1. Presenter : r98942058 余芝融 1 EE lab.530

2. Overview  Introduction to image compression  Wavelet transform concepts  Subband Coding  Haar Wavelet  Embedded Zerotree Coder  References 2 EE lab.530

3. Introduction to image compression  Why image compression?  Ex: 3504X2336 (full color) image : 3504X2336 x24/8 = 24,556,032 Byte = 23.418 Mbyte  Objective Reduce the redundancy of the image data in order to be able to store or transmit data in an efficient form. 3 EE lab.530

4. Introduction to image compression  For human eyes, the image will still seems to be the same even when the Compression ratio is equal 10  Human eyes are less sensitive to those high frequency signals  Our eyes will average fine details within the small area and record only the overall intensity of the area, which is regarded as a lowpass filter. EE lab.530 4

5. Quick Review  Fourier Transform  Does not give access to the signal’s spectral variations  To circumvent the lack of locality in time → STFT 5 EE lab.530

6. Quick Review  The time-frequency plane for STFT is uniform Constant resolution at all frequencies 6 EE lab.530

7. Continuous Wavelet Transform  FT &STFT use “wave” to analyze signal  WT use “wavelet of finite energy” to analyze signal  Signal to be analyzed is multiplied to a wavelet function, the transform is computed for each segment.  The width changes with each spectral component 7 EE lab.530

8. Continuous Wavelet Transform  Wavelet: finite interval function with zero mean(suited to analysis transient signals)  Utilize the combination of wavelets(basis func.) to analyze arbitrary function  Mother wavelet Ψ(t):by scaling and translating the mother wavelet, we can obtain the rest of the function for the transformation(child wavelet, Ψ a,b (t)) 8 EE lab.530

9. Continuous Wavelet Transform  Performing the inner product of the child wavelet and f(t), we can attain the wavelet coefficient  We can reconstruct f(t) with the wavelet coefficient by 9 EE lab.530

10. Continuous Wavelet Transform  Adaptive signal analysis -At higher frequency, the window is narrow, value of a must be small  The time-frequency plane for WT(Heisenberg) multi-resolution diff. freq. analyze with diff. resolution 10 EE lab.530

11. window a  Low freq. large  High freq. small EE lab.530 11

12. Gaussian Window for S-Transform EE lab.530 12 High Frequency Low Frequency Time Shifted SKC-2009

13. Discrete Wavelet Transform  Advantage over CWT: reduce the computational complexity(separate into H & L freq.)  Inner product of f(t)and discrete parameters a & b  If a 0 =2,b 0 =1, the set of the wavelet 13 EE lab.530

14. Discrete Wavelet Transform  The DWT coefficient  We can reconstruct f(t) with the wavelet coefficient by 14 EE lab.530

15. Subband Coding 15 EE lab.530

16. 16 EE lab.530 WT compression

17. 2-point Haar Wavelet (oldest & simplest) h[0] = 1/2, h[−1] = −1/2, h[n] = 0 otherwise g[n] = 1/2 for n = −1, 0 g[n] = 0 otherwise n g[n] g[n] -3 -2 -1 0 1 2 3 ½ n h[n] h[n] -3 -2 -1 0 1 2 3 ½ -½ then (Average of 2-point) (difference of 2-point) 17 EE lab.530

18. Haar Transform  2-steps 1.Separate Horizontally 2. Separate Vertically 18 EE lab.530

19. 2-Dimension(analysis) EE lab.530 19 Diagonal Horizontal Edge Vertical Edge Approximatio n

20. Haar Transform ABCDA+BC+DA-BC-D LH (0,0)(0,1)(0,2)(0,3)(0,0)(0,1)(0,2)(0,3) (1,0)(1,1)(1,2)(1,3)(1,0)(1,1)(1,2)(1,3) (2,0)(2,1)(2,2)(2,3)(2,0)(2,1)(2,2)(2,3) (3,0)(3,1)(3,2)(3,3)(3,0)(3,1)(3,2)(3,3) Step 1: 20 EE lab.530

