Joint Source-Channel Coding of GGD Sources with Allpass Filtering ...

Joint Source-Channel Coding of GGD Sources with Allpass Filtering Source Reshaping 1 Jianfei Cai2 Nanyang Technological University School of Computer Engineering E-mail: [email protected] and Chang Wen Chen Florida Institute of Technology Dept. of Electrical and Computer Engineering E-mail: [email protected] Version: Nov. 28, 2003 An optimal joint source-channel coding (OJSCC) scheme is developed for encoding memoryless generalized Gaussian distribution (GGD) sources and transmission over noisy channels, including memoryless channels and bursty channels. In particular, we are able to incorporate both transition probability and bit error rate of a bursty channel model into an end-to-end rate-distortion (R-D) function. This R-D function results in an optimum tradeoff between source coding accuracy and channel error protection under a fixed transmission rate. Based on the results of OJSCC, we further proposed a robust joint source-channel coding (RJSCC) scheme based on a combination of OJSCC with allpass filtering source reshaping. Experimental results based on a broad class of GGD sources suggest that the RJSCC scheme consistently outperforms the OJSCC scheme when the shape factor, a parameter of GGD, is less than 2.0. The comparison of RJSCC with the state-of-the-art robust quantization (RQ) [1] shows that RJSCC is comparable with RQ at low bit rate and significantly better than RQ at high bit rate. In addition, an image transmission system using RJSCC demonstrates that the RJSCC system can be efficiently employed in practical applications.

Keywords: J oint source-channel coding; source reshaping; bit allocation. 1. INTRODUCTION In an end-to-end communication system designed for image or video over noisy channels, the encoder usually consists of a source coder and a channel coder. The source coder, typically including transform, quantization and entropy coding, is to reduce the redundancy of input sources. The channel coder, usually employing error correction codes, is to provide error protection by adding controlled redundancy. 1 This paper is submitted to Journal of Visual Communication and Image Representation for peer review. This paper has been presented in part at the 1999 IEEE Wireless Communications and Networking Conference (WCNC’99), New Orleans, LA, U.S.A., Sept. 1999. 2 Send correspondence to Jianfei Cai: Nanyang Technological University, School of Computer Engineering, Singapore 639798, Phone: (+65)6790-6150, Fax: (+65)6792-6559, Email: [email protected].

1

According to Shannon’s information theory, any coding scheme can be implemented by separately designing source and channel coders without sacrificing the overall optimality. However, such design may lead to intolerable long delay and extremely high complexity. In practice, joint source-channel coding (JSCC) is often employed to achieve an optimum coding performance while maintaining moderate delay and complexity. In fact, any scheme jointly considering source coding efficiency and transportation reliability can be considered as a JSCC scheme. A class of the JSCC schemes is to jointly select source coding and transportation parameters such as in [2, 3, 4, 5] while still keeping the independence between source coding and transportation modules. The second class of the JSCC schemes has more connections between source coding and transportation, which is termed as priority-based transmission, i.e., providing different priorities for information with different importance. However, how to define the importance of different information and how to give them different priorities are very challenging tasks. There are many ways to provide transportation priorities. For example, the transportation priorities are provided by different FEC (forward error correction) codes in [6, 7, 8, 9, 10], different ARQ (auto repeat request) schemes in [11], different modulations in [12], different network QoS classes in [13], and different network routing paths in [14]. Another class of the JSCC schemes is to directly consider the reliability issues in source coding by adding controlled redundancy, which is called error-resilient schemes, such as in [15, 16]. In this paper, we focus on the priority-based JSCC schemes by using different FEC codes, which is commonly termed as unequal error protection (UEP). A key problem of this type of the JSCC is how to optimally allocate bits between source coding and channel coding under a fixed overall coding rate so that end-to-end distortion can be minimized. It is well-known that the total distortion comes from two sources: quantization errors and channel errors. If an end-to-end R-D function can be set up, an optimum result can be achieved. However, for most existing image or video codecs based on variable-length coding such as JPEG and MPEG, it is very difficult to obtain an analytically tractable overall R-D function. Most of the prior JSCC work [17, 5, 18, 19] experimentally assigns bits between source coding and channel coding. Typically the optimal bit allocation results are obtained through exhaustive search, a very complicated and time-consuming operation. On the other hand, it is possible to derive an overall R-D function for fixed-length coding schemes. In [20], Ruf and Modestino proposed a fixed-length JSCC scheme for image transmission over additive white Gaussian noise (AWGN) channels. In their scheme, different channel codes are applied to different bits according to their respective importance on the reconstructed data. The most interesting characteristic of Ruf and Modestino’s system is the explicit R-D function. With this R-D function, the optimal bit allocation can be easily implemented without exhaustive search. In particular, in this paper we developed an optimal joint source and channel coding (OJSCC) scheme for generalized Gaussian distribution (GGD) sources over memoryless and bursty channels. The scheme is similar to the approach proposed by Ruf and Modestino [20]. However, we consider a full range of GGD shape factors of the sources and bursty characteristics of channels. This scheme enables us to conduct extensive study on the behavior of OJSCC under different source distributions and channel conditions. Based on the study of OJSCC, we further proposed a robust joint source-channel coding (RJSCC) scheme based on a combination of OJSCC with allpass filtering source reshaping. The idea of source reshaping is 2

