TRANSFORMED VECTOR QUANTIZATION BASED - Shodhganga

CHAPTER 3

Transformed Vector Quantization with Orthogonal

Polynomials

3.1. Introduction

In the previous chapter, a new integer image coding technique based on

orthogonal polynomials for monochrome images was proposed. After the

proposed transformation the coefficients are scalar quantized and entropy coded

in order to obtain a higher compression ratio. This technique proved to be better

in the sense that it gives higher PSNR value. However, the quality of the

reconstructed image degrades when the quality factor increases due to the scalar

quantization step effect. To overcome this problem, in this chapter, a new vector

quantization technique has been proposed in the transformed domain. The

rationale behind the introduction of vector quantization is that the vector

quantization of signal reduces the coding bit rate significantly with good quality

of reconstruction picture. The proposed transformed vector quantization exploits

the combined features of energy preservation by the proposed transform coding

and high compression ratio of the vector quantization.

3. 1. 1 Vector quantization

In the current scenario, Vector Quantization [Alle92] has been found to be

an efficient data compression technique for speech and image as it provides

many attractive features for image compression. A vector quantizer Q of

dimension k and size N is mapped from a point in k-dimensional Euclidean space

Rk, into a finite set C containing N output or reproduction points that exist in the

same Euclidean space as the original point. These reproduction points are known

as codewords and these set of codewords are called codebook C with N distinct

codewords in the set. Thus, the mapping function Q is defined as,

Q: Rk → C ... (3.1)

CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS

36

The rate of the vector quantizer or the number of bits (r) used to express each

quantized vector is given by the relation,

r = log2 N / k … (3.2)

This rate equation is very useful as it gives the amount of compression that can

be expected in a particular VQ coding scheme.

Vector quantization can be considered as a pattern recognizer where an

input pattern is approximated by a predetermined set of standard patterns

[Robe02]. Experiments have shown that vector quantization produces superior

performance over scalar quantization even when the components of the input

vectors are statistically independent. Vector quantization exploits the linear and

nonlinear dependence among vectors. It also provides flexibility in choosing

multi-dimensional quantizer cell shapes and in choosing a desired codebook size.

If scalar quantization is extended to k dimensional vectors using N levels, then

the codebook would contain N x k codewords. In the case of vector quantization

there could be arbitrary partitions with integer number of codewords N. Another

advantage of vector quantization over scalar quantization is the fractional value

of resolution that is achievable. This is very important for low bit rate

applications where low resolution is sufficient.

The number of codewords in the codebook decides the quality of the

reconstructed vectors. If the number of code words is large, the output vectors

would be close to the input vectors. The dimension (i.e the number of elements

present in each vector) of the input vectors and code words also play a crucial

role in quality of reconstruction. Ideally, the compression performance improves

as the vector dimension increases but the tradeoff is the increased coding

complexity. Besides dimension, the difficult task in any VQ scheme is the

generation of codewords that best represent the input vectors [Lind80, Gray84].

The performance of the quantizer is assessed using a suitable statistical distortion


37

measure. The generic statistical distortion measure as applied to vectors is

represented as:

))(( , iiixQxdD … (3.3)

where xi is the input vector and Q(xi) is the approximation of xi and

))(,( ii xQxd represents the squared Euclidean Distance between the input vector

ix and its approximation )( ixQ .

3. 2 Literature survey

Vector quantization is a very powerful method for lossy compression of

data such as images and speech. The lossy compression scheme can be analyzed

using rate distortion theory [Alle92]. In this scheme the decompressed data will

not be a replica of the original. Instead, it will be distorted by an amount D.

