guaranteed QoS, the high bit error rate (BER) in wireless chan- nel demands .... Assume the total available transmission rate is Rmax and de- fine R = [R1, R2, ...
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➡ OPTIMAL RATE ALLOCATION IN PROGRESSIVE JOINT SOURCE-CHANNEL CODING FOR IMAGE TRANSMISSION OVER CDMA NETWORKS Jianping Hua and Zixiang Xiong Department of Electrical Engineering, Texas A&M University, College Station, TX 77843, USA ABSTRACT This paper presents an optimal rate allocation scheme in joint sourcechannel coding (JSCC) for image transmission over CDMA networks. Our scheme first uses progressive joint source-channel codes [1] to generate operational transmission rate-distortion (TRD) functions of different multiple-access channels with different BERs. The Lagrange multiplier is then employed to set the optimal rate allocation among different channels, subject to a total transmission rate constraint. Progressive JSCC offers the scalability feature that is very attractive for handling bandwidth variations in wireless communications. By extending the rate-distortion function in source coding to the TR-D function in JSCC, our work borrows the “equal slope” argument [2] to effectively solve the rate allocation problem in JSCC for multi-channel image transmission. Experiments show that our scheme runs in real time and provides competitive performance at all intermediate transmission rates. 1. INTRODUCTION Recent advances in wireless communications have made it possible to transmit multimedia data over CDMA networks. In practice, two main issues must be resolved. One is error control. Compressed images or video sequences are vulnerable to transmission errors. Since the current wireless network still cannot provide guaranteed QoS, the high bit error rate (BER) in wireless channel demands good error control solutions. Besides error control, power allocation is another issue. Although battery can last longer than before, limited power is always a constraint to portable handsets. Thus efficient power allocation is also critical to wireless multimedia data transmission. Different from traditional voice communication over a single channel, modern communication networks such as CDMA network now allow users to transmit multimedia data over multiple channels simultaneously. Although transmission over multipleaccess channels provides more flexibility for multimedia data communication, it also raises new issues: how to allocate power and how to allocate rate among all channels? To solve these problems, one approach is to consider rate and power allocation jointly [3]. Given the total transmission rate and power level, Zhao et al. [3] proposed a joint error control and power allocation scheme to minimize the expected distortion. However, this approach is timeconsuming and always exhausts all available resources in order to maximize the performance. In this paper, we take another approach which decouples the combined optimization problem into one of power allocation and another of error control and let the BER constraint to be the connection between these two problems. For the former problem, we Work supported in part by the NSF, ARO and ONR.
0-7803-7965-9/03/$17.00 ©2003 IEEE
formulate a power minimization problem under QoS constraint. This is different from the power allocation problem treated in [3]. Suppose we transmit k embedded bitstreams through different CDMA channels. For each bitstream, we set a QoS constraint (e.g., a maximum BER) which must be satisfied according to the transmission requirement of the bitstream. Given the channel information, we want to assign the power levels on these k channels to satisfy the BER requirement with minimum total power consumption. For the latter problem, under a given total transmission rate for all bitstreams to be transmitted, we want to find the best rate allocation with associated error control schemes for different channels that minimizes the expected distortion at the receiver subject to the BER constraint. Since standard approaches like the distributed power allocation algorithm of Wu and Bertsekas [4] for wireless networks can be modified to solve the power minimization problem, we focus on error control in this paper. Many powerful joint source-channel coding (JSCC) schemes for error control have already been proposed for multimedia data transmission over a single channel [5, 6, 7, 8, 9]. The drawback of these schemes is that they are optimized for one target rate only, thus the performance is not guaranteed at intermediate rates. Progressive JSCC schemes have recently been proposed [1, 10] to optimize the expected distortion over a set of rates. Using progressive JSCC for error control has the advantage that re-optimization and channel code packet re-assembly are not needed when the transmission rate changes. However, the schemes in [1, 10] are designed for multimedia data communication over a single channel, so they do not have the rate allocation problem encountered in the multi-channel communication’s scenario over CDMA networks. In this paper, we present a practical scheme for optimal rate allocation in JSCC for image transmission over CDMA networks. Our scheme first uses the progressive joint source-channel codes [1] to generate the operational transmission rate-distortion (TRD) functions of different multiple-access channels with different BERs. The Lagrange multiplier is then employed to set the optimal rate allocation among all channels under a total transmission rate constraint. Experiments show that our scheme runs in real time and provide competitive performances at all rates. 2. MULTI-CHANNEL RATE ALLOCATION AND ERROR CONTROL To exploit the potential of multi-channel transmission, we use the embedded block coding with optimized truncation (EBCOT) [11] algorithm, which is now being used in JPEG-2000, as the source encoder. EBCOT’s block-based coding scheme can encode each subband separately. Thus we can obtain independent decodable source bitstreams for subbands of different decomposition levels. The rate allocation scheme will apply different source bitstreams