motivated by the scheme of robust quantization (RQ) presented in [1]. In that paper, allpass filtering is combined with a channel optimized scalar quantization. There are several fundamental differences between our proposed scheme and the scheme presented in [1]. First, channel coding and optimal bit allocation were not considered in [1]. Second, bursty characteristics of the channel was not addressed. Third, we show that source reshaping is applicable to the cases beyond Gaussian distribution, including the extreme case of uniform distribution. The contribution of this paper is two fold. First, we extend Ruf and Modestino’s work to more general sources and channel models. Especially, we derive an explicit R-D function for bursty channels modeled by finite-state Markov process. This will be applicable to wireless transmission since wireless channels exhibit bursty characteristics. Second, we proposed the RJSCC scheme, an integration of allpass filtering with the OJSCC scheme, to improve the performance. Experimental results based on a broad class of GGD sources suggest that the RJSCC scheme consistently outperforms the OJSCC scheme when the shape factor v < 2.0. This is particularly useful for image and video transmission as these sources can be modeled as GGD with shape factor v < 2.0. The comparison of RJSCC with the state-of-the-art robust quantization (RQ) [1] shows that the RJSCC is comparable with RQ at low bit rate and significantly better than RQ at high bit rate. We would like to acknowledge that our related work in [21] shows the applications of RJSCC in image transmission. However, the research presented in this paper addresses fundamental aspects of general data sources. We expect this work will be applicable to any data transmission applications so long as the input sources can be modeled by GGD. The rest of this paper is organized as follows. Section 2 introduces the OJSCC system and presents the experimental results. Section 3 describes the RJSCC system and shows the corresponding experimental results. Section 4 gives an example of applying the proposed system in image transmission. Finally, Section 5 concludes this paper with some discussions. 2. OJSCC SYSTEM The OJSCC system structure is shown in Fig. 1. Each input data unit is mapped into an index i by an m-bit quantizer. Then different channel codes are applied to different bits of each index according to their relative importance on the reconstructed data. The encoded data are transmitted over a noisy channel. The receiver is essentially an inverse process. Input Data

Quantizer

m bits Channel Encoder i

Noisy Channel

Channel Decoder

Optimal Bit Allocation

FIG. 1 OJSCC system structure.

3

m bits Dequantizer j

Reconstructed Data

2.1.

Source Description

GGD data is used as the input sources. This is because GGD has been shown to match the distribution of many decorrelated signal sources, especially the subband coefficients after applying wavelet transform to image and video sources [18]. The probability density function of GGD is given by   vη(v, σ) exp{−[η(v, σ)|x|]v }, x(−∞, ∞) (1) p(x) = 2Γ(1/v) where η(v, σ) = σ −1



Γ(3/v) Γ(1/v)

1/2

(2)

in which v denotes the exponential rate of decay, also called shape factor, and σ 2 denotes the variance of the random variable. Eqn. (1) becomes Gaussian distribution when v = 2, and Laplacian distribution when v = 1. The GGD with values of v in the range 0.1 < v < 1.0 provides a useful model for broad-tailed processes. Notice that for very large values of v, the distribution tends to a uniform distribution [22]. 2.2.

Channel Model

Two types of channels, memoryless channels and bursty channels, are considered in this research. Specifically, binary symmetric channel (BSC) is adopted to model memoryless channels; finite-state Markov channel [23] is adopted to model bursty channels. For BSC, there is only one parameter, bit error rate (BER), to describe the channel condition. For finite-state Markov channels, we consider the simplest case, the two-state Markov channel, which is called Gilbert-Elliott channel (GEC). The GEC model [23], as shown in Fig. 2, has two states: Good state and Bad state. Each state can be considered as a BSC. The parameters of GEC include Tgb and Tbg , which represent the transition probabilities from one state to the other. Parameters eG and eB are the BERs at Good state and Bad State, respectively. PG and PB are the probabilities staying at Good state and Bad State, respectively. The average BER of such channel can be written as BER = PG × eG + PB × eB . It is evident that this GEC model is able to characterize some bursty nature of wireless channels. 1- Tgb

Tgb

G

1- Tbg

B

Tbg

FIG. 2 The Gilbert-Elliott model.

4

2.3.

