EFFICIENT IMAGE COMPRESSION ALGORITHMS USING EVOLVED WAVELETS

International Journal of ISSN 0974-2107 Systems and Technologies IJST Vol.4, No.2, pp 127-146 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:Krishna.rakesh4@gmail.com 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 (), 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. 127

G.Chenchu Krishnaiah 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 filters with length kand l respectively, and are the low pass and high pass analysis 128

Efficient Image. 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 ( and G(z) = (-z)) we can rewrite the perfect reconstruction conditions as: (z) = H(-z) As shown in [4], writing the non distortion condition 3 on h and trigonometric polynomials, it becomes: in terms of 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 equation 6 becomes =4. When R 0, k=4 and 8 The term cos( / 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 3 1 20 4 20x 10 / 20x x 0 (9) 129

G.Chenchu Krishnaiah and b. Equation 8 leads to: cos( / 2) H a b a 4 2 4 ( ) [ sin( / 2) sin( / 2) ] (10) Developing equations 10 and 11 we can build filters coefficients as shown in table 1 b 3 1 a r 1 1 1 where K 1 = 2 8 b, K 2 2, k3, J1 and J2. 2a 8a 32a 2r 8r 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 130

Efficient Image. properties, alternating lifting and dual lifting steps are applied, in which samples from one path are filtered by L i (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 x 0 to x 7. 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 131

G.Chenchu Krishnaiah 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 = x 2m+1 + α(x 2m +x 2m+2 ) + 0.5 a m =x 2m + β (d m 1 + d m ) + 0.5 d m =d m + γ (a m + a m+1 ) + 0.5 (12) a m =a m + δ(d m-1 + d m ) + 0.5 The result after all lifting steps is an interleaved sequence of low-pass filter output a m (approximation signal) and the high-pass filter output d m (detail signal). Figure 3 also shows the reconstruction of the original signal x n by performing the lifting steps in the reverse order and using the opposite signs a m = a m δ (d m-1 + d m ) + 0.5 d m = d m γ (a m + a m+1 ) + 0.5 x = 2m a m β (d m-1 + d m ) + 0.5 (13) x = 2m+1 d m α (x 2m +x 2m+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 a m or detail sample d m, respectively. For the moment we will disregard the rounding operations. The resulting symmetric 7-tap impulse response of the analysis high-pass filter is h 1 [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 h 0 [n] = { αβγδ βγδ 132

Efficient Image. {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 x n. In this particular lifting structure, it turns out that they are directly related to the analysis filters by g 0 [n] = ( 1) n +1 h 1 [n] n = 0, 1, 2,... (16) g 1 [n] = ( 1) n h 0 [n]. (17) The frequency response (in z-domain) of a t-tap filter h[n] is t 1 H(z) = h[n]z n. (18) n=0 Since h 0 [n] and g 0 [n] should be real low-pass filters, their magnitude responses at sampling frequency must be equal to zero: G 0 (z) z = 1 = 0 and H 0 (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) 133

G.Chenchu Krishnaiah [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 H 0 (z) z=-1 and G 0 (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 G 0 (z) (and accordingly for H 1 (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 H 0 (z) and G 1 (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 134

Efficient Image. 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 h [ n] { (1 2 ) } 0 h[ n] { 1 } 1 (26) 135

G.Chenchu Krishnaiah The required conditions are in z-domain H 0 (z) z=-1 = 0 = αβ β + (1+2 αβ ) - β + αβ H 1 (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 (), 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. 136

Efficient Image. PERFORMANCE COMPARISIONS OF 9/7 & 5/3 LIFTING BASED INTEGER TO INTEGER WAVELET TRANSFORMS INPUT IMAGE PERFORMANCE 5/3-9/7- Cameraman Rice Bird Encoding Time (sec) Decoding Time (sec) PSNR (db) Encoding Time(sec) Decoding Time(sec) Encoding Time(bits) Decoding Time(bits) 0.49088 2.0138 2.2291 23.1718 12.1399 43187 0.46749 1.4128 2.4905 27.8369 13.8906 37744 0.45374 1.0453 0.49863 31.1828 14.2427 0.39729 0.84589 3.263 43.0286 9.724 53917 0.24723 0.43202 1.8889 45.4028 11.6584 44971 0.3353 0.44641 2.5553 44.0904 11.9717 43794 137

ENCODING TIME (SEC) G.Chenchu Krishnaiah SAMPLE GRAPHS: 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 9/7transform 5/3 transform TYPE OF WAVELETS Figure 1: Encoding time values of 9/7 &5/3 wavelet transforms for Cameraman image (monochrome) 138

M S E (db) Decoding TIME (SEC ) Efficient Image. 2.5 2 1.5 1 0.5 0 9/7transform 5/3 transform TYPE OF WAVELETS Figure 2: Decoding time values of 9/7 &5/3 wavelet transforms for Cameraman image (monochrome) 3.5 3 2.5 2 1.5 1 0.5 0 9/7transform 5/3 transform 139 TYPE OF WAVELETS

C R (bpp) P S N R (db) G.Chenchu Krishnaiah Figure 4: PSNR values of 9/7 &5/3 wavelet transforms for 50 40 30 20 10 0 9/7transform 5/3 transform TYPE OF WAVELETS 13 12 11 10 9 8 7 6 5 4 3 2 1 0 9/7transform 5/3 transform TYPE OF WAVELETS Cameraman image (monochrome) Figure 5: Compression Ratio () values of 9/7 &5/3 wavelet transforms for Cameraman image (monochrome) 140