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➡ to different channels. Given a total transmission rate and the BER constraint, the objective of rate allocation is to find an optimized rate allocation and the corresponding JSCC schemes for different channels that minimize the expected distortion at the receiver. 2.1. Problem Formulation Consider a handset that uses k channels for multimedia data communication. The BER constraint set for all k channels is given as = [ρ1 , ρ2 , ..., ρk ]. depends on the CDMA channel model (e.g., single-path fading channel [12] or multipath fading channel [13]), the receiver, and most importantly, the signal-to-interference ratio. The latter can be controlled by power allocation. Assume the total available transmission rate is Rmax and define R = [R1 , R2 , ..., Rk ] as the transmission rate allocation vector for all k channels. Then the optimization problem becomes min D(R, )
s.t.
k X
Ri ≤ Rmax ,
(1)
i=1
where D(R, ), the expected total distortion at the receiver, is the sum of expected distortions of all k channels D(R, ) =
k X
di (Ri , ρi ).
(2)
i=1
Here it’s worthwhile to point out that the rate allocation problem is very similar to “reverse waterfilling” in rate distortion theory [14]. While “reverse waterfilling” allocates the available bit rate among different sources to minimize the total distortion, rate allocation assigns transmission rates to different channels to minimize the expected total distortion. We define di (Ri , ρi ) in Eqn. (2) as the TR-D function of the i-th channel, which corresponds to the rate-distortion (R-D) function in source coding, and use Lagrange multiplier to solve our rate allocation problem. Thus the problem is equivalent to minimizing k X
(di (Ri , ρi ) + λRi )
(3)
i=1
where λ is the Lagrange multiplier. The optimal λ is searched P bisectionally [2] to satisfy ki=1 Ri ≤ Rmax . So it is crucial to find the TR-D function of each channel. For any source coder with R-D function D(·) and a noisy binary channel of capacity C, Shanon’s “Separation Theorem” says that one can transmit R bits per source sample through the channel with distortion arbitrarily close to D(CR) by independently designing the source and channel coders. However, the TR-D function D(CR) only serves as a theoretical bound. Operational TR-D functions are needed for practical rate allocation. Operational TR-D function can be generated with an existing JSCC scheme by running it at each rate. Although many powerful schemes [5, 6, 7, 8, 9] have been proposed, they are not suitable for generating operational TR-D function. To choose proper scheme, two criteria must be considered. One is the complexity of the algorithm. Wireless communications call for fast algorithms which can work in real time. Another is the convexity of the resulting operational TR-D function. The Lagrange multiplier requires that operational TR-D function is strictly convex. So points not on the convex hull of the operational TR-D function must be discarded. If the discarding rate is high, the available rate candidates for rate allocation will be much limited. Based on the above observations, we chose the progressive JSCC scheme in [1] for image transmission.
2.2. Progressive JSCC In this section, we give a brief description of the progressive JSCC scheme in [1] and discuss the properties of the operational TR-D function it generates for single channel. Assume the JSCC scheme encodes the source bitstream into a sequence of channel codewords (packets) with fixed length of L bits. We can then denote the transmission rate by the number of packets to be transmitted. In each packet, the source bits are protected by a CRC coder with fixed length of LCRC bits concatenated with a rate-compatible punctured convolutional (RCPC) channel coder. Let < = {r1 , r2 , ..., rm } be the set of available code rates of the RCPC code, where r1 < r2 < ... < rm . The probability of decoding error of a packet protected by code rate ri is denoted as p(ri ), and there is p(r1 ) < p(r2 ) < ... < p(rm ). Define rN = [rk1 , rk2 , ..., rkN ] as the error protection scheme (EPS) for transmission rate N , where rki ∈ < is the channel code rate set for the i-th packet. We assume all errors can be detected and whenever an error is detected in one packet, this and the following packets are all useless. If the maximal transmission rate is N , then the probability that no error in first i packets with an error in the next packet is 8 < p(rk1 ) Q p(rk ) ij=1 (1 − p(rkj )) Pi (rN ) = : QN i+1 j=1 (1 − p(rkj ))
i=0 1≤i≤N −1 i = N.