According to Shannon’s rate distortion theory [Jude76], vector quantization of

signals reduce the coding bit rate significantly when compared to scalar

quantization. Vector Quantization takes M number of multi dimensional vectors

and reduces their representation to k number of code words, where k < M. The

key to Vector Quantization is to construct a good codebook of representative

vectors. The most popular method for designing a codebook was proposed by

Linde, Buzo and Gray in [Lind80, Gray84]. This method is now commonly

referred to as LBG algorithm. In this algorithm, all the training vectors are

clustered, using the minimum distortion principle, around trial code vectors. The

centroids of these clusters then become the new trial code vectors at the next

iteration. This procedure continues until there is no significant change in the total

distortion between cluster members and the code vectors around which they are

clustered. Then the training vectors are compared with codebook that is

generated by LBG algorithm. The result is an index position of codebook with

minimum distortion. This algorithm works directly on the image pixels and it

uses the full search technique in the encoding process. So it takes longer time to


38

construct the codebook and each codeword contains every pixel of a block. Due

to the enormous size of the code book the search time to find the best match

vector in the encoding process increases drastically. The methods available in the

literature to alleviate this problem are presented below.

Generation of fast codebooks for Vector Quantization of images, based on

the features of training vectors has been reported by Hsieh. C.H et. al [Hsie91].

This method uses the good energy compaction capability of the Discrete Cosine

Transform and uses certain significant components of the feature space to

construct the binary tree. Design of codebook for vector quantization with the

discrete cosine transform Coefficients as training vector feature has been

reported by Hsieh. C. H. [Hsie92]. In this work, the energy preserving property

of the DCT has been used to reduce the dimension of the feature training vector.

From these works, research activities on design of transformed vector

quantization (TVQ) that combines the features of transform coding and vector

quantization have gained popularity.

Timo Kaukoranta, Pasi Fanti and Olli Nevalanen [Timo00] have reported

a scheme for reducing the number of distance calculations in the LBG algorithm

and are included in several fast LBG variants reducing their running time by over

50% on average. A scheme based on Vector Quantization in transformed domain

has been reported in [Robe02] by Roberts et al. This scheme uses a fast Kohonen

self-organizing neural network algorithm to achieve reduction in codebook

construction time and transformed vector quantization to obtain better

reconstructed images. Hsien-Wen Tseng et al. [Hsie05] have reported a classified

vector quantization (CVQ) in the DCT transform domain. In this method DC

coefficients are coded by difference pulse code modulation and only the most

important AC coefficients are coded using classified vector quantization (CVQ).

These AC coefficients are selected to train the codebook according to the energy

packing region of different block classes. Evaluation of TVQ as low bit rate


39

image coding has been reported by Clyde Shamers et al. [Clyd04]. This coding

technique which is based on the combined use of discrete cosine transform and

Vector Quantization eliminates the artifacts generated in JPEG compression.

Use of wavelet transformation in the design of TVQ has been reported in

[Min05]. Here the relationship between the input vector and codeword, as well as

the relationship among code words and characteristics of code words in wavelet

domain are utilized. Another scheme for image compression with transform

vector quantization of the wavelet coefficients has been reported by Momotaz

Begum et al. [Momo03]. This scheme utilizes mean-squared error and variance

based selection for good clustering of data vectors in the training space. The two

major drawbacks of the LBG algorithm namely, the choice of initial codebook

and the huge computational burden have been alleviated by this scheme. Fast

search algorithm for vector quantization with multiple triangle inequalities in the

wavelet domain has been reported by Chaur H.H and Liu. Y.J. [Chau00]. The

multiple triangle inequalities confine a search range using the intersection of

search areas generated from several control vectors. Also a systematic way for

designing the control vectors is reported. The wavelet transform combined with

the partial distance elimination is used to reduce the computational complexity of

the distance calculation of vectors. A fast codeword searching algorithm based

on mean-variance pyramids of codewords [Lu00] and Hadamard Transformation

[Lu00a] have also been found in the literature. Given initial vectors, two design

techniques for adaptive orthogonal block transforms based on VQ codebooks are

presented in [Cagl98]. Both the techniques start from reference vectors that are

adapted to the characteristics of the signal to be coded, while using different

methods to create orthogonal bases. The resulting transforms represent a signal

coding tool that stands between a pure VQ scheme on one hand and signal-

independent, fixed block transformation like discrete cosine transform (DCT) on

the other.