R-D Function

Following [20], we can define the R-D function for memoryless channels as D = Em +

m−1 X

(j)

A(j) m Pb ,

(3)

j=0

where Em is the distortion caused by quantization, (j) (j) Am , Pb

Pm−1 j=0

(j)

(j)

Am Pb

is the distortion

caused by channel noise, and are the bit error sensitivity (BES) and the equivalent channel bit error rate after channel decoding for the jth bit. When a finite-state Markov model [23] is adopted to model bursty channels, the general R-D function for bursty channels can be derived as D = Em +

m−1 X

A(j) m

N −1 X

(j)

Pn Pb (en ),

(4)

n=0

j=0

where N is the number of channel states, Pn is the probability of channel staying (j) at the n-th state, en is the bit error rate for the n-th state, and Pb (en ) is the equivalent bit error rate of the j-th bit for the n-th state after channel decoding. For BSC, Eqn. (4) becomes exactly the same as Eqn. (3). For GEC model, by PN −1 (j) (j) (j) replacing n=0 Pn Pb (en ) with PG Pb (eG ) + PB Pb (eB ), we generate the R-D function for GEC model. The overall rate (bits/sample) can be written as R=

m−1 X j=0

1 , r(j)

(5)

where r(j) is the channel code rate assigned to the j-th bit. We would like to point out that the OJSCC system shown in Fig. 1 is a flexible system. Various scalar quantization schemes and channel codes can be employed in the system without changing the analytical form of the R-D function shown in Eqn. (4) and (5). 2.4.

Optimal Bit Allocation (j)

Suppose the channel condition (Pn , en ) is known. Pb (en ) can be determined (j) by the channel codes. Em and Am are determined by the quantization scheme. In this research, we employ two different quantization schemes: source optimized uniform quantization (SOUQ) and source optimized generalized Gaussian quantization (SOGGQ). For these two quantization schemes, we can derive close-form expressions of the bit error sensitivities (BES). The details on the derivation of the R-D function and BES can be found in Appendix. Thus, the optimal bit allocation problem can be summarized as: given the channel condition (Pn , en ) and the constraint of a fixed total bandwidth, how to find the optimal values of m and R(j) so that the expected distortion in Eqn. (4) can be minimized. From Eqn. (4), we can see that given a value of m, the problem of the FEC selection is the same as the problem of bit allocation among m independent units, which can be easily solved by processing the falling convex hull of the R-D slopes. The details can be found in [24, 25]. Then, we compare the local minimum distortion values obtained under different values of m and choose the one with the smallest distortion. 5

2.5.

Experimental Results of OJSCC

We use rate-compatible punctured convolutional (RCPC) codes [26] to provide unequal error protection. This is because RCPC can easily change coding rates without changing the basic codec structure. The available channel coding rates of the RCPC codes (with memory M = 4, puncture period P = 8) are {1/1, 8/9, 4/5, 2/3, 4/7, 1/2, 4/9, 4/10/, 4/11, 1/3, 4/13, 4/15, 1/4}. Other channel coding techniques, including the combination of several channel coding schemes, can also be applied. For a given quantization scheme, quantization bits m, and GGD shape factor v (assuming GGD variance is normalized), we can analytically calculate the quanti(j) zation error Em , bit error sensitivities Am , j = 0, 1, . . . , m − 1, quantization levels and reconstruction levels. Fig. 3 shows the analytical results of quantization errors and bit error sensitivities of SOUQ for different GGD sources. For the BES part, there are m points in the interval [m, m + 1), which correspond to the m bits of a quantization index. In general, the larger the shape factor v, the smaller the quantization error Em and BES for each bit. This suggests that, with equal quantization bits, GGD sources with larger v will result in lower quantization errors and are (j) less sensitive to channel noise. In addition to Em and Am , the equivalent channel bit error rate Pb for a given RCPC channel coding rate can also be obtained as a function of the real channel BER. These computations can be off-line, and can be used to generate several look-up tables. As a result, the analytical solutions for Eqns. (3), (4) and (5) can be easily obtained by simply looking up the tables and finding the best combination. 2.5.1.

Results with BSC

The results of OJSCC using SOUQ for Gaussian sources are presented as an example, shown in Table 1, to demonstrate the desired consistency between the analysis and the simulation. Notice that, the analytical results are obtained when we use the proposed analytical R-D functions to calculate the distortions. The simulation results are obtained when we apply the optimal bit allocation results to the simulation system, and compare the reconstructed data with the original data. Table 1 shows the difference between actual SNR computed from the simulation and the SNR computed from the analysis is insignificant, not larger than 0.05 dB with one exception, which is 0.12 dB. Therefore, without specification, analytical results are used in the subsequent study to simplify the transmission system implementation. Table 2 summarizes the comparison between OJSCC and the locally optimum system (LOSFBC) for Gaussian sources. LOSFBC is the best channel optimized scalar quantization (COSQ) scheme reported in [22]. OJSCC/SOUQ indicates that SOUQ is used in the OJSCC system, while OJSCC/SOGGQ indicates that SOGGQ is adopted. For 1 bit/sample rate, SOUQ is the same as SOGGQ, so the quantization scheme is not specified. The table shows that at low bit rate, the performance of OJSCC is inferior to that of LOSFBC, but at high bit rate (say larger than 3 bits/sample), OJSCC achieves the better performance. This is because, at high bit rate, the OJSCC enables optimal channel coding when there is sufficient bit rate available for optimization. Fig. 4 shows a summary of SNR performance of the OJSCC system for coding memoryless Uniform, Gaussian, Laplacian, and GGD-0.5 sources with 3 bits/sample

