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CHAPTER 5

IMAGE COMPRESSION TECHNIQUE USING

NONSUBSAMPLED CONTOURLET TRANSFORM

5.1 INTRODUCTION

Image compression is currently a well-known topic for

researchers. Due to fast growth of digital media and the following need for

reduced storage and to convey the image in an efficient way, better image

compression approach is required.

Among the rapid growth of digital technology in consumer

electronics, their requirement to protect raw image data for future editing

or repeated compression goes on increasing. Thus, image compression has

become a dynamic area of research in the field of image processing to

lessen file size. Wavelet and curvelet transformations are widely used

transformation techniques to carry out compression. But both have their

own limitations which affects overall performance of the compression

process. This research work focuses on presenting a non-linear image

compression technique that compresses image both radically and angularly.

Nonsubsampled Contourlet Transformation (NSCT) has the potential to

approximate the natural images comprising contours and oscillatory

patterns. In addition to this transformation, vector quantization technique is

used to remove the redundancies in the images. Finally, this technique uses

Hybrid Set Partitioning in Hierarchical Trees (HSPIHT) with Huffman for

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efficient encoding process. The experimental results of Non-sub sampled

contourlet transformation, with vector quantization and HSPIHT are better

when compared with existing transformations techniques.

5.2 PROBLEMS IN WAVELET BASED CONTOURLET

TRANSFORM

A non-redundant transform is needed for image coding. Certain

types of non-redundant geometrical image transforms that depends on the

DFB have been presented by the researchers. The octave-band directional

filter banks which are introduced by Paul S Hong & Mark JT Smith (2002)

correspond to a new family of the DFB that attains radial decomposition as

well. The CRISP-contourlet is another transform proposed by Lu & Do

(2003), which is implemented based on the contourlet formation, other than

nonseparable filter banks are merely used. Truong & Oraintara (2005)

presented a Non-uniform DFB which is a modified form of CRISP-

contourlets and is the other non-redundant directional transform, which

also gives multiresolution.

Neither of the above non-redundant directional schemes have

been used in a practical image processing application. Ramin Eslami &

Hayder Radha (2008) introduced a Wavelet-Based Contourlet Transform

(WBCT), in which DFB is applied to all the feature subbands of wavelets

in an analogous way that one constructs contourlets. The major difference

is that, wavelets are used instead of the Laplacian pyramids in contourlets.

As a result, the WBCT is non-redundant and can be adapted for some

resourceful wavelet-based image coding methods.

The main drawbacks of the WBCT and other contourlet-based

transforms contain artifacts that are occurred by setting some transform

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coefficients to zero for nonlinear approximation and also owing to

quantizing of the coefficients for coding.

The image focusing multiple objects contains more information

than those which just focus one object. This type of image is broadly used

in various areas such as remote sensing, medical imaging, computer vision

and robotics. Actually, all cameras used in computer visualization systems

are not pin-hole devices but consist of convex lenses. Therefore, they

endure from the problem of partial depth of field (Chunming Li et al 2004).

It is often not possible to get an image which contains all relevant objects

in focus. The objects in front of or behind the focus plane would be

blurred. To overcome this problem, image fusion is introduced by the

author in several images with different focal points are shared together to

create an image with all objects fully focused (Pan et al 1999). Over past

years, numbers of techniques were proposed by the researchers for

multifocus image fusion. The easiest multifocus image fusion technique in

spatial domain is to take the average of the source images pixel by pixel,

would lead to numerous unpredicted side effects like reduced difference

Yuancheng & Yang (2008). Various methods depends on multiscale

decomposition (MSD) as presented by Pajares & Cruz (2004).

Wavelet transform for 1-D piecewise smooth signals are better,

though it has severe limitations with high dimensional signals. 2D split

wavelet performs better at isolating the discontinuities at object edges, but

cannot effectively signify the line and the curve discontinuities. In contrast,

it can only capture restricted directional information. Accordingly, WT

based method cannot conserve the relevant features in source images well

and will possibly introduce some artifacts and variation in the combined

results (Yuancheng & Yang 2008). Minh N Do & Vetterli (2005) presented

a Contourlet Transform (CT) is a two dimensional image representation

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method. It is attained by combining the directional filter bank (DFB) as

discussed by Burt & Adelson (1983); Bamberger & Smith (1992).

