Signal Processing: Image Communicat

Multiple description imafor wireless channels

when submitted to different kinds of channels noise.

The use of mobile communication and multi-media communication knew an enormous increase

ARTICLE IN PRESS

*Corresponding author.

E-mail addresses: pereira@i3s.unice.fr (M. Pereira), am@in the latest decade, being the wireless channelsconsidered as a transport medium for varioustypes of multimedia information. However, thescarcity of wireless bandwidth, the time-varyingcharacteristics of the channel, and the power

i3s.unice.fr (M. Antonini), barlaud@i3s.unice.fr (M. Barlaud).

URLs: http://www/i3s.unice.fr/pereira, http://www.i3s.uni

ce.fr/barlaud.1On leave from Universidade da Beira Interior, Portugal.

Research partially supported by PRAXIS XXI Grant SFRH/

BD/1234/2000.

0923-5965/$ - see front matter r 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.imar 2003 Elsevier B.V. All rights reserved.

Keywords: Joint source channel coding; Multiple description coding; Bit allocation; Adaptability to channel model and state; Error

resilience video coding; Video over 3G wireless systems

1. IntroductionManuela Pereira* , Marc Antonini, Michel Barlaud

I3S Laboratory of CNRS, University of Nice-Sophia Antipolis, Algorithmes/Euclide B, 2000 route des Lucioles BP-121 06903

Sophia Antipolis Cedex, France

Abstract

We consider the problem of efcient image/video transmission over wireless channels. Such a problem involves good

compression rates and effectiveness in presence of channel failures. In this work, we use the multiple description coding

(MDC) techniques, based on Wavelet Transforms, that have been shown to be powerful against channel failures. We

propose a bit allocation procedure that dispatches the source redundancy between the different channels when

compressing to a target bit rate with a bounded side distortion. In this way, we develop an MDC scheme well adapted

to channel noise. Then, we extend this bit allocation to video. This extension uses the 3D scan-based DWT of Parisot

et al. (MMSP, IEEE, New York, 2001); that involves scan-based MDC with rate or quality control. In this work we

propose to adapt redundancy between the descriptors in function of channel model and state (BER). Furthermore, the

channels can have time-varying states. We consider BSC channels, Gaussian channels, and UMTS channels. The

method can be efciently extended to other channels when the channel model that matches the channel behavior is

known.

We evaluate the performance of the proposed MDC image/video coder for two descriptions. In a rst group of

simulations, we compare our application with some different MDC techniques presented by Vaishampayan (IEEE

Trans. Inform. Theory 39 (3) (1993) 821) and Servetto et al. (ICIP, IEEE, Chicago, USA, 1998; IEEE Trans. Image

Process. 9 (5) (2000) 813826). In a second group of simulations, we evaluate the robustness of the proposed MDC,1ge.2003.08.009ion 18 (2003) 925945

ge and video coding

ARTICLE IN PRESS

Imaglimitations of wireless devices, imposes tremen-dous challenges on wireless multimedia commu-nications. The present work involves image andvideo transmission over wireless channels.Given the bandwidth limitations, compression

will be required for image and video transmissionover wireless channels. Wireless video transmis-sion is specially difcult because the huge volumeof data required to describe a video greatly slowsdown transmission and then, involves the use oflossy compression at low bit rates. The transmis-sion delay is very important since information thatarrives too late at decoder is considered as lost.Moreover, data are corrupted by channel noise.

The channel noise can occur in the form ofrandom bit errors, bursty bit errors or packetlosses. Both problems result on losses of percep-tual quality at decoder. This made robust com-pression schemes very useful especially fortransmission at low bit rates.

1.1. Error control techniques or multiple

description coding (MDC)?

The classic technique to combat transmissionerrors is forward error correction (FEC)[5,6,8,27,34,35,38]. FEC involves the addition ofredundant data to the compressed signal, whichallows the decoder to correct errors up to a certainlevel. This redundancy increases the total numberof bits required and thus reduces compression.Moreover, FEC code must be designed with aworst case channel scenario in mind. For channelsthat have a highly variable quality, this worst casemay imply the need for a very powerful code, andhence highly or even prohibitive amount ofredundancy, which will severally reduce thecompression performance. In the case of bursterrors, the error correction capability is oftenexceeded or the block is error-free in which caseadditional redundancy is wasted. To overcome thislimitation, FEC is often enhanced by a techniqueknown as interleaving. For burst errors, thiseffectively reduces concentration of errors in singlecode words, more precisely, a burst of b con-secutive symbol errors causes a maximum of b=Msymbol errors in each code word. Thought