40

Review of early works on VQ can be found in [Nasr98]. S. Esakkirajan et

al. [Esak06] have proposed an image coding scheme based on contourlet

transform and multiscale VQ. Recently filter banks approach for VQ has been

developed by Brislawn and Wohlberg [Chri06] to overcome obstructions for a

class of half-sample symmetric filter banks. They employ lattice vector

quantization to ensure symmetry preserving rounding in reversible

implementations. Z. Liu et al. [Liu07] reported the use of biorthogonal wavelet

filter banks (BWFBs) for image coding with lower computational costs. Here a

new class of Integer Wavelet Transforms (IWT) parameterized simply by one

parameter, obtained by introducing a free variable to the lifting based

factorization of a Deslauriers-Dubuc interpolating filter, is introduced. The exact

one-parameter expressions for this class of IWTs are deduced. In this technique,

different IWTs are obtained by adjusting the free parameter and several IWTs

with binary coefficients are constructed.

In this chapter we explore the possibility of introducing a new VQ with

the integer transform coding proposed in chapter 2. This new integer

Transformed Vector Quantization takes the advantage of decorrelation and

energy compaction properties of Orthogonal Polynomials based Transform

coding and the superior rate distortion performance of VQ in the orthogonal

polynomials transformed domain. In this work, the energy preserving property of

the proposed transformation coding scheme is analyzed to truncate higher

frequency components. This truncated sub-image is then subjected to vector

quantization for effective codebook design with less complexity in sample space.

The important steps involved in the work presented in this chapter are

highlighted below.

Analysis of the point spread operator introduced in chapter 2 that defines the

proposed coding and shows its completeness, for the purpose of perfect

reconstruction of the original image by proposing difference operators.


41

Formations of training vectors with scale quantized high energy transform

coefficients.

Design of codebook on training vectors as in LBG algorithm.

Identification of index values by performing vector quantization on training

vectors and entropy coding of the identified indices.

Inverse transformation with proposed basis operators after carrying out a

simple look-up process in the codebook.

3.3 Proposed orthogonal polynomials based framework for vector

quantization

3.3.1 Completeness of the proposed transformation

Before presenting the new Vector Quantization in the orthogonal

polynomials based transformation domain, we first prove its completeness and

the same is presented in this subsection.

The point spread operator in equation (2.3) described in chapter 2 that

defines the linear orthogonal transformation for gray scale images is obtained as

|M| |M|, where |M| is computed and the elements are scaled to make them

integers as follows.

222120

121110

020100

xuxuxu

xuxuxu

xuxuxu

M =

111

201

111

... (3.5)

The set of polynomial operators Oijn (0 ≤ i, j ≤ n-1) can be computed as

Oijn = ûi ûj

t

where ûi is the (i + 1)st

column vector of |M| and is the outer product.

In this subsection we prove that the proposed polynomials based

difference operators is complete and hence the reconstruction of the image under


42

analysis after the said 2-D transformation is possible in terms of linear

combination of basis operators Oij and the transform coefficients ji .

The following symmetric finite differences for estimating partial

derivatives at (x, y) position of the gray level image I are analogous to the eight

finite difference operators Oijs excluding O00.

1

1

, 1,1,i

yx yixIyixIy

I

1

1

, ,1,1i

yx iyxIiyxIx

I

1

1

,2

2

1,,21,i

yx yixIyixIyixIy

I

1

1

,2

2

,1,2,1i

yx iyxIiyxIiyxIx

I … (3.6)

and so on.

In general,

ijji

ji

Oyx

and

02,0,,

jiandjiIOyx

Ijiijji

ji

… (3.7)

where | | indicates the arrangement in dictionary sequence and ( , ) indicates the

inner product. Hence, Oijs are symmetric finite difference operators. ji s are the

coefficients of the linear transformations and are defined as follows.

ji | = |M|t |I| … (3.8)

where | M | is the 2-D point-spread operator defined as | M | = |M| |M|.


43

Now we will show that the orthogonal transformation defined in equation (3.8)

by the orthogonal system | M | is complete. We may obtain an orthogonal system

| H | by normalizing | M | as follows.

|H| = | M | (|M |t | M |)

-½

Consider the following orthonormal transformations

|Z|= | H |t |I| = (|M |

t | M |)

½ | M |

t |I| = (|M)

t |M |)

-½ | |

Since, | H | is unitary,

|I| = |H| |Z| = | M |

2

0

2

0i j

ijij O … (3.9)

where | | = (|M |t | M |)

-1 | |.