(4)
Then the expected distortion at the receiver with transmission rate N is d(N, ρ, rN ) =
N X
Pi (rN )di (rN ),
(5)
i=0
where di (rN ) is the reconstruction error using first i packets. The non-progressive EPS is obtained by minimizing (5) at the target rate. For the progressive JSCC scheme, in order to optimize the performance of EPS at different rates, it does not optimize d at one target rate, but the average d of all rates: N 1 X DN (ρ, rN ) = d(n, ρ, rN ). (6) N n=1 The N -packet EPS rN obtained by minimizing cost function (6) is called progressive EPS. For the progressive EPS rN , its first n items will automatically be the progressive EPS for rate n ≤ N , i.e., rn = [rk1 , ..., rkn ]. Experiments in [1] have shown that the progressive EPS has better performance at most intermediate rates and only slightly worse performance near the target rate. Three progressive EPS’s, progressive rate optimization (PRO), progressive distortion optimization (PDO) and progressive local search (PLS), have been introduced in [1]. Using number of correctly decoded source bits rather than reconstruction error as the performance measure, PRO runs fast and does not depend on source R-D function. Compared to PRO, PDO and PLS, based on full range search and local search respectively, take longer time but have better performance in higher rates. The generation of operational TR-D function from the progressive EPS is straightforward. With the EPS obtained, one can generate the TR-D points through Eqn. (5). Further more, note that the components of DN (ρ, rN ) in Eqn. (6) are just the expected distortion at all rates, so if PDO or PLS is employed, the TR-D points are obtained along with the minimization procedure. We now take a look at the convexity of the operational TR-D function generated by progressive EPS. Since all packets have the same length, the TR-D slope can be represented by the difference
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➡ of expected distortion between consecutive rates. The TR-D slope between rates i and i + 1 is ∆d(i) |λi | = L
= = =
3. EXPERIMENTAL RESULTS
d(i, ri , ρ) − d(i + 1, ri+1 , ρ) L Pi Pi+1 P (r )d j i j (ri ) − j=0 j=0 Pj (ri+1 )dj (ri+1 ) L Q ( i+1 (1 − p(r )))(d i (ri ) − di+1 (ri+1 )) kj j=1 L
, (7)
where we use the property that di (ri ) = di (rj ) when j ≥ i, which can be deduced from the definition of progressive EPS. Q In Eqn. (7), i+1 j=1 (1 − p(rkj )) is strictly decreasing with i. For di (ri ) − di+1 (ri+1 ), it corresponds to the MSE reduction brought by the source bits in (i + 1)-th packet, which are rki+1 (L − LCRC ) bits. If rki+1 = rki , there is rki+1 (L − LCRC ) = rki (L − LCRC ). Then if the source R-D function is strictly convex, di (ri ) − di+1 (ri+1 ) is strictly decreasing. So the only possible exception is when rki+1 > rki . Thus there might be points not lying on the convex hull around the rates where channel code rate changes. Fig. 1 illustrates this case. From the figure we can see the generated operational TR-D function has points not on the convex hull only around three rate-changing points.
Fig. 1. An operational TR-D function obtained from real source bitstream using PLS. Assume BER=0.1 and four RCPC code rates are available. The black regions correspond to the points not on the convex hull. Eqn. (7) does not tell the discarding rate of operational TR-D function. But several observations can be made as below: 1) If all packets have the same rate, i.e. rk1 = rk2 = . . . = rkN , and the source R-D function is strictly convex, then the generated operational TR-D function is strictly convex. 2) If rki+1 > rki , then p(rki+1 ) > p(rki ). Thus the item 1 − p(rki+1 ) will become smaller and give more reduction on ∆d(i) to compensate the gain obtained from the rate change. So the number of points being discarded may not be very large.
