EFFICIENT IMAGE COMPRESSION ALGORITHMS USING EVOLVED WAVELETS · EFFICIENT IMAGE COMPRESSION...

International Journal of ISSN 0974-2107

Systems and Technologies

Vol.4, No.2, pp 127-146

127

IJST

KLEF 2011

EFFICIENT IMAGE COMPRESSION

ALGORITHMS USING EVOLVED WAVELETS

1.G.CHENCHU KRISHNAIAH*

2.T.JAYACHANDRAPRASAD

3.M.N.GIRI PRASAD

1.ECE DEPT.,GKCE,Sullurpet-524121,A.P,India.

Email:[email protected] 2.ECEDEPT.,RGMCET,Nandyal-528502,A.P,India

3.ECEDEPT.,JNTUCE,Pulivendula-515002,A.P,India

Abstract: The 9/7 and 5/3 lifting based wavelet filters are widely used in different

image compression schemes, such as JPEG 2000 image compression standard. The

performance of a hardware implementation of the 9/7 and 5/3 filter banks depends on

the accuracy with which filter coefficients are represented. In this paper an attempt

has been made to study the performance of 9/7 and 5/3 wavelets on photographic

images (monochrome and color) and estimated Peak Signal to Noise Ratio (PSNR),

Compression Ratio (CR), Mean Square Error (MSE), Encoding Time, Decoding

Time, Transforming Time or Decomposition Time…etc. This study shows that the

5/3 wavelet transform out perform the 9/7 wavelet transform.

Key words: Low complexity, efficient, 5/3 wavelet filter, 9/7 wavelet filter,

implementation, image compression, Algorithms, Evolved Wavelets.

Introduction

The Discrete Wavelet Transform (DWT) has gained wide popularity due to its

excellent decorrelation property[1], as a consequence many modern image and video

compression systems embody the DWT as the transform stage [2], [3]. It is widely

recognized that the 9/7 filters [4] are among the best filters for wavelet based image

compression [5]. In fact the JPEG2000 image coding standard [6] employs the 9/7

filters as the default wavelet filters for lossy compression, fostering many research

efforts in the development of fast and efficient hardware codecs.

G.Chenchu Krishnaiah

128

The performance of a hardware implementation of the 9/7 filter bank depends

on the accuracy with which filter co-efficient are represented. However high

precision representation increases hardware resources and processing time. To reduce

the complexity of the 9/7 filters the lifting scheme [7] can be adopted. Unfortunately

the lifting scheme increases hardware timing accumulation due to its serial nature [8],

so that for certain applications it cannot be employed. The flipping structure [8] is an

attractive alternative to the standard lifting scheme DWT, since it reduces timing

accumulation, however it still requires multiplications.

Complexity reduction can be achieved resorting to a filter bank

implementation; in particular very good results can be obtained with the cascaded

method proposed in [9]. The basic idea described in [9] is to minimize the number of

bit required to represent the 9/7 coefficients. Since this operation would move filters

zeros from their original position, the authors modify some terms to account for zeros

compensation. Currently the compatibility of low complexity 9/7 filters

implementation with floating point ones has not been stressed yet. The aim of this

paper is to show that great complexity reduction can be achieved analyzing the 9/7

filters directly from their analytical derivation [4]. In particular the proposed solution

shows negligible quality loss if employed in the direct DWT, with the floating point

9/7 filter bank in the inverse DWT.

1. THEORETICAL DERIVATION

Let’s consider the filter bank shown in figure 1, where

are the low pass and high pass analysis

filters with length kand l respectively, and

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the low pass and high pass synthesis

ones with length and . It is well known that wavelet filter banks ought to satisfy

the perfect reconstruction conditions [10]:

Imposing the biorthogonality condition together with filters symmetry ( (z) = H(-z)

and G(z) = (-z)) we can rewrite the perfect reconstruction conditions as:

As shown in [4], writing the non distortion condition 3 on h and in terms of

trigonometric polynomials, it becomes:

Moreover, together with divisibility of H and respectively by and

[4] it leads to:

Where ( )R is an odd polynomial in ( )Cos and 2l=k+ .