As per equation (3.9) the image region | I | can be expressed as a linear

combination of the nine basis operators of which |O00| is the local averaging

operator and the remaining eight are finite difference operators (equation 3.7).

From equation (3.9) we obtain the completeness relation or Bessel's equality as

follows.

2

0

2

0

22

0

2

0

2..,,i j

ij

i j

ij ZIeiZZII ... (3.10)

Thus it is proved that the proposed transformation is complete and hence the

transformed image can be reconstructed perfectly. In the next section, Vector

Quantization on the proposed orthogonal polynomials based transformation

coefficients is presented.


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3. 4. Proposed vector quantization

In this section, a new Transformed Vector Quantization scheme that

facilitates the image coding using Orthogonal Polynomials is proposed. This

proposed scheme combines both transform coding and VQ technique. The

advantage of combining both the proposed transformation and VQ is that, when a

linear transform is applied to the vector signal, the information is compacted into

a subset of the vector components. In the frequency domain, the high energy

components are concentrated in the low frequency region. This means that the

transformed vector components in the high frequency regions have very little

information and so these low energy components can be entirely discarded. The

procedure involved in the proposed transformed vector quantization is presented

hereunder.

3.4.1 Formation of training vector

The proposed TVQ starts by portioning the original image of size (R x C)

into non-overlapping sub-blocks of size (n x n) and mapping them to the

frequency domain by applying the proposed orthogonal polynomials based

transformation as described in section 2.2.2. The resultant transform coefficients

ji {i, j = 0, 1, 2, …, n-1} are subjected to scale quantization using the default

quantization table of JPEG baseline system. The aim of using the scale

quantization is to obtain a suboptimal VQ codebook with reduced reconstruction.

As the proposed orthogonal polynomials based transform and DCT based JPEG

are both unitary, without loss of generosity, the default quantization table of

JPEG is utilized here. The scale quantized transform coefficients ji are then

arranged into 1-D zig-zag sequence to form a k-dimensional transformed input

vector Y and it is mapped into a p-dimensional (p < k) training vector T by

considering only the energy preserving components due to the proposed


45

orthogonal polynomials transformation. The energy preservation by the proposed

transform is extracted as follows.

The energy preserving property of the proposed transformation is based

on the estimates of the variances 2

, jiZ s corresponding to the mean squared

amplitude responses of the basis, difference operators Oi,j. These variances are

computed as

jjii

ji

jiWW

Z,,

2

,2

,

… (3.11)

where MMWt

and IMMt

ji

The F-ratio test [Fish87] is then conducted on the variances 2

, jiZ s to

identify the significant responses towards signal compared to a threshold. The

fact that a variance passes the test implies that considerable energy is compacted

in the transform coefficients ji corresponding to that variance. After applying

the F-ratio test on every 2

, jiZ the insignificant responses that do not contain much

energy can be discarded.

The aforementioned process is repeated to form the training vectors of all

the partitioned sub-blocks and the codebook is designed as described in the

following section using the training vectors.

3.4.2 Codebook design

The selection of the initial set of codewords is a very tricky problem in

any VQ design. A variety of techniques are available in the literature for the

initial selection of codewords [Alle92]. The proposed technique uses the splitting

technique for choosing the initial set of codewords. In the splitting technique, the


46

initial codeword C0 is chosen by taking the centroid of the entire training vectors

T. Then, this codeword is split into two, namely C0 + ε and C0 - ε, where ε is any

Euclidean norm and indicates the optimization precision. This process of

iteratively splitting each codeword into two continues until the desired number of

codewords of the codebook is obtained. These codewords do not qualify as final

codewords for quantizing the input vectors as they do not satisfy the necessary

conditions of optimality. However these can be used as the initial codewords. To

optimize the codewords, the proposed technique makes use of Linde Buzo Gray

(LBG) Algorithm [Lind80]. Here the training vectors T are clustered by

computing the minimum distortion of the training vectors against the initial

codewords C. The centroids of the clusters thus formed become the new

codewords for the next iteration. This procedure continues until there is no

significant change in the total distortion between cluster members and the

codewords around which they are clustered. The final set of code vectors

obtained constitutes the codebook. The steps involved in the design of codebook

in the proposed transform domain are presented below:

Step 1. Initialize the initial codeword C0 with the mean of the entire set of

training vectors Z and perturbation value ε to a fixed value.