Compared with the traditional DWT, the CT is not only with multi-scale

and localization, but also with mulitdirection and anisotropy.

Consequently, the CT can symbolize edges and further singularities along

curves. On the other hand, CT possesses shift-invariance and results in

artifacts along the edges to some level. Arthur et al (2006) proposed a

transform known as nonsubsampled contourlet transform (NSCT).

The NSCT come into the perfect properties of the CT, and in the meantime

consist of shift invariance. In image fusion, the NSCT has more

information fusion together and is obtained and the influence of

misregistration on the fused information which can also be reduced. As a

result, the NSCT is more appropriate for image fusion and also in image

compression etc.

5.3 NONSUBSAMPLED CONTOURLET TRANSFORM:

FILTER DESIGN AND APPLICATIONS

Several image processing tasks are executed in a particular

domain other than the pixel domain, frequently using invertible linear

transformation. This linear transformation can be superfluous or not, based

on whether the set of basic functions is linear independent. By means of

redundancy, it is feasible to improve the set of fundamental functions so

that the representation is more resourceful in capturing several signal

behavior. Applications of imaging are of edge detection, contour detection,

denoising and image restoration that can significantly assistance from

redundant representations. In the situation of multiscale expansions

employed with filter banks, dropping the fundamental requirement

provides the chance of an expansion that is shift-invariant, a vital property

in a number of applications. For example in image denoising ,by means of

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thresholding in the wavelet domain, the lack of shift-invariance reasons

pseudo-Gibbs phenomena in the region of singularities (Coifman &

Donoho 1995). Thus, most of the modern wavelet denoising routines

employ an expansion with low shift sensitivity than the standard maximally

decimated wavelet decomposition as explained in Chang et al (2000);

Sendur & Selesnick (2002) and Minh Do & Vetterli (2005) proposed a

contourlet transform which is a directional multiscale transform that is built

by combining the Laplacian pyramid (LP) and the directional filter bank

(DFB). Due to downsamplers and upsamplers present in both the LP and

DFB, the contourlet transform is not shift-invariant. The nonsubsampled

contourlet transform (NSCT) is obtained by coupling a nonsubsampled

pyramid structure with the nonsubsampled DFB.

Figure 5.1 Idealized frequency partitioning obtained with the NSCT

5.3.1 Nonsubsampled Contourlet Transform

The idea behind a fully shift invariant multiscale directional

expansion similar to contourlets is to obtain the frequency partitioning of

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Figure 5.1 without resorting to critically sampled structures that have

periodically time-varying units such as downsamplers and upsamplers. The

NSCT construction can thus be divided into two parts: (1) A

nonsubsampled pyramid structure which ensures the multiscale property

and (2) A nonsubsampled DFB structure which gives directionality.

5.3.2 Nonsubsampled Pyramid

The shift sensitivity of the LP can be remedied by replacing it

with a 2-channel nonsubsampled 2-D filter bank structure. Such expansion

is similar to the 1-D a trous wavelet expansion Shensa (1993) and has a

redundancy of J + 1 when J is the number of decomposition stages.

The ideal frequency support of the low-pass filter at the j-th stage is the

region jjjj 2,

2x

2,

2. Accordingly, the support of the high-pass

filter is the complement of the low-pass support region on the

1j1j1j1j 2,

2x

2,

2square. The proposed structure is thus different

from the tensor product, a trous algorithm. It has j+1 redundancy.

By contrast, the 2-D a trous algorithm has 3j+1 redundancy.

5.3.3 Nonsubsampled Directional Filter Bank

The directional filter bank is constructed by combining critically

sampled fan filter banks and pre/post re-sampling operations. The result is

a tree-structured filter bank which splits the frequency plane into

directional wedges.

