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

Outline

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

3

IntroductionUEP – 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

C1 C3 C4C2

4

IntroductionIrregular 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

IntroductionIrregular LDPC Codes

6

IntroductionOptimization 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

IntroductionOptimization 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

IntroductionUEP 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

Scalable Image Transmission

Consider an JPEG2000 codestream compressed into Nc – 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 di

fferent protection levels.

Different schemes will be compared.

10

Scalable Image Transmission

Equal Error Protection (EEP) Scheme: EEP: Entire JPEG2000 bitstream is directly encoded b

y a systematic LDPC encoder, block by block.bitstream

codewords

11

Scalable Image Transmission

Unequal Error Protection Scheme: (UEP)-AWGN opt: Use the irregularity of the code.

Nc – 1 quality layers are distributed over all the codewords.

bitstream

codewords

12

Scalable Image Transmission

Unequal Error Protection Scheme: (UEP)-UEP opt: Use the irregularity of the code.

Nc – 1 quality layers are distributed over all the codewords.

bitstream

codewords

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: σ2

14

LDPC Code Design for UEP Channel

UEP parameter description and notations A channel codeword of a rate R LDPC code divided into Nc classes o

rdered in decreasing order of their error sensitivity.

Considering the set of Nc classes:

C1 : highest required protection level

C Nc : highest required protection level

:1 | Ck NkC

C1 C3 C4C2

Information bits Redundancy bits

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LDPC Code Design for UEP Channel

UEP parameter description and notations A channel codeword of a rate R LDPC code divided into Nc classes o

rdered in decreasing order of their error sensitivity.

Let the proportions be the normalized len

gths 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:

}1:1|{ C

Nkk

1

1

1CN

kk

C

NkCk :1|

} )1( , ..., , { 11 RRRp Nc

C1 C3 C4C2

Information bits Redundancy bits

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.)(x

max

2

1)(rt

j

jj xx

maxrtj

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:

max

2

1)(ct

i

ii xx

maxcti

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LDPC Code Design for UEP Channel

UEP parameter description and notations Define and optimize variable node distribution

for each class Ck :

Maximum variable node connection degree in class Ck :

)(

max)(

2

1)( )(

kc

kCk

t

i

ii

C xx

)(

max

kct

19

LDPC Code Design for UEP Channel

UEP parameter description and notations Generating function of variable nodes degree

distribution:

)()(

...

...)(

)()(

2)(3

2)(3

)(2

)(2

232

21

2121

xx

xxxx

xxx

CC

CCCC

max

2

1)(ct

i

ii xx

20

LDPC Code Design for UEP Channel

UEP parameter description and notations

} 5.0 ,3.0 ,2.0 {

21

LDPC Code Design for UEP Channel

UEP parameter description and notations

32)(

21

4

21

3)(1 xxxC

22

LDPC Code Design for UEP Channel

UEP parameter description and notations

1)(

21

6)(2 xxC

23

LDPC Code Design for UEP Channel

UEP parameter description and notations

10)(

21

6

21

2)(3 xxxC

24

LDPC Code Design for UEP Channel

UEP parameter description and notations

3210

3210

10132

)()()(

0.19050.14290.57140.0952

21

4

21

3

21

66

21

2

)21

6

21

2()

21

6()

21

4

21

3(

)()()()( 321

xxxx

xxxx

xxxxx

xxxx CCC

25

LDPC Code Design for UEP Channel

UEP parameter description and notations

43

4321

2

1

2321.07619.0

21

5

21

1600

)(max

xx

xxxx

xxrt

j

jj

26

LDPC Code Design for UEP Channel

UEP parameter description and notations Some others notations (1/2):

CNktt k

cc ..., ,1 ,)max( )(

maxmax

1: one valued vector

TT ] 1

..., ,2

1 [

1 ,]

1..., ,

2

1 [

1

maxmax rrcc tttt

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 .

T)()(3

)(2

)( ] ..., , , [ )(

max

kk

c

kkk C

t

CCC )()( xkC

)(x

T)()()( ] ..., , , [ 21 cNCCC )(x

) , , ( p

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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

0N

Eb

the code threshold

(worst threshold)

29

LDPC Code Design for UEP Channel

Hierarchical optimization algorithm To ensure the UEP constraints would not lead to a too-la

rge 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.

] ,[

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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: C1: Rate constraint

C2: Proportion distribution constraints

C3: Convergence constraint

C4: Stability condition

C5: Minimum variable node degree constraint

C6: Previous optimizations constraints

Linear programming.

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 Ck:

fulfilling constraints C1 to C6

end while end for

k = k+1

maxmin

)(C

kC tt

start

k=1

yes

T)(max kC

C1 to C6

success?

