Combined Source-Channel Coding Schemes for Video Transmission

Combined Source-Channel Coding Schemes for Video Transmission over an Additive White Gaussian Noise Channel y

M.Bystrom and J.W.Modestino Electrical, Computer and Systems Engineering Department and Center for Image Processing Research Rensselaer Polytechnic Institute Troy, New York 12180

Abstract

In recent years there has been an increased interest in the transmission of digital video over real-world transmission media, such as the direct broadcast satellite (DBS) channel. Video transmitted over such a channel is subject to degradation due, in part, to additive white Gaussian noise (AWGN). Some form of forward error-control (FEC) coding may be applied in order to reduce the eect of the noise on the transmitted bitstream; however, determination of the appropriate level of FEC coding is generally an unwieldy and computationally intensive problem, as it may depend upon a variety of parameters such as the type of video, the available bandwidth, and the channel SNR. More speci cally, a combined source-channel coding approach is necessary in optimally allocating rate between source and channel coding subject to a xed constraint on overall transmission bandwidth. In this work we develop a method of optimal bit allocation under the assumption that the distortion is additive and independent on a frame-by-frame basis. A set of universal operational distortion-rate characteristics is developed which balances the tradeo between source coding accuracy and channel error protection for a xed overall transmission rate and provides the basis for the optimal bit allocation approach. The results for speci c source and channel coding schemes show marked improvement over suboptimum choices of channel error protection. In addition, we show that our results approach information-theoretic performance bounds which are developed in this work.

This work was supported in part by USAF Rome Laboratories under Contract No. F30602-95-C-0183 and a grant from GTE Government Systems Division. y

I. Introduction

Digital transmission of multimedia and, more speci cally, digital transmission of video has gained increasing importance in recent years. In particular, the problem of transmission of video over real-world channels, such as the direct broadcast satellite (DBS) channel and other additive white Gaussian noise (AWGN) channels, has become a topic of signi cant interest. It has been shown that purely passive error recovery techniques generally provide inadequate reconstructed video quality and therefore some level of forward error-control (FEC) coding may be necessary to provide a reasonable quality of service (QoS) [1]. In addition, there is a strong trend towards operating in a multiresolution framework with digital multimedia transmission in order to provide dierent levels of service at dierent receiver or reception conditions [2]. In this work we consider a multiresolution delivery system in which coded video information is classi ed into a number of priority classes. The high-priority class contains the most important information, while intermediate- and low-priority classes consist of information that contributes less to the quality of the reconstructed video. A user with good reception conditions will reliably receive all priority classes and therefore can reconstruct the video sequence at the highest resolution, while a user with reduced reception conditions may reliably receive only the highest priority class and therefore can reconstruct only the base-level resolution. Each priority class may be transmitted separately on dierent channels and therefore may have dierent channel and degradation characteristics. Recent research has determined that, although in theory source and channel coding may be considered separately, in practice for a transmission scheme to be both ecient and robust, a combined source and channel coding approach is necessary [3]-[5]. An eective combined source-channel coding scheme takes advantage of the dierential sensitivities of the video encoder output components while compensating for the dierent channel characteristics associated with dierent priority classes. It is clear that, since the high-priority information contributes more to the quality of the reconstructed video than does the lowpriority information, when the channel is bandwidth limited, as would be the case in practical applications, in order to maximize reconstructed quality the high-priority information would require a higher rate allocation than would the low-priority information. While this appears obvious, it is not clear how the rate should be allocated; what percentage of the total rate should be allocated to high-priority information and what percentage to lower priority. Furthermore, it is not obvious what relative percentages of the overall bit budget should be allocated to source coding and channel coding. To answer these questions we present a bit allocation methodology based upon the use of universal operational distortionrate characteristics. These characteristics, de ned for each component of the video encoder output, allow for optimal distribution of overall rate across priority classes and to source and channel coding. We show how these universal characteristics, which depend upon the selected source coding scheme and its corresponding sensitivity to channel errors, can be developed for individual sequences and then be used to determine optimal source-channel coding tradeos for dierent modulation and channel coding schemes. While the approach to optimal bit allocation developed in this work is applicable to arbitrary source coding schemes, we concentrate on the subband-based entropy-constrained subband coding (ECSBC) system developed by Kim and Modestino [6], and the discrete cosine transform (DCT) based MPEG-2 coding scheme [7]. Likewise, any channel coding scheme may be utilized; however, here we consider only the class of binary rate-compatible punctured convolutional (RCPC) codes due to their ease of implementation in a multiresolu1

Xi

VARIABLE RATE

PRIORITY

BIT STREAM

CLASSES

{b(1) } i

{ci (1) }

{b(2) } i

{c (2) } i

TO CHANNEL

PRIORITIZATION AND TRANSPORT ENCODER

IMAGE/VIDEO ENCODER

MODULATOR

{b(s) } i

{c i(K)}

^ {b(1) } i

{ci (1) }

^ {b(2) } i

{ci (2) }

^

^

^

X i’

FROM CHANNEL PRIORITIZATION AND TRANSPORT DECODER

IMAGE/VIDEO DECODER

^ {b(s) } i

Figure 1:

DEMODULATOR

^

{c i(K)}

A generic block diagram of a system for transmission of video over a wireless channel.

tion framework. These source and channel coding schemes are discussed in detail in Section II. In Section III the rate allocation problem is considered and a methodology based upon universal distortion-rate characteristics for simplifying the solution to this problem is presented. In addition, information-theoretic bounds are developed using the cuto rate of the various delivery channels. Results are presented for a representative video sequence in Section IV. II. Preliminaries

A. Introduction Consider a general video transmission system model as in Fig. 1 which is suciently broad enough to encompass many source and channel coding schemes. In this system model an input video sequence fXi g is supplied to the video coder block which outputs one or more variable-rate coded bitstreams, fbi(l)g; l = 1; 2; : : :; s. In the case of the ECSBC video coder, the bitstreams represent coded subbands and s = 16, whereas in the case of the MPEG-2 video coder s = 1, and the single bitstream consists of coded DCT coecients and coded motion vectors 1 . Following source encoding, the prioritization and transport encoder assigns each of the s source coding components to a priority class PC (k) ; k = 1; 2; : : :; K . The corresponding prioritized bitstreams, fc(i k)g; k = 1; 2; : : :; K , are then individually channel encoded. The nal block in the encoder, labeled modulator, modulates the priority 1

The MPEG-2 encoder used in this work provides only single resolution capability.