21. Haar Transform Step 2: ACA+BC+D BDLLHL LH A-BC-D LHHH (0,0)(0,1)(0,2)(0,3)(0,0)(0,1)(0,2)(0,3) (1,0)(1,1)(1,2)(1,3)(1,0)(1,1)(1,2)(1,3) (2,0)(2,1)(2,2)(2,3)(2,0)(2,1)(2,2)(2,3) (3,0)(3,1)(3,2)(3,3)(3,0)(3,1)(3,2)(3,3) L H LH HH LL HL 21 EE lab.530

22. LL1HL1 LL2HL2 HL1 LH2HH2 LH1HH1LH1HH1 LL3HL3 HL2 HL1 LH3HH3 LH2HH2 LH1HH1 First levelSecond level Third level Most important part of the image 22 EE lab.530

23. Example: 68103619326-38619 7679-4-716-322-7 2-3412 41 -105-2-9-105-2-9 201530203550510 17163122335319 151817253342-3-8 2122191843371 Original image O 1 st horizontal separation 1 st vertical separation 2 nd level DWT result 23 EE lab.530

24. 24 Original Image LH HL HH LL

25. EE lab.530 25 LL2HL2 LH2HH2 LH HL HH LH HL HH HL2 LH2HH2 LL3HL3 HH3LH3

26. Embedded Zerotree Wavelet Coder EE lab.530 26

27. Structure of EZW  Root: a  Descendants: a1, a2, a3 EE lab.530 27 …

28. 3-level Quantizer(Dominant) EE lab.530 28 sp sn

29. EZW Scanning Order EE lab.530 29 LL3HL3 HL2 HL1 LH3HH3 LH2 HH2 LH1HH1 scan order of the transmission band

30. EZW Scanning Order EE lab.530 30 scan order of the transmission coefficient

31. Scanning Order EE lab.530 31 sp: significant positive sn: significant negative zr: zerotree root is: isolated zero

32. Example:  Get the maximum coefficient=26  Initial threshold : 1. 26>16 → sp 2. 6<16 & 13,10,6, 4 all less than 16 → zr 3. -7<16 & 4,-4, 2,-2 all less than 16 → zr 4. 7<16 & 4,-3, 2, 0 all less than 16 → zr EE lab.530 32

33.  Each symbol needs 2-bit: 8 bits  The significant coefficient is 26, thus put it into the refinement label : Ls= {26}  To reconstruct the coefficient: 1.5T 0 =24  Difference:26-24=2  Threshold for the 2-level quantizer:  The new reconstructed value: 24+4=28 EE lab.530 33

34. 2-level Quantizer(For Refinement) EE lab.530 34

35.  New Threshold: T 1 =8  iz zr zr sp sp iz iz → 14-bit EE lab.530 35

36. Important feature of EZW  It’s possible to stop the compression algorithm at any time and obtain an approximate of the original image  The compression is a series of decision, the same algorithm can be run at the decoder to reconstruct the coefficients, but according to the incoming but stream. EE lab.530 36

37. References [1] C.Gargour,M.Gabrea,V.Ramachandran,J.M.Lina, ”A short introduction to wavelets and their applications,” Circuits and Systems Magazine, IEEE, Vol. 9, No. 2. (05 June 2009), pp. 57-68. [2] R. C. Gonzales and R. E. Woods, Digital Image Processing. Reading, MA, Addison-Wesley, 1992. [3] NancyA. Breaux and Chee-Hung Henry Chu,” Wavelet methods for compression, rendering, and descreening in digital halftoning,” SPIE proceedings series, vol. 3078, pp. 656-667, 1997. [4] M. Barlaud et al., "Image Coding Using Wavelet Transform" IEEE Trans. on Image Processing 1, No. 2, 205-220 (April, 1992). [5] J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Acous., Speech, Signal Processing, vol. 41, no. 12, pp. 3445-3462, Dec. 1993. EE lab.530 37