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 Future work Future work

3. Introduction SPIHT coding performance is associated with: SPIHT coding performance is associated with: properties of wavelet filter energy compaction implementation Our goal is to make filters achieve the best SPIHT coding performance. Our goal is to make filters achieve the best SPIHT coding performance.

4. Properties of wavelet filter Orthogonality Orthogonality Complexity Complexity integer-to-integer (reversible) real-to-real (irreversible)

5. 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. Dot products between the two filter masks and the signal.

6. Wavelet filters for evaluation 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)

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

8. Complexity Real-to-real: Real-to-real:Haar:9/7: i Low-Pass Filter High-Pass Filter 00.5-0.5 10.50.5 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

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

10. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Future work Future work

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

12. 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

13. 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

14. Energy of LL subband (%,5 level decomposition ) 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

15. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Future work Future work

16. Scaling Optimal scaling factor Optimal scaling factor ->fixed scaling ->variable scaling Scaling strategy Scaling strategy ->Low frequency subband scaling ->High frequency subband scaling

17. 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 filters. With proper scaling, the compression performance is much better for all wavelet filters.

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

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

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

21. Energy distribution with best fixed scaling and without scaling(“Lena”) 5/313/7-C 9/7-F

22. Energy distribution with best fixed scaling and without scaling(“Lena”) 10/189/7 Haar

23. Scaling strategy :reduce highband coefficients Reduce the high frequency component can make energy compact in the lowest frequency component. Reduce the high frequency component can make energy compact in the lowest frequency component. Optimal scaling for reducing highband coefficients is still to be investigated. Optimal scaling for reducing highband coefficients is still to be investigated.

24. Scaling strategy :fixed scaling

25. Scaling strategy :reduce high frequency component

26. Performance (reducing high frequency, “Lena”) 5/313/7-C

27. Performance (reducing high frequency, “Lena”) 9/7-F

28. Performance (reducing high frequency, “Lena”) 9/710/18

29. Energy distribution (reducing high frequency, “Lena”) 5/313/7-C 9/7-F

30. Energy distribution (reducing high frequency, “Lena”) 9/710/18 Haar

31. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Future work Future work

32. 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.

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

34. Extension affects coding performance Symmetric extension periodic extension

35. Extension affects coding performance Point-Symmetric extension

36. Performance (with proper and improper extension ) 5/39/7-F9/7-M5/11A5/11-C13/7C13/7-T10/189/7HaarD4D6 1/12826.48 /26.90 27.12 /27.66 26.40 /27.08 26.57 /26.92 26.51 /26.82 26.51 /26.96 26.51 /27.21 27.48 /27.92 27.14 /27.59 25.42 /25.42 26.4226.83 1/6429.20 /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 30.20 /30.68 30.06 /30.53 27.53 /27.53 28.9729.38 1/3232.12 /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 33.75 /33.75 33.21 /33.58 30.21 /30.21 31.8532.35 1/1635.50 /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 36.46 /36.86 36.52 /36.74 33.50 /33.50 35.2435.75 1/838.76 /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 39.75 /39.96 39.77 /39.92 37.47 /37.47 38.9239.26 1/442.38 /42.46 41.70 /41.78 42.43 /42.54 42.42 /42.51 42.28 /42.38 42.59 /42.72 42.58 /42.69 44.29 /44.29 44.17 /44.31 42.55 /42.55 43.6043.78 1/247.98 /48.02 45.44 /45.41 48.37 /48.93 48.22 /48.24 48.09 /48.12 48.27 /48.31 48.33 /48.34 54.92 /54.92 54.93 /55.11 53.25 /53.25 54.3054.28

37. Outline Introduction Introduction Compression performance Compression performance Scaling Scaling Finite length signal analysis Finite length signal analysis Future work Future work

38. Future work Investigate best scaling strategy Investigate best scaling strategy Best implementation scheme Best implementation scheme Achieve totally the best performance of SPIHT Achieve totally the best performance of SPIHT