2

classes onto appropriate carriers for transmission over the channel. Although the individual priority classes may be subject to dierent modulation schemes, in this paper we consider only the case where the dierent prioritized bitstreams are multiplexed onto a common binary phase-shift keyed (BPSK) modulator. In the decoder the received waveforms are demodulated into the respective priority classes, which are then separated into their corresponding bitstreams, fb^ i(l)g; l = 1; 2; : : :; s. ^ 0i g which The bitstreams are then decoded to form the reconstructed video sequence fX diers from the original sequence due to source coding errors and possible channel errors. In the following sections we discuss the source and channel coding schemes considered in this work as well as the distortion metric used to measure the quality of the reconstructed video. B. Source Coding Schemes B.1 Entropy-Constrained Subband Coder A block diagram of a generic backward 2 motion-compensated predictive interframe image sequence coding system is illustrated in Fig. 2. The input frame sequence fXig to the encoder in Fig. 2a is motion-compensated predicted, based upon the locally reconstructed ^ ig, in order to produce the predicted sequence fX^ +i g. The residual sequence sequence fX fEig, representing the motion-compensated displaced frame dierences, Ei = Xi ? X^ +i , is then encoded by a 2-D intraframe encoder. The intraframe encoding includes entropy coding, speci cally arithmetic encoding, resulting in the parallel variable-rate encoded output sequences fbi(l)g; l = 1; 2; : : :; s. The decoder in Fig. 2b performs the inverse operations of the encoder based upon the received bitstreams fb^ i(l)g; l = 1; 2; : : :; s, which, again, may dier from the corresponding transmitted bitstreams due to channel errors. The particular combination of pyramid-based hierarchical motion-compensated prediction and 2-D entropy-constrained subband intraframe coding which we employ is described in detail in [8], [9]. In short, a residual frame, Ei , is rst decomposed into 16 uniform subbands (s = 16 in Fig. 2) which are then independently applied to an equal number of optimized entropy-constrained memoryless scalar quantizers. The quantizer output indices are then arithmetically encoded to produce the variable-rate outputs, fbi(l) g; l = 1; 2; :::; 16. In order to operate eciently in the presence of channel errors, the entropy coder is modi ed as in [10] to be less sensitive to error propagation eects. As direct use of the arithmetic decoder in the presence of channel errors may potentially result in loss of synchronization, resynchronization ags are periodically added to the bitstream. The actual positioning of the ags was optimized empirically for a SIF (352x240) size sequence and resulted in a resynchronization period of 176 subband samples for subbands one through four and 352 subband samples for the remaining high-frequency subbands. Further details on the resynchronization process may be found in [11].

B.2 MPEG-2 Coder The MPEG-2 coder is similar to the ECSBC coder in that a motion-compensated prediction scheme is utilized to remove temporal redundancy. However, the prediction scheme is 2 Backward indicates that the motion-compensated prediction is based upon the encoded video sequence and does not require separate transmission of motion vectors.

3

Xi

+

(l)

Ei

Σ

bi

Intraframe Encoder

To Prioritization and Transport Decoder

s

^ +

Xi

Intraframe Decoder ^

Ei

+ +

Σ ^

Xi

MotionCompensated Predictor

(a) Encoder

From Prioritization and Transport Encoder

^ (l) bi

s

^ /

Intraframe Decoder

Ei

^ /

+

Xi

Σ + / ^ + Xi

MotionCompensated Predictor

(b) Decoder

Figure 2: A block diagram of a generic backward motion-compensated predictive interframe image sequence coding system.

4

I

Figure 3:

B

B

P

B

B

I

Illustration of the MPEG-2 prediction scheme for one group of pictures (GOP).

now a bilateral scheme in which motion vectors are used to estimate the current frame from the previous one or to interpolate between future and past frames. Figure 3 illustrates a group of pictures (GOP) consisting of I-, B-, and P-frames. The I-frames are not predicted, but are individually coded (intraframes). Neglecting the fact that macroblocks in the Por B- frames may be intracoded, we can regard the prediction as follows: each P-frame is predicted from the previous I- or P-frame, while each B-frame is bilaterally predicted from the neighboring two I- or P-frames. In order to further reduce the source rate, the input images are separated into their luminance and chrominance components and the chrominance components are subsampled before coding. The block-based motion compensated prediction is performed on only the luminance components of the images. The MPEG-2 standard allows for use of a base layer, which provides the coarsest resolution, as well as for scalable extensions which allow for multiresolution transmission. In the multiresolution case, the motion vectors resulting from the prediction could then represent one of the bitstreams, fbi(l)g, in Fig. 1; however, in this work we use only the base layer so that the motion vectors are transmitted as part of the coded information. Nevertheless, the methodology presented here may be easily adapted to a multiresolution MPEG-2 system. A diagram of a generic encoder for a forward motion compensated prediction scheme, such as MPEG-2, is shown in Fig. 4. More speci cally, in MPEG-2 each residual frame, Ei, is subdivided into slices each of which consists of a number of macroblocks such that the slice length in pixels does not exceed the picture width; we have chosen a slice length of 4 macroblocks as discussed in [11] 3 . Macroblocks are composed of a designated number of blocks, each 8x8 squares of pixels. In order to remove spatial correlation, the blocks are individually compressed using a DCT. The resulting DCT coecients are then quantized and variable-length encoded using a combination of run-length and Human encoding to form the parallel bitstreams, fbi(l)g; l = 1; 2; : : :; s, of Fig. 1. In this work, we use only the base layer of the MPEG-2 system, hence s = 1 and there is a single priority class as well, i.e., K = 1. In order to protect against error propagation, the MPEG-2 standard inherently includes resynchronization following each slice, frame, and group of pictures, so that in the face of errors and incorrect decoding, once a resynchronization code is received, the decoder may begin anew and error propagation is halted. 3 It was found that in an ATM network with a packet loss rate of 1%, a slice length of 4 macroblocks minimizes the error propagation with only a slight increase in bit-rate. It is expected that the results will be similar for an AWGN channel.

5

(l)

Xi

+

Ei

Σ

b i

Intraframe Encoder

s

^+ Xi

To Prioritization and Transport Encoder

Intraframe Decoder +

-

^ Ei

Σ ^ Xi

MotionCompensated Predictor

Motion Vector Estimator

Motion Vector Encoder

(a) Encoder

(l)

b i From Prioritization and Transport Encoder

s

Intraframe Decoder

^ E’ i ^ +’ Xi

+

^ X i’ Σ +

MotionCompensated Predictor Motion Vector Decoder

(b) Decoder

Figure 4: A block diagram of the encoder of a generic forward motion-compensated predictive interframe image sequence coding system.

6

1

1

1

1

1

1

0

1

P

(a) An RCPC encoder of rate R = 4=7. 0

1

1

1

1

0

0

1

P

(b) A rate-compatible puncturing table for rate R = 4=5. Figure 5: An illustration of the derivation of a rate R = 4=7 code from a mother code of rate R = 1=2 through puncturing, and an alternate puncturing table which produces a code of rate R = 4=5.