A fully shift-invariant directional expansion is obtained by

simply switching off the downsamplers and upsamplers in the DFB

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equivalent filter bank. Due to multirate identities, this is equivalent to

switching off each of the downsamplers in the tree structure, while still

keeping the re-sampling operations that can be absorbed by the filters. This

results in a tree structure composed of two-channel nonsubsampled filter

banks. The NSCT is obtained by carefully combining the 2-D

nonsubsampled pyramid and the nonsubsampled DFB (NSDFB) Arthur et

al (2006).

The resulting filtering structure approximates the ideal partition

of the frequency plane displayed in Figure 5.2. It must be noted that,

different from the contourlet expansion, the NSCT has a redundancy given

by J0j j

l2R where jl2 is the number of directions at scale j.

Figure 5.2 Two kinds of desired responses (a) The pyramid desired

response (b) The fan desired response

The directional filter bank of Bamberger & Smith (1992) is

constructed by combining critically-sampled two-channel fan filter banks

and resampling operations. The result is a tree-structured filter bank that

splits the frequency plane in the directional wedges shown in Figure 5.3(a).

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Figure 5.3 The directional filter bank (a) Ideal partitioning of the 2-

D frequency plane into 23 = 8 wedges (b) Equivalent

multi-channel filter bank structure. The number of

channels is L = 2l, where l is the number of stages in the

tree structure

Using multirate identities Bamberger & Smith (1992), the tree-

structured DFB can be put into the equivalent form shown in Figure 5.3

(b), where the downsampling/upsampling matrices Sk for 0 2l-1 are

given by Equation (5.1)

;12k2for;12k0for

)2,2(diag)2,2(diag

S 1l1l

1l

1l

1l

k (5.1)

with ‘l’ denoting the number of stages in the tree structure. It is

clear from the above that the DFB is not shift-invariant. A shift-invariant

directional expansion is obtained with a nonsubsampled DFB (NSDFB).

The NSDFB is constructed by eliminating the downsamplers and

upsamplers as in Figure 5.3(b). This is equivalent to switching off the

downsamplers in each two-channel filter bank in the DFB tree structure

and upsampling the filters accordingly. This results in a tree consisting of

two-channel nonsubsampled filter banks. The equivalent filter in the kth

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channel of the analysis NSDFB tree is the filter )z(Ueqk as in Figure 5.3 (b).

Thus, each filter )z(Ueqk is a product of l simpler filters. Just like the DFB,

all NSFB’s in the NSDFB tree structure are obtained from a single NSFB

with fan filters (Figure 5.4b) which illustrates a four channel

decomposition. Note that in the second level, the upsampled fan filters

1,0i),z(U Qi have checker-board frequency support, and when combined

with the filters in the first level, gives the four directional frequency

decomposition that is shown in Figure 5.4. The synthesis filter bank is

obtained similarly. Just like the NSP case, each filter bank in the NSDFB

tree has the same computational complexity as that of the prototype NSFB.

Figure 5.4 A four-channel nonsubsampled directional filter bank

constructed with two-channel fan filter banks

(a) Filtering structure (b) Corresponding frequency

decomposition

Figure 5.4(a) shows a four-channel filtering structure the

equivalent filter in each channel is given by )z(U)z(U)z(U Qji

eqk .

Figure 5.4(b) shows the corresponding frequency decomposition.

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5.4 APPLICATIONS OF NON-SUBSAMPLED

CONTOURLET TRANSFORM

5.4.1 Image Denoising

In order to illustrate the potential of the NSCT designed using

the techniques previously discussed an Additive White Gaussian Noise

(AWGN) removal is implemented from the images by means of

thresholding estimators, and tested the NSCT fewer than two denoising

schemes described below:

1) Hard Threshold

For the hard threshold estimator, a global threshold

ijnj,i KT for each directional subband is chosen. This has been termed K-

sigma thresholding (Starck et al 2002). Set K = 4 for the finer scale and

K = 3 for the remaining ones. This method known as NSWT-HT when

applied to NSWT coefficients and NSCT-HT when applied to NSCT

coefficients. Five scales of decomposition are used for both NSCT and

NSWT. For the NSCT 4, 8, 8, 16, 16 directions in the scales are used from

coarser to finer respectively.