M. Pereira et al. / Signal Processing:926interleaving can be implemented with low com-plexity it suffers from increased delay, dependingon the number of interleaved blocks M : Thereforeinterleaving is a frequently used technique forbursty channels if additional delay is acceptable.These problems can be solved if used unequal lossprotection [20].Closed-loop error control techniques like Auto-

matic Repeat reQuest (ARQ) have been shown tobe more effective than FEC and successfully towireless video transmission [17]. Retransmission ofcorrupted data frames, however, introduces addi-tional delay. Moreover, it is possible to combineFECARQ to be successfully to wireless videotransmission [4,12,18].The frameworks for wireless communication

should join optimization of source coding andchannel coding, should present great robustnessand adaptability to adverse transmission conditionand should make efcient use of limited wirelessnetwork resources. In this way, a particular jointsource and channel coding method, known asMDC, has proven to be an effective way toprovide error resilience with a relatively smallreduction in compression ratio. This codingscheme assumes that there are several parallelchannels between the source and destination, andthat each channel may be temporarily down orsuffering from long burst errors. Furthermore, theerror events of different channels are independentso that the probability that all channels simulta-neously experience losses is small. These channelscould be physically distinct paths between thesource and destination in, for example, a wirelessmultipath network or a packet switched network.Even when only a single physical path existsbetween the source and destination, the path canbe divided into several virtual channels by usingtime interleaving frequency division, etc.In the MD problem (reduced to the simplest

case of two descriptions), a source is described bytwo descriptors with side rates R1 and R2: Thesetwo descriptions individually lead to reconstruc-tions with side distortions D1 and D2; respectively;the two descriptions together yield a reconstruc-tion with central distortion D0pD1 (and D2).Almost all multiple description codes to date

assume the existence of multiple independent

e Communication 18 (2003) 925945onoff channels between the transmitter and the

ARTICLE IN PRESS

Imagreceiver (e.g. Internet). When a link is broken, allof the symbols or packets passing through thatchannel are lost; when it is functioning properly,the symbols are transmitted error free. Someexceptions are presented in the following.In [14] the authors replace the onoff channel

model with wireless channel, where they assumethat the system will employ multiple transmit andmultiple receive antennas. In this work, theyobserve that it is possible to improve averagetransmission error probability by a proper choiceof the correlating transform. In [3] they introducefading models, such as Rayleigh, Rician, orNakagami channel, within the context of MDC.Their simulations show the efciency of MDC forfading channels with multiple antennas.Vaishampayan in [46] showed that for transmit-

ting information from memoryless Gaussiansource over a Rayleigh fading channel, the multi-ple description approach results in good perfor-mances at low interleaving delays as compared tostandard channel coding approaches. This conclu-sion was extended to sources with memory in [15]where, on an equal interleaving delay basis,signicant performance improvements are ob-tained over channels codes for speech transmissionon Rayleigh fading channels.The great amount of works dedicated to MDC

for onoff channels vs. the very reduced amount ofwork dedicated to MDC for wireless channels isexplained by the relatively large overhead asso-ciated with MDC that implies that when channelloss rate is small, the reconstruction performancein the error-free case dominates and the singledescription coding (SDC) perform best. So, ascited in [45] A challenging task is how to designthe MDC coder that can automatically adapt theamount of added redundancy according to under-lying channel error characteristics. It is thatchallenge we propose to respond in the presentwork.

1.2. Prior work on MDC

We can nd three distinctly different approachesof MD quantization. In the rst approach,pioneered by Vaishampayan, MD scalar, vector,

M. Pereira et al. / Signal Processing:or trellis quantizers are designed to produce twodescriptions, using a generalized Lloyd-like cluster-ing algorithm that minimizes the Lagrangian of therates and expected distortions R1; R2; D1; D2; D0[31,32,4042]: In the second approach, pioneered byWang, Orchard, and Reibman [44], MD quantizersare constructed by separately describing (i.e.,quantizing and coding) the N coefcients of an N N block linear transform, which has been designedto introduce a controlled amount of correlationbetween the transform coefcients [10,9]. In thethird approach, pioneered by Goyal, Kovacevic,and Vetterli, MD quantizers are constructed byseparately describing the N coefcients of an over-complete N K tight frame expansion [10,11].Other works were developed by Jiang et al. whopropose an MD extension to SPIHT coderpresented in [30], by separating Zerotrees intopolyphase components [13]. Rogers et al. proposeto rearrange bits at the output of one congurationof the SPIHT coder, in such a way that the loss ofone packet results in an error that does notpropagate beyond the image region contained inthat packet [29]. Mohr et al. propose the use oferror correcting codes of different strengths appliedto different portions of a progressive bit streamsuch as that generated by SPIHT coder [20].Previous MD coding dedicated to video was

proposed by Vaishampayan in [43]. A predictiveMD system was applied along with transformcoding to construct an interframe balanced MDvideo coder based on the H.263 standard. In [2],Apostolopoulos and Wee show that MD codingand path diversity provide improved reliability insystems with multiple paths with equal or un-equal bandwidths. In [28], Reibman et al. pro-posed MD video coders which use motioncompensated predictions. In [23], we propose anextension of [25] for video that uses the 3D scan-based DWT.

