Wavelet-based Coding And its application in JPEG2000

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

2. Introduction • Signal decomposition • Fourier Transform • Frequency domain  • Temporal domain  Time information?

3. What is wavelet transform? • 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

4. Haar wavelet What are wavelets • Wavelets are functions defined over a finite interval and having an average value of zero.

5. Haar Wavelet Transform • Example: Haar Wavelet

6. Haar Wavelet Transform 1. Find the average of each pair of samples. 2. Find the difference between the average and the samples. 3. Fill the first half of the array with averages. 4. Normalize 5. Fill the second half of the array with differences. 6. Repeat the process on the first half of the array. • 1+3 / 2 = 2 • 1 - 2 = -1 • Insert • Normalize • Insert • Repeat Signal 1. Iteration 2. Iteration 2 6 -1 -1 4 -2 -1 -1

7. Haar Wavelet Transform Signal 1 3 5 7 Signal [ 1 3 5 7 ] Signal recreated from 2 coefficients 2. Iteration 4 -2 -1 -1 [ 2 2 6 6 ]

8. Haar Basis Lenna Haar Basis

9. Frequency domain 2D Mexican Hat wavelet Time domain

10. 2D Mexican Hat wavelet (Movie)low frequency  high frequency <Time Domain Wavelet> <Fourier Domain Wavelet>

11. Scale = 38

12. Scale =2

13. Scale =1

14. Wavelet Transform • Continuous Wavelet Transform (CWT) • Discrete Wavelet Transform (DWT)

15. Continuous Wavelet Transform • continuous wavelet transform (CWT) of 1D signal is defined as • the a,b is computed from the mother wavelet by translation and dilation

16. Discrete Wavelet Transform • CWT cannot be directly applied to analyze discrete signals • CWT equation can be discretised by restraining a and b to a discrete lattice • transform should be non-redundant, complete and constitute multiresolution representation of the discrete signal

17. Discrete Wavelet Transform • Discrete wavelets • In reality, we often choose

18. Discrete Wavelet Transform In the discrete signal case we compute the Discrete Wavelet Transform by successive low pass and high pass filtering of the discrete time-domain signal. This is called the Mallat algorithm or Mallat-tree decomposition.

19. Pyramidal Wavelet Decomposition

20. Wavelet Decomposition The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.

21. Lenna Image Source: http://sipi.usc.edu/database/

22. Lenna DWT

23. Lenna DWT DC Level Shifted +70

24. Restored Image Can you tell which is the original and which is the restored image after removal of the lower right?

25. DWT for Image Compression • Block Diagram 2D Discrete Wavelet Transform Entropy Coding Quantization 2D discrete wavelet transform (1D DWT applied alternatively to vertical and horizontal direction line by line ) converts images into “sub-bands” Upper left is the DC coefficient Lower right are higher frequency sub-bands.

26. LL1 HL1 LH1 HH1 DWT for Image Compression • Image Decomposition • Scale 1 • 4 subbands: • Each coeff. a 2*2 area in the original image • Low frequencies: • High frequencies:

27. LL2 HL2 HL1 LH2 HH2 LH1 HH1 DWT for Image Compression • Image Decomposition • Scale 2 • 4 subbands: • Each coeff. a 2*2 area in scale 1 image • Low Frequency: • High frequencies:

28. LL3 HL3 HL2 LH3 HH3 HL1 LH2 HH2 LH1 HH1 DWT for Image Compression • Image Decomposition • Parent • Children • Descendants: corresponding coeff. at finer scales • Ancestors: corresponding coeff. at coarser scales

29. LL3 HL3 HL2 LH3 HH3 HL1 LH2 HH2 LH1 HH1 DWT for Image Compression • Image Decomposition • Feature 1: • Energy distribution similar to other TC: Concentrated in low frequencies • Feature 2: • Spatial self-similarity across subbands The scanning order of the subbands for encoding the significance map.

30. JPEG2000 • JPEG2000 (J2K) is an emerging standard for image compression • Achieves state-of-the-art low bit rate compression and has a rate distortion advantage over the original JPEG. • Allows to extract various sub-images from a single compressed image codestream, the so called “Compress Once, Decompress Many Ways”. • ISO/IEC JTC 29/WG1 Security Working Setup in 2002

31. JPEG 2000 • Not only better efficiency, but also more functionality • Superior low bit-rate performance • Lossless and lossy compression • Multiple resolution • Range of interest(ROI)

32. JPEG2000 • Can be both lossless and lossy • Improves image quality • Uses a layered file structure : • Progressive transmission • Progressive rendering • File structure flexibility: • Could use for a variety of applications • Many functionalities

33. Why another standard? • Low bit-rate compression • Lossless and lossy compression • Large images • Single decompression architecture • Transmission in noisy environments • Computer generated imaginary

34. By layers By resolutions Region of Interest “Compress Once, Decompress Many Ways” A Single Original Codestream

35. Components • Each image is decomposed into one or more components, such as R, G, B. • Denote components as Ci, i = 1, 2, …, nC.

36. JPEG2000 EncoderBlock Diagram • Key Technologies: • Discrete Wavelet Transform (DWT) • Embedded Block Coding with Optimized Truncation (EBCOT) transform quantize coding

37. Resolution & Resolution-Increments J2K uses 2-D Discrete Wavelet Transformation (DWT) 1-level DWT

38. Resolution and Resolution-Increments 1-level DWT 2-level DWT

39. Discrete Wavelet Transform LL2 HL2 HL1 LH2 HH2 LH1 HH1

40. Layers & Layer-Increments {L0, L1} {L0, L1, L2} L0 All layer- increments

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

42. JPEG2K - Quality Scalability • Improve decoding quality as receiving more bits:

43. Spatial Scalability • Multi-resolution decoding from one bit-stream:

44. ROI (range of interest)