Step 2. Initialize iteration number n to zero and distortion D-1 to ∞.

Step 3. Form the desired number of codewords for the new codebook by

splitting each codeword into two using the binary splitting

operation.

Step 4. Optimize the new codebook formed in step 3 using the centroid

condition.

Step 5. Compute r = (D n-1 - D n)/ Dn. where D n-1 distortion before

optimization and D n is the distortion after optimization.

Step 6. Repeat steps 3 through 5 until r ≤ ε.


47

Then the training vectors T are compared with codebook, and index

positions of code vectors that give minimum distortion, are identified. These

index values are entropy coded as in JPEG baseline system and are transmitted to

the receiver.

3.4.3 Reconstruction

The receiver decodes the received bits to get the index values. It then

initiates the lookup process in order to get the p-dimensional transformed

coefficients vector from the codebook which is identical to the one at the sender.

Then (k–p) additional components with value zero are appended to the vectors,

producing the k-dimensional vectors Y and scale dequantization is performed on

the elements of Y to get the transformed coefficients ji . Finally these

coefficients are subjected to inverse transform with the help of basis functions of

the proposed orthogonal polynomials as described in section 2.3 to get back the

decompressed image.

3.4.4 Time minimization

The goal of the proposed TVQ scheme is threefold. First, the proposed

scheme tries to minimize the time taken for construction of the codebook. The

second goal is to reduce the size of the codebook. Thirdly, the proposed scheme

aims to reduce the time consumption for the encoding process. To perceive how

these goals are achieved, let us consider a k-dimensional input vector X with a

resolution of r-bits per component constituting a total bit allocation of r x k bits.

Normally in VQ, the codebook size would be N = 2rk

. With the proposed TVQ,

using the same r-bits the maximum possible codebook size is reduced to N = 2rp

,

which can be of smaller magnitude since p < k. Also, the time taken for

constructing the codebook is reduced drastically, as the proposed scheme uses

only p-dimension vectors instead of k-dimension vectors. In a generic VQ, the


48

number of comparisons required during the encoding process is N x k whereas

the proposed scheme requires only N x p comparisons. Hence, it is evident that

the proposed TVQ technique consumes less time and takes less storage for image

coding. Alternatively, for the same time and space, the resolution or codebook

size can be increased to obtain better performance. The steps involved in the

proposed TVQ technique, are presented below.

3. 4. 4. TVQ Algorithm

Input: Gray-level image of size (R x C). [ ] denotes the matrix and the suffix

denotes the elements of the matrix. Let [I] be the (n x n) non-overlapping image

region (block) extracted from the image.

Begin

Step 1. Divide the given input image into number of non-overlapping image

regions [I] of size (n x n).

Step 2. Repeat the steps 3 to 7 for all the image regions.

Step 3. Compute the orthogonal polynomials based transform coefficients

[] as described in section 2.2.2.

Step 4. Apply scale quantization on the transform coefficients [].

Step 5. Arrange the scale quantized [] in 1-D zig-zag sequence.

Step 6. Truncate the low energy components from the scale quantized []

based on the energy preserving property of the proposed

Orthogonal Polynomials based transformation as described in

section 3.4.1.

Step 7. Form a vector T using the truncated low frequency coefficients.

Step 8. Design the codebook with LBG algorithm on the vectors T as

described in section 3.4.2.


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Step 9. Perform VQ operation as explained in section 3.1.1 and obtain the

index values.

Step 10. The index values are subjected to entropy coding as in JPEG and

the coded value is transmitted to the receiver through channel.

