1. 1 Scalable Image Transmission Using UEP Optimized LDPC Codes Charly Poulliat, Inbar Fijalkow, David Declercq International Symposium on Image/Video Communications over Fixed and Mobile Networks (ISIVC), July 2004

2. 2 Outline Introduction Scalable Image Transmission LDPC Code Design for UEP Channels Simulation Results Conclusion

3. 3 Introduction UEP – Unequal Error Protection Data of different importance from source encoder Multimedia:  Network/Transport Layer Headers  I/P/B Frames & Motion Vectors Different protection classes/levels provided by:  Physical Layer: modulation  Network Layer: protocols  Channel Coding C1C1 C3C3 C4C4 C2C2

4. 4 Introduction Irregular LDPC Codes Inherent UEP:  Highly connected nodes are protected better. More parity-check equations.  Most connected variable nodes assigned to most important class.  Maximizes average connection degree of variable nodes in each class.

5. 5 Introduction Irregular LDPC Codes

6. 6 Introduction Optimization of Irregularity Optimized for a specific channel:  Binary Erasure Channel (BEC)  Additive White Gaussian Noise Channel (AWGN Channel)  Optimized globally.  Every bit in the codeword has the same average error probability.  Do not necessarily ensure a good UEP capacity.

7. 7 Introduction Optimization of Irregularity Optimized by modeling the UEP transmission scheme as a specific channel.  Optimized locally (within class).  Provides a better UEP capacity. Enhanced UEP:  The error probability within a class is minimized by: maximizing the average connection degree. maximizing the minimum degree of its variable nodes.

8. 8 Introduction UEP properties Interprets the UEP properties of LDPC code as different local convergence speeds. The most protected class:  Be assigned to the bits in the codeword which converge to their right value in the minimum number of decoding iterations.

9. 9 Scalable Image Transmission Consider an JPEG2000 codestream compressed into N c – 1 progressive quality layers.  Source bitstream is encoded into codewords: Length N, each containing K information bits.  Codewords are transmitted over an AWGN channel. Do not consider joint source. Consider a source with fixed classes number that requires different protection levels.  Different schemes will be compared.

10. 10 Scalable Image Transmission Equal Error Protection (EEP) Scheme:  EEP: Entire JPEG2000 bitstream is directly encoded by a systematic LDPC encoder, block by block. bitstream codewords

11. 11 Scalable Image Transmission Unequal Error Protection Scheme:  (UEP)-AWGN opt: Use the irregularity of the code. N c – 1 quality layers are distributed over all the codewords. bitstream codewords

12. 12 Scalable Image Transmission Unequal Error Protection Scheme:  (UEP)-UEP opt: Use the irregularity of the code. N c – 1 quality layers are distributed over all the codewords. bitstream codewords

13. 13 LDPC Code Design for UEP Channel UEP parameter description and notations The transmission scheme consists of sending a UEP coded bitstream:  Under given UEP constraints  Over AWGN channel  Binary input  Noise variance parameter: σ 2

14. 14 LDPC Code Design for UEP Channel UEP parameter description and notations A channel codeword of a rate R LDPC code divided into N c classes ordered in decreasing order of their error sensitivity. Considering the set of N c classes:  C 1 : highest required protection level  C N c : highest required protection level C1C1 C3C3 C4C4 C2C2 Information bits Redundancy bits

15. 15 LDPC Code Design for UEP Channel UEP parameter description and notations A channel codeword of a rate R LDPC code divided into N c classes ordered in decreasing order of their error sensitivity. Let the proportions be the normalized lengths of each class, corresponding to the info bit with:  The proportions distribution of the bits in the channel codewords belonging to each classes is given by: C1C1 C3C3 C4C4 C2C2 Information bits Redundancy bits

16. 16 LDPC Code Design for UEP Channel UEP parameter description and notations Generating function of check nodes degree distribution: Fraction of edges emanating from variable nodes of degree i : Maximum check node connection degree: Assuming is the same for each class.

