1. A Comprehensive Study of Wavelet Transforms for SPIHT 台北科技大學資工所指導教授：楊士萱學生：廖武傑 2003/03/27

2. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Conclusion Conclusion

3. Introduction Transforms Transforms integer-to-integer (reversible) real-to-real(irreversibel) SPIHT SPIHT wavelet domain coding zero-tree coding

4. Transforms Integer-to-integer transform : Integer-to-integer transform : Real-to-real transform : Real-to-real transform :  Dot products between the two filter masks and the signal.

5. Wavelet filters for evaluation of coding Integer-to-integer: Integer-to-integer: 5/3, 9/7-M, 5/11-A, 5/11-C,13/7-T, 13/7-C, 9/7-F (biorthogonal) Real-to-real: Real-to-real: 9/7, 10/18 (biothogonal) Haar, Daubechies 4 taps, 6 taps(orthogonal)

6. Complexity Integer-to-integer: Integer-to-integer:5/3:9/7-F:

7. Complexity Real-to-real: Real-to-real:Haar:9/7: i Low-Pass Filter High-Pass Filter 01 111 i Low-Pass Filter High-Pass Filter 00.60294901823635791.115087052456994 ±10.2668641184428723 -0.5912717631142470 -0.5912717631142470 ±2-0.078223266652898785-0.05754352622849957 ±3-0.016864118442874950.09127176311424948 ±40.02674875741080976

8. SPIHT(set partitioning in hierarchical trees) Zero-tree coding: Zero-tree coding: ->inter-scaling correlation ->energy distribution

9. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis conclusion conclusion

10. Compression performance Test images: Test images: lena baboon pepper F16

11. Compression performance 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12826.9027.6627.0826.9226.8226.9627.2127.9227.5925.4226.4226.83 1/6429.7130.2529.7829.8429.7929.9429.9030.6830.5327.5328.9729.38 1/3232.6033.2432.8732.8132.8833.0433.0733.7533.5830.2131.8532.35 1/1635.7536.1735.9335.9235.8936.1436.1336.8636.7433.5035.2435.75 1/838.8738.8438.8038.8938.8039.0339.0039.9639.9237.4738.9239.26 1/442.4641.7842.5442.5142.3842.7242.6944.2944.3142.5543.6043.78 1/248.0245.4148.9348.2448.1248.3148.3454.9255.1153.2554.3054.28 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12820.2420.5520.1220.2120.1520.3220.2620.6420.5620.2220.4020.47 1/6420.9621.4220.8520.9220.8721.0520.9821.6021.4920.9721.2821.37 1/3222.2522.8022.1822.2322.1722.4022.3522.9722.8822.1422.5422.64 1/1624.2225.0724.2824.2524.2324.4924.4725.1325.1124.60 24.79 1/827.7128.3727.8027.7927.7628.0227.9828.6128.6227.97 28.21 1/433.1333.6333.1533.1433.0933.3533.3234.0234.1032.7433.4333.66 1/242.5841.9042.7342.6142.5142.9042.8644.7244.8843.5144.2644.44

12. Compression performance 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12825.6526.5025.5625.6725.6125.7025.6926.4726.3924.5425.5725.75 1/6428.3128.9528.4928.4128.3928.5628.5729.2629.2826.7827.9728.18 1/3231.4532.1431.8231.6631.7231.8531.8832.5632.5129.5730.9531.40 1/1635.2735.8535.7535.5535.6135.8435.8736.4536.4233.4534.7535.27 1/839.8439.6639.8239.8639.7439.99 40.92 38.4239.6139.59 1/444.5443.0144.7244.5944.4744.8344.8446.5246.5844.6945.6445.82 1/248.5145.4648.5848.6248.4648.4748.6057.1157.2954.6156.2155.86 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12825.4225.8625.2725.4325.3925.4525.4126.1426.0723.2624.4724.74 1/6428.3128.7728.3028.3628.3028.4428.4329.0129.0225.7827.2227.56 1/3230.6831.4231.0730.8330.8631.1931.1531.6731.7428.7630.2030.46 1/1633.0933.5133.1633.0732.9833.3333.3033.7833.8031.8132.9533.06 1/835.3835.8435.4835.3435.1835.6935.6636.2136.1835.0135.7135.77 1/438.9339.4439.4138.8838.6939.6439.5840.6940.7539.6340.2240.29 1/244.2644.8347.1244.2944.1447.1447.1751.9151.9750.3551.3251.30