C. Channel Coding The class of channel codes considered in this work is the set of binary rate-compatible punctured convolutional (RCPC) codes obtained from a speci ed mother code which is, in general, a binary convolutional code of rate R = 1=n. In this section the derivation of the RCPC codes from mother codes and the resulting bound on bit error probability are discussed. It is well-known (cf., [12] or [13]) that the bit error probability of a convolutional code may be bounded by 1 Pb < cd Pd ; (1)

X

d=d

f ree

where dfree is the free distance of the code, Pd is the pairwise error probability in choosing between two paths of mutual distance d, and cd is the average number of bit errors resulting from an erroneous choice between two distance d paths. Therefore, the decoded bit error probability for any channel using a speci ed convolutional code may be bounded if Pd can be computed for the channel. In [14] Hagenauer introduced the class of rate-compatible punctured convolutional (RCPC) codes. By deleting, or puncturing, bits from the coded bitstream, higher-rate codes are produced from lower-rate codes. The puncturing is controlled by a puncturing table which indicates which of the coded bits are to be transmitted and which are punctured. Puncturing some bits in the code serves to increase the rate of the code in a controlled manner so that the level of protection oered by the coding may be adapted to channel conditions. This is shown, for example, in Fig. 5(a) in which the output of a convolutional encoder with memory M = 2 and rate R = 1=2 is punctured according to puncturing table P. In this table, a \1" represents a bit which is transmitted while a \0" represents a bit which is punctured. Therefore, this puncturing table produces a rate R = 4=7 code from the rate R = 1=2 mother code. With P representing the puncturing period of the code, the rates of the codes that may be generated by puncturing a rate R = 1=n mother code 7

are R = P=(P + j ); j = 1; 2; : : :; (n ? 1)P . The bound on bit error probability for these punctured codes is then determined from (1), with the coecients cd calculated for each punctured code. By placing a restriction on the punctured codes, namely that all of the code bits used for high-rate codes must be used for low-rate codes of the same family, Hagenauer [14] developed the class of binary rate-compatible punctured convolutional (RCPC) codes. This process is illustrated in Fig. 5(b), which shows a puncturing table for a rate R = 4=5 code that was derived from the same mother code as the rate R = 4=7 code of Fig. 5(a). Observe that the transmitted bits of the higher-rate code of Fig. 5(b) are a subset of those transmitted for the lower-rate code of Fig. 5(a). Therefore, these two codes are examples of rate-compatible codes of the same family. The advantage of binary RCPC codes is that dierent codes in a family may all be decoded with the same Viterbi decoder. The class of RCPC codes is especially well suited for a multiresolution transmission system, as the dierent priority classes may be provided dierent levels of protection. By using a family of RCPC codes these dierent levels of protection may be obtained from a given mother code using dierent puncturing tables. Hence, by switching between puncturing tables the level of channel protection may be adapted to suit channel conditions with minimal coder as well as decoder complexity. D. Distortion Metric Although subjective criteria may be utilized in performance evaluation of coding schemes, there is no standard subjective delity metric; hence, we employ an objective delity metric. The normalized mean-square distortion between the frame, Xi , input to the encoder ^ 0i, output following decoding, was selected as an objective delity criterion and the frame, X due to its adequate representation of reconstructed quality and ease of computation4 . The corresponding distortion metric is given as 4

XN NX? 1 NX? NX? [Xi;j (m; n) ? X^i;j0 (m; n)] ;

1

t

DS+C = N N N N t f v h j =1

f

1

i=0

v

i 2

1

h

1

m=0 n=0

2

(2)

where (m; n) represents pixel location within each frame of size Nh  Nv and the subscript S + C indicates that the distortion is due to both source coding and channel error eects. To provide a more meaningful metric, the distortion of the ith frame is normalized by its variance, i2 . In order to eliminate statistical idiosyncrasies when computing distortion for video, a sequence of Nf 5 input frames is encoded, channel errors are simulated and the resulting distortion is averaged. The seed on the random number generator is then changed and this process is repeated for Nt iterations. By taking empirical averages with Nt suciently large, statistical con dence in the resulting distortion can be achieved. Observe that in (2) we have made the simplifying assumption that the distortion of the frames is additive, i.e., that the eect of channel errors is independent among frames. This is generally not the case due to error propagation eects; however, the problem of dependent coding is much more dicult and will be discussed later in Section V.

In the ECSBC coder the frame under consideration is the entire fullband frame, while in the case of the MPEG-2 coder the frame would be the luminance component of the original image. 5 In a subband-based system N would be the distance between anchor frames, while in an MPEG-2 system N would be the size of a GOP. 4

f

f

8

III. Optimal Rate Allocation

In this section we discuss the problem of optimally allocating rate between source and channel coding subject to an overall bit budget. It is shown that this allocation is dependent on both the source and channel coding schemes, as well as the chosen modulation scheme and the channel conditions, and that a direct evaluation of the optimal rate allocation cannot generally be performed as it would be too computationally intensive. Therefore, we propose a less computationally intensive method of optimally allocating rate through the generation and use of a set of universal distortion-rate characteristics. These characteristics are families of curves, particular to each source coder, indicating the sensitivity to channel errors and which need not be recomputed when channel coders or channel conditions change. In the following we present the rate allocation problem and, for illustrative purposes, decompose the problem into a two-step process. We then present the methodology for bit allocation through the use of universal operational distortion-rate characteristics. In addition, we develop information-theoretic bounds on distortion-rate performance and show that our operational results closely approach these bounds. The optimal rate allocation problem is reasonably straightforward: optimally allocate bits to source and channel coding for each source coding component, fbi(l)g, subject to a total bit budget, RS +C , in channel uses/sample, such that the overall distortion, DS +C , is minimized. Due to the assumption of independency, for a particular component l the same bit allocation will be made for all frames in the sequence. The outcome of optimal rate allocation is the computation of the optimal operational distortion-rate function, DS +C (RS +C ), for a particular sequence using a selected source coder, set of channel codes, and subject to given channel conditions. The obvious method of evaluating this function is to do an exhaustive search over all source coding rates, channel coding rates, and channel conditions of interest; however, as the coded video is composed of components fbi(l)g; l = 1; 2; : : :; s, each of which may be source coded at a dierent rate, and then sorted into priority classes PC (k) ; k = 1; 2; : : :; K , each of which may be channel coded at a dierent rate 6, the search space is extensive. The computational complexity of this problem arises from the fact that the distortion from source coding and channel errors must be evaluated for each combination of source and channel rate allocations for every component l before the minimization is performed. In addition, the computation process must be repeated as the channel conditions vary since the resulting distortions will be aected. Even with a small number of admissible source and channel coding rates, this problem quickly becomes intractable. However, the computational complexities can be reduced considerably through the utilization of universal distortion-rate characteristics. This is perhaps best illustrated by rst decomposing the rate allocation procedure into a two-step process. Let us suppose that it is possible to determine the optimal class-conditional distortion-rate characteristics, DS +C;k (RS +C;k ); k = 1; 2; : : :; K . The corresponding optimal overall distortion-rate characteristic may then be obtained as

DS +C (RS+C ) = R

min ; k=1;2;:::;K

XK DS C;k (RS C;k ) ; +

+

(3)

k=1 6 We make the assumption that the channel coding rate is xed for all encoder outputs associated with a particular priority class. If we denote (the index set of all encoder outputs which comprise priority class k by S , where S  f1; 2; : : : ; sg, then R ) = R bits/channel use for l 2 S . S +C;k

l

k

k

C

C;k

k

9

. * D S+C,1

. ...