2.3. Rate allocation With the progressive JSCC scheme, each channel generates its operational TR-D function. The search for optimal λ can be done via a fast bisectional search [2] for a given total transmission rate. The search has been proved to be extremely fast comparing to the generation of operational TR-D functions. Note that not all rates in the operational TR-D functions are available as rate candidates, so it may not be able to find a λ with the sum of rates exactly equal to Rmax .
We have conducted our experiments based on the source R-D functions obtained from the standard 512×512 Lena image. The EBCOT algorithm is implemented based on JPEG2000 verification Model VM3A [15]. In our experiments, we assumed that there are four channels for image transmission. The source encoder uses threelevel wavelet decomposition and assembles the encoded bits from the subbands of same decomposition level into an independent decodable source bitstream. Bitstream of highest decomposition level, i.e. the low pass subband, is denoted as S1 and assigned to first channel, bitstream of second highest decomposition level is denoted as S2 and assigned to second channel, and so on to S4 . Our experiments have been conducted on two different cases. In the first case, we assumed that all four channels have the same BER with = [0.05, 0.05, 0.05, 0.05]. This corresponds to the case of uniform power allocation over all four channels. In the second case, we assumed four channels have different BERs with = [0.05, 0.1, 0.1, 0.1]. This corresponds to a case of nonuniform power allocation. For the progressive JSCC scheme, each packet has a length of L = 512 bits. The source bits are protected by a 16-bit CRC code used in [5] concatenated with an RCPC code from [16] with mother code memory length six, generator polynomial (147, 163, 135, 135), and a code rate 41 . The puncturing period is eight. The operational TR-D functions are computed at rates from 1 to 512 packets, which correspond to the transmission rates up to 1bpp. Table 1 shows the comparison of the operational TR-D functions generated by different EPS’s. Three schemes are compared here: PRO, PLS and a non-progressive EPS denoted as NPDO. NPDO is obtained from PLS by replacing the cost function (6) with the expected distortion (5). To generate operational TR-D function, NPDO must be run at each rate. The running time and discarding rate of all seven operational TR-D functions generated by each EPS in the experiments are compared in the table. From the table we can see that PRO takes least time to run, and PLS a little longer. NPDO takes much longer which is unacceptable for real-time applications. As for the discarding rate, most functions generated by PRO and PLS have their discarding rates at around 20% to 30%, while functions generated by NPDO have much higher discarding rates, which are normally higher than 50%. Fig. 2 and 3 show the PSNR optimized at different total transmission rates based on the operational TR-D functions generated by different EPS’s for two BER constraints. The curves based on PRO and PLS are almost identical in both cases, so we use one curve to represent both of them. Besides curves based on PRO, PLS and NPDO, the theoretical curve is obtained by using the ideal TR-D function, and uniform rate allocation curve is obtained by splitting the available transmission rate equally among all four channels and using NPDO to optimize each channel at the available rate. From both figures we can see that the curves based on PRO, PLS and NPDO have very close performances. Actually in Fig. 2 the largest gap between them is only 0.05dB and in Fig. 3 only 0.1dB. Although NPDO optimizes its operational TR-D function at each rate, its high discarding rate makes its performance only slightly better than others. Uniform rate allocation performs poorly at low rates and approaches the curves based on PRO, PLS and NPDO at higher rates with a gap of 0.2dB and 0.4dB in Fig. 2 and 3, respectively. All curves still have a big gap from the theoretical one. It is obvious that PRO-based rate allocation scheme is the best choice for practical multimedia data communication.