The 9/7 filters have been proposed in [4] as a particular case of trigonometric

polynomial that satisfy equation 6 with R 0, k=4 and =4. When R 0, k=4 and

=4 equation 6 becomes

The term cos8( / 2) can be split into two equal parts with degree 4. The polynomial in

sin ( / 2) can be considered as a third order equation and factorized into two

polynomials with degree 2 and 4 respectively in order to obtain:

Where r is the real solution of the third order equation

2 31 20 4 20 10 / 20 0 (9)x x x


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and b. Equation 8 leads to:

42 4cos( / 2)

( ) [ sin( / 2) sin( / 2) ] (10)H a ba

Developing equations 10 and 11 we can build filters coefficients as shown in table 1

where K1 = 2 3 1 2

3 11 1 12 8 2, , , .

2 8 32 2 8

ba r

bK k J and J

a a a r r

2. THE LIFTING SCHEME

The conventional structure of two-channel filter banks based on lifting steps is shown

in Figure 2. The incoming signal (z-domain) is split into two paths (polyphase

transform) containing the values at even or odd sample positions. In order to devise

the filter

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properties, alternating lifting and dual lifting steps are applied, in which samples from

one path are filtered by Li(z) and added to a sample of the other path. With respect to

the design method to be proposed, however, the illustration with a different flow chart

is more helpful. Figure 3 depicts a lifting cascade suitable for representing a 9-tap

low-pass and a 7-tap high-pass filter pair. Essentially, it shows the signal flow for

processing a signal with eight samples x0 to x7. A pair of samples at even positions is

weighted by (typically negative) coefficients α and added to the sample in between.

The next lifting step combines the results of the summations in pairs using the

coefficients β. The third and fourth lifting step act in the same manner using the

weights γ and δ. The property of integer-to-integer mapping, which will be essential

for lossless compression, is simply imposed


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by properly rounding the intermediate values to integer values [12] (not shown

in the flow diagram). The arithmetic calculations are (with m= 0, 1, 2, . . . )

d′m= x2m+1 + ⌊α(x2m+x2m+2) + 0.5⌋

a′m=x2m+⌊β (d′m−1+ d′m) + 0.5⌋

dm=d′m+ ⌊γ (a′m+ a′m+1) + 0.5⌋ (12)

am=a′m+ ⌊δ(dm-1+ dm) + 0.5⌋

The result after all lifting steps is an interleaved sequence of low-pass filter output

am(approximation signal) and the high-pass filter output dm (detail signal). Figure 3

also shows the reconstruction of the original signal xn by performing the lifting steps

in the reverse order and using the opposite signs

a′m= am− ⌊δ (dm-1+ dm) + 0.5⌋

d′m= dm − ⌊γ (a′m+ a′m+1) + 0.5⌋

x2m=a′m− ⌊β (d′m-1+ d′m) + 0.5⌋ (13)

x2m+1=d′m− ⌊α (x2m+x2m+2) + 0.5⌋ .

Furthermore, the flow diagram simply explains the exception handling at the signal

borders. When applied to signals with an odd length, the handling has to be changed

slightly.

3. FILTER DESIGN

3.1 Filters with maximum number of vanishing moments

The flow diagram, as depicted in Figure 3, allows the derivation of the

analysis filters of the corresponding two-channel filter bank simply by considering all

paths from the input samples to a particular approximation sample amor detail sample

dm, respectively. For the moment we will disregard the rounding operations. The

resulting symmetric 7-tap impulse response of the analysis high-pass filter is

h1[n] = { αβγ βγ [γ (2αβ + 1) + α (1+γβ)] (2βγ + 1)

[γ (2αβ + 1) + α (1+γβ)] βγ αβγ } (14)

and the analysis 9-tap low-pass filter reads as

h0[n] = { αβγδ βγδ

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{δ [γ (2αβ + 1) + α (1 + γβ)] + αβ (1 + γδ)}

[δ (2βγ + 1) + β(1 + γδ)]

{Α[δ (2βγ + 1) + β(1 + γδ)] + (1 + 2γδ)+

α [δ (2βγ + 1) + β(1 + γδ)]} (15)