. .

. . .

D* S+C,K

. .

.

.

. R S+C,K c.u./sample

R S+C,1 c.u./sample

(a) Sketches of the optimal class-conditional distortion-rate characteristics for the priority classes k = 1; 2; : : :; K . .

D S+C

. . . .. . . . . . . . . . . . . .. . . . . . . .. . . . . .. . . . . . D *S+C R S+C c.u./sample

(b) Possible overall distortion-rate allocations shown with the convex hull of these allocations, the optimal overall distortion-rate characteristic for the sequence. Illustration of the calculation of the overall optimal distortion-rate characteristic from the K optimal operational class-conditional distortion-rate characteristics.

Figure 6:

in which the minimization is over all RS +C;k ; k = 1; 2; : : :; K , subject to the constraint that

RS+C =

XK Nk RS C;k ; c:u:=sample ;

k=1

N

+

(4)

where N = Nh  Nv , and Nk is the number of samples per frame associated with priority class k; k = 1; 2; : : :; K . Once the distortion-rate functions for the K priority classes are known, the minimization in (3) may easily be performed by a number of rate-allocation algorithms; in this work we use the Westerink Algorithm [15] for bit allocation as is described in Appendix A. The process of computation of the optimal overall distortion-rate characteristic for the sequence from the operational distortion-rate characteristics for each of the K priority classes is illustrated in Fig. 6. A set of K optimal class-conditional distortion-rate characteristics is shown in Fig. 6(a). By selecting a distortion-rate point from each of the K functions, summing these rates and distortions, and then normalizing the total rate as 10

in (4), a possible distortion-rate allocation for the sequence is determined. The set of all such combinations of operating points, with one point drawn from each class-conditional distortion-rate characteristic, is precisely the set of possible distortion-rate allocations for the sequence, as illustrated in Fig. 6(b). The convex hull of these points is then the optimal overall distortion-rate characteristic, DS +C (RS +C ), as given by (3). Now let us show that it is possible to construct the optimal class-conditional distortionrate characteristics from the set of s component distortion-rate characteristics. By recalling that each priority class, k, is composed of source coding components fbi(l)g; l 2 Sk and that the channel coding rate is held xed for each priority class (i.e., RC(l) = RC;k ; bits=c:u:; l 2 Sk ), it can readily be seen that the optimal class-conditional distortion-rate characteristic may be computed in a manner similar to that of the computation of the overall distortionrate characteristic, namely,

DS +C;k (RS+C;k ) = min R ;R

(

(l) S

C;k )

X DSl C (RSl ; RC;k) :

l2S

() +

()

(5)

k

The above minimization is taken over all RS(l); l 2 Sk , and RC;k , where RS(l) is the source coding rate of the lth source coding component, and RC;k is the channel coding rate allocated to the kth priority class, k = 1; 2; : : :; K , such that

RS+C;k = R1

X N l RSl ; c:u:=sample ; ()

C;k l2S Nk

()

(6)

k

where N (l) denotes the number of samples per frame associated with coding component l. The computation of the optimal class-conditional distortion-rate characteristics from the component distortion-rate characteristics is illustrated in Fig. 7. First, for each component l 2 Sk , a distortion-rate characteristic, such as is shown in Fig. 7(a), is generated. For each source coding rate and channel coding rate of interest, the combined source-channel rate is determined as RS(l+) C = RS(l) =RC;k ; l 2 Sk , and the corresponding distortion is calculated for this overall rate. In this manner a family of distortion-rate curves for the particular source coding component l is determined. Each curve in the family, as shown in Fig. 7(a), is the result of xing the source coding rate while varying the channel coding rate. The lower envelope of the points on these curves is then the distortion-rate characteristic of source coding component l, DS(l+) C (RS(l+) C ), as illustrated by the bold line in Fig. 7(a). The process is repeated for each source coding component, l = 1; 2; : : :; s, in order to generate the s component distortion-rate characteristics. Next, as in the derivation of the overall distortion-rate characteristic, the set of possible distortion-rate allocations for the kth priority class is generated through appropriate summation of points drawn from each operational component distortion-rate characteristic, l; l 2 Sk . Representative allocations are shown in Fig. 7(b). As before, the convex hull of these points forms the optimal class-conditional distortion-rate characteristic. In summary, the evaluation of the optimal overall distortion-rate characteristic for a sequence utilizing K priority classes is obtained through a two-step process: First, determine DS +C;k (RS +C;k ); k = 1; 2; : : :; K , as a function of RS +C;k from (5). Then determine DS +C (RS+C ) as a function of RS+C from (3). The values of DS(l+) C (RS(l); RC;k ); l 2 Sk , required for evaluation of DS +C;k (RS +C;k ); k = 1; 2; : : :; K , can be determined either analytically or empirically; however, analytic determination is extremely dicult due to the 11

. R . . . .. . . . . .. . . . ... . . . . . .. .

.

S

(l)

D S+C

D S+C,k

. .

. . . . . . . . . .. . . . . . . . . . . . D *S+C,k

(l)

(l)

R S+C =

RS

RC,k

c.u./sample

R S+C,k c.u./sample

(a) Possible distortion-rate allocations (b) Possible distortion-rate allocations for the lth coding component. The for the kth priority class shown with lower envelope of the allocations is the the optimal class-conditional distortion-rate distortion-rate characteristic for the characteristic. component. Illustration of the calculation of the optimal distortion-rate characteristic for one priority class from the coding component distortion-rate characteristics corresponding to the particular priority class.