Step 11. At the receiving end, decode the index values and form the

truncated code words using index values as a table look-up process.

Step 12. Obtain all the n2 coefficients by substituting zero values in

truncated high frequency coefficients.

Step 13. These n2

coefficients are then subjected to the scale dequantization

to form an approximation to the original transform coefficients [].

Step 14. Reconstruct the input image region [I] using the polynomial basis

functions as discussed section 2.3.

Step 15. Repeat the steps 11 to 14 until all the image regions [I] are

reconstructed.

End.

The above said proposed algorithm is presented in diagrammatic way in Figure

3. 1.


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Figure 3. 1: Flow diagram of the proposed TVQ technique

Index Value

Index Value

1001 …

1001 …

Choose the

Energy

Preserving

Coefficients

Channel

Symbol Decoder

Nearest Neighbor

principle

Symbol Encoder

Table Lookup

Scale

Quantizer

Generate

codebook with

LBG

Add zero values

to the decoded

code words to

compensate 16

coefficients

Scale

Dequantizer

Inverse

Transformation

XXX

XXX

Code book

Proposed

Transformation

Original Image

4 x 4 blocks

Reconstructed

Image


51

3. 5 Experiments and results

The proposed orthogonal polynomials based Transformed Vector

Quantization has been experimented with 2000 test images, having different low

level primitives. For illustration two test images viz, Lena and Pepper, both of

size (256 x 256) with gray scale values in the range (0 – 255) are shown in

figures 3.2(a) and 3.2(b) respectively. The input images are partitioned into

various non-overlapping sub-images of size (4 x 4). We then apply the proposed

orthogonal polynomials based transformation on each of these image blocks and

obtain the transform coefficients ji . All these ji s of each block are then

subjected to scale quantization. The resulting coefficients are re-ordered to 1-D

zig-zag sequence and a subset of corresponding ji s due to energy compaction

of the proposed orthogonal polynomials based transformation are extracted as

described in section 3.4.1. These resulting frequency coefficients are treated as a

vector Ti with dimensionality six and the experiment is repeated for all the sub-

images to form a set of vectors T = {Ti , i = 1, 2,…, k} where ‘i’ represents the

sub-images and ‘k’ represents total number of sub-images.

Then the codebook is designed on truncated scale quantized transform

coefficients with LBG algorithm as described in section 3.4.2. The vectors and

the codebook thus generated are subjected to Vector Quantization to obtain the

index value for each vector Ti corresponding to the sub-images under analysis.

These index values are subjected to entropy coding and are transmitted to the

receiver side. In the decompression process, these index values are decoded and

are used to generate the approximated truncated transform coefficients, with the

help of the codebook that were generated in the earlier stage. These 1-D

transform coefficients are reordered to the original 2-D array after compensating

zeros to the truncated high frequency components. Then the decompressed

original image is obtained with the orthogonal polynomials basis functions as

described in section 2.3.


52

(a) (b)

Figure 3.2: Original test images considered for proposed TVQ

(a) (b)

Figure 3.3: Results of proposed TVQ when bpp = 0.25


53

Table 3.1: PSNR values obtained with proposed TVQ, DCT based TVQ and the

JPEG baseline for different bpps.

Bit rate

(bpp)

Proposed TVQ

(Dimensionality 6)

DCT based TVQ

(Dimensionality 6)

JPEG baseline

system

Lena Pepper Lena Pepper Lena Pepper

0.25

0.20

0.18

0.16

0.14

0.12

33.09

30.14

29.51

29.02

28.67

27.64

33.22

30.90

29.80

29.47

28.92

28.03

32.49

29.71

28.64

28.14

27.42

26.04

32.53

29.88

29.02

28.44

27.49

26.53

31.62

29.47

28.28

27.02

24.99

21.48

31.86

29.81

28.69

27.26

25.29

21.58

The bit per pixel (bpp) scheme is used to estimate the transmission bit

rate. The performance of the proposed TVQ scheme is measured with the

standard measure Peak-Signal-to-Noise-Ratio (PSNR) as described in section 2.5

with the proposed TVQ. We obtain PSNR values of 33.09dB and 33.22dB for a

bit rate of 0.25 for the input images 3.2(a) and 3.2(b) respectively and the

corresponding resulting images are shown in figures 3.3(a) and 3.3(b)

respectively. The experiment is repeated by varying the bpp for all the 2000

images and the results for the Lena and Pepper images are presented in table 3.1.

