Model-Based Real-Time Progressive Transmission of

Youssef Charfi,Raouf Hamzaoui,Dietmar Saupe

University of Konstanz

Department of Computer and Information Science

D-78457Konstanz

charfi,hamzaoui,saupe@fmi.uni-konstanz.de

Abstract—Many unequal error protection algorithms used in image communication systems need the operational distortion-rate (D/R)curve of the source coder whose computation is time-consuming.We study the use of parametric models instead of the true D/R curves for wavelet-based embedded image and video coders.We propose a Weibull model and show its superiority to the previous models for real-time applications.For unequal error protection over binary symmetric and packet erasure channels,the Weibull model yielded performance similar to the one obtained with the true D/R curve while satisfying the real-time constraint.

I.I NTRODUCTION

Because of the increasing popularity of the Internet and wireless multimedia products,a lot of work has recently been dedicated to the design of efficient systems for the transmis-sion of images and video over noisy channels.Traditional error control systems,which are based on error detection and retransmission,suffer from delays that may be unacceptable in real-time applications.An alternative is to use forward error correction(FEC).Some of the best FEC systems for multimedia data generate an embedded wavelet bitstream and protect it in an optimal way with unequal error protection[1], [2],[3],[4],[5],[6],[7],[8].

In this paper,we focus on the real-time ability of these FEC systems.The real-time constraint is important in wireless communications.For example,suppose that a user wants to take a picture with a digital camera of a3G mobile phone and immediately send it to a receiver.The encoder must compress the image,protect the source code,and send the source-channel bitstream in real-time.Whereas the source coding with an appropriate embedded wavelet coder is very fast, the best unequal error protection algorithms require a time-consuming preprocessing step,which consists of computing the operational distortion-rate curve of the original image. Although an embedded coder has the desirable property that the source code at the highest available bitrate can be used to generate the distortion-rate points at all lower rates,the computation time is still prohibitive for real-time applications. One faster solution is to estimate the distortion-rate points in the wavelet domain during the encoding.This gives very good results for orthogonal or nearly orthogonal waveletfilters. Another solution is to use a parametric model instead of the true distortion-rate function.An important advantage of the model approach is that the encoder need not send overhead bits to specify the error protection solution to the receiver. Indeed,only the model parameters(a few real numbers)have to be transmitted,allowing the receiver to compute the solution on its side.

Mallat and Falzon[9]introduced a parametric model for wavelet-based coders but they did not discuss applications. This model was used by Huang and Liang[10]for joint-source channel coding with MPEG2.A different parametric model for the distortion-rate function of the SPIHT coder[11] was used for joint-source channel coding by Appadwedula, Jones,Ramchandran,and Kosintzev[12].However,in all these works the complexity aspects,which are essential in real-time applications,were not studied.

We propose a parametric Weibull model for the operational distortion-rate curves of the SPIHT,JPEG2000[13],and 3D SPIHT[14]wavelet coders,which are the most popular embedded wavelet coders.We show that the Weibull model is more appropriate for real-time joint source-channel coding than the models of[9],[12].We apply the parametric models to unequal error protection in binary symmetric and packet erasure channels and show that they allow the computation of a solution in real-time while ensuring reconstruction quality similar to the one obtained with the true distortion-rate func-tion.

The rest of the paper is organized as follows.In Section II, we give an overview of unequal error protection algorithms of embedded codes in binary symmetric and packet erasure channels.In Section III,we propose a parametric model that approximates the operational distortion-rate function of the SPIHT,JPEG2000,and3D SPIHT source coders and show its superiority over the previous models of[9],[12].In Section IV,we present simulation results which show the relevance of a parametric model for real-time joint source-channel coding in binary symmetric and packet erasure channels.

II.A LGORITHMS FOR ERROR PROTECTION

A.Binary symmetric channels

One of the most successful systems for the robust progres-sive transmission of images over binary symmetric channels was introduced by Sherwood and Zeger[1].The basic idea is to use an embedded wavelet coder as a source coder and a con-catenation of an outer cyclic redundancy-check(CRC)coder and an inner rate-compatible punctured convolutional(RCPC) coder as a channel coder.Error propagation is avoided by stopping the decoding when thefirst packet error is detected. Similar systems were also efficiently used by Banister,Belzer,and Fischer[15]for protecting JPEG2000[13]coded images and by Xiong,Kim,and Pearlman[16]for video transmission. The fastest rate-distortion based unequal error protection algorithm for this system is the local search algorithm of Hamzaoui,Stankovi´c,and Xiong[17].The algorithm starts from a solution that maximizes the expected number of received source bits and iteratively searches for a solution with a lower expected distortion using the distortion-rate function of the source coder.Experimental results for a binary symmetric channel with the SPIHT coder and JPEG2000show that the solution given by the algorithm is almost optimal[17].

