. For Example compressed Video signal is different from the original.

However, we can generally obtain the higher compression ratio than the Lossless

compression methods. Some of the common techniques for lossy compression are Lossy

predictive coding, Transform coding, Zonal coding, Wavelet coding, Image compression

standard [4]. Lossy compression techniques are much more effective at compression than

the Lossless methods.

And also these techniques give substantial image compression with very good quality

reconstruction. The higher compression ratio, the more noise added to the data [9-19].

2. 2 Image Compression Methodologies

Many methods have been presented over the past years to perform image compression

having one common goal: to alter the representation of information contained in an

image, so that it can be represented sufficiently well with less information, regardless of

the details of each compression method.

In Current methods for lossless image compression, such as the one used in Graphical

Interchange Format (GIF), image standard typically uses some form of Huffman or

Arithmetic Coder or Integer-to-Integer Wavelet Transform [12]. Unfortunately, current

lossless algorithms provide relatively, small compression factors compared with Lossy

methods. To achieve a high compression factor, a lossy method must be used. The most

popular current lossy image compression methods use a transform-based scheme as

shown in figure-2.2. It consists of three components namely: 1) Source Encoder 2)

Quantizer 3) Entropy Encoder.

42

Figure-2.2: Lossy Compression System

Source Encoder (or linear transformer) is intended to decorrelate the input signal by

transforming its representation in which the set of data values is sparse, thereby

compacting the information content of the signal into smaller number of coefficients. A

good Quantizer tries to assign more bits for coefficients with more information content or

perceptual significance, and fewer bits for coefficients with less information contents

bases on a given fixed bit budget entropy coding which removes redundancy from the

output of the Quantizer.

2.2.1 Source Encoder.

Over the past years, a variety of linear transforms have been developed, and the choice of

the use of transform used depended on a number of factors, in particular computational

complexity and coding gain [12]. The most commonly used transforms today are Discrete

Fourier Transform (DFT), Discrete Cosine Transform (DCT), Discrete Wavelet

Transform (DWT), Continuous Wavelet Transform (CWT), Generalized Lapped

Orthogonal Transform (Gen LOT)[2,12]. Noisy images can be compressed by removing

Gaussian noise and this can be removed using decomposition technique. Discrete

Wavelet Transform (DWT) and Discrete Cosine Transform (DCT) are used to

decompose noisy images [13]. The performance of DCT and Wavelet transforms is

Source Encoder

Quantizer

Entropy Encoder

Input Image

Output Compressed Image

Original Image

Compressed Image

43

presented and it is also noted that DCT used in JPEG standard is less computationally

complex than wavelet transform for a number of image samples [2,12]. However, it is

also recognized that wavelet transform, GenLOT can achieve coding gain superior to that

of DCT. Wavelet based transforms perform slightly better than GenLOT at medium bit

rate, but it is difficult to select one over the other at high bit rates [12].

2.2.2 Quantizer:

A Quantizer simply reduces the number of bits needed to store transformed coefficients

by reducing the precision of those values. Quantization can be performed on each

individual coefficient i.e. Scalar Quantization (SQ) or it can be performed on a group of

coefficients together i.e. Vector Quantization (VQ). Many methods be proposed to

perform quantization of the transform coefficients [16][17]. Even so, quantization

remains an active field of research and some new results show a greater promise for

wavelet based image compression [12][18][19], the choice of a good Quantizer depends

on the transform that is selected while transforming. The Quantizer can be �mixed and

match� to a certain degree, some quantization methods perform better with particular

transform. Also, perceptual weighting of coefficients in different sub bands can be used

to improve subjective image quality.

Quantization methods used with wavelet transforms fall into two general categories:

Embedded and Non Embedded quantizers. They determine bit allocation based on a

specified bit budget, allocating bit across a set of quantizers. Corresponding to the image

sub bands embedded quantization scheme [12].