6

1 GGD with v=0.5 GGD with v=1.0 GGD with v=2.0 0.1

Em/Variance (dB)

0.01

0.001

0.0001

1e-05

1e-06 0

2

4

6 Quantization Bits (m)

8

10

12

100 GGD with v=0.5 GGD with v=1.0 GGD with v=2.0 10

BES/Variance (dB)

1

0.1

0.01

0.001

0.0001 1

2

3

4

5 6 Quantization Bits (m)

7

8

9

FIG. 3 Top: Quantization error of SOUQ; Bottom: Bit error sensitivity of SOUQ. over BSC, where GGD-0.5 denotes GGD data with v = 0.5. An interesting observation is that, the larger the shape factor of the input data becomes, the better SNR performance the OJSCC scheme achieves. Other coding rates produce similar observations. 2.5.2.

Results with GEC

A particular GEC model is adopted to generate bursty errors. The parameters of the channel are Tgb = 0.00125, Tbg = 0.005, eG = 0.001, eB = 0.12, PG = 0.80, PB = 0.20. The average BER is 0.0248. For this channel, we compare three different optimal bit allocation strategies: 1. Strategy A: The optimization is based on our proposed R-D function shown in Eqn. (4). 2. Strategy B: The channel is treated as a BSC with the average BER of the GEC model and the optimization is based on Eqn. (3). 7

TABLE 1 The SNR (dB) comparison between the analysis and the simulation results of OJSCC using SOUQ for Gaussian sources. BSC BER 0. 005 0.01 0.05 0.1

1 bit/sample Analy. Sim. 4.25 4.25 4.10 4.12 3.09 3.08 2.09 2.10

2 bit/sample Analy. Sim. 8.49 8.50 7.85 7.87 4.63 4.61 2.47 2.48

3 bit/sample Analy. Sim. 11.85 11.88 10.31 10.28 7.31 7.26 4.25 4.24

4 bit/sample Analy. Sim. 14.02 14.02 13.58 13.55 8.51 8.52 6.29 6.17

TABLE 2 The SNR (dB) Comparison of different systems for Gaussian sources Bit Rate 1 bit/sample 2 bit/sample

3 bit/sample

4 bit/sample

Scheme OJSCC LOSFBC OJSCC/SOUQ OJSCC/SOGGQ LOSFBC OJSCC/SOUQ OJSCC/SOGGQ LOSFBC OJSCC/SOUQ OJSCC/SOGGQ LOSFBC

0.005 4.25 4.25 8.49 8.52 8.52 11.85 11.99 12.04 14.02 14.32 14.14

BSC BER 0.01 0.05 4.10 3.09 4.11 3.15 7.85 4.63 7.85 4.56 7.88 5.20 10.31 7.31 10.36 7.20 10.50 6.47 13.58 8.51 13.83 8.53 12.30 7.81

0.1 2.09 2.27 2.47 2.47 3.63 4.25 4.25 4.67 6.29 6.09 5.60

3. Strategy C: The channel is also treated as a BSC with the BER of the Bad state, and the optimization is based on the R-D function shown in Eqn. (3). Table 3 shows the SNR results of OJSCC for different quantization schemes and different optimization strategies. The results show that the optimal design based on the fully considered R-D function, Strategy A, outperforms the other two strategies which are based on the simple R-D function. It demonstrates that our proposed RD function better characterizes the bursty channel, because the channel transition probability is incorporated. We have also demonstrated that, at the same bit rate, GGD sources with larger shape factor v achieve better SNR performance. 3. RJSCC SYSTEM Based on the experimental results of OJSCC, an important conclusion is that, for either BSC or GEC, the performance of OJSCC for GGD sources with larger shape factor v is better than that for GGD sources with smaller v. Therefore, if a source of smaller shape factor can be reshaped into a source of larger shape factor, 8

14 OJSCC (Uniform) OJSCC (Gaussian) OJSCC (Laplacian) OJSCC (GGD-0.5)

12

SNR (dB)

10

8

6

4

2

0 0.01

0.1 Channel BER

FIG. 4 SNR performance for 3 bit/sample encoding of memoryless sources.