Figure 7:

motion-compensated prediction in the source coder. Therefore, these values are perhaps best determined through simulation. Unfortunately, these simulations must be repeated for each choice of codes and channel conditions, which represents a formidable computational challenge. In the next section we describe how this problem can be managed through the use of universal operational distortion-rate characteristics. A. Utilization of Universal Distortion-Rate Characteristics As stated previously, the distortion-rate characteristics DS(l+) C (RS(l+) C ) are functions of the selected source coder, channel coder, coding rates and channel parameters. For each choice of these quantities, these characteristics must be recomputed, a signi cant task even if only a small subset of the available choices are to be examined. Motivated by this, we have developed a more computationally manageable approach based upon the use of universal distortion-rate characteristics which are speci c to the source encoder in use, but are independent of the channel coder and channel conditions. First, we recall that RS(l+) C = RS(l)=RC;k ; l 2 Sk , while noting that DS(l+) C (RS(l+) C ) is inherently a function of the channel bit error probability. It follows that, as illustrated in Fig. 8(a), the combined source-channel distortion of a particular source coding component can be evaluated for a xed source coding rate, RS(l), as a function of the inverse bit error probability, 1=Pb . The resulting family of curves, indicated as DS(l+) C (RS(l); Pb), provide a set of universal distortion-rate characteristics for the selected source coder. Next, the bit error probability of a speci ed channel code is computed as a function of channel parameters for a selected set of channel coding rates. In this work, as we are

12

RC (l)

D S+C

(l)

RS

1/Pb

Pb

E b / N0

(a) Universal distortion-rate characteristics (b) The channel bit error probability as a function for a single coding component. of code rates and channel parameters. Illustration of universal operational distortion-rate characteristics for the l source coding component and the channel bit error probability as a function of the channel codes and parameters for a selected channel code.

Figure 8:

th

assuming an AWGN channel with BPSK modulation, the single channel parameter is the channel SNR. If a dierent channel were utilized, for example, a fading channel, a larger number of channel parameters, such as the channel SNR and the ratio of specular-to-diuse energy, must be speci ed. An illustration of the typical bit error probability behavior for a xed family of channel codes as a function of channel SNR, Eb =N0, is illustrated in Fig. 8(b). Once the universal characteristics, DS(l+) C (RS(l); Pb ), have been obtained for each source coding component l as a function of the channel bit error probability for a xed source coding rate, RS(l), it is a simple matter to evaluate the component distortion-rate characteristics, DS(l+) C (RS(l+) C ). For a selected channel parameter, such as Eb =N0, and a speci ed class of codes, nd the corresponding channel bit error probability for each admissible channel coding rate, RC;k , from an appropriate bound as in (1). Then, from the set of universal curves, this bit error probability yields a combined source-channel distortion, DS(l+) C (RS(l); RC;k ), for each given source coding rate and channel coding rate. The advantage in the use of universal characteristics is that they need only be generated once for each source coder and set of source coding rates of interest, and then are utilized in evaluation of the component distortion-rate characteristics, DS(l+) C (RS(l+) C ); l = 1; 2; : : :; s, for any speci ed set of channel codes, channel coding rates, or channel conditions of interest. Although a signi cant amount of computation is required for calculation of the universal characteristics, there is still a signi cant reduction in computational complexity compared to direct evaluation of the component distortion-rate characteristics, and hence the use of universal characteristics makes the problem of optimal rate allocation tractable. B. Information-Theoretic Performance Bounds Finally, it is of some interest to examine bounds on the achievable performance of a combined source-channel coding scheme. Recall that each priority class, k = 1; 2; : : :; K , may be transmitted over a separate channel and hence be subject to dierent channel

13

conditions. With this in mind, one method of obtaining information-theoretic bounds on performance is to characterize the coding channels by their corresponding cuto rates. The channel cuto rate is generally considered the largest possible signaling rate for which arbitrarily high transmission reliability may be acheived at a reasonable level of system complexity [16]. In this sense it represents the largest practical signaling rate for which arbitrarily small error probability may be assumed. A cuto rate, R0;k , in bits/channel use, can then be calculated for each priority class. The cuto rate depends on the modulation scheme and channel conditions, but is independent of the particular channel coding scheme. It is assumed that transmission is distortion-free for RC;k  R0;k ; k = 1; 2; : : :; K , which implies that (7) DS(l+) C (RS(l); RC;k) = Ds(l)(Rs(l)) ; for RC;k  R0;k : It follows that the optimized distortion for priority class k is then

DS +C;k (RS+C;k ) = min

()

R ;R (l) S

X DSl (RSl ) ;

C;k

l2S

()

(8)

k

subject to the constraint of (6). The minimum occurs when RC;k is maximized, namely when RC;k = R0;k ; 8k. Setting RC;k = R0;k in (6) yields the constraint

RS+C;k = R1

X N l RSl ; c:u:=sample :

;k l2S

0

()

k

Nk

()

(9)

Then an information-theoretic bound on the achievable distortion for the system may be found by utilizing (8) and (9) in (3), subject to the constraint of (4). Bounds for chosen transmission schemes will be presented and discussed in the following section. IV. Results and Discussion

In this section we present results for the two coders, the ECSBC and the MPEG-2 coders, for a representative sequence. The ECSBC encoder is as described in Section II, with the number of coding components s = 16, the number of priority classes K = 2, and the high-priority class consisting of the four lowest subbands, i.e., S1 = f1; 2; 3; 4g. The value of Nt was chosen to be 30. The video sequence used is a grayscale SIF (352x240) sequence, and no passive error recovery was employed. The MPEG-2 encoder is a constant bit-rate encoder with no passive error concealment. The encoded sequence is a color CCIR601 (704x480) sequence, with the value of Nt selected to be 200. This value of Nt is signi cantly larger than the value used for the ECSBC system as the structure of the MPEG-2 coder requires that a larger number of iterations be performed in order to assure statistical con dence in the resulting distortions. This is due, in the most part, to the use of slices as a fundamental resynchronization unit. If errors occur in a slice header, the entire slice may be lost, whereas if errors occur within an ECSBC resynchronization period, loss propagates only until the next resynchronization ag is decoded. As far fewer pixels are contained in a resynchronization period than are in a slice, the loss in an ECSBC system is therefore less severe and fewer iterations are required. In this work only the base layer of the MPEG-2 coder was used, hence s = K = 1, resulting in a single coding component and a single priority class. All results for both encoders are for an AWGN channel with BPSK modulation. 14

Rate 1/2 1/3 133 133 171 171 145 Table 1:

1/4 133 171 145 133

Generating polynomials (in octal) for three rate R = 1=n mother codes.