2.2.3 Entropy Coding:

Entropy coding substitutes a sequence of codes for a sequence of symbols, where the

codes are chosen to have fewer bits when the probability of occurrence of the

corresponding symbol is higher. This process removes redundancy in the form of

repeated bit patterns in the output of the Quantizer. The most commonly used entropy

coders are the Huffman Coding, Arithmetic Coding, Run Length Encoding (RLE) and

Lempel-Ziv(LZ) algorithm[12]., although for applications requiring fast execution,

44

simple Run Length (RLE) has proved very effective[2]. Consequently arithmetic codes

are most commonly used in wavelet-based algorithms [20]-[24].

2.3 Performance criteria in Image Compression

The aim of image compression is to transform an image into compressed form so that the

information content is preserved as much as possible. Compression efficiency is the

principal parameter of a compression technique, but it is not sufficient by itself. It is

simple to design a compression algorithm that achieves a low bit rate, but the challenge is

how to preserve the quality of the reconstructed image at the same time. The two main

criteria of measuring the performance of an image compression algorithm thus are

compression efficiency and distortion caused by the compression algorithm. The standard

way to measure them is to fix a certain bit rate and then compare the distortion caused by

different methods.

The third feature of importance is the speed of the compression and decompression

process. In on-line applications the waiting times of the user are often critical factors. In

the extreme case, a compression algorithm is useless if its processing time causes an

intolerable delay in the image processing application.

In an image archiving system one can tolerate longer compression times if the

compression can be done as a background task. However, fast decompression is usually

desired. Among other interesting features of the compression techniques we may mention

the robustness against transmission errors, and memory requirements of the algorithm.

The compressed image file is normally an object of a data transmission operation. The

transmission is in the simplest form between internal memory and secondary storage but

it can as well be between two remote sites via transmission lines. The data transmission

systems commonly contain fault tolerant internal data formats so that this property is not

always obligatory. The memory requirements are often of secondary importance;

however, they may be a crucial factor in hardware implementations.

From the practical point of view the last but often not the least feature is complexity of the

algorithm itself, i.e. the ease of implementation. Reliability of the software often highly

45

depends on the complexity of the algorithm. Let us next examine how these criteria can

be measured.

2.3.1 Compression efficiency

The most obvious measure of the compression efficiency is the bit rate, which gives the

average number of bits per stored pixel of the image:

Bit rate = size of the compressed file

pixels in the image

C

N (bits per pixel)����.��.(2.7)

Where C is the number of bits in the compressed file, and N (=XY) is the number of

pixels in the image. If the bit rate is very low, compression ratio might be a more

practical measure:

Compression ratio = size of the original file

size of the compressed file

N k

C�����.��.(2.8)

Where, k is the number of bits per pixel in the original image. The overhead information

(header) of the files is ignored here [1][3][4].

2.3.2 Distortion

Distortion measures can be divided into two categories: subjective and objective

measures. A distortion measure is said to be subjective, if the quality is evaluated by

humans. The use of human analysts, however, is quite impractical and therefore rarely

used. The weakest point of this method is the subjectivity in the first place. It is

impossible to establish a single group of humans (preferably experts in the field) that

everyone could consult to get a quality evaluation of their pictures. Moreover, the

definition of distortion highly depends on the application, i.e. the best quality evaluation

is not always made by people at all. There are two kinds of quality measures: Objective

and Subjective quality measures and the details are discussed in Chapter 4.

46

2.4 Noise Pattern in Image The principal sources of noise in digital images arise during image acquisitions

(digitization) and / or transmission and these images are often degraded by some error

and this error for the variation in the brightness of a displayed image even when no image

detail is present. The variation is usually random and has no particular pattern reducing

the image quality specifically when the images are small and have relatively low contrast,

this random variation in image brightness known as noise.[1][3][4][8]

Image sensors are affected by environmental condition during image digitization and by

quality of elements. In acquiring image with a CCD Camera, light levels and sensors

temperature are major factors affecting the amount of noise in the resulting images.

Noises may dependent or independent of the image content. Images are corrupted during

the transmission due to interference in the channel used for transmission.

There are two basic types of noise models: noise in the spatial domain (described by the

noise probability density functions), and noise in the frequency domain, described by the

various Fourier properties of the noise [1][3][4][8][25].

