1. Wavelet-based Coding And its application in JPEG2000 Monia Ghobadi CSC561 project monia@cs.uvic.ca

2. 2 Agenda Why Wavelet Transform Continuous & Discrete Wavelet Transform Haar Wavelet Transform Application of wavelet transform is JPEG2000: EZW coding

3. 3 Introduction Multimedia Transformations are applied to signals to obtain further information. Most of the signals in practice, are time- domain signals in their raw format. Not always the best representation of the signal. The most distinguished information is hidden in the frequency content.

4. 4 Fourier Transform The frequency spectrum of the signal shows what frequencies exist in the signal FT Frequency domain Temporal domain  No frequency information is available in time-domain No time information is available in frequency-domain signal

5. 5 Stationary Signals x(t)=cos(2 π *10t)+cos(2 π *25t)+cos(2 π *50t)+cos(2 π *100t) FT Four spectral components corresponding to the frequencies 10, 25, 50, 100 Hz

6. 6 Non-stationary Signals FT Four different frequency components at four different time intervals

7. 7 Comparison of two examples Two spectrums are similar! Four spectral components at exactly the same frequencies The corresponding time domain signals are not even close

8. 8 What is wavelet transform? Provides time-frequency representation Wavelet transform decomposes a signal into a set of basis functions (wavelets) Wavelets are obtained from a single prototype wavelet Ψ(t) called mother wavelet by dilations and shifting: where a is the scaling parameter and b is the shifting parameter

9. 9 Wavelet Transform Continuous Wavelet Transform (CWT) Discrete Wavelet Transform (DWT)

10. 10 CWT Continuous wavelet transform (CWT) of 1D signal is defined as The  a,b is computed from the mother wavelet by translation and dilation

11. 11  Separates the high and low-frequency portions of a signal through the use of filters  One level of transform:  Signal is passed through G & H filters.  Down sample by a factor of two  Multiple levels (scales) are made by repeating the filtering and decimation process on lowpass outputs 1D Discrete Wavelet Transform

12. 12 Haar Wavelet Transform Find the average of each pair of samples Find the difference between the average and sample Fill the first half with averages Fill the second half with differences Repeat the process on the first half Step 1: [3 5 4 8 13 7 5 3] [4 6 10 4 -1 -2 3 1] Averaging Differencing

13. 13 Haar Wavelet Transform Step 2 [4 6 10 4 -1 -2 3 1] [5 7 -1 3 -1 -2 3 1] ex. (4 + 6)/2 = 5 4 - 5 = -1 Averaging Differencing

14. 14 Haar Wavelet Transform Step 3 [5 7 -1 3 -1 -2 3 1] [6 -1 -1 3 -1 -2 3 1] ex. (5 + 7)/2 = 6 5 - 6 = -1 Averaging Differencing row average

15. 15 Image representation [33 32 33 32 31 -29 27 -25] [32.5 32.5 0.5 0.5 31 -29 27 -25] [32.5 0 0.5 0.5 31 -29 27 -25]

16. 16 Applying on rows row average detail coefficients

17. 17 Applying on columns Choose a threshold δ δ = 5

18. 18 Decompressing apply the inverse of the averaging the differencing operations

19. 19 Result Decompressed ImageOriginal Image

20. 20 2-D DWT Step 1: replace each row with its 1-D DWT. Step 2: Replace each column with its 1-D DWT Step 3: Repeat steps 1 & 2 on the lowest subband for the next scale. Step 4: Repeat step 3 until as many scales as desired original LH LHHH HLLL LHHH HL One scaletwo scales

21. 21 Discrete Wavelet Transform LL 2 HL 2 LH 2 HH 2 HL 1 LH 1 HH 1

22. 22 JPEG2000 (J2K) is an emerging standard for image compression Achieves low bit rate compression Not only better efficiency, but also more functionality Lossless and lossy compression JPEG2000

23. 23 JPEG2000 v.s. JPEG low bit-rate performance

24. 24 Embedded Zero Tree Wavelet Coding The era of modern lossy wavelet coding began in 1993 when Jerry Shapiro introduced EZW coding Improved performance at low bit rates relative to the existing JPEG standard. Much of the energy in the wavelet transform is concentrated into the LL k band.

25. 25 Significance map An indication of whether a particular coefficient is zero or nonzero relative to a given quantization level. EZW determined a very efficient way to code significance maps. A wavelet coefficient is insignificant if |x| < T. By coding the location of zeros.

26. 26 EZW algorithm If a wavelet coefficient at a coarse scale is insignificant, then all wavelet coefficients of the same orientation in the same spatial location at finer scales are likely to be insignificant. Tree Structure: Recognizing the coefficients of the same spatial location Zero tree: set of insignificant coefficients

27. 27 DWT for Image decomposition

28. 28 Zero Tree A coefficient is part of a zero tree if it ’ s zero and all of its descendents are zero Efficient for coding: by declaring only one coefficient a zero tree root, all descendants are known to be zero

29. 29 Implementation Implementing 2D DWT image compression algorithm A JPEG2000 like implementation: EZW coding Haar wavelet transform

30. 30 Question