image compression methodologies -...
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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
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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
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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
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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)
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(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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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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