2.4.1 Histogram, PMF and PDF

Histogram displays the number of samples which are in the signal having each of these

possible values. The histogram is what is formed an acquired signal. The corresponding

curve for the underlying process is called the Probability mass functions (PMF). A

histogram is always calculated using finite number of samples, while the PMF is what

would be obtained with a finite number of samples. The PMF can be estimated (inferred)

from the histogram, or it may be deduced by some mathematical technique, such as in the

coin example. The each value in the histogram is divided by the total number of samples

to approximate the PMF. This means that each value in the PMF must be between 0 and

1, and that the sum of all of the values in the PMF must be equal to 1. The PMF is

important because it describes the probability that certain value will be generated. For

example what is the probability that any one sample will have between 0 and 255?

Summing all of the values in the PMF produces the probability of 1.00, which is a

certainty that this will occur. The histogram and PMF can only be used with discrete data,

such as a digitized signal residing in the computer. A similar concept applies to the

47

continuous signal, such as voltage appearing in analog electronics. The probability

density functions (PDF) also called the probability distribution functions, as to

continuous signals what the probability mass function is to discrete signals. A PDF has

two theoretical properties: The PDF is zero or positive for every possible outcome. The

integral of a PDF over its entire range of values is one [8].

2.4.2 Normal Distribution:

Signals formed form random process usually have a bell shaped PDF. This is called the

normal distribution, a Gaussian distribution or a Gaussian after the great German

mathematician, Karl Freindrcih Gauss( 1777-1855). The basic shape of the curve is

generated from a negative squared exponent.

2

)( xexy ������������������������(2.9)

This raw curve can be converted into the complete Gaussian by adding an adjustable

mean and standard deviation F. In addition, the equation must be normalized so that

the total area under the curve is equal to one, a requirement of all probability distribution

functions. This results in the general form of the normal distribution, one of the most

important relations in the statistics and probability:

22 2/)(

2

1)(

xexP �������������������(2.10)

The mean centers the curve over the particular value, while the standard deviation

controls the width of the bell shape [8][26[27][28].

2.4.3 Digital Noise Generation:

The heart of the digital noise generation is the random number generator. Most

programming languages have this as standard function. The BASIC statement x=rnd,

loads the variable x with new random number each time the command is encountered.

Each random number has value between 0 and 1, with equal an equal probability of being

anywhere between these two extremes. There are two methods for generating the random

number are: First one is x= rnd + rnd, since each of the random numbers can run from 0

to 1, the sum can run from 0 to 2. The mean is now 1 and the standard deviation is 1/6.

48

The PDF has changed from a uniform distribution to a triangular distribution. That is, the

signal spends more of its time around a value of 1. What is important is PDF virtually

becomes a Gaussian. This procedure is used to create normally distributed noise signal

with an arbitrary mean and standard deviation. In second method for generating normally

distributed random numbers, the random number generator is invoked twice, to obtain R1

and R2. Where R1 and R2 are the two random variables used to generate the random

numbers. A normally distributed random number, X can be found:

)22cos()1log2( 2/1 RRX ����������������...(2.11)

This equation generates the normally distributed random signals with an arbitrary mean

and standard deviation, Take each number generated by this equation, multiply it by the

desired standard deviation, and add the desired mean[8][26]-[29].

2.4.4 Spatial Random Noise with Specified Distribution:

Spatial noise values are random numbers, characterized by a probability density functions

(PDF) or equivalently by the corresponding Cumulative Distribution Function (CDF).

Random number generation for the type of distributions in which we are interested follow

some fairly simple rules from probability theory. Numerous random number generators

are based on expressing the generation problem in terms of random numbers with

uniform CDF in the interval [0, 1]. In some instances the base random generator choice is

a generator of Gaussian random numbers with 0 mean and 1 variance. The foundation of

the approach described is a well known result from probability which states tat if w is

uniformly distributed random variable in the interval [0, 1] then we can obtain variable z

with a specified CDF, Fz by solving the equation

)(1 wzFZ ����������������������������(2.12)

This is yet simple and powerful result can be stated equivalently as finding a solution for

the equation.

wZFz )( ����������������������������..(2.13)