B.Packet erasure channels

Systematic Reed-Solomon(RS)codes allow efficient packet loss protection of embedded bitstreams in packet erasure chan-nels.The best RS-based unequal loss protection algorithms are due to Puri and Ramchandran[3]and Stankovi´c,Hamzaoui, and Xiong[18].Thefirst algorithm computes the convex hull of the operational distortion-rate curve in a preprocessing step. The local search technique of[18]is similar to that of[17] and also needs the distortion-rate function of the source coder.

III.M ODELING THE DISTORTION-RATE FUNCTION The operational distortion-rate function of the source coder gives the reconstructionfidelity at a given source rate.The reconstructionfidelity is commonly measured by the mean-square error(MSE)or the peak signal-to-noise ratio(PSNR), which is defined in dB as

P SNR=10log102552

MSE

.(1)

All unequal error protection algorithms of the previous section require the operational distortion-rate curve of the source coder for the original image.To determine p distortion-rate points,one needs in principle p encodings and p decodings of the source coder.Since the source code is embedded,one encoding at the highest rate gives the bitstream at the p−1 lower rates.But p decodings are still required.Moreover, these algorithms work best under the ideal assumption that the operational distortion-rate curve is convex.

An alternative to the true distortion-rate function is a para-metric model.Mallat and Falzon[9]proposed the model

y=Cr1−2γ,(2) where C is a positive number andγis of the order of1,to approximate the MSE of a zerotree-based wavelet coder at bit rates r under1bit-per-pixel(bpp).

In[12],the operational MSE-rate function of the SPIHT coder was modeled by the sum of four exponential terms

y=

4

k=1

c k e−l k r.(3)

We show in the following that a better modeling can be obtained with a Weibull model

y=a−be−cr d,(4)

100

200

300

400

0.010.20.40.60.81

M

S

E

Source rate [bpp]

[9]

True function

Weibull model

[12]

Fig.1.Comparison between the Weibull model,the model of[9],and the model of[12]with the true MSE-rate function of the SPIHT coder for the 512×512Lenna image.

where the real numbers a,b,c,and d are parameters that depend on the image and the source coder.

Suppose that we are given N MSE-rate points (r i,MSE(r i)),1≤i≤N,corresponding to N equidistant rates in the range[0.001,R].Tofit these points to the above models,we used linear least squares(applied to the points(log r i,log MSE(r i)),i=1,...,N)for(2)and the Levenberg-Marquardt nonlinear regression method[19]for (3)and(4).

In the paper,we give numerical results for the standard8 bpp512×512Lenna image.We obtained similar results for other images.

For the SPIHT source coder,a maximum source rate R of 1bpp,and N=1024data points,the root-mean square(rms) of the difference in MSE between the models and the true MSE-rate function was107.99,1.11,and3.07for(2),(3), and(4)respectively.For JPEG2000and the same settings,the rms error was121.39,32.95,and29.7for(2),(3),and(4) respectively.

These results may indicate that(3)and(4)should be preferred to(2).However,the relevance of a parametric model for our real-time unequal error protection problem depends not only on the accuracy of the datafitting but also on both the number N of data points used for thefitting and the number of parameters in a model.The number of parameters should be kept small because both the size of the overhead and the fitting time grow with the number of parameters.The second issue is the number of data points used for thefitting.To be meaningful,it should be at least equal to the number of parameters.On the other hand,it should be as small as possible to limit the time needed for the decoding and for determining the model parameters.

To take the real-time constraint under consideration,we now compare the three models when the number of data points used for thefitting is limited to4,8,and4for(2),(3),and (4)respectively.Figure1shows the resulting modeling of the operational MSE-rate function of the SPIHT coder.Here the Weibull model was y=1422.99−1424.e−0.0053r−0.9. Figure2complements Figure1by showing the difference in MSE between the models and the true MSE-rate function. The rms error was125.22,.74,and24.32for(2),(3),and (4),respectively.

Thefigures show that under the real-time constraint,the

01020304050607080901000

0.2

0.40.60.8

1

M S E

Source rate [bpp]

[12][9]

Weibull model

Fig.2.Difference in MSE between the parametric models and the true MSE-rate function of the SPIHT coder for the 512×512Lenna image.