an improved transmission performance can be obtained. It is this observation that motivates us to develop the RJSCC scheme, an integration of the OJSCC scheme with allpass filtering source reshaping. The allpass filtering is able to shape a wide range of input sources into Gaussian distributed sources based on, intuitively, the central limit theorem. Notice that if we can design a filter to shape input sources into uniform distributed sources, we can achieve even better performance. However, it is non-trivial to design such a practical filter capable of non-linear mapping. Therefore, the allpass filtering method is adopted for its implementation advantage. We would like to point out that reshaping a source with v > 2.0 into Gaussian distribution will increase the distortion. Therefore, applying allpass filtering to any source with v > 2.0 is not appropriate. However, for different sources with v > 2.0, different system parameters should be chosen in order to achieve the optimal performance. This will greatly increase the complexity of the entire system. Therefore, only two sets of system parameters are employed in the RJSCC system. One is designed for Gaussian sources, while the other is designed for uniform sources. Therefore, a threshold T is introduced to determine how to switch between two configurations. The RJSCC system structure is shown in Fig. 5. First, the shape factor v of the input GGD data is compared with the threshold T . If v < T , a prefilter is applied to shape the input data into approximately Gaussian distribution. Then these modified data are transmitted through an OJSCC system designed for Gaussian sources. The output of OJSCC is filtered by a postfilter to recover the data. If v ≥ T , the input data are directly transmitted through an OJSCC system designed for uniform distributed sources. In this way, for a broad class of GGD sources with v < T , RJSCC can achieve the same performance as OJSCC for memoryless Gaussian sources; while for GGD sources with v ≥ T , uniform OJSCC is applied to achieve the high SNR performance. In addition, the results of OJSCC show that for GGD sources with v ≥ 2.0, the performance of OJSCC using SOUQ is nearly the same as that of using SOGGQ. Therefore, only SOUQ is employed in RJSCC so that the complexity of quantization is greatly reduced. 9

TABLE 3 SNR (dB) of OJSCC for GGD sources over the bursty channel. Shape Factor v=0.5

Quanti. Scheme SOUQ

Optimal Strategy A B C A B C A B C A B C A B C A B C

SOGGQ

v=1.0

SOUQ

SOGGQ

v=2.0

SOUQ

SOGGQ

x(n)

Input

Prefilter h(n) (Allpass Filtering)

u(n)

1 1.49 1.49 1.49 1.49 1.49 1.49 2.62 2.62 2.62 2.62 2.62 2.62 3.65 3.65 3.65 3.65 3.65 3.65

Gaussian OJSCC

Bit 2 3.33 3.33 3.33 3.46 3.46 1.49 5.10 5.10 5.10 5.07 5.07 2.62 6.44 6.44 6.44 6.37 6.37 3.65

v(n)

Rate (bit/sample) 3 4 5 4.30 4.30 5.05 4.30 2.39 3.18 1.66 3.84 3.94 3.48 4.65 6.52 3.64 4.09 3.98 1.66 4.65 4.77 6.39 6.39 8.16 6.39 6.40 6.56 2.98 6.21 6.35 6.26 6.32 8.15 5.24 6.09 6.29 2.98 6.32 6.48 7.49 8.06 10.36 6.23 7.47 7.70 4.31 8.06 8.34 7.54 8.11 10.43 6.34 7.53 7.52 4.32 8.11 8.39

Postfilter g(n) (Allpass Filtering)

Y

6 6.37 2.37 4.32 6.99 3.38 5.58 8.14 4.36 6.88 8.36 4.07 7.31 10.89 7.81 8.95 11.07 8.06 8.95

y(n) Output

V 2.0. Even if there is a source with v > 2.0, in which case the allpass filtering may increase the mean square error (MSE), it is still worth to apply allpass filtering because allpass filtering will improve perceptual quality of the reconstructed images [1]. We use the same image coding structure as the A-RQ system in [1] for comparison purpose. In that image coding structure, an input image is decomposed into 13 hierarchical subbands using DWT with the Daubechies’ 9/7 biorthogonal filterbanks [28]. Each subband is treated as a subsource. There are totally 13 subsources. Then, each subsource is fed into the RJSCC system. 11

4.1.

Optimal Bit Allocation for Image Transmission

In our proposed image transmission system, the total bit rate needs to be optimally allocated not only between source coding and channel coding of each subsource, but also among subsources. We solve this problem as follows. First we establish the R-D functions for each subsource. Since the bit allocation in one subsource is the same as that problem in Section 2.4, given the total bit rate Ri for the i-th subsource, we can find the optimal quantization bits, FEC codes and the corresponding distortion Di . For different values of Ri , we obtain different values of Di . In this way we can establish the R-D functions for all the subsources. Second, based on these R-D functions, we solve the bit allocation among subsources by using the same method as [25], i.e., processing the falling convex hull of the R-D slopes, because each subsource is independent of others. 4.2.

Experimental Results of Image Transmission

Table 5 shows how bits are allocated among different subbands, and between source and channel for the 512 × 512 monochrome Lena image. It is evident that the lower frequency the subband, the more bits it is assigned. For instance, the 0-th subband, LFS, is assigned 12.75 bits/sample, while 8 bits/sample is used for quantization and the 4.75 bits/sample left is for channel coding. Such bit allocation for LFS leads to an average distortion D0 of 117.592. Finally, for the entire image, the actual bit rate is about 0.50293 bpp, a little higher than the designed bit rate, with the predict average distortion 37.6235. TABLE 5 Bit allocation for Lena image at 0.5 bpp with BER = 0.01.