As described previously, the selected channel codes are families of RCPC codes. Results will be presented for codes of memory length M = 6 derived from mother codes of rate R = 1=2, R = 1=3, and R = 1=4, whose generating polynomials are given in Table 1. As the set of code connection vectors of each higher-rate code is a subset of the set of connection vectors of the next lower-rate code, by using a puncturing approach as described in [14], the rate R = 1=3 and R = 1=2 codes can be derived from the rate R = 1=4 mother code. Therefore, we may essentially restrict our attention to the rate R = 1=4 mother code. It should be noted, however, that the generating polynomials of Table 1 do not necessarily produce the best rate R = 1=3 and R = 1=4 codes in the sense of maximizing dfree ; nevertheless, the rate R = 1=2 code punctured from the rate R = 1=4 code is the optimum code at this rate. On the other hand, by optimally puncturing better (in terms of free distance) codes of rate R = 1=3 and R = 1=4, a signi cant improvement in asymptotic bit error probability performance can be obtained over that of codes punctured from the suboptimal mother codes of Table 1; however, at the relatively low channel SNRs considered in this work, the suboptimal codes actually produced punctured codes with better error probability performance than did the optimal codes. A plot of DS(l+) C (RS(l); Pb ) versus 1=Pb for the ECSBC coder for source coding component l = 1 and three selected rates is shown in Fig. 9. The corresponding plot for the MPEG-2 encoder for four selected rates is shown in Fig. 10. These are operational results determined through simulation of a binary symmetric channel with loss probability, Pb ; they correspond to the schematic curves illustrated in Fig. 8(a). An illustration of the behavior of RCPC codes is presented in Fig. 11. These results show the bit error probability for each code as a function of Eb =N0 for an AWGN channel with soft-decision decoding. The bit error probability performance for all possible codes which can be derived with a puncturing period P = 8 from the rate R = 1=4 mother code are shown in addition to the uncoded (RC = 1 bit=c:u:) bit error probability. In the region of low SNRs, when the bound on bit error probability exceeds the uncoded bit error probability, the uncoded bit error probability is used in place of the bound. It would be expected that in this gure the bounds would be ordered monotonically in terms of the free distance of the corresponding punctured code, and hence in order of increasing code rate. If the error probability bounds were extended to include much lower bit error probabilities, the expected asymptotic performance would result; namely, that at high SNRs the bounds are strictly ordered according to free distances. However, as can be seen from Fig. 11, this is not necessarily the case for low channel SNRs, as the weight spectra contribute signi cantly to the bounds. Therefore, at low values of Eb=N0 , a high-rate code may yield the same or a lower bit error probability than a lower-rate code of the same family. 15

Rs = 0.25 bpp Rs = 0.5 bpp Rs = 1.0 bpp

Ds+c

0.005

0.002

0.001

0.000

0

2000

4000

6000

8000

10000

1/Pb

Distortion, D(1)+ (R(1); P ), plotted vs. inverse probability of bit error for three selected source coding rates. These operational results are for the lowest subband of the ECSBC system.

Figure 9:

S

C

S

b

16

0.50

Ds+c

Rs = 0.006 bpp Rs = 0.12 bpp Rs = 0.25 bpp Rs = 0.39 bpp

0.05

0.00

0

2000000

4000000

6000000

1/Pb

Distortion, D(1)+ (R(1); P ), plotted vs. inverse probability of bit error for four selected source coding rates. These operational results are for the single coding component of the MPEG-2 system. Figure 10:

S

C

b

S

0

10

R

-2

C

Pb, Bit Error Probability

10

uncoded

-4

10

-6

10

-2

0

2

4 Eb/N0 (dB)

6

8

10

Probability of bit error vs. E =N0 for a selected RCPC code with puncturing rate P = 8, memory M = 6, derived from a mother code of rate R = 1=4. These results are for an AWGN channel with soft-decision decoding.

Figure 11:

b

17

In Fig. 12 the optimum allocation for priority class k = 1, DS +C;1 (RS +C;1 ), is illustrated for an ECSBC system with the remaining possible distortion-rate allocations achievable with the set of selected source and channel rates. The optimal distortion-rate characteristic was found from (5). The possible distortion-rate points for the sequence, along with the optimal overall distortion-rate characteristic, DS +C (RS +C ), are shown in Fig. 13. Any point on the optimal distortion-rate characteristic results in the optimal rate allocation to source and channel coding over the priority classes and coding components for the selected overall rate, given the set of available source and channel coding rates. Figure 14 presents the optimal distortion-rate characteristic together with the informationtheoretic bound on performance calculated as described in Section III.B. These results are for the ECSBC coder with BPSK modulation on an AWGN channel with Eb =N0 = 5 dB. For this choice of modulation scheme, channel conditions and set of admissible coding rates, the optimal operational distortion-rate characteristic is reasonably close to the informationtheoretic bound. For example, for the overall rate RS +C = 0:5 c:u:=sample the results would suggest that the performance resulting from the optimal rate allocation is approximately 1 dB less than that of the information-theoretic bound when averaged over the sequence. If a larger range of source and channel coding rates were admissible, it is conjectured that the operational distortion-rate characteristic would approach the theoretic bound more closely. Both subjective and objective results for the ECSBC coder are shown in Fig. 15. Figure 15(a) shows the original 12th frame of a sequence, while Fig.'s 15(b) and 15(c) present the 12th frame following coding at the channel cuto rate and at the optimal rate allocation, respectively, for an overall rate of RS +C = 0:5 c:u:=sample. At this transmission rate, the result of the optimal rate allocation is close, both subjectively and objectively, to the information-theoretic bound. Objectively, there is a dierence of 2:6 dB between the PSNR of the optimal allocation and that of the information-theoretic bound, whereas from examining the distortion-rate curves a dierence of approximately 1 dB in PSNR averaged over the sequence was predicted. This dierence can be explained due to the fact that we are using dierent performance measures, PSNR vs. average PSNR; the PSNR values given in Fig. 15 are the PSNRs of only the nal frames in the sequence rather than the average PSNRs of the reconstructed sequences. In Fig. 15(d) the overall transmission rate is held at RS +C = 0:5 c:u:=sample; however, no channel coding is applied so that RC = 1 bit=c:u: and all of the rate is allocated to source coding. Again, the 12th frame of the sequence following simulated transmission is shown. It is readily observed that there is a decrease of more than 7 dB in reconstructed quality from the preceding results. In Fig. 16 the set of possible distortion-rate points for the single priority class of the MPEG-2 system is shown along with the optimal distortion-rate characteristic. Since the MPEG-2 system used in this work consists of a single coding component, the optimal distortion-rate characteristics for the coding component, the priority class, and the overall ()  sequence are equivalent, i.e., DS(1)+C (R(1) S +C ) = DS +C;1 (RS +C;1 ) = DS +C (RS +C ). This gure then corresponds to Fig. 7 of Section III and the construction of the operational distortionrate characteristics for the single coding component can be readily observed. Each of the curves in Fig. 16, excluding the optimal characteristic, results from holding the source coding rate xed and varying the channel coding rate. The convex hull of all of the rate allocations yields the optimal operational rate allocation for the selected set of source and channel coding rates for the given modulation scheme and channel. An interesting feature of the constant-rate curves is that they each have a slight cusp 18

0.006

Optimal Distortion-Rate Characteristic

0.005

Ds+c

0.004

0.003

0.002

0.001 0.0

0.2

0.4

0.6 0.8 Rs+c (c.u./sample)

1.0

1.2

1.4

Possible distortion-rate allocations for the high-priority class of an ECSBC system, shown with the optimal operational distortion-rate characteristic, D + 1 (R + 1), for this priority class. Channel SNR E =N0 = 5 dB.