In order to measure the efficiency of the proposed orthogonal polynomials

based TVQ, we conduct experiments with discrete cosine transform based

transformed vector quantization. For this comparison, the proposed orthogonal

polynomials based transformation is replaced with discrete cosine transformation

and the other steps are kept unaltered. The experiments are conducted for

different bpp and the corresponding PSNR values obtained are incorporated in

the same table 3.1, for both the input images and the corresponding results for

0.25 bpp are shown in figure 3.4(a) and 3.4(b) respectively. The proposed


54

transformed vector quantization algorithm is also compared with the

international standard JPEG type compression algorithm where discrete cosine

transformation and scale quantization are used. For this experiment

normalization and quantization arrays are scaled in JPEG algorithm, to adjust the

compression ratio to the desired level. The experiments are carried out for

varying bit rates for different images and the corresponding results are

incorporated in the same table 3.1. The results of JPEG algorithm for the input

images shown in figures 3.2(a) and 3.2(b) when the bpp is 0.25 are shown in

figures 3.5(a) and 3.5(b) respectively. The graphs of PSNR vs. bpp for Lena and

Pepper images are plotted and the same are shown in figure 3.6(a) and 3.6(b)

respectively.

(a) (b)

Figure 3.4: Results of DCT based TVQ when bpp = 0.25


55

(a) (b)

Figure 3.5: Results of JPEG when bpp=0.25

Lena

20

22

24

26

28

30

32

34

.12 .14 .16 .18 .2 .25

Bit rate (bpp)

PS

NR

(d

B)

Proposed TVQ

DCT based TVQ

JPEG

(a)


56

Pepper

20

22

24

26

28

30

32

34

.12 .14 .16 .18 .2 .25

Bit rate (bpp)

PS

NR

(d

B)

Proposed TVQ

DCT based TVQ

JPEG

(b)

Figure 3.6: Bit rate versus PSNR comparison of the proposed TVQ with DCT

based TVQ and JPEG

From table 3.1 and figures 3.3, 3.4, 3.5 and 3.6, it is evident that the

proposed TVQ outperforms JPEG and discrete cosine transformation based

TVQ. From these outputs, it is clear that the proposed scheme gives higher

PSNR value with a reasonably good reconstruction quality. It can also be

observed that the quality obtained with JPEG image coding is found to have

degradation at low bit rates. But the proposed Transformed Vector Quantization

produces a stable quality of picture even at low bit rates. [Observe that the PSNR

value that ranges from 31.62dB to 21.48dB and 31.86 to 21.58dB for Lena and

Pepper images respectively for JPEG scheme (from table 3.1)].

3.6 Conclusion

In this chapter a new transformed vector quantization based on orthogonal

polynomials has been proposed for 2-D gray scale images. This technique

combines the features of both transform coding and vector quantization. The

proposed transform coding is based on a set of orthogonal polynomials. The code

book is designed with LBG algorithm that utilizes only few transformed


57

coefficients due to the proposed transformation. Training vectors are then formed

as a subset from the image data in frequency domain and is compared with the

code book, to result in the index position of the code book and sent to the

decoder after entropy coding. The decoder has the code book identical to the

encoder and decoding mechanism is a simple table look-up process with

additional null values added to the high frequency samples. These coefficients

are subjected to inverse transform with the help of basis functions of the

proposed orthogonal polynomials transformation to get back the decompressed

image. The performance of the proposed scheme is measured with standard

PSNR value and is compared with DCT based TVQ and JPEG type algorithms.

However, the encoder phase of the proposed VQ scheme uses the full search

algorithm for finding the best match vectors and it leads to increase in searching

time. To overcome this problem, a binary tree based codebook design is

presented in the next chapter.

TRANSFORMED VECTOR QUANTIZATION BASED - Shodhganga

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