subband i 0 1 2 3 4 5 6 7 8 9 10 11 12

Result of the 13-band RJSCC σi2 ri mi Di 451866. 12.75 8 117.592 23748.4 9 6 47.3967 6017.54 5 4 122.993 6144.84 6 4 82.3409 3485.47 5 4 71.2395 1092.22 3.5 3 62.0434 1064.25 3.5 3 60.455 443.604 2 2 72.7537 170.284 1 1 66.2145 115.986 0 0 115.986 42.9661 0 0 42.9661 17.1912 0 0 17.1912 8.70312 P 0 0 8.70312

scheme si S ri 0.0498047 0.0351562 0.0195312 0.0234375 0.078125 0.0546875 0.0546875 0.125 0.0625 0 0 0 0 0.50293 bpp

si S Di

0.459344 0.185143 0.480441 0.321644 1.11312 0.969428 0.944609 4.54711 4.13841 7.24913 10.7415 4.2978 2.17578 37.6235

Table 6 shows the source coding rate allocation of RJSCC and OJSCC. We can see that the noisier the channel becomes, the more bits we need to assign for the channel coding. Comparing the schemes between RJSCC and OJSCC, we can 12

conclude that, with allpass filtering, more bits are available for the source coding to achieve an improved image reconstruction at the receiver. TABLE 6 Source coding rate allocation.

Schemes RJSCC OJSCC

Rs (%) for Lena coded at 0.5 bpp e=0 e=10−3 e=10−2 e=5 × 10−2 100 97.66 85.44 61.66 100 92.95 82.97 47.91

e=10−1 46.88 38.78

The comparisons among RJSCC, OJSCC and A-RQ [1] for image transmission are tabulated in Table 7. Notice that the results of RJSCC and OJSCC in Table 7 are the average simulation results over 20 trials. The results of A-RQ are adopted directly from [1]. We show that RJSCC outperforms OJSCC at all cases. Comparing RJSCC with A-RQ, we show that RJSCC performs slightly worse than A-RQ for noise free cases. However, RJSCC outperforms A-RQ for up to 5.86 dB for moderate and high BER channel conditions. The reconstructed images of Lena at 0.5 bpp with various channel BERs are shown in Fig. 6. These images show that the perceptual quality is still quite good for highly corrupted channels with BER = 0.1. TABLE 7 PSNR (in dB) of transmitting 512 × 512 Lena over BSC. Rate (bpp) 1

0.5

0.25

Scheme RJSCC OJSCC A-RQ RJSCC OJSCC A-RQ RJSCC OJSCC A-RQ

0 36.22 33.80 36.72 33.02 31.21 33.45 30.40 28.93 30.73

10−3 35.85 33.52 36.02 32.78 30.64 33.00 30.20 28.59 30.34

BER 10−2 5 × 10−2 34.07 32.74 32.35 30.73 33.57 N/A 31.94 30.29 30.03 28.53 31.42 N/A 29.43 27.94 28.04 26.87 29.24 N/A

10−1 31.26 28.94 25.40 28.68 27.33 24.82 26.85 25.79 24.12

5. CONCLUSION AND DISCUSSION In this paper, we have introduced the OJSCC system, an extension of Ruf and Modestino’s system, for transmitting memoryless GGD sources over noisy channels modeled by BSC and GEC. Experimental results show that, for BSC, OJSCC outperforms LOSFBC reported in [22] at high bit rate. For GEC, the proposed optimal bit allocation strategy based on the derived R-D function, shown in Eqn. 13

FIG. 6 Reconstructed 512 × 512 Lena images using RJSCC coded at 0.5bpp. From left to right and from top to bottom: BER = 0; BER = 10−3 ; BER = 10−2 ; BER = 10−1 .

(4), outperforms the popular designs based on either the average BER or the worst BER at Bad state. Moreover, based on the various results of OJSCC, we developed the RJSCC system based on a combination of OJSCC with allpass filtering source reshaping. Experimental results based on a broad class of GGD sources suggest that the RJSCC consistently outperforms the OJSCC when the shape factor v < 2.0. The comparison between RJSCC and the state-of-the-art robust quantization (RQ) [1] shows that RJSCC is comparable with RQ at low bit rate and significantly better than RQ at high bit rate. We also present an example of applying RJSCC for image transmission. Experimental results show that RJSCC is able to achieve very good PSNR performance as well as perceptual quality. The proposed scheme is essentially a fixed-length coding scheme. In terms of data compression applications, especially image and video compressions, variablelength coding (VLC) has been widely adopted because of its high efficiency. However, considering highly noisy and bursty wireless channels, VLC-based systems may encounter catastrophic error propagation problems. It is also difficult to derive an overall R-D function for a VLC-based system. On the other hand, as we 14

have shown in this paper, it is able to derive an overall R-D function for fixed-length coding schemes, with which the optimal bit allocation can be easily implemented. This is very useful for time-varying channels, since if we can obtain channel information through feedback channels, we can quickly reconfigure the system based on the overall R-D function. Therefore, we believe fixed-length coding schemes have some potentials for wireless communications. A hybrid FLC/VLC system may be able to provide better tradeoff between efficiency and reliability. In addition, the proposed application of allpass filtering is also interesting. Allpass filtering not only improves the performance of the proposed system but also has an excellent property that can enhance the perceptual quality for image/video transmission over noisy channels. APPENDIX A: DERIVATION OF R-D FUNCTION AND BIT ERROR SENSITIVITIES Two different quantization schemes, the source optimized uniform quantization (SOUQ) and the source optimized generalized Gaussian quantization (SOGGQ), are adopted as source coders in this research. For an m-bit quantizer, let M = 2m , and vl , l = 0, 1, . . . , M − 1, denote the reconstruction levels and uk , k = 0, 1, . . . , M , denote the decision levels. The quantization error can be written as Em