The Table.2.4.4 (a) shows the random variables along with their PDF�s and CDF�s and

random generator equations. With the Rayleigh and Exponential variables, it is possible

to find a closed form solution for the CDF and its inverse. In the case of Gaussian and

49

lognormal densities, closed form solutions for the CDF do not exist and it becomes

necessary to find the alternatives ways to generate the desired random numbers. In the

lognormal case, for instance, we make use of the knowledge that a lognormal random

variable z is such that ln(z) has Gaussian distribution. In other cases, it is advantageous to

reformulate the problem to obtain an easier solution. Gaussian noise is used

approximately in case such as imaging sensors operating at low light levels. Salt and

pepper noise arise in faulty switching devices. Rayleigh noise arises in rang imaging,

while exponential and Erlang noise are useful in describing noise in laser imaging

[28][29]

Name PDF Mean and Variance CDF Uniform

abzPz

1)(

if bza , 0 otherwise

2

bam

,

12

)( 22 ab

Fz(z)=

1

0

ab

az

Z<a, a<=z<=b, z>b Gaussian

Pz(z)=

z

bazeb

222)(

2

1

M=a, 2 =b2

Fz(z)= dvvPzz

)(

Salt & Pepper Pz(z)=

0

Pb

Pa

for z = a , z = b otherwise b>a

M=aPa +bPb 2Pa )( ma + (b-m)2Pb

Fz(z)=

PbPa

Pa

0

Lognormal Pz(z) = bazInebz 2

2))(2

1

2

Z > 0

M= ea+(b2/2), 2=[eb2-1] e2a+b2

Fz(z) = z

dvvPz0

)(

Rayleigh Pz(z) =

0

2)()(2

bazeazb

z a , Z < a

M=a + ,4/b

2 =4

)4( b

Fz(z) =

0

2)(1 baze

z a, Z< a

Exponential Pz(z) =

0

azae

0

0

z

z M=

a

1, 2=

2

1

a

Fz(z)=

0

1 aze

Z<0 0z Erlang Abzb-1

Pz(z)= )!1( b e-az

0z

M=a

b, 2 = b/a2

Fz(z)=

1

0 !

)(1

b

n n

nazaze

0z

Table.2.4.4 (a): shows the random variables along with their PDF�s and CDF�s and random generator equations.

50

Type of Noise

Noise Pattern Original Image Noisy Image

Gaussian

Poisson

salt & pepper

Speckle

Uniform

Figure-2.4.4 (b): Shows the some common Noise patterns on Cameraman Image.

51

2.5 Summary

This chapter describes the motivation for Lossy compression originates from the inability

of the Lossless algorithms to produce as low bit rates as desired to achieve high

compression ratio.

It also describes principles behind the Digital Image compression and various image

compression methodologies. This chapter has mainly focused on lossy compression

techniques, which are based on Transform Coding. Further this chapter has examined the

effects produced on image quality and compression ratio by varying physical size and

number of bits per pixel independently. The effect shown in introduction part is suitable

to get the high compression ratio by selecting appropriate physical size and number of

bits per pixel according to the requirement of application. It also describes the undersized

theory regarding noise generation and presence of noise in the digital images. The various

theoretical views on the latest transform coding techniques used compression and

decompression for noisy and noiseless images are discussed in Chapter 3.

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References:

01 R.C. Gonzalez, R.E. Woods �Digital Image Processing�, Second Edition, Pearson

Education, 2004. 02 Subhasis Saha, 21 Jan 2001, �Image Compression from DCT to Wavelet: A review�,

http://www.acm.org/crossroads/xrds6-3/sahaimgcoding.html 03 Khalid Sayood �Introduction to Data Compression", Second Edition, Morgan

Kaufman Publisher, 2003. 04 David Salomon, �Data Compression the Complete Reference�, 2

nd Ed. Springer 05 R.C. Gonzalez, R.E. Woods and Steven L Eddins �Digital Image Processing using

MATLAB�, Second Edition, Pearson Education, 2004 06 Pasi Fränti, �Image Compression�, Lecture Notes, University of joensuu, dept. of

Computer Science, 2002 07 K.P.Soman, K.Ramchandran "Insight into Wavelets from Theory to Practice", PHI