2) Local Adaptive Shrinkage

Perform soft thresholding (shrinkage) independently in each

subband. The threshold (Grace Chang et al 2000) is chosen by

Equation (5.2).

n,j,i

ijN2

j,iT (5.2)

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where n,j,i denotes the variance of the n-th coefficient at the ith

directional subband of the jth scale, and ijN2 is the noise variance at scale j

and direction i. It is shown in Grace Chang et al et al (2000) that shrinkage

estimation with x

2T , and assuming X generalized Gaussian distributed,

yields a risk within 5% of the optimal Bayes risk. Po & Do (2006) studied

that contourlet coefficients are well modelled by generalized Gaussian

distributions. The signal variances are estimated locally using the

neighbouring coefficients contained in a square window within each

subband and a maximum likelihood estimator. The noise variance in each

subband is inferred using a Monte-Carlo technique where the variances are

computed for a few normalized noise images and then averaged to stabilize

the results. This method is known as local adaptive shrinkage (LAS).

Effectively, LAS method is a simplified version of the denoising method

proposed in Grace Chang et al (2000) that works in the NSCT or NSWT

domain.

The other applications of NSCT is image enhancement, image

fusion, image compression, applications of medical image analysis, video

compression etc.

5.4.2 Advantages of NSCT

The nonsubsampled contourlet transform (NSCT) not only has

multiresolution but has multidirectional properties

By using non-subsampled contourlet transform ,coefficients in

small-scale and gradient information of each colour channel is

identified efficiently

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The NSCT is a fully shift-invariant, multi scale, and multi

direction expansion that have a fast implementation. Here,

filters are designed with better frequency selectivity, thereby

achieving better sub band decomposition

5.5 PROPOSED METHOD USING NON-SUBSAMPLED

CONTOURLET TRANSFORM

The basic steps of the proposed image compression algorithm

are shown in Figure 5.5. All these steps are invertible, therefore lossless,

except for the Quantize step. Quantizing is the process of reduction of the

precision of the floating point values of the WBC transforms.

Figure 5.5 Process of the proposed approach

5.5.1 Nonsubsampled Contourlet Transform

First, contourlet transform down samplers and up samplers are

formed in both the laplacian pyramid and the Directional Filter Bank

(DFB). Thus, it is not shift-invariant, which causes pseudo-Gibbs

phenomena around singularities. NSCT is an improved form of contourlet

transform. It is employed in some applications, in which redundancy is not

a major issue, i.e. image fusion. In contrast with contourlet transform, non-

subsampled pyramid structure and non-subsampled directional filter banks

are employed in NSCT. The non-subsampled pyramid structure is achieved

Non sub sampled

Contourlet Coefficient

Dead zone Quantization

HSPIHT Encoding

with Huffman

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by using two-channel nonsubsampled 2-D filter banks. The DFB is

achieved by switching off the down samplers/up samplers in each two-

channel filter bank in the DFB tree structure and up sampling the filters

accordingly. As a result, NSCT is shift-invariant and leads to better

frequency selectivity and regularity than contourlet transform. Figure.5.6

shows the decomposition framework of contourlet transform and NSCT

Arthur L da Cuncha (2006).

Figure 5.6 Nonsubsampled contourlet transform (a) Nonsubsampled

filter bank structure that implements the NSCT

The NSCT structure consists of a bank of filters that splits the 2-

d frequency plane in the subband; these are a nonsubsampled pyramid

structure that ensures the multiscale property and a nonsubsampled

directional filter bank structure that gives directionality.

5.5.2 Dead Zone Quantization

The dead zone quantization discussed in the chapter 3 and 4 is

used in this chapter. It is observed from the chapters 3 and 4 that dead zone

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quantization provides better results when compared with scalar and vector

quantization. So, in this approach dead zone quantization has been used.

5.5.3 Hybrid Encoding Approach using SPIHT Algorithm with

Huffman Encoder for Image Compression

According to statistical analysis of the output binary stream of

SPIHT encoding, a simple and effective method combining SPHIT with

Huffman encoding is proposed for further compression in this thesis work.