Efficient Image. 0.51696 0.43424 1.242 0.49488 1.1571 1.1571 Dog 27.8874 11.5515 47.5311 9.8587 Compressed image Size(bits) 45387 53180 0.50847 0.3646 1.3234 0.63403 3.263 2.4905 Barbara 28.7809 44.202 12.4638 11.6571 Compressed image Size(bits) 42065 44976 0.54101 0.36707 1.1259 0.50336 1.8889 0.49863 Rose 28.8329 13.5489 51.187 12.2666 Compressed image Size(bits) 38696 42741 0.50626 0.35049 1.2363 0.51646 0.79579 0.79579 26.7498 49.1568 13.8767 11.7955 Circuit Original Image Size (bits) Compressed image Size(bits) 37782 4444 141

G.Chenchu Krishnaiah 0.46708 0.49609 1.082 0.54546 0.16154 4.8219 Pepper 22.9909 41.3327 11.4819 10.0519 Compressed image Size(bits) 45662 52158 0.45743 0.36332 1.2707 0.53267 12.1021 2.4154 Gold Hill 25.0914 10.4162 44.3349 8.89 Compressed image Size(bits) 50334 58975 0.52949 0.3885 1.3649 0.55014 3.2536 3.2536 Lena 26.8885 43.0412 12.0479 10.4222 Compressed image Size(bits) 43517 50305 0.50596 0.37665 Gray Granite 1.3436 0.5806 2.8973 2.8973 142

Efficient Image. 29.0242 43.5449 11.4792 10.7858 Compressed image Size(bits) 45673 48609 0.48915 0.44463 1.2873 0.57181 0.79579 0.11275 Circle 23.9129 57.6437 45.1233 32.0117 Compressed image 11619 16378 Size(bits) 0.54289 0.45352 1.1755 0.63435 2.8973 3.4723 Hill 23.3348 42.7586 10.3089 8.8802 Compressed image 50858 59040 Size(bits) 0.5161 0.36887 1.1777 0.53379 1.1571 12.1021 GKCE Font 19.9739 37.3362 87.5272 19.3094 143

G.Chenchu Krishnaiah Compressed image 5990 27152 Size(bits) 0.58224 0.43825 1.1192 0.56097 12.1021 12.1021 GKCE Logo 24.6747 37.3362 20.8034 12.2744 Compressed image 25202 42714 Size(bits) 0.40889 0.32063 1.065 0.50548 2.5553 2.8421 Bridge 28.7507 43.6284 12.6594 10.6565 Original Image Size(dB) Compressed image 41415 49199 Size(dB) 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. 144

Efficient Image. We have presented a new biorthogonal 9/7 tap wavelet with simple coefficients, so computational complexity is reduced greatly compared to the wellknown 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. REFERENCES 1. M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, SignalProcessing, Prentice Hall, Englewood, Cliff. NJ, 1995. 2. D. Taubman, High performance scalable image compression with EBCOT, IEEE Trans. On Image Processing, vol.9, No.7, pp.1158-1170, Jul.2000. 3. A. Said and W.A. PLearlman, a new, fast, and efficient image codec based on Set Partitioning In Hierarchical Trees, IEEE Trans. On Circuits and Systems for Video Technology, Vol.6, no.3, pp.243-250, Jun.1996. 4. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image coading using the wavelet transform, IEEE Tran. On Image Processing, Vo.1, No.2,,pp.205-220, Apr.1992. 5. J.D. Villasenor, B. Belzer, and J. Liao, wavelet filter evaluation for image compression, IEEE Tran. On Image Processing, Vol.4, No.8, pp.1053-1060, Aug.1995. 6. M. Boliek, JPEG 2000 Final Committee Draft, http://www.jpeg.org/public/fcd15444-1.pdf, 2000. 7. I. Daubechies and W. Sweldens, Factoring Wavelet Transforms into Lifting Steps,: Tech.Rep.Bell Laboratories, Lucent Technologies, 1996. 145

G.Chenchu Krishnaiah 8. C.T. Haung, P.C. Tseng, and L.G. Chen, Flipping Structure: an efficient VLSI architecture for liftingbased discrete wavelet transform, IEEE Tran. On Signal Processing, vol.52, no.4, pp.1080-1089, Apr.2004. 9. K.A. Kotteri, A.E. Bell, ad J.e. Carletta, Design of multiplierless, highperformance, wavelet filter banks with image compression applications, IEEE Tran. On Circuits and Systems-I, vol.51, no.3, pp.483-494, Mar.2004. 10. G. Strang and T.Q.Nguyen, Wavelets and Filter Banks, Wellesley, Wellesley- Combridge, MA, 1996. 11. Sweldens, W; The lifting scheme: A new philosophy in biorthogonal wavelet construction on proc. Of SPIEE, Vo..2569, Sar Diego, USA, July 1995, 68-79. 12. Calgerbank, A.R.; Daubechies, I.; Sweledens, W.; Yeo, B.L,: Wavelet Transform that maps integers to integers. Applied Computational and harmonic analysis, Vol.5, No.3, 1998, 332-369. 13. Strutz, T.: Muller, E.: Wavelet filter design for image compression. IEEE-SP Int. Symposium on Time-Frequency and Time-scale analysis, Paris, 18-21 June 1996, 273-276. 14. Cohen, A.; Daubechies, I.; Feauveau, J.-C.: Biorthogonal Bases of compactly supported Wavelets. Comm. On Pure and Applied Mathematics, Vol.45, 1992, 485-560. 146

Efficient Image. 147