17. 17 LDPC Code Design for UEP Channel UEP parameter description and notations Generating function of variable nodes degree distribution: Fraction of edges emanating from variable nodes of degree i : Maximum variable node connection degree:

18. 18 LDPC Code Design for UEP Channel UEP parameter description and notations Define and optimize variable node distribution for each class C k : Maximum variable node connection degree in class C k :

19. 19 LDPC Code Design for UEP Channel UEP parameter description and notations Generating function of variable nodes degree distribution:

20. 20 LDPC Code Design for UEP Channel UEP parameter description and notations

21. 21 LDPC Code Design for UEP Channel UEP parameter description and notations

22. 22 LDPC Code Design for UEP Channel UEP parameter description and notations

23. 23 LDPC Code Design for UEP Channel UEP parameter description and notations

24. 24 LDPC Code Design for UEP Channel UEP parameter description and notations

25. 25 LDPC Code Design for UEP Channel UEP parameter description and notations

26. 26 LDPC Code Design for UEP Channel UEP parameter description and notations Some others notations (1/2): 1: one valued vector

27. 27 LDPC Code Design for UEP Channel UEP parameter description and notations Some others notations (2/2):  Vector form association:  A LDPC Code is then parameterized by.

28. 28 LDPC Code Design for UEP Channel Hierarchical optimization algorithm We will consider the LDPC codes that converge to a vanishing bit error probability at a given threshold:  The threshold of the optimized LDPC irregularity without UEP constraints:  The threshold of the optimized LDPC irregularity with UEP constraints: greater than the code threshold (worst threshold)

29. 29 LDPC Code Design for UEP Channel Hierarchical optimization algorithm To ensure the UEP constraints would not lead to a too- large degradation of the threshold, we limit the set of possible LDPC codes to those whose convergence threshold lies within:  : a small constant fixed in the optimization algorithm.

30. 30 LDPC Code Design for UEP Channel Hierarchical optimization algorithm Optimization is done class after class.  Most important class first. Objective function:  Maximize the average variable node’s connection degree within a class, subjecting to a minimum degree of its variable nodes within a class. Constraints:  C 1 : Rate constraint  C 2 : Proportion distribution constraints  C 3 : Convergence constraint  C 4 : Stability condition  C 5 : Minimum variable node degree constraint  C 6 : Previous optimizations constraints Linear programming.

31. 31 LDPC Code Design for UEP Channel Hierarchical optimization algorithm Given parameters: for each class k, starting with the most important class  initialization:  while optimization failure (any constraints is not fulfilled) maximize average connection degree of C k : fulfilling constraints C 1 to C 6  end while end for k = k+1 start k=1 yes C 1 to C 6 success? no k == N C -1 yes stop no

32. 32 LDPC Code Design for UEP Channel Hierarchical optimization algorithm Constraints (1/2):  [C 1 ] Rate constraint: (global constraint)  [C 2 ] Proportion distribution constraints: (global constraint) (i) (ii)

33. 33 LDPC Code Design for UEP Channel Hierarchical optimization algorithm Constraints (2/2):  [C 3 ] Convergence constraint: (global constraint)  [C 4 ] Stability condition: (global constraint)  [C 5 ] Minimum variable node degree constraint: (class constraint)  [C 6 ] Previous optimizations constraints:

34. 34 LDPC Code Design for UEP Channel Hierarchical optimization algorithm example:

35. 35 LDPC Code Design for UEP Channel Performance analysis for short length codewords Simulation results for finite length codewords are given for the decoding iteration. Optimization parameters: The offset is arbitrary set to 0.05 dB.

36. 36 LDPC Code Design for UEP Channel Performance analysis for short length codewords Designed codes have the following parameters:  (K = 2048, N = 4094) : (UEP)-AWGN opt code.  (K = 2047, N = 4095) : (UEP)- UEP opt code.  These codes are both used in the following when scalable image transmission is considered. For (UEP)-AWGN opt code:  Assign the information bits which belong to the class to the most connected variable nodes.  Assign the information bits which belong to the class to the most connected variable nodes … and so up to class.  The redundancy bits are associated to the remaining ( 1 – R ) variable nodes.

