1. Evaluation of SPIHT Coding Parameters Shih-Hsuan Yang and Wu-Jie Liao Department of Computer Science and Information Engineering National Taipei University of Technology Taipei, Taiwan, ROC December 15, 2003

2. Outline Wavelet Transform SPIHT Quantization Experimental Result Conclusion

3. Wavelet Transform Time domain pixelsTransform domain coefficients

4. Wavelet Transform (1D) h0 h1 2 2 g0 g1 2 2 LPF HPF X(n) y(n) If X(n) = y(n), this called perfect reconstruction

5. Effects of Wavelet Filters Properties of wavelets:  Desirable time-frequency localization.  Compact support.  Orthogonality.  Smoothness, regularity, or vanishing moments.  Symmetry (linear-phase constraint).

6. Wavelet Filters for Evaluation Real to real transform : (irreversible)  5/3, 9/7-F, 9/7-M, 5/11-A, 5/11-C, 13/7-T, 13/7-C (biorthogonal) Integer to integer transform : (reversible)  Haar wavelet (D2, orthogonal)  Daubechies 4 and 6 tap (D4, D6, orthogonal)  9/7, 10/18 (biorthogonal)

7. Real to Real Transform(RWT) Conventional transform (convolves the input signal with the wavelet filter kernel.) Computational complexity is proportion to the length of filter kernel.

8. Real to Real Transform (RWT) indexD2(h 0 )D4(h 0 )D6(h 0 ) 00.70710.48300.332705 10.70710.83650.806915 20.22410.459877 3-0.1294-0.135011 4-0.0854412 50.0352263 index9/7(h 0 )9/7(g 0 )10/18(h 0 )10/18(g 0 ) 00.8526990.7884860.758907730.62335964 10.3774020.418092.076790490.163368 2-0.110624-0.040689-0.157526-0.0856619 3-0.023849-0.0645390.0000824478-0.013765 40.0378280.02885250.03083373 5-0.002528037 6-0.0094524629 7-0.00000272719 80.0009544

9. Integer to Integer Transform (IWT) Fixed-point approximation to conventional transform (RWT). Suitable for lossy and lossless coding. Computational complexity is proportion to lifting steps required.

10. Integer to Integer Transform (IWT) Lifting step 5/3: 9/7-F:

11. Computational complexity RWT D2D4D69/710/18 1.001.562.062.033.59 IWT 5/39/7-F9/7-M5/11A5/11-C13/7C13/7-T 1.001.941.011.52 1.13 Relative computation time required for transformation The simulation is conducted on Pentium-4 2.4GHz PC

12. Effects of Extension Types Periodic extension Odd-symmetric (for odd-tap filter) even-symmetric (for even-tap filter)anti-symmetric (for even-tap filter)

13. SPIHT Quantization Wavelet coefficients c[i]Bit plane of c[i] Significant : | c[i] | >= k=0,1,2,…,n

14. SPIHT Quantization(cont.) Example of Parent-Offspring dependencies (i,j) root O(i,j) offspring of root D(i,j) descendant of root L(i,j) = D(i,j) - O(i,j) Type A Type B

15. SPIHT Algorithm

16. Experiments Images  Lena and baboon. Wavelet filters  IWT and RWT. Extension types  Periodic and symmetric.

17. Test images & Visual Quality Measurement (MSE, PSNR) lena baboon

18. Compression Results (“lena”) bpp RWT D2D4D69/710/18 0.12527.5328.9729.3830.5330.68 0.2530.2131.8532.3533.5833.75 0.533.5035.2435.7536.7436.86 1.037.4738.9239.2639.9239.96 bpp IWT 5/39/7F9/7M5/11A5/11C13/7C13/7T 0.12529.7130.2529.7829.8429.7929.9429.90 0.2532.6033.2432.8732.8132.8833.0433.07 0.535.7536.1735.9335.9235.8936.1436.13 1.038.8738.8438.8038.8938.8039.0339.00

19. Compression Results (“baboon”) bpp RWT D2D4D69/710/18 0.12520.9721.2821.3721.4921.60 0.2522.1422.5422.6422.8822.97 0.524.0824.6024.7925.1125.13 1.027.3127.9728.2128.6228.61 bpp IWT 5/39/7F9/7M5/11A5/11C13/7C13/7T 0.12520.9621.4220.8520.9220.8721.0520.98 0.2522.2522.8022.1822.2322.1722.4022.35 0.524.2225.0724.2824.2524.2324.4924.47 1.027.7128.3727.8027.7927.7628.0227.98

20. Energy Compaction (“lena”) RWTIWT D2D4D69/710/185/39/7-F9/7-M5/11A5/11C13/7C13/7T 97 78968278778182 Energy percentage of DC subband (%,5 level decomposition) 9/7-F 5/3

21. Energy Compaction (“baboon”) RWTIWT D2D4D69/710/185/39/7-F9/7-M5/11A5/11C13/7C13/7T 98 999888989188 91 Energy percentage of DC subband (%,5 level decomposition) 9/7-F5/3

22. Compression Results for Period/Symmetric Extension (lena) bpp RWT D2D4D69/710/18 0.12527.5328.9729.3830.06/30.5330.20/30.68 0.2530.2131.8532.3533.21/33.5833.58/33.75 0.533.5035.2435.7536.52/36.7436.46/36.86 1.037.4738.9239.2639.77/39.9239.75/39.96 bpp IWT 5/39/7F9/7M5/11A5/11C13/7C13/7T 0.125 29.20 /29.71 29.86 /30.25 29.40 /29.78 29.33 /29.84 29.32 /29.79 29.53 /29.94 29.53 /29.90 0.25 32.12 /32.60 32.88 /33.24 32.53 /32.87 32.36 /32.81 32.41 /32.88 32.70 /33.04 32.69 /33.07 0.5 35.50 /35.75 35.96 /36.17 35.73 /35.93 35.72 /35.92 35.72 /35.89 35.90 /36.14 35.89 /36.13 1.0 38.76 /38.87 38.74 /38.84 38.71 /38.80 38.78 /38.89 38.70 /38.80 38.92 /39.03 38.88 /39.00

23. Conclusions 9/7 and 10/18 biothogonal wavelets with symmetric extension provide the best compression performance, but highest complexity. 5/3 filter may be reasonably choice for low complexity codecs.

24. Conclusions This work investigate several parameters of wavelet transform for SPIHT. Provide guidelines for the best tradeoff of a SPIHT-based image compression system.