Huffman coding is an entropy encoding algorithm used for

lossless data compression. The term refers to the use of a variable-length

code table for encoding a source symbol (such as a character in a life)

where the variable-length code table has been derived in a particular way

based on the estimated probability of occurrence for each possible value of

the source symbol. It uses a specific method for choosing the

representation for each symbol, resulting in a prefix code that expresses the

most common source symbols using shorter strings of bits than are used for

less common source symbols.

The Huffman algorithm is based on statistical coding, which

means that the probability of a symbol has a direct bearing on the length of

its representation. The more probable the occurrence of a symbol, the

shorter will be its bit-size representation. In any file, certain characters are

used more than others. Using binary representation, the number of bits

required to represent each character depends upon the number of characters

that have to be represented. Using one bit we can represent two characters,

i.e., 0 represents the first character and 1 represent the second character.

Using two bits we can represent four characters, and so on (Tripatjot Singh

et al 2010).

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Unlike ASCII code, which is a fixed-length code using seven

bits per character, Huffman compression is a variable length code system

that assigns smaller codes for more frequently used characters and larger

codes for less frequently used characters in order to reduce the size of files

being compressed and transferred (Kharate & Patil 2010).

Huffman coding is a popular lossless Variable Length Coding

(VLC) scheme (Arjun Nichal et al 2013), based on the following principles

(a) Shorter code words are assigned to more probable symbols and longer

code words are assigned to less probable symbols, (b) No code word of a

symbol is a prefix of another code word. This makes Huffman coding

uniquely decidable and (c) Every source symbol must have a unique code

word assigned to it. In image compression systems, Huffman coding is

performed on the quantized symbols. Quite often, Huffman coding is used

in conjunction with other lossless coding schemes, such as run-length

coding. In terms of Shannon’s noiseless coding theorem, Huffman coding

is optimal for a fixed alphabet size, subject to the constraint that the source

symbols are coded one at a time.

Huffman codes are assigned to a set of source symbols of

known probability. If the probabilities are not known a priori, it should be

estimated from a sufficiently large set of samples, and the average code

word length is computed.

After obtaining the Huffman codes for each symbol (Table 5.1).

It is easy to construct the encoded bit stream for a string of symbols.

For example, inorder to encode a string of symbols

a4 a3 a5 a4 a1 a4 a2

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It can be started from the left, taking one symbol at a time.

The code corresponding to the first symbol a4 is 0, the second symbol a3

has a code 1110 and so on. Proceeding as above, the encoded bit stream is

obtained as 0111011110100110.

Table 5.1 Huffman coding

Symbol Probability Assigned code

a4 0.6 0

a1 0.2 10

a2 0.1 110

a3 0.05 1110

a5 0.05 1111

In this example, 16 bits were used to encode the string of

7 symbols. A straight binary encoding of 7 symbols, chosen from an

alphabet of 5 symbols would have required 21 bits (3 bits/symbol) and this

encoding scheme therefore demonstrates substantial compression.

Description of the Algorithm

SPIHT is one of the most advanced schemes available that

outperforms even the state-of-the-art JPEG 2000 in some situations.

The Set-Partitioning in Hierarchical Trees (SPIHT) coding operates by

exploiting the relationships among the wavelet coefficients across the

different scales at the same spatial location in the wavelet subbands.

In general, SPIHT coding involves the coding of the position of significant

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wavelet coefficients and the coding of the position of zero trees in the

wavelet subbands.

The SPIHT coder exploits the following image characteristics:

The majority of an image’s energy is concentrated in the low

frequency components and a decrease in variance is observed when moved

from the highest to the lowest levels of the sub band pyramid. It has been

observed that there is a spatial self-similarity among the sub bands, and the

coefficients are likely to be better magnitude-ordered if moved downward

in the pyramid along the same spatial orientation.

A tree structure, termed spatial orientation tree, clearly describes

the spatial relationship on the hierarchical pyramid. Figure 5.7 shows how

the spatial orientation tree is defined in a pyramid, constructed with

recursive four-sub band splitting. Every pixel in the image signifies a node

in the tree and is determined by its corresponding pixel coordinate.

Its direct descendants (offspring) symbolize the pixels of the same spatial

orientation in the next finer level of the pyramid. The tree is defined in

such a manner that each node has either no offspring (the leaves) or four

offsprings, which all the times form a group of 2 X 2 adjacent pixels.