0100

200

300

400

0.01

0.2

0.40.60.8

1

M S E

Source rate [bpp]

[9]

True function Weibull model

[12]

Fig.3.Comparison between the Weibull model,the model of [9],and the model of [12]with the true MSE-rate function of the JPEG2000coder for the 512×512Lenna image.

Weibull model significantly outperformed the models of [12]and [9].In particular,it was better than that of [12]although it has half as many parameters.

We obtained similar results for JPEG2000(see Figure 3and Figure 4).Here the Weibull model was y =2550.34−

2550.e −0.0025r −0.99

.

No model was previously proposed for the operational PSNR-rate function of an embedded wavelet source coder.For example,model (3)is always convex when the c k s are positive,thus it cannot provide an appropriate fit for the PSNR-rate curve,which is concave.Using relation (1),one could convert the MSE-rate curves found with (2)and (3)into PSNR-rate curves.However,better results are obtained by fitting the operational PSNR-rate points to the models.Figures 5and 6show the curves obtained by converting the MSE-rate points and those by fitting the PSNR-rate points to (2)and (4)for the SPIHT and JPEG2000coders.Here again the Weibull model yielded the best approximation.The

models given by (4)were y =120.99−120.25e −0.4r

0.15

and y =105.11−117.41e −0.6r 0.12

for SPIHT and JPEG2000,respectively.

Figure 7compares model (2)and the Weibull model (4)to the true PSNR-rate function of the 3D SPIHT bitstream corresponding to the first 16frames of the QCIF YUV foreman video sequence.The PSNR was computed for the luminance component and averaged over the 16decoded frames.The frame rate was 30frames per second.

Finally we studied the quality of the fitting for various values of the maximum rate R .Our experimental results

01020304050607080901000

0.2

0.40.60.8

1

M S E

Source rate [bpp]

[12][9]

Weibull model

Fig.4.Difference in MSE between the parametric models and the true MSE-rate function of the JPEG2000coder for the 512×512Lenna image.

15

20

25

30

35

40

0.2

0.40.60.8

1

P S N R [d B ]

Source rate [bpp]

[12](4)

True function

(2)[9]

Fig.5.Comparison between the true PSNR-rate curve of the SPIHT bitstream of the Lenna image,the modeling of this curve with the Weibull model (4),and its modeling with (2).The curves denoted by [9]and [12]are derived from the modeling of the MSE-rate curve with (2)and (3),respectively.

showed that in contrast to the other models,the Weibull model was not penalized by decreasing the maximum rate R to values as low as 0.125bpp.This is an advantage because the decoding is faster when the maximum rate is lower.

IV.E XPERIMENTAL R ESULTS

In this section,we provide numerical results which show the suitability of the Weibull model to unequal error protection of image and video codes in a binary symmetric channel and unequal loss protection in a packet erasure channel.The state-of-the-art protection algorithms of [17],[3],[18]were used for the study.All simulations were run on a PC with a Linux operating system having an AMD Athlon (TM)XP 16001400MHz processor with a main memory of 1Gbyte.The programs were written in C and compiled with the -O3optimization option.

For a given set of channel code rates and a target transmis-sion rate,we determined the highest possible source rate R for an unequal error protection solution.Then we computed the model parameters by using equidistant rates in the range [0.001,R ](see the previous section).

We first give results for unequal error protection over a binary symmetric channel.We used the local search algorithm of [17]to minimize the expected MSE.The source coders were SPIHT and JPEG2000.

The time needed to encode the Lenna image at R =1bpp was 0.11s for the SPIHT coder and 0.09s for JPEG2000.This source rate was the highest one used in our experiments.

15

20

25

30

35

40

0.2

0.40.60.8

1

P S N R [d B ]

Source rate [bpp]

[12](4)

True function

(2)[9]

Fig.6.Comparison between the true PSNR-rate curve of the JPEG2000bitstream of the Lenna image,the modeling of this curve with the Weibull model (4),and its modeling with (2).The curves denoted by [9]and [12]are derived from the modeling of the MSE-rate curve with (2)and (3),respectively.

1520

25

30

35

150

300450600

750

Y -P S N R [d B ]

Source rate [kbps]

(2)

True function

(4)

Fig.7.Modeling of the 3D SPIHT PSNR-rate curve with the Weibull model (4)and model (2).The rate is given in kbits per second (kbps).

For the SPIHT coder (resp.JPEG2000),the time required to compute the MSE-rate points and the model parameters was 0.27s (resp.0.14s)for the Weibull model (four data points,four parameters)and 0.51s (resp.0.28s)for the model of [12](eight data points,eight parameters).