1.3. Main contributions of the paper

In this paper, we propose a joint source andchannel coding method that presents robustnessand adaptability to channel characteristics andstate. For its compression and synchronizationcapabilities, it is suitable for real-time transmis-

e Communication 18 (2003) 925945 927sion. This is specially true for the video case.

The proposed method uses an MD schemebased on the DWT and an efcient bit allocationtechnique. Our goal is to nd an optimal trade-offbetween efcient compression and robustness fromlosses due to communications using unreliablechannels. The new bit allocation technique isderived from [25] and adapted to the 3D scan-based DWT video coder presented in [23]. Here,we propose to control automatically the amount ofredundancy dispatched on the different descrip-tions by taking into account the channel modeland state. The 3D scan-based DWT transformallows us to develop a stripe-based MDC and so touse different redundancies to take into account

The paper is organized as follows: Section 2presents the problem statement and is followed bythe proposed bit allocation for multiple descrip-tion scheme in Sections 3 and 4. Section 5 presentsthe proposed algorithm and Section 6 presentsconcisely the channels models used in this work.Finally, results are presented in Section 7. Weconclude and propose some future works inSection 8.

2. Problem statement

Our MD scheme focus on the special case in

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Decoder

Contro

ontro

M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945928changes in channels state while coding.We show that the proposed MDC automatically

adapted to different channels and states, is efcientfor transmission over time varying channels. Inthis perspective, the proposed method is analternative to methods that use error controlschemes, such as FEC or ARQ.Our simulations are divided into two main

parts. The rst one serves to prove the effective-ness of our MDC scheme when compared to otherMDC schemes. The results presented show thatthe proposed MDC overcomes previous MDCschemes in the relation side PSNR vs. centralPSNR. The second group of simulations show therobustness of the proposed MDC when submittedto different kinds of channel noise. Simulations areperformed using the BSC channel, the Gaussianchannel, and the standard UMTS channel.

Rl( Dl ),

R1( D1),

R2( D2),

SequenceVideo

3DScan Based

DWT

BER

MDCQuality

orCRateFig. 1. 3D scan-basedR0( D0),

R2( D2),

channelnoisyCentral

Decoder

SideDecoder

bitstreaml

l bitstreamwhich there are two channels of equal capacitybetween a transmitter and a receiver. In such ascheme, a sequence of source symbols is given toan encoder to produce two independent bitstreamsof equal importance. These bitstreams are trans-mitted to three decoders over two noisy channels.One decoder (the central decoder) receives infor-mation sent over both channels while the remain-ing two decoders (the side decoders) receiveinformation only from their respective channel.The amount of redundancy is dispatched on thedifferent descriptions by taking into accountthe channel model and state (or bit error rate).The general coder is presented in Fig. 1.Given the side rate Rl and the side distortion

Dl ; the generation of the two descriptions isconstrained to some conditions that we detail inthe following.

R1( D1),SideMDC scheme.

solve in Section 3.

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Imag3. Proposed bit allocation for MDC

3.1. Proposed scheme for DWT

The problem is to nd, for a given redundancybetween the descriptors, which combination ofscalar quantizers across the various waveletcoefcients subbands will produce the minimumtotal central distortion while satisfying the side bitrates, and side distortions constraints.Then, the purpose of our bit allocation for MD

scheme is to determine the optimal sets ofquantization steps fqi;1; i 1;y;#SBg; fqi;2; i 1;y;#SBg; for descriptors 1 and 2. Moreprecisely, the bit allocation nds which combina-tion of quantizers in the various subbands of thetwo descriptors considered jointly minimizes thecentral distortion D0 (Condition 1) for a bit rate2Rl (Condition 2). This goal should be met whilethe side distortion is kept below a given distortionDl (Condition 3). The parameters Rl and Dl areCondition 1. The central decoder has to recon-struct the original sequence from the two descrip-tors with minimal central distortion D0:

Condition 2. An MDC coder must generate twodescriptors each with a side rate R1 R2 Rl :

Condition 3. The side decoders must reconstructthe original sequence from a single descriptor witha side distortion D1pDl and D2pDl :