Figure 12:

S

S

C;

C;

b

0.014 Optimal Distortion-Rate Characteristic

0.012

Ds+c

0.010

0.008

0.006

0.004

0.002

0.000 0.0

0.2

0.4

0.6 0.8 Rs+c (c.u./sample)

1.0

1.2

1.4

Possible overall distortion-rate allocations for the sequence, shown with the optimal operational distortion-rate characteristic for this sequence. These results are using the ECSBC system with channel SNR E =N0 = 5 dB.

Figure 13:

b

19

0.014 Information-Theoretic Bound Optimal Distortion-Rate Characteristic 0.012

Ds+c

0.010

0.008

0.006

0.004

0.002

0.000 0.0

0.2

0.4

0.6 0.8 Rs+c (c.u./sample)

1.0

1.2

1.4

The information-theoretic bound on rate shown with the optimal operational rate allocation for the ECSBC system. These results are for an AWGN channel with BPSK modulation and a channel SNR E =N0 = 5 dB.

Figure 14:

b

for increasing values of RS +C . This is a result of the behavior of the channel codes observed earlier in Fig. 11, in that at lower values of the channel SNR, such as Eb =N0 = 5 dB as we are assuming in this work, the lower-rate codes may actually provide worse bit error probability performance than higher-rate codes due to the corresponding dierences in their weight spectra. The points on the optimal curve in Fig. 16 then correspond to those channel coding rates which provide the best error performance at the given SNR, while the points on the cusp correspond to lower-rate codes which perform worse at this SNR. Figure 17 presents the optimal distortion-rate characteristic for the sequence along with the information-theoretic bound for the AWGN channel with channel Eb =N0 = 5 dB. At low and high overall transmission rates the operational distortion-rate curve closely approaches the bound. However, at intermediate transmission rates, the distortion-rate curve diverges somewhat from the bound. For instance, for the overall rate of RS +C = 0:87 c:u:=sample, the distortion-rate results indicate that the PSNR given the optimal rate allocation would dier on the average from the PSNR of the information-theoretic bound by almost 3 dB. It is expected that this divergence results from the use of a single source coding component and priority class, and that use of a scalable MPEG-2 coder would result in performance close to the information-theoretic bound. Further objective as well as subjective results for the MPEG-2 coder are presented in Fig. 18. Figure 18(a) shows the original image, the 12th frame in a GOP of length Nf = 12. Figures 18(b) and 18(c) are the information-theoretic bound and the optimal rate allocation results for this frame coded at a xed overall rate of RS +C = 0:87 c:u:=sample, respectively. The result of the optimal operational rate allocation closely approaches that of the information-theoretic bound in both subjective and objective quality. Again, as the 20

(a) Original image.

(b) Information-theoretic bound. PSNR = 39.3 dB

(c) Optimum rate allocation. PSNR = 36.7 dB

(d) Source coding alone. PSNR = 29.1 dB

The 12 frame of a sequence with overall rate held constant at R and E =N0 = 5 dB for an ECSBC system.

Figure 15:

th

S +C

= 0:5 c:u:=sample

b

21

0.06

0.05 Optimal Distortion-Rate Characteristic

Ds+c

0.04

0.03

0.02

0.01

0.00 0.0

1.0

2.0

3.0

Rs+c (c.u./sample)

Figure 16: Possible distortion-rate allocations for the single priority class of the MPEG-2 system, shown with the optimal operational distortion-rate characteristic for this priority class. Channel SNR E =N0 = 5 dB. b

0.06

0.05 Information-Theoretic Bound Optimal Distortion-Rate Characteristic

Ds+c

0.04

0.03

0.02

0.01

0.00 0.0

0.5

1.0 Rs+c (c.u./sample)

1.5

2.0

The optimal operational distortion-rate characteristic with the information-theoretic bound for the MPEG-2 system. Channel SNR E =N0 = 5 dB.

Figure 17:

b

22

(a) Original image.

(b) Information-theoretic bound. PSNR = 41.3 dB

(c) Optimum rate allocation. PSNR = 40.0 dB

(d) Suboptimum rate allocation. PSNR = 20.5 dB

The 12 frame of a GOP (N = 12) with overall rate held constant at R 0:87 c:u:=sample and E =N0 = 5 dB for the MPEG-2 system.

Figure 18:

th

f

S +C

=

b

PSNR of 40:0 dB of Fig. 18(c) is the PSNR of only the nal frame in the GOP rather than the PSNR averaged over the GOP, the results of the optimal allocation are closer to those of the information-theoretic bound than was expected through comparison of the corresponding distortion-rate functions. The nal image in this sequence, Fig. 18(d), shows results for the 12th frame of the sequence coded at a non-optimal rate allocation with RS+C = 0:87 c:u:=sample. Signi cant degradation can be seen in this image; this degradation is due to errors in slice headers and error propagation through the GOP and yields a dramatic decrease in both subjective and objective quality from that of the informationtheoretic bound and the optimal operational rate allocation. Finally, in Fig. 19, the optimal operational distortion-rate function for a base-layer MPEG-2 system is presented along with the information-theoretic bound on performance for the AWGN channel with SNR of 10 dB. These results indicate that in the case of higher channel SNRs multiresolution transmission may not be necessary; rather a combined source-channel approach is sucient to operate close to performance bounds. 23

0.05

0.04 Information-Theoretic Bound Optimal Distortion-Rate Characteristic

Ds+c

0.03

0.02

0.01

0.00 0.0

0.5

1.0

1.5

Rs+c (c.u./sample)

The optimal operational distortion-rate characteristic with the information-theoretic bound for the MPEG-2 system. Channel SNR E =N0 = 10 dB.