[δ (2βγ + 1) + β(1 + γδ)]

{δ[γ(2αβ + 1) + α (1 + γβ)] + αβ(1 + γδ)}

βγδ α βγδ}

The synthesis filters are derived by following all paths from a particular

approximation (or detail) sample to the reconstructed signal values xn. In this

particular lifting structure, it turns out that they are directly related to the analysis

filters by

g0[n] = (−1)n+1

h1[n] n = 0, 1, 2, . . . (16)

g1[n] = (−1)n h0[n] . (17)

The frequency response (in z-domain) of a t-tap filter h[n] is

t−1

H(z) = ∑ h[n]zn. (18)

n=0

Since h0[n] and g0[n] should be real low-pass filters, their magnitude responses at

sampling frequency must be equal to zero: G0(z)|z = −1 = 0 and H0(z)|z=−1=0. This leads

to following conditions in the spatial domain.

0 = αβγ + βγ + [γ (2αβ + 1) + α (1 + γβ)] +(2βγ + 1) + [γ (2αβ + 1) + α (1 + γβ)]

+βγ + αβγ . (19)

and

0 = αβγδ − βγδ +{δ [γ(2αβ + 1) + α (1 + γβ)] + αβ (1 + γδ)}

−[δ (2βγ + 1) + β(1 + γδ)]

+{α [δ (2βγ + 1) + β(1 + γδ)] + (1 + 2γδ)

+ α [δ (2βγ + 1) + β(1 + γδ)]} (20)


134

−[δ (2βγ + 1) + β(1 + γδ)]

+{δ[γ(2αβ + 1) + α (1 + γβ)] + αβ(1 + γδ)}

−βγδ + αβγδ

The original aim of filter design in [13] was to create low-pass filters with frequency

responses that are as flat as possible at sampling frequency by imposing a maximum

number of so-called vanishing moments, i.e. multiple zeros at H0(z)|z=-1and G0(z)|z=-1.

Multiple vanishing moments at z = -1 can be incorporated by substituting z with n√n

p

(p = 0, 1, 2, . . . ) in equation (18). The second zero for G0(z) (and accordingly for

H1(z) at z = 1), for example, is included using z =n√n leading to the condition

0 = 0 αβγ + 1 βγ + 2 [γ (2αβ + 1) + α (1 + γβ)]

+3 (2βγ + 1) + 4 [γ (2αβ + 1) + α (1 + γβ)]

+5 βγ + 6αβγ . (21)

A different interpretation of this approach is based on the approximation of

signal segments by polynomials of increasing order [13]. The condition for the second

zero for H0(z) and G1(z) reads as

0 = 0 αβγδ − 1 βγδ

+2{δ [γ (2αβ + 1) + α (1 + γβ)] + αβ (1 + γδ)}

−3 [δ (2βγ + 1) + β(1 + γδ)] +

4{α [δ (2βγ + 1) + β(1 + γδ)] + (1 + 2γδ) +

α [δ (2βγ + 1) + β (1 + γδ)]}

−5 [δ (2βγ + 1) + β(1 + γδ)]

+6{δ [γ (2αβ + 1) + α (1 + γβ)] + αβ(1 + γδ)}

−7 βγδ + 8 αβγδ . (22)

The conditions (21) and (21) are, however, not independent from (19) and (20). Two

more constraints are necessary for the determination of the four weights α. . . δ.

Choosing z = n√n

2 imposes another vanishing moment. The corresponding conditions

are

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0 = 0 αβγ + 1 βγ + 4 [γ (2αβ + 1) + α (1 + γβ)]

+9 (2βγ + 1) + 16 [γ (2αβ + 1) + α (1 + γβ)]

+25 βγ + 36 αβγ . (23)

0 = 0 αβγδ − 1 βγδ

+4{δ [γ(2αβ + 1) + α (1 + γβ)] + αβ (1 + γδ)}

−9 [δ (2βγ + 1) + β(1 + γδ)] +

16{α [δ (2βγ + 1) + β(1 + γδ)] + (1 + 2γδ) +

α [δ (2βγ + 1) + β (1 + γδ)]}

−25 [δ (2βγ + 1) + β(1 + γδ)]

+36{δ [γ (2αβ + 1) + α (1 + γβ)] + αβ (1 + γδ)}

−49 βγδ + 64 αβγδ . (24)

Equations (21) – (24) form a system of non-linear equations resulting to the irrational

weights

α ≈−1.58613434206

β ≈−0.05298011857

γ ≈ 0.88291107553 (25)

δ ≈ 0.44350685204 .