The problem is then to minimize the centraldistortion D0 (Condition 1) when Conditions 2and 3 are veried. That is, for # SB the number ofspatio-temporal subbands, we have to nd thesets of bit rates fRi;1g; fRi;2g that minimizes thecentral distortion D0; where Ri;j is the bit rate ofsubband iAf1;y;#SBg for descriptor jAf1; 2g:More precisely, we have to nd the set ofquantization steps fqi;1g; fqi;2g that produce thesets of bit rates fRi;1g; fRi;2g: This problem isknown as the bit allocation problem we propose to

M. Pereira et al. / Signal Processing:given for the bit allocation (Fig. 1).This allocation problem is a constrained pro-blem which can be solved by introducing theLagrange operators. The Lagrangian functionalfor the constrained optimization problem, isgiven by

Jfqi;1; qi;2g D0 X2j1

ljRjpRl

X2j1

mjDjpDl; 1

where, for a source with generalized Gaussiandistribution, D0 can be written as [25]:

D0 X#SBi1

Dis2i;0Di;0qi;1

si;1;qi;2

si;2

; 2

where Di is an optional weight for frequencyselection.The expected central distortion is estimated

based on the channels states and the a priorichannels models as we will see in the next sections.

3.2. Central distortion modeling

Recall that the central distortion is the distor-tion of the decoded image when using bothdescriptors at decoding. When the decoder receivesboth descriptors, each subband appears twice,with different bit rates (different associated quan-tization steps). In this case, if the subbands arenoiseless, the central decoder chooses the sub-bands with the smaller quantization step and theothers, that we will call the redundant ones, areonly considered for side decoders. Hence, we cancalculate, the central distortion of the decodedimage as

P#SBi1 minDi;1; Di;2; where Di;1; Di;2

are the distortions of subband i for descriptors1 and 2, respectively. In the general case wehave to take into account channel noise. So, wecannot despise the redundant subbands as inthe noiseless case. Actually, the level of redun-dancy should increase when the BER increases,such in the case of very noisy channels where theredundant subbands are as important as theothers. In this case the central distortion cansimply be written as

P#SBi1 minDi;1; Di;2 P

e Communication 18 (2003) 925945 929maxDi;1; Di;2 #SBi1 Di;1 Di;2:

ARTICLE IN PRESS

ImagThen, introducing a weighting parameter rN tothe redundant subbands, called redundancy para-meter, we propose to write the central distortionfor a subband as

Di;0qi;1

si;1;qi;2

si;2

1

s2i;0

1

rN 1min s2i;1Di;1

qi;1

si;1

;

s2i;2Di;2qi;2

si;2

rN max s2i;1Di;1qi;1

si;1

;s2i;2Di;2

qi;2

si;2

;

3

where s2i;jDi;jqi;j=si;j is the mean square error forthe ith subband, in the case of a generalizedGaussian distribution.Eq. (3) can be simplied as

s2i;1s2i;0

1

rN 1Di;1

qi;1

si;1

s2i;2s2i;0

rN

rN 1Di;2

qi;2

si;2

if mins2i;1Di;1; s

2i;2Di;2

s2i;1 Di;1;

s2i;2s2i;0

1

rN 1Di;2

qi;2

si;2

s2i;1s2i;0

rN

rN 1Di;1

qi;1

si;1

otherwise:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

4

The amount of redundancy, i.e., the importance ofthe redundant subbands, depends on the channelBER and we propose in the following somestrategies for the choice of rN :

3.3. Redundancy parameter

Taking into consideration what we said above, itis easy to conclude that the redundancy parameterdomain is [0, 1]. rN 0 when the channel isnoiseless and rN 1 when is expected a very noisychannel. The problem is how to choose inter-mediate redundancies, and implicitly intermediatevalues of rN : Taking into account the Shannontheorem, [33, Theorem 10], we propose to solvethis problem using the equivocation Hyx: Indeed,in this theorem, Shannon states that the equivoca-tion Hyx is the amount of redundancy that the

M. Pereira et al. / Signal Processing:930decoder needs to correct the received message.Then, we propose to use (5) to compute theredundancy parameter:

rN Hyx

maxpxHx; 5

where px stands for the distribution of the inputchannel symbols.

Hyx is unknown at encoding; however, we cannd some bounds for equivocation by stating thefollowing proposition.