The overhead that specifies a model consisted of 16bytes (four single-precision floating point numbers)for the Weibull model and 32bytes for the model of [12].If the true distortion-rate curve is computed or if it is estimated in the wavelet domain during the encoding,the encoder needs to send an overhead of N log 2m bits,where m is the number of channel code rates and N is the number of channel packets.Since m <The channel coder was a concatenation of a 32-bit CRC coder and a rate-compatible punctured turbo coder.The turbo coder consisted of two identical recursive systematic convolu-tional encoders with memory length 4and generators (31,27)(octal).The mother code was 20/60=1/3,and the puncturing rate was 20,yielding 41possible channel code rates.The length of a packet was equal to L =2048bits,consisting of a variable number of source bits,32CRC bits,4bits to set the turbo encoder into a state of all zeroes,and protection bits.We used iterative maximum a posteriori decoding,which was stopped if no correct sequence was found after 20iterations.

Rate Weibull [12]True function

(bpp)MSE Time MSE

Time MSE 0.2566.29<0.0166.59<0.0166.320.534.<0.0134.63<0.0134.630.7523.20<0.0123.270.0123.261.017.33<0.0117.330.0117.331.2513.950.0113.950.0313.951.511.630.02

11.63

0.0511.65

TABLE I

CPU TIME IN SECONDS AND EXPECTED MSE AT VARIOUS TRANSMISSION RATES FOR THE LOCAL SEARCH ALGORITHM OF

[17].R ESULTS ARE

GIVEN FOR THE SPIHT BITSTREAM OF THE 512×512L ENNA IMAGE .

T HE BER OF THE BSC IS 0.1.

Rate Weibull [12]True function

(bpp)MSE Time MSE

Time MSE 0.2570.42<0.0171.04<0.0171.870.536.07<0.0136.15<0.0136.960.7523.56<0.0123.460.0123.771.017.57<0.0117.600.0217.811.2514.120.0114.110.0314.081.511.720.02

11.72

0.0511.78

TABLE II

CPU TIME IN SECONDS AND EXPECTED MSE AT VARIOUS TRANSMISSION

RATES FOR THE LOCAL SEARCH ALGORITHM OF

[17].R ESULTS ARE

GIVEN FOR THE JPEG2000BITSTREAM OF THE 512×512L ENNA

IMAGE .T HE BER OF THE BSC IS 0.1.

Table I and Table II give for SPIHT and JPEG2000the CPU time and the expected MSE at various transmission rates when the unequal error protection solutions were determined with the Weibull model and the model of [12].Note that the expected MSE is shown for the true MSE-rate function.The tables also provide the expected MSE when the solution was computed with the true MSE-rate curve.

The performance of the solution found with the Weibull model was similar to the one obtained with the model of [12].However,the Weibull model allowed a faster computation of the solution.This was due to two reasons.First,(3)is more complex than (4).Second,the local search algorithm applied to (4)needed fewer iterations to converge.The results also show that our model can be used for real-time applications.For JPEG2000,for example,even by adding the 0.23s needed for the encoding and the modeling of the MSE-rate curve,the Weibull model allows a joint source-channel coding in less than 0.25s at all transmission rates.

On the other hand,the Weibull model provided almost the same or a lower MSE than the true MSE-rate curve.The gain in performance is due to the fact that the local search algorithm works best when the distortion-rate curve is convex.This condition is fulfilled by models (2),(3),and (4)but is only an assumption for a true distortion-rate curve.

We now consider unequal loss protection.We suppose that N packets of L bytes each are sent over a packet erasure channel.We assume an exponential packet loss model with a mean loss rate of 20%.We used the unequal loss protection algorithms of [3]and [18]to maximize the expected PSNR.We compare the solutions obtained with the best parametric

Table III and Table IV show the expected PSNR in dB and the time in seconds for the SPIHT bitstream of the Lenna image.The algorithm of[3]computes the vertices of the convex-hull of the PSNR-rate points in a preprocessing step. Since both the Weibull model and model(2)are concave for the PSNR-rate data points,this step is not necessary when the parametric model is used.

Here also using the parametric models instead of the true PSNR-rate function did not cause a significant loss in expected PSNR.Moreover,the time complexity was acceptable for real-time applications.The CPU time of the two models was almost the same.The Weibull model yielded slightly better PSNR results.Table V and Table VI show the results for JPEG2000.