Figure 19:

b

V. Summary and Conclusions

A method for optimal rate allocation between source and channel coding in a video multiresolution transmission system was developed and it was shown that a direct evaluation of the optimal allocation is an extremely computationally intensive problem. Motivated by this complexity, a simpli ed two-step optimal rate allocation algorithm based on the utilization of universal operational distortion-rate characteristics was presented. Both subjective and objective results were shown for two coders, the ECSBC and the MPEG-2 coders, with BPSK modulation on an AWGN channel. In addition, information-theoretic bounds, based upon the channel cuto rate, were derived and it was shown that the optimal rate allocation results approach these bounds for selected overall transmission rates. It was further shown that the optimal rate allocation results in improved reconstructed quality in both a subjective and an objective sense over a non-optimal rate allocation. In this work we made the simplifying assumption of independent frame distortion, i.e., that distortion of frames is additive. This is not necessarily an accurate assumption, especially for the cases of motion-compensated predictive coding schemes such as the ECSBC and MPEG-2 encoders due to the fact that quantization and channel errors in earlier frames in the sequence may aect later frames in the sequence. In short, errors propagate through the frames in a sequence, so that the distortion in a particular frame is dependent upon the distortion in preceding frames. In this case, allocations must be performed on a frameby-frame basis and distortions may no longer be averaged over a sequence; therefore, direct computation of the optimal rate allocations is prohibitively computationally intensive. Future work will be directed toward the study of dependent combined source-channel rate allocation and the development of methodologies to make this problem tractable. 24

Appendix A

Optimal Rate Allocation Algorithm Given the large sample space, solving (5) is a problem of signi cant computational complexity. For each bitstream, l, composing a priority class, k, there is a large number of operational distortion-rate points denoting possible bit allocations. A simple search over all of the possible rate allocations for a particular priority class k would involve examining all of the distortion-rate points in each l 2 Sk , a procedure which is obviously computationally intensive. However, it is well-known from rate-distortion theory that rate-distortion curves are convex [17]. Therefore, the optimal bit allocation for a set of rate-distortion points can be de ned as in [15] as the lower convex hull of set of points. Following the algorithm presented in [15], we note that a line drawn through two points on the convex hull of a set of points has the property that all other points in the set must be on the line or on one side of the line. By denoting each possible choice of coding schemes (including both source and channel coding) by ci , each particular assignment of a channel coding scheme to the L bitstreams forming the priority class k can be denoted by c = (c1; : : :; cl; : : :; cL) l 2 Sk . The total rate and distortion for the priority class k given that particular assignment of coding schemes is then DS+C;k (c) = DS(l+) C (cl ) ; and;

X l2S X RS C;k (c) = RSl C (cl) ; k

+

l2S

() +

k

respectively. With this notation the convex hull of a set of points is described as

RS+C (c) ? RS+C (v)  S (w; v)[DS+C (c) ? DS+C (v)] 8c ; where S (w; v) is the slope of the line through w and v, two points on the convex hull of all points under consideration. Given an initial vector v, which is a point on the convex hull, it is possible to nd another point on the convex hull w by solving [15]

X [s(wl; vl) ? max s(cl ; vl )][dl(wl) ? dl(vl)] = 0 ;

l2S

l 2S 0

k

0

0

k

where s(wl; vl ) is the slope of the line between two rate-distortion points of one bitstream, i.e., rl (vl) s(wl ; vl) = drl ((wwl)) ? ? d (v ) : l

l

l l

This algorithm greatly reduces the computational intensity of the problem, as points whose distortions are greater than dl(vl ) need not be considered when searching for wl. By repeatedly solving for points on the convex hull of one priority class, the entire convex hull for the particular priority class may be found. This yields the class-conditional distortion-rate characteristic. By repeating this process for each priority class, the optimal rate assignments for the K priority classes are computed. This algorithm is then applied using these optimal class-conditional distortion-rate characteristics to solve for the optimal overall distortionrate characteristic as in (3).

25

References

[1] M.Bystrom, V.Parthasarathy and J.W.Modestino, \Hybrid Error Concealment Schemes for Broadcast Video Transmission Over ATM Networks." Submitted to IEEE Trans. on Circuits and Systems for Video Tech., 1997. [2] W.F.Schreiber, \Advanced Television Systems for Terrestrial Broadcasting: Some Problems and Some Proposed Solutions," Procedings of the IEEE, vol. 83, pp. 958{981, June 1995. [3] M.J.Ruf and J.W.Modestino, \Rate-Distortion Performance for Joint Source and Channel Coding of Images." submitted to IEEE Trans. on Image Proc., 1995. [4] J.W.Modestino, \Combined Source-Channel Coding Schemes for Multiresolution Digital Video Transmission." Internal Report, ECSE Department, Rensselaer Polytechnic Institute, 1995. [5] J.W.Modestino and D.G.Daut, \Combined Source-Channel Coding of Images," IEEE Trans. Commun., vol. COM-29, pp. 1261{1274, Nov. 1970. [6] Y.H.Kim and J.W.Modestino, \Adaptive Entropy-Coded Subband Coding of Image Sequences," IEEE Trans. Commun., vol. 41, pp. 975{987, June 1993. [7] ISO/IEC, \Information Technology - Generic Coding of Moving Pictures and Associated Audio Information: Video." Recommendation H.262, ISO/IEC 13818-2, Draft International Standard, May 1994. [8] Y. H. Kim and J. W. Modestino, \Adaptive entropy coded subband coding of images," IEEE Trans. Image Process., vol. IP-1, pp. 31{48, Jan. 1992. [9] V.Parthasarathy, J.W.Modestino and K.S.Vastola, \Reliable Transmission of HighQuality Video over ATM Networks." submitted to IEEE Trans. on Image Proc. [10] G.Karlsson and M.Vetterli, \Packet Video and its Integration into the Network Architecture," IEEE J. Select Areas Commun., vol. SAC, pp. 739{751, June 1989. [11] V.Parthasarathy, \Transport Protocols for Multi-Resolution Transmission over ATM Networks." Ph.D. thesis, ECSE Dept., Rensselaer Polytechnic Institute, Troy, NY, 1995. [12] A.J.Viterbi and J.K.Omura, Principles of Digital Communication and Coding. New York: McGraw-Hill Book Company, 1979. [13] J.G. Proakis, Digital Communications. New York: McGraw-Hill Book Compan, Second ed., 1989. [14] J. Hagenauer, \Rate-Compatible Punctured Convolutional Codes (RCPC Codes) and their Applications," IEEE Trans. Commun., vol. COM-36, pp. 389{400, April 1988. [15] P.H.Westerink, J.Biemond, and D.E.Boekee, \An Optimal Bit Allocation Algorithm for Sub-band Coding," in ICASSP`88, (New York City), pp. 757{760, April 1988. [16] J.L.Massey, \Coding and Modulation in Digital Communications," in Proceedings Int. Zurich Seminar on Digital Communications, pp. E2(1) { E2(4), 1974. [17] T.Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression. Englewood Clis, NJ: Prentice Hall, 1971.

26