In Figure 5.7, the arrows are directed from the parent node to its four

offsprings. The pixels in the highest level of the pyramid are the tree roots

and are also grouped in 2 X 2 adjacent pixels. Nevertheless, their offspring

branching rule is different, and in each group, one of them has no

descendants.

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Figure 5.7 Tree structure used in SPIHT

The following are the sets of coordinates used to represent the

coding method:

O(i,j) in the tree structures is the set of offspring (direct

descendants) of a tree node defined by pixel location (i,j).

D(i,j) is the set of descendants of node defined by pixel location

(i,j) .

L(i,j) is the set defined by Equation (5.3)

L(i,j) = D(i,j) - O(i,j)

Except for the highest and lowest pyramid levels, the set

partitioning trees has,

O(i,j) = {(2i,2j),(2i,2j+1),(2i+1,2j)(2i+1,2j+1)} (5.4)

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The rules for splitting the set (e.g. when found significant) are

as follows :

1) The initial partition is formed with the sets (i,j) and D(i,j) for all

2) If D(i,j) is significant, then it is partitioned into L(i,j) plus the

four single-element sets with (k,l) O(i,j).

3) If L(i,j) is significant, then it is partitioned into the four sets

D(k,l) with (k,l) O(i,j). The significant values of the wavelet

coefficients modeled in the spatial orientation tree are stored in

three ordered lists namely: LIS, LIP and LSP.

4) List of Insignificant Sets (LIS) : contains the set of wavelet

coefficients defined by tree structures, and found to have

magnitude smaller than a threshold (are insignificant). The sets

exclude the coefficient corresponding to the tree or all sub tree

roots, and have at least four elements. The entries in LIS are sets

of the type D( i,j) (type A) or type L( i,j) (type B).

5) List of Insignificant Pixels (LIP) : contains the individual

coefficients that have magnitude smaller than the threshold.

6) List of Significant Pixels (LSP) : contains the pixels that are

found to have magnitude larger than the threshold (are

significant). During the sorting pass, the pixels in the LIP that

were insignificant in the previous pass are tested, and those that

emerge significant are moved to the LSP. Then, the sets are

sequentially assessed along the LIS order, and when a set is

found significant it is removed from the list and partitioned.

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The new sets with more than one element are added back to

LIS, while the one element sets are added to the end of LIP or

LSP, according to their being significant.

A) Algorithm

In the algorithm, initialization is done for output, LSP, LIP and

each entry in LIS is set as type A. Then sorting pass is done for LIP and

output for each LIS. For each output LSP is appended. Based on the sign of

the output, sign of LIP is appended. Then the output is moved to end of

LIS as type B, each of LIS is appended to the end of LIS and type A is

removed from LSP. The refinement is done for each LSP those have not

included in the last sorting process output of nth MSB is set. There will

quantization pass after this. Then the process will move to sorting process.

The process is repeated for entire output.

Some of the advantages of SPIHT encoding include:

i. Allows a variable bit rate and rate distortion control as well as

progressive transmission

ii. An intensive progressive capability - the decoding (or coding)

can be interrupted at any time and a result of maximum possible

detail can be reconstructed with one-bit precision.

iii. Very compact output bit stream with large bit variability and no

additional entropy coding or scrambling has to be applied.

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B) Analysis of SPIHT algorithm

Here a concrete example to analyze the output binary stream of

SPIHT encoding is considered. A 3-level wavelet decomposition

coefficients of SPIHT encoding is assumed as

n = [log 2 max{c (i, j)}] =4, with the initial threshold

Value: T0=24, for T0, the output binary stream:

111000000000111100111011111111100011101110100, total 42 bits in all.

By the SPIHT encoding results, the output bit stream has a large number of

series of "0" situations Along with the gradual deepening of quantification,

the situation will become much more severity and so there will be great

redundancy in the output.

C) Modified SPIHT Algorithm

For the output bit stream of SPIHT encoding with a large

number of seriate "0" situation, ‘000’ appears with the greatest probability

value, usually will be about 1/4. Therefore, divide the binary output stream

of SPIHT every 3 bits as a group, every group recorded as a symbol.