37. 37 LDPC Code Design for UEP Channel Hierarchical optimization algorithm example:

38. 38 LDPC Code Design for UEP Channel Hierarchical optimization algorithm

39. 39 Simulation Results Consider the image “Lena” compressed into a JPEG2000 bitstream with three progressive quality layers.  Progressive bit rates: B = (0.125 bpp, 0.250 bpp, 0.5 bpp) We then consider the (UEP) UEP-opt code with optimization parameters: Channel coding rate: R = 1/2 The results are given for 100 independent Monte-Carlo runs using Verification Model version 8.6 as source decoder.

40. 40 Simulation Results Two performance criteria:  Decoding failure: Headers are not protected by any additional forward error protection code, they may be erroneous and decoding failure can occur.  PSNR: The average PSNR of the reconstructed image versus the E B /N 0 for a given iteration number would be studied.

41. 41 Simulation Results Decoding failure

42. 42 Simulation Results PSNR vs. E B /N 0

43. 43 Conclusion Evaluate performance improvement by UEP optimized LDPC codes.  In terms of average PSNR and decoding failure for a low iteration number. Underline the importance for data block interleaving into codeword to fully benefit from LDPC irregularity.

44. 44 Thank you References:  C. Poulliat, D. Declercq, and I. Fijalkow, “Enhancement of Unequal Error Protection Properties of LDPC Codes,” EURASIP Journal on Wireless Communication and networking, 2007.  Neele von Deetzen, “Unequal Error Protection Turbo and LDPC Codes, ” Class Note for Summer Academy, School of Engineer and Science, Jacobs University Bremen, Germany, 2007.  P. S. Guinand, D. Boudreau, and R. Kerr, “Construction of UEP Codes Suitable for Iterative Decoding,” in Proceedings of the 6th Canadian Workshop on Information Theory, pp. 17-20, Kingston, Ontario, Canada, June, 1999.  T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of Capacity- Approaching Irregular Low-Density Parity-Check Codes,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp.619-637, 2001.

45. 45 Supplementary Variable node degree distribution: Fraction of edges emanating from variable nodes of degree i : Assume the code has n variable nodes, the number of variable nodes of degree i is then:

46. 46 Supplementary So E, the total number of edges emanating from all variable nodes, is equal to: Also, assuming the code has m check nodes, total number of edges emanating from all check nodes, is equal to:

47. 47 Supplementary Equating these two expressions for E, we conclude that: We see that the design rate is equal to:

48. 48 Supplementary We see that the design rate R is equal to: Rate constraints:

49. 49 Supplementary We see that the design rate R is equal to:

50. 50 Supplementary [C 1 ] Rate constraints:

51. 51 Supplementary Number of degree 2 variable nodes :

52. 52 Supplementary By using:  Gaussian assumption for Log Density Ratio (LDR) message  Independence assumption between LDR messages give the evolution of the Mutual Information (MI) associated with the mean of the LDR messages for one decoding iteration. We denote the Mutual Information associated with LDR messages at the input of :  variable nodes:  check nodes: at the lth decoding iteration.

53. 53 Supplementary Assuming Gaussian approximation:  check node message update:  variable node message update: with being the Mutual Information function: of a Gaussian random variable: (2) (1)

54. 54 Supplementary Combining (1) and (2) gives the EXIT Chart of the LDPC code: The initial condition is given by. The condition: ensures the convergence of BP algorithm to an error-free codeword.

55. 55 Supplementary EXIT Chart associated with LDPC code:  EXtrinsic Information Transfer Chart, a technique to aid the construction of iteratively-decoded error-correcting codes (LDPC codes and Turbo codes).  Depicted the explicit relation of the MI from iteration l – 1 to iteration l. If there are two components which exchange messages, the behavior of the decoder can be plotted on a two-dimensional chart.  One component: Input: horizontal axis Output: vertical axis  The other component: Input: vertical axis Output: horizontal axis

56. 56 Supplementary

57. 57 Supplementary For a successful decoding, there must be a clear swath between the curves so that:  Iterative decoding can proceed from 0 bits of extrinsic information to 1 bit of extrinsic information.