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➠ NPDO-based scheme is not a good choice due to its complexity and the lack of progressive coding property. S1 0.05 1.8s 20.0 2.6s 20.0 163s 15.3
Source channel BER time PRO % time PLS % time NPDO %
S2 0.05 1.8s 34.6 5.1s 34.5 157s 46.7
S3 0.1 2.2s 10.5 5.7s 18.9 457s 63.1
0.05 1.6s 45.3 3.5s 50.4 34s 56.6
S4 0.1 2.3s 28.1 4.7s 34.6 376s 61.5
0.05 1.9s 49.4 2.8s 49.4 20s 62.1
0.1 2.4s 31.8 3.9s 27.9 124s 62.3
with competitive performance at all rates and runs in real time. Future work includes extending the current work to different channel models and combining it with power minimization. 5. ACKNOWLEDGMENTS The authors would like to thank Vladimir Stankovic for providing us with the C-codes introduced in [1]. 6. REFERENCES
Table 1. Time needed to generate the operational TR-D functions using different EPS’s and the percentage of discarding rate in traversing the convex hulls. 40
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Fig. 2. PSNR vs. total transmission rate with BER constraint = [0.05, 0.05, 0.05, 0.05]. 38
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Fig. 3. PSNR vs. total transmission rate with BER constraint = [0.05, 0.1, 0.1, 0.1].
4. CONCLUSION AND FUTURE RESEARCH In this paper, we propose an optimal rate allocation scheme in JSCC for image transmission over CDMA networks under a total transmission rate constraint. The progressive JSCC scheme is used to generate the operational TR-D function of each channel for optimal rate allocation through the Lagrange multiplier. By extending the R-D function in source coding to the TR-D function in JSCC, our work borrows the “equal slope” argument to effectively solve the rate allocation problem in JSCC for multi-channel image transmission. Compared to a non-progressive scheme, experiments show that our scheme provides progressive transmission
[1] V. Stankovic, R. Hamzaoui, Y. Charfi and Z. Xiong, “Realtime unequal error protection algorithms for progressive image transmission,” submitted to IEEE Journal on Sel. Areas in Comm., Sep. 2002. [2] Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, pp. 1445-1453, Sep. 1988. [3] S. Zhao, Z. Xiong and X. Wang, “Joint error control and power allocation for video transmission over CDMA networks with multiuser detection,” IEEE Trans. Circuits and System for Video Tech., vol. 12, pp.425-437, Jun. 2002. [4] C. Wu and D. P. Bertsekas, “Distributed power control algorithms for wireless networks,” IEEE Trans. on Vehicular Tech., vol. 50, pp.504-514, Mar. 2001. [5] P. G. Sherwood and K. Zeger, “Progressive image coding for noisy channels,” IEEE Signal Proc. Letters, vol. 4, pp. 191198, Jul. 1997. [6] V. Chande and N. Farvardin, “Progressive transimission of images over memoryless noisy channels,” IEEE Journal on Sel. Areas in Comm., vol. 18, pp. 850-861, Jun. 2000. [7] R. Hamzaoui, V. Stankovic, and Z. Xiong, “Rate-based versus distortion-based optimal joint source-channel coding,” Proc. DCC’02, pp. 66-72, Snowbird, UT, Apr. 2002. [8] V. Stankovic, R. Hamzaoui, and Z. Xiong, “Joint product code optimization for scalable multimedia transmission over wireless channels,” Proc. ICME’02, Lausanne, Switzerland, Aug. 2002. [9] V. Stankovic, R. Hamzaoui, and Z. Xiong, “Packet loss protection of embedded data with fast local search,” Proc. ICIP’02, Rochester, NY, Sep. 2002. [10] P. G. Sherwood, X. Tian and K. Zeger, “Channel code blocklength and rate optimization for progressive image transmission,” Proc. WCNC 1999 IEEE Wireless Communications and Networking Conference, pp. 978-982, 1999. [11] D. Taubman, “High performance scalable image compression with EBCOT,” IEEE Trans. Image Processing, vol. 9, pp. 1158-1170, Jul. 2000. [12] A. Host-Madsen and X. Wang, “Performance of blind and group-blind multiuser detection,” IEEE Trans. Inform. Theory, vol. 48, pp.1849-1872, Jul. 2002. [13] J. Evans and D. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 2059-2078, Sep. 2000. [14] T. M. Cover and J. A. Thomas, Elements of Information Theory, New York: Wiley, 1991. [15] JPEG2000 Verification Model VM3A, ISO/IEC JTC1/SC29/WG1 N1143, Feb. 1999. [16] S. Lin and D. J. Costello, Error Control Coding, Englewood Cliffs, New Jersey: Pentice-Hall, 1983.
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