Weibull(2)True function N PSNR Time PSNR Time PSNR

5031.10<0.0131.10<0.0131.11

10034.05<0.0134.05<0.0134.06

15035.770.0135.770.0135.79

20037.040.0237.040.0237.04

25038.030.0338.020.0338.02

TABLE III

CPU TIME IN SECONDS AND EXPECTED PSNR IN D B FOR THE ALGORITHM OF[3].T HE RESULTS ARE FOR THE SPIHT BITSTREAM OF THE512×512L ENNA IMAGE,N PACKETS OF L=200BYTES EACH, AND AN ERASURE CHANNEL WITH PACKET MEAN LOSS RATE0.2.

Weibull(2)True function

N PSNR Time PSNR Time PSNR

5031.110.0231.110.0231.13

10034.060.0434.050.0434.06

15035.780.0735.780.0635.78

20037.020.0937.020.0837.01

25038.030.1138.030.1138.04

TABLE IV

CPU TIME IN SECONDS AND EXPECTED PSNR IN D B FOR THE ALGORITHM OF[18].T HE RESULTS ARE FOR THE SPIHT BITSTREAM OF THE512×512L ENNA IMAGE,N PACKETS OF L=200BYTES EACH, AND AN ERASURE CHANNEL WITH PACKET MEAN LOSS RATE0.2.

Finally,Table VII shows that the Weibull model is also successful for source-channel coding of video.Here0.31s were spent for the encoding and0.8s were needed to model the PSNR-rate function with the Weibull model.

V.CONCLUSION

We studied optimal unequal error protection of embedded wavelet image and video bitstreams using parametric distortion models of the source coders.We showed that the operational MSE-rate and PSNR-rate functions of the SPIHT,JPEG2000, and3D SPIHT coders are well approximated by a Weibull

Weibull(2)True function N PSNR Time PSNR Time PSNR

5030.83<0.0130.81<0.0130.91

10033.<0.0133.86<0.0133.92

15035.710.0135.670.0135.74

20036.950.0236.900.0236.98

25037.860.0337.850.0337.92

TABLE V

CPU TIME IN SECONDS AND EXPECTED PSNR IN D B FOR THE ALGORITHM OF[3].R ESULTS ARE GIVEN FOR THE JPEG2000BITSTREAM OF THE512×512L ENNA IMAGE,N PACKETS OF L=200BYTES EACH, AND AN ERASURE CHANNEL WITH PACKET MEAN LOSS RATE0.2.

Weibull(2)True function

N PSNR Time PSNR Time PSNR

5030.840.0230.830.0230.93

10033.900.0533.880.0533.92

15035.720.0735.680.0735.75

20036.920.0936.910.0936.96

25037.820.1137.830.1137.

TABLE VI

CPU TIME IN SECONDS AND EXPECTED PSNR IN D B FOR THE ALGORITHM OF[18].R ESULTS ARE FOR THE JPEG2000BITSTREAM OF THE512×512L ENNA IMAGE,N PACKETS OF L=200BYTES EACH, AND AN ERASURE CHANNEL WITH PACKET MEAN LOSS RATE0.2.

model,which outperforms previously proposed models under a real-time constraint.Experimental results showed that the Weibull model is suitable to unequal error protection in binary symmetric channels and unequal loss protection in packet erasure channels.

The main contribution of the paper was to show that parametric distortion-rate models allow real-time unequal error protection and yield PSNR(or MSE)performance comparable to and sometimes better than the one obtained with the true operational distortion-rate curves.

The parametric modeling approach is not limited to the source-channel coding systems considered in this paper;they can also be used with all systems that exploit the distortion-rate function of the source coder,including the powerful product code systems of[6],[7],which were designed for fading channels.

Weibull(2)True function N Y-PSNR Time Y-PSNR Time Y-PSNR

5028.96<0.0128.97<0.0129.03

10031.570.0131.570.0131.58

15033.480.0233.490.0233.49

20034.990.0234.990.0234.99

25036.330.0336.330.0336.34

TABLE VII

CPU TIME IN SECONDS AND EXPECTED Y-PSNR IN D B FOR THE ALGORITHM OF[3].Y-PSNR DENOTES THE PSNR OF THE LUMINANCE COMPONENT.R ESULTS ARE GIVEN FOR THE3D SPIHT BITSTREAM OF THE FIRST16FRAMES OF THE F OREMAN SEQUENCE,N PACKETS OF L=200BYTES EACH,AND AN ERASURE CHANNEL WITH A PACKET

MEAN LOSS RATE OF0.05.Acknowledgment.We thank Daniel Sachs and Kannan Ramchandran for providing us with the C-code of[3].We also thank Zixiang Xiong,Vladimir Stankovi´c,and Jim Fowler for helpful comments.Youssef Charfiis supported by the German academic exchange service(DAAD).

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