Due to the inherent structure of the filter bank, each of the conditions impose double

zeros, i.e. each filter shows four vanishing moments in total. The result is exactly the

same as derived from the factorisation of a polyphase matrix presented in [14].

4. DESIGN OF 5/3 WAVELET FILTERS

Setting the factors γ and δ equal to zero shortens the work in signal

decomposition. The lengths of the impulse responses are reduced to 5 taps for the

low-pass filter and 3 taps for the high pass, respectively

0

1

[ ] { (1 2 ) }

[ ] { 1 }

h n

h n

(26)


136

The required conditions are in z-domain

H0(z)|z=-1 = 0

= αβ – β + (1+2 αβ ) - β + αβ

H1(z)|z=1 = 0=α+1+α (27)

Leading to the unique solution of α=-1/2 and β=1/4. This is accordance with the

original solution in [11].

5. SIMULATION RESULTS

The proposed low-complexity architecture has been tested on the standard

images ‘Lena’ 256 x 256 and ‘Barbara’ varying the number of decomposition levels.

The proposed architecture has been employed for the direct transform whereas a

floating point, standard 9/7and 5/3 implementation has been used for the inverse

transform. The values of Peak Signal to Noise Ratio (PSNR), Compression Ratio

(CR), Mean Square Error (MSE), Original image size, Compressed image size,

Encoding time, Decoding time and transforming time or decomposition time were

obtained from the experimental results, and summarized in tables below. The 5/3

filters have lower computational complexicity than the 9/7’s. However the

performance gain of the 9/7’s over the 5/3’s is quite large for JPEG-2000.

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0.3353

0.44641

2.5553

44.0904

11.9717

524288

43794

0.45374

1.0453

0.49863

31.1828

14.2427

524288

36811

Encoding

Time(bits)

Decoding

Time(bits)

MSE(dB)

PSNR(dB)

CR

Original Image

Size(bits)

Bird

0.24723

0.43202

1.8889

45.4028

11.6584

524288

44971

0.46749

1.4128

2.4905

27.8369

13.8906

524288

37744

Encoding

Time(sec)

Decoding

Time(sec)

MSE(dB)

PSNR(dB)

CR

Original Image

Size(bits)

Compressed image

Size(bits)

Rice

0.39729

0.84589

3.263

43.0286

9.724

524288

53917

0.49088

2.0138

2.2291

23.1718

12.1399

524288

43187

Encoding Time

(sec)

Decoding Time

(sec)

MSE(dB)

PSNR (dB)

CR

Original Image

Size(bits)

Compressed image

Size(bits)

Cameraman

5/3-TRANSFORM

9/7-TRANSFORM

PERFORMANCE CRITERION

INPUT IMAGE

PERFORMANCE COMPARISIONS OF 9/7 & 5/3 LIFTING

BASED INTEGER TO INTEGER WAVELET TRANSFORMS


138

SAMPLE GRAPHS:

Figure 1: Encoding time values of 9/7 &5/3 wavelet transforms for

Cameraman image (monochrome)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

9/7t

ransf

orm

5/3

tran

sform

TYPE OF WAVELETS

EN

CO

DIN

G T

IME

(S

EC

)

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Figure 2: Decoding time values of 9/7 &5/3 wavelet transforms for


0

0.5

1

1.5

2

2.5

9/7t

ransf

orm

5/3

tran

sform

TYPE OF WAVELETS

Deco

din

g

TIM

E (

SE

C )

0

0.5

1

1.5

2

2.5

3

3.5

9/7t

ransf

orm

5/3

tran

sform

TYPE OF WAVELETS

M S

E (

dB

)