1)()(

minmin k

Ck

C tt

no

k == NC -1

yes

stop

maxmin

)(C

kC tt

no

, , ),( ,max

xtC

maxmin

)(C

kC tt

T)(max kC

1)()(

minmin k

Ck

C tt

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LDPC Code Design for UEP Channel

Hierarchical optimization algorithm Constraints (1/2):

[C1] Rate constraint: (global constraint)

[C2] Proportion distribution constraints: (global constraint)

(i)

(ii)

rCk

C

tRtk

1

1

11 TT)(

k

Ck 11T)(

} 1..., ,1 { CNk

rk

C

C

tR

R

tk

1

1

1 TT)(

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LDPC Code Design for UEP Channel

Hierarchical optimization algorithm Constraints (2/2):

[C3] Convergence constraint: (global constraint)

[C4] Stability condition: (global constraint)

[C5] Minimum variable node degree constraint: (class constraint)

[C6] Previous optimizations constraints:

xx ) , ,F( 2

k

t

jj

Cr

k jemax

2

2

2

1)(

2 )1(/

0 , )()(

min kC

ik

cti

fixed is , ' )( 'kCkk

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LDPC Code Design for UEP Channel

Hierarchical optimization algorithm example:

35

LDPC Code Design for UEP Channel

Performance analysis for short length codewords Simulation results for finite length codewords are given f

or the decoding iteration . Optimization parameters:

The offset is arbitrary set to 0.05 dB.

6l

30

9563.00.0437)(

dB 05.0

) 5.0 ,25.0 ,25.0 (

4

max

87

c

C

t

xxx

N

0N

Eb

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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 transm

ission is considered.

For (UEP)-AWGN opt code: Assign the information bits which belong to the class to the mos

t connected variable nodes. Assign the information bits which belong to the class to the mos

t connected variable nodes … and so up to class.

The redundancy bits are associated to the remaining (1 – R) variable nodes.

2

1R

R1

R2

1C

2C1CN

) class (CNC

37

LDPC Code Design for UEP Channel

Hierarchical optimization algorithm example:

297642 2943.01587.00870.00541.01945.02114.0)( xxxxxxx

38

LDPC Code Design for UEP Channel

Hierarchical optimization algorithm

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.

) 5.0 ,25.0 ,25.0 (

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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 EB/N0 for a given iteration number would be studied.

41

Simulation ResultsDecoding failure

42

Simulation ResultsPSNR vs. EB/N0

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

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

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:

max

2

1)(ct

i

ii xx

i

1

02)(

/

/

/

dxx

in

j

in i

jj

i

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:

1

02

1

0)(

1

)(

/

dxxn

dxx

ini

i

i

1

0)(

1

dxxm

47

Supplementary

Equating these two expressions for E, we conclude that:

We see that the design rate is equal to:

1

0

1

0

)(

)(

dxx

dxxnm

1

0

1

0

)(

)(11

dxx

dxx

n

m

n

mnR

48

Supplementary

We see that the design rate R is equal to:

Rate constraints:

1

0

1

0

)(

)(11

dxx

dxx

n

m

n

mnR

rCk

C

tRtk

1

1

11 TT)(

49

Supplementary

We see that the design rate R is equal to:

1

0

1

0

)(

)(11

dxx

dxx

n

m

n

mnR

1

0

1

0

1

0

1

0

)(1

1)(

1)(

)(

dxxR

dxx

Rdxx

dxx

50

Supplementary

22

)(

22

1

0

1

0

/1

1/

/1

1/

)(1

1)(

jj

k j

C

jj

jj

jR

j

jR

j

dxxR

dxx

k

j

rCk

C

tRtk

1

1

11 TT)(

[C1] Rate constraints:

51

Supplementary

3 5, 12

12)(

12

6

12

6)(

3

21

mn

xx

xxx

3

61

41

41

5

312/6

212/6

212/6

5

Number of degree 2 variable nodes:cc

cxx

xx

dxxjj

j

6

1

4

1

3

1

2

1

2

1

2

1

)2

1

2

1(

)(/

1

0

32

1

0

21

1

02

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.

)(lux

)(lvx

53

Supplementary

Assuming Gaussian approximation:

check node message update:

variable node message update:

with being the Mutual Information function:

max

2

11-)1( ) )1(J )1( J(1rt

j

lvj

lu xjx

C c

k

N

k

t

i

lu

Ci

lv xix

1 2

)1(1-2

)()(max

) )(J)1(2

J(

) (J ) )1(log E(1)(J 2

xem

of a Gaussian random variable:

) 2 , (~ mmx

(2)

(1)

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.

) , , ( 2)1()( lv

lv xFx

0)0( vx

1) ,0 [ ,) , , ( 2 xxxF

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

Supplementary

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.