A total of eight kinds of symbols are encoded using variable-length

encoding and further compressed. For variable-length encoding Huffman

encoding is used.

Using the output bit stream of above example the new encoding

method is processed as follows:

1) First, divide the binary output stream every 3 bits as a group:

110 000 000 001 111 001 110 111 111 111 000 111 011 101. In

this process, there will be remainder 0, 1, 2 bits that cannot

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participate. So, in order for unity, in the head of the output bit

stream of Huffman encoding cost ,two bits are used to record

the number of bits that do not participate in group and those

remainder bits arrive in the output at the end.

2) The emergence of statistical probability of each symbol

grouping results is as follows:

P (‘000’) = 0.2144 P (‘001’) =0.14

P (‘010’) = 0 P (‘011’) =0.07

P (‘100’) = 0 P (‘101’) = 0.07

P (‘110’) = 0.14 P (‘111’) = 0.35

3) According to the probability of the above results, using

Huffman encoding, code word book is obtained as shown in

Table 5.2.

Table 5.2 Code word comparison table

000 00 100 011

001 01 101 0011

010 11 110 01111

011 010 111 111111

Through the above code book ,one can get the corresponding

output stream as : 00 01 11 010 011 0011 01111 111111, a total of

27 bits. The ‘10’ in the head is binary of remainder bits’ number. The last

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two bits ‘00’ are the result of directly outputting remainder bits. Compared

with the original bit stream 15 bits are saved. Decoding is the inverse

process of the above-mentioned scheme.

Decompression

This process is the reverse to the compression technique. After

SPIHT, it is necessary to transform data to the original domain (spatial

domain). To do this, the inverse fractional fourier transform is applied first

in columns and secondly in rows.

Proposed Algorithm

The proposed image compression technique is very effective in

overcoming the limitations of the wavelet transform based coding

techniques when applied to images with linear Curves. The steps involved

in the proposed approach are:

i. Represent the image data as intensity values of pixels in the

spatial co-ordinates.

ii. Apply nonsubsample contourlet Transform on the image matrix

and get the contourlet coefficients of the image.

iii. Quantize the available coefficients using dead zone quantization

algorithm.

iv. Hybrid SPIHT encoding algorithm with Huffman encoding on

the bit stream.

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5.6 EXPERIMENTAL RESULTS

In order to evaluate proposed algorithm, certain non-linear

approximation experiments are carried out in comparison with curvelets

and contourlets. The objective image quality metrics like compression

ratio, Peak Signal to Noise Ratio (PSNR) and Root Mean Squared Error

(RMSE), cross correlation.

Table 5.3 Comparison of various parameters for wavelet based

contourlet transform and nonsubsampled contourlet

transform with SPIHT, MSPIHT and hybrid coding

Parameters Wavelet based Contourlet

Transform

Nonsubsampled

Transform

SPIHT MSPIHT HYBRID SPIHT MSPIHT HYBRID

PSNR 31.7497 33.013 33.6687 39.9126 40.6146 41.2701

MSE 6.5926 5.7002 5.2857 2.5758 2.3758 2.2031

CR 72.7491 84.6424 88.0727 78.5964 88.2472 90.3085

Correlation 0.9962 0.9952 0.9942 0.9967 0.9956 0.9946

Table 5.3 shows the comparison performance of the proposed

WBCT and NSCT for dead zone quantization with SPIHT, modified

SPIHT and hybrid coding. The result of PSNR for NSCT with MSPIHT is

1.75% higher than with SPIHT and for hybrid coding, it is 3.40 % higher.

Thus, from the results, it is observed that the proposed approach provides

significant results.

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5.7 SUMMARY

This chapter clearly shows and discusses about the problems in

wavelet based contourlet transform. Due to the drawbacks of wavelet based

contourlet transform, proposed approach uses a nonsubsampled contourlet

transformation with dead zone quantization. Moreover, for the encoding

purpose, SPIHT, modified SPIHT and hybrid coding of MSPIHT with

Huffman coding are used.