140

Figure 4: PSNR values of 9/7 &5/3 wavelet transforms for


Figure 5: Compression Ratio (CR) values of 9/7 &5/3 wavelet

transforms for Cameraman image (monochrome)

0

10

20

30

40

50

9/7t

rans

form

5/3

tran

sform

TYPE OF WAVELETS

P S

N R

(d

B)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

9/7t

ransf

orm

5/3

tran

sform

TYPE OF WAVELETS

C R

(b

pp

)

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Dog

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR

Original Image Size(bits)

Compressed image Size(bits)

0.51696

1.242

1.1571

27.8874

11.5515

524288

45387

0.43424

0.49488

1.1571

47.5311

9.8587

524288

53180

Barbara

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR



0.50847

1.3234

3.263

28.7809

12.4638

524288

42065

0.3646

0.63403

2.4905

44.202

11.6571

524288

44976

Rose

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR



0.54101

1.1259

1.8889

28.8329

13.5489

524288

38696

0.36707

0.50336

0.49863

51.187

12.2666

524288

42741

Circuit

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR

Original Image Size (bits)


0.50626

1.2363

0.79579

26.7498

13.8767

524288

37782

0.35049

0.51646

0.79579

49.1568

11.7955

524288

4444


142

Pepper

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR



0.46708

1.082

0.16154

22.9909

11.4819

524288

45662

0.49609

0.54546

4.8219

41.3327

10.0519

524288

52158

Gold Hill

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR



0.45743

1.2707

12.1021

25.0914

10.4162

524288

50334

0.36332

0.53267

2.4154

44.3349

8.89

524288

58975

Lena

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR



0.52949

1.3649

3.2536

26.8885

12.0479

524288

43517

0.3885

0.55014

3.2536

43.0412

10.4222

524288

50305

Gray Granite

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

0.50596

1.3436

2.8973

0.37665

0.5806

2.8973

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PSNR(dB)

CR



29.0242

11.4792

524288

45673

43.5449

10.7858

524288

48609

Circle

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR


Compressed image

Size(bits)

0.48915

1.2873

0.79579

23.9129

45.1233

524288

11619

0.44463

0.57181

0.11275

57.6437

32.0117

524288

16378

Hill

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR


Compressed image

Size(bits)

0.54289

1.1755

2.8973

23.3348

10.3089

524288

50858

0.45352

0.63435

3.4723

42.7586

8.8802

524288

59040

GKCE Font

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR

0.5161

1.1777

1.1571

19.9739

87.5272

0.36887

0.53379

12.1021

37.3362

19.3094


144


Compressed image

Size(bits)

524288

5990

524288

27152

GKCE Logo

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR


Compressed image

Size(bits)

0.58224

1.1192

12.1021

24.6747

20.8034

524288

25202

0.43825

0.56097

12.1021

37.3362

12.2744

524288

42714

Bridge

Encoding Time(sec)

Decoding Time(sec)

MSE(dB)

PSNR(dB)

CR

Original Image Size(dB)

Compressed image

Size(dB)

0.40889

1.065

2.5553

28.7507

12.6594

524288

41415

0.32063

0.50548

2.8421

43.6284

10.6565

524288

49199

CONCLUSION

In this paper a low-complexity, efficient 9/7 wavelet filters implementation,

has been derived. A detailed analysis of the proposed solution architectural impact

has been shown with performance and comparisons with the direct implementation.

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We have presented a new biorthogonal 9/7 – tap wavelet with simple

coefficients, so computational complexity is reduced greatly compared to the well-

known CDF 9/7 – tapwavelet. The simulation shows that the new 9/7-tap wavelet is

very ideal alternative to CDF 9/7 – tap wavelet.

The other wavelet transform, 5/3 wavelet is very efficient in lossless

compression and is low complex. From all the above factors, we can conclude that

simple 9/7 and 5/3 wavelet transforms are very efficient than the conventional

wavelets/ traditional wavelets/ hand designed wavelets presently used to compress the

images.

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3. A. Said and W.A. PLearlman, “a new, fast, and efficient image codec based on

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for Video Technology, Vol.6, no.3, pp.243-250, Jun.1996.

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