= E[(x − x ˆ)2 ] M−1 X Z uk+1 (x − vk )2 px (x)dx, = k=0

(6)

uk

with x and x ˆ the input and reconstructed data, respectively, and px (x) the probability density function of the input. When channel noise is considered, the total distortion becomes M−1 X Z uk+1 (x − vk0 )2 px (x)dx D = k=0

=

k=0

=

uk

M−1 XZ

uk+1

uk

M−1 X Z uk+1 k=0

+

uk

[(x − vk ) + (vk − vk0 )]2 px (x)dx (x − vk )2 px (x)dx +

M−1 X Z uk+1 k=0

uk

M−1 X Z uk+1 k=0

(vk − vk0 )2 px (x)dx.

uk

2(x − vk )(vk − vk0 )px (x)dx (7)

As shown in Eqn. (7), the total distortion consists of three parts. The first part, Ds , is the quantization error Em , the second part is the mutual error, and the third part, Dc , is the error caused by channel noise. If R uk+1 xpx (x)dx vk = Rukuk+1 , (8) px (x)dx uk

the mutual error becomes zero. This is indeed true in the case of SOGGQ. Although it is not exactly true for SOUQ, we assume the mutual error is zero for simplicity. 15

If we consider only one bit error per quantization index, as assumed in [20], the j-th bit error sensitivity (BES) is defined as A(j) m

=

M−1 X Z uk+1 k=0

uk

(vk − vk0 )2 px (x)dx,

(9)

where vk0 is the corresponding erroneous reconstruction level due to an error in the Pm−1 (j) (j) (j) j-th bit of the index for vk . Therefore, Dc = j=0 Am Pb , where Pb is the equivalent bit error rate for the j-th bit after channel coding. If we assume that any wireless channel can be modeled by a finite state Markov model, with N the number of channel states, Pn the probability of channel staying at the n-th state, PN −1 (j) (j) and en the bit error rate for the n-th state, we can derive Pb = n=0 Pn Pb (en ). Therefore, the total distortion function becomes D = Em +

m−1 X

A(j) m

N −1 X

(j)

Pn Pb (en ).

(10)

n=0

j=0

As for the derivation of the bit error sensitivities, they depend on the quantization scheme and the quantization index assignment scheme. For the SOUQ case, it is essentially a uniform quantization. Therefore, M −1 )r, l = 0, 1, . . . , M − 1 2 M uk = (k − )r, k = 1, . . . , M − 1 2 u0 = −∞, uM = +∞.

vl = (l −

(11)

It is clear that SOUQ is a one dimensional optimization problem to find the optimal r so that Em is minimized. Fig. 7 shows a 3-bit quantizer index assignment example. The far left bit, j = m − 1, of an index represents the sign bit, while the other bits represent magnitude bits. From Fig. 7, we can see that for each magnitude bit, the distance between vk and vk0 is a constant. Therefore, according to Eqn. (9) and the index assignment scheme shown in Fig. 7, the BES of magnitude bits for SOUQ can be simplified as: j 2 A(j) m = (2 r) ,

j = 0, 1, . . . , m − 2.

(12)

The derivation of BES for sign bit is more complicated. However, the final expression can be written as = 2[(M − 1)r]2 A(m−1) m

Z

M/2−2

∞

px (x)dx + 2

(M/2−1)r

X

k=0

[(2k + 1)r]2

Z

(k+1)r

px (x)dx.

kr

(13)

If px (x) is Gaussian distributed, then the BES for sign bit can be written as M/2−2 X (M/2 − 1)r kr (k + 1)r √ = [(M −1)r] erfc( )+ [(2k+1)r]2 [erfc( √ )−erfc( √ )]. 2σ 2σ 2σ k=0 (14) For the SOGGQ case, because it is exactly the quantization scheme employed in [20], the corresponding BES formulas can be found in that paper.

A(m−1) m

2

16

Y X 111

110

101

100

000

001

010

011

FIG. 7 Quantization index assignment.