13. Energy of LL subband(%) 5/39/7-F9/7-M5/11-A5/11-C13/7-C13/7-T10/189/7HaarD4D6 Lena78.2796.1282.1478.0977.7481.8882.1097.4297.9897.7997.3797.28 Babo o n 88.6998.4091.2188.5888.3991.6991.6498.6899.1898.8598.9698.90 F1688.6598.0691.1088.6088.4590.9190.1098.3398.9898.7798.7698.74 peppe r 68.5392.1473.4668.7168.6272.5473.1094.2795.7194.7394.7294.08

14. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Conclusion Conclusion

15. Scaling Optimal scaling factor Optimal scaling factor ->fixed scaling ->variable scaling Modify SPIHT coding algorithm Modify SPIHT coding algorithm ->variable sorting threshold

16. Fixed scaling Optimal scaling factor for all wavelet decomposition is 1.41421,except 9/7-F(1.1496) Optimal scaling factor for all wavelet decomposition is 1.41421,except 9/7-F(1.1496) With proper scaling, the compression performance is much better for all wavelet filter. With proper scaling, the compression performance is much better for all wavelet filter.

17. Coding with or without scaling (“Lena”) 5/39/7-F

18. Coding with or without scaling (“Lena”) 13/7-T13/7-C

19. Coding with or without scaling (“Lena”) 5/11-A5/11-C

20. Finite length signal analysis Optimal signal extension Optimal signal extension ->minimal the distortion of the reconstructive signal Restriction of signal extension Restriction of signal extension ->extension must match the filter-bank.

21. Extensions for various filters –Odd symmetric extension for odd taps filter. –Even symmetric extension and anti-symmetric for even taps filter. –periodic extension for asymmetric filter. (circular convolution) Only guarantee the forward-backward transform works. Only guarantee the forward-backward transform works.

22. Extension affects performance Symmetric extension periodic extension

23. Performance (with proper and improper extension ) 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12826.9027.6627.0826.9226.8226.9627.2127.9227.5925.4226.4226.83 1/6429.7130.2529.7829.8429.7929.9429.9030.6830.5327.5328.9729.38 1/3232.6033.2432.8732.8132.8833.0433.0733.7533.5830.2131.8532.35 1/1635.7536.1735.9335.9235.8936.1436.1336.8636.7433.5035.2435.75 1/838.8738.8438.8038.8938.8039.0339.0039.9639.9237.4738.9239.26 1/442.4641.7842.5442.5142.3842.7242.6944.2944.3142.5543.6043.78 1/248.0245.4148.9348.2448.1248.3148.3454.9255.1153.2554.3054.28 5/39/7-F9/7-M5/11A5/11- C 13/7C13/7- T 10/189/7HaarD4D6 1/12826.4827.1226.4026.5726.51 27.59 1/6429.2029.8629.4029.3329.3229.53 30.53 1/3232.1232.8832.5332.3632.4132.7032.6933.58 1/1635.5035.9635.7335.72 35.9035.8936.74 1/838.7638.7438.7138.7838.7038.9238.8839.92 1/442.3841.7042.4342.4242.2842.5942.5844.31 1/247.9845.4448.3748.2248.0948.2748.3355.11

24. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Conclusion Conclusion

25. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Conclusion Conclusion

26. Conclusion Coding performance associated with filter: Coding performance associated with filter: –Properties of filter –Energy distribution of wavelet coefficients –Some issues of implementation The differences between fixed and floating point filtering computation. The differences between fixed and floating point filtering computation.