New Delhi-2004 08 Steven W. Smith �Digital Signal Processing�, The Scientist and Engineers Guide,

Second Edition, California Technical Publishing San Diego, California. 09 Zhou Wang, Alan C. Bovik, and Ligang Lu �Wavelet Based Foveated Image Quality

Measurement for Region of Interest Image Coding", © 2001 IEEE.� 10 Zhou Wang, Alan C. Bovik, Hamid Rahim Sheikh and Eero P. Simonelli �Image

Quality Assessment: From Error Visibility to Structural Similarity�, IEEE

Transactions on Image Processing Vol. 13. No. 4. April-2004. 11 S. Arivazhagan, D. Gnanaduri, K.M. Sarojani, L. Ganesan "Evaluation of Zero Tree

Wavelet Coders" Proceedings of the International Conference on information Technology: Computers and Communications. © 2003 IEEE.

12 Michael B. Martin �Applications of Multiwavelets to image Compression�, M.S.,

Thesis, June 1999 13 G. Panda, S. K. Meher and B. Majhi, �De-noising of Corrupted data using Discrete

Wavelet Transform�, The Journal of CSI, Vol. 30, No.3, September-2000. 14 Debargha Mukherjee and Sanjit K. Mitra �Vector SPHIT for embedded Wavelet

Video and Image Coding�, IEEE Transactions on Circuits and Systems for Video Technology Vol.13. No. 3, March-2003

53

15 Kai Bao and Xiang Genxia �Image Compression using a new Discrete Multiwavelet

Transform and a new embedded Vector Quantization�, IEEE Transactions on

Circuits and Systems for Video Technology Vol.10. No. 6, Sept-2000 16 Sonja Grigic, Mislav Grigic, Bronka zovko �Optimal Decomposition for Wavelet

Image Compression � First International Workshop on Image and Signal Processing

And Analysis, June 14-15, 2000, Pula, Coatia 17 Sonja Grigic, Mislav Grigic, Branko Zovko-Cihlar, � Performance Analysis of

Image Compression using Wavelets �, IEEE Transactions on Industrial

electronics PP.682- 695, Vol.48. N0. 3, June 2001 18 Amir Said and William A Pearlman, �A New fast and efficient Image codec based

on Set Partitioning in Hierarchical Trees (SPHIT)�, IEEE Transactions on Circuits

and Systems for Video Tech 6(3): PP 243-250, June 1996. 19 J.M.Shapiro �Embedded Image Coding Zero tree of Wavelet Coefficients�, IEEE

Transactions on Image Processing, 41(12): PP 3445-3462, Dec-1993. 20 Zixing Xiong, Kannan Ramachandran , Michel T Orchad, "Wavelet Packet Image

Coding using Space Frequency Quantization", IEEE Transaction on Image Processing Vol.7, No.6, June-2001

21 O.O.Khalifa � Fast Adaptive for VQ based Wavelet Coding System�, Palmstones

North Nov-2003 22 Debargha Mukherjee and Sanjit K. Mitra �Vector SPHIT for embedded Wavelet

Video and Image Coding� IEEE Transactions on Circuits and Systems 23 Jo Yew Tham, Lixin shen et.el. �A General approach for Analysis and Application

of Discrete Multiwavelet Transform� IEEE Transaction on Signal Processing

Vol.48,No.2, Feb-2000 24 K.W.Cheurg and et.el. �Spatial Coefficients partitioning for Lossless Wavelet Image

Coding �, IEEE Transaction on Signal Processing Vol.149, No.6, Dec-2002. 25 Kenneth R Castelman, � Digital Image Processing�, LPE, Pearson education, 26 Stevan C Chapra, Raymound P Canale, �Numerical Methods for Engineers with

Programming and Software Applications�, 3rd Edition Mcgraw Hill International

Editors General Engineering Series 1998 27 P Kandasmy K Thilagavathy, K Gunvathy, � Numerical Mehtods�, S Chand and

Company Ltd, New Delhi, 2001

54

28 www.mathworks.com 29 Rafeal C Gonzalez and Richard E Woods, �Digital Image Processing using

MATALAB�, Second Edition, Pearson education, 2004