REFERENCES [1] Q. Chen and T. R. Fischer, “Image coding using robust quantization for noisy digital transmission,” IEEE Trans. on Image Processing., vol. 7, pp. 496–505, Apr. 1998. [2] K. Stuhlmuller, N. Farber, M. Link, and B. Girod, “Analysis of video transmission over lossy channels,” IEEE Journal on Selected Areas in Communications, vol. 18, pp. 1012–1032, June 2000. [3] Z. He, J. Cai, and C. W. Chen, “Joint source channel rate-distortion analysis for adaptive mode selection and rate control in wireless video coding,” IEEE Trans. on Circuits and Systems for Video Technology, vol. 12, pp. 511–523, June 2002. [4] A. Nosratinia, J. Lu, and B. Aazhang, “Source-channel rate allocation for progressive transmission of images,” IEEE Trans. on Communications, pp. 186– 196, Feb. 2003. [5] P. G. Sherwood and K. Zeger, “Progressive image coding for noisy channels,” IEEE Signal Processing Letters, vol. 4, pp. 189–191, July 1997. [6] A. E. Mohr, E. A. Riskin, and R. E. Ladner, “Unequal loss protection: graceful degradation of image quality over packet erasure channels through forward error correction,” IEEE JSAC speical issue on Error-Resilient Image and Video Transmission, vol. 18, pp. 819–828, June. 2000. [7] J. Kim, R. M. Mersereau, and Y. Altunbasak, “Error-resilient image and video transmission over the Internet using unequal error protection,” IEEE Trans. on Image Processing, pp. 121–131, Feb. 2003. [8] K. Stuhlmuller, M. Link, and B. Girod, “Scalable internet video streaming with unequal error protection,” in Packet Video Workshop 99, April 1999. [9] V. Chande and N. Farvardin, “Progressive transmission of images over memoryless noisy channnels,” IEEE Journal on Selected Areas in Communications, pp. 850–860, June 2000. [10] V. Stankovic, Y. Charfi, R. Hamzaoui, and Z. Xiong, “Read-time unequal error protection for distortion-optimal progressive image transmission,” in Proceedings of IEEE WCNC’03, (New Orleans, LA), March 2003.

17

[11] S. Wang, H. Zheng, and J. A.Copeland, “Video delivery over a wireless channel with dynamic qos control,” in Proceedings of SPIE Electronic Imaging 2000, Jan. 2000. San Jose, CA. [12] H. Zheng and K. J. R. Liu, “Image and video transmission over wireless channel: a subband modulation approach,” in Proceedings of IEEE ICIP98, 1998. [13] J. Shin, J. Kim, and C.-C. J. Kuo, “Quality-of-service mapping mechanism for packet video in differentiated services network,” IEEE Trans. on Multimedia, pp. 219–231, June 2001. [14] B. Girod and et al., “Advances in network-adaptive video streaming,” in Proceedings of 2002 Tyrrhenian International Workshop on Digital Communications (IWDC 2002), Sept. 2002. [15] D. W. Redmill and N. G. Kingsbury, “The EREC: an error-resilient technique for coding variable-length blocks of data,” IEEE Trans. on Image Processing, vol. 5, April 1996. [16] Y. Wang and S. Lin, “Error-resilient video coding using multiple description motion compensation,” IEEE Trans. on Circuits and Systems for Video Technology, pp. 438–452, June 2002. [17] J. Cai, C. W. Chen, and Z. Sun, “Error resilient image coding with ratecompatible punctured convolutional codes,” in Proceedings of IEEE ISCAS98, June 1998. Monterey, CA. [18] N. Tanabe and N. Farvadin, “Subband image coding using entropy-coded quantization over noisy channels,” IEEE Journal on Selected Areas in Commun., vol. 10, June 1992. [19] H. Li and C. W. Chen, “Bi-directional synchronization and hierarchical error correction for robust image transmission,” in Proceedings of SPIE Visual Communication and Image Processing’99, pp. 63–72, Jan. 1999. San Jose, CA. [20] M. J. Ruf and J. W. Modestino, “Operational rate-distortion performance for joint source and channel coding of images,” IEEE Trans. on Image Processing, vol. 8, pp. 305–320, March 1999. [21] J. Cai and C. W. Chen, “Robust joint source-channel coding for image transmission over wireless channels,” IEEE Trans. on Circuits and Systems for Video Technology, pp. 962–966, Sept. 2000. [22] N. Farvardin and V. Vaishampayan, “Optimal quantizer design for noisy channels: An approach to combined source-channel coding,” IEEE Trans. on Info. Theory, vol. 33, pp. 827–838, Nov. 1987. [23] H. S. Wang and N. Moayeri, “Finite-state Markov channel – a useful model for radio communication channels,” IEEE Trans. on Vehicular Technology, vol. 44, pp. 163–171, Feb. 1995. [24] Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers,” IEEE Trans. on ASSP, vol. 36, sept. 1988.

18

[25] P. H. Westerink, J. Biemond, and D. E. Boekee, “An optimal bit allication algorithm for subband coding,” in Proceedings of ICASSP’88, pp. 757–760, 1988. [26] J. Hagenauer, “Rate-compatible punctured convolutional codes and their applications,” IEEE Trans. Commun., vol. 36, pp. 389–400, Apr. 1988. [27] A. C. Popat and K. Zeger, “Robust quantization of memoryless sources using dispersive FIR filters,” IEEE Trans. on Commun., vol. 40, pp. 1670–1674, Nov. 1992. [28] P. M. M. Antonini, M. Barlaud and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. on Image Processing, vol. 1, pp. 205–220, 1992.

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