Proposition 1.

minpx

Hyxpmaxpx

Hx Cpmaxpx

Hyx; 6

where C is the channel capacity defined by C maxpxHx Hyx:

Proof. Being C maxpxHx Hyx we caninfer that CpmaxpxHx minpxHyx:Thus, minpxHyxpmaxpxHx C: For theright bound, we start also from the channelcapacity denition, C maxpxHx Hyxand we infer that CXmaxpxHx maxpxHyx maxpxHx maxpxHyx:Thus, we can conclude that maxpxHyxXmaxpxHx C: &

Therefore, instead of (5), we use (7) to compute theredundancy parameter.

rN maxpxHX C

maxpxHx: 7

We know that 0pHyxpHx: Thus,0pCpmaxpxHx: Using (7) we can concludethat 0prNp1 as pretended.We will show (in Section 6) that the channel

capacity is known for several interesting channelmodels. Thus, more details about the calculi of rNparameters for different channel models will bedone in that section.

3.4. Constraints

Conditions 2 Rj1;2pRl and 3 Dj1;2pDlhave to be dened for each descriptor. For the

e Communication 18 (2003) 925945different descriptors j 1; 2; we write Condition 2

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Imagas a constraint F1 such that

F1 X#SBi1

aiRi;jqi;j

si;j

Rl

!; 8

where ai is the quotient of the size of the subbanddivided by the size of the whole image (e.g., ai 1=22i in the dyadic case) and Ri;jqi;j; is the bit ratein bits per sample for the ith subband.Condition 3 is forced using a penalty. The side

distortions D1; D2 are dened by

Dj X#SBi1

Dis2i;jDi;jqi;j

si;j

; for all jAf1; 2g: 9

The penalty method is simple and efcient.Consider a constraint x > 0 the penalty is writtenas Px jxj x=22: If the constraint is veriedthen xX0 and Px 0: Otherwise, xo0 andPx x2: Considering the side distortions D1 andD2 dened by (9), the constraint is Dj Dlp0:The penalty is then written as

P1 jDj Dl j Dj Dl

2

2for all jAf1; 2g: 10

This penalty function allows us to nd a solutionwith

P#SBi1 Dis

2i;jDi;jqi;j=si;jpDl : In this case we

say that the penalty is veried.

3.5. Solution of the problem

Considering the central distortion given by (3)the constraints (8) and (10), the Lagrangianfunctional (1) can be rewritten as

Jfqi;1; qi;2g

X#SBi1

Dis2i;0Di;0qi;1

si;1;qi;2

si;2

X2j1

ljX#SBi1

aiRi;jqi;j

si;j

Rl

!

X2j1

mjjDj Dl j

2

Dj Dl2

2: 11

Solution of (11) is obtained when@Jfqi;1; qi;2g=@qi;1 0; @Jfqi;1; qi;2g=@qi;2 0

M. Pereira et al. / Signal Processing:and @Jfqi;1; qi;2g=l 0: In the following, wedetail the derivative with respect to qi;1:

@Jfqi;1; qi;2g@qi;1

Dis2i;0@

@qi;1Di;0

qi;1

si;1;qi;2

si;2

12a

l1ai@

@qi;1Ri;1

qi;1

si;1

12b

m1@

@qi;1

jD1 Dl j2

D1 Dl

2

2 0: 12c

From Eq. (4), (12a) can be derived as

s2i;1s2i;0

1

rN 1@

@qi;1Di;1

qi;1

si;1

if mins2i;1Di;1;s

2i;2Di;2

s2i;1Di;1;

s2i;1s2i;0

rN

rN 1@

@qi;1Di;1

qi;1

si;1

otherwise:

8>>>>>>>>>>>>>:

13

In this way, the derivation of the Lagrangianfunctional (12) becomes

@Jfqi;1; qi;2g@qi;1

Ci;1

1 rNDis2i;1

@

@qi;1Di;1

qi;1

si;1

l1ai@

@qi;1Ri;1

qi;1

si;1

m1@

@qi;1

jD1 Dl j2

D1 Dl

2

2 0; 14

where, from (13), Ci;1 is dened by

Ci;1 1 if mins2i;1Di;1; s

2i;2Di;2 s

2i;1Di;1;

rN otherwise:

(

15

Furthermore, using (9), @=@qi;1 P1 can be written as

0 if D1qi;1

si;1

pDl

2 D1 DlDis2i;1@

@qi;1

Di;1qi;1

si;1

otherwise:

8>>>>>>>>>>>>>:

e Communication 18 (2003) 925945 93116

i;11 rN

Dis2i;1 @qi;1Di;1

i;1

si;1

m1E1Dis2i;1

@

@qi;1Di;1

qi;1

si;1

0; 17

where, from (16), E1 2 D1 Dl if D1 > Dl or0 otherwise.Then, Eq. (17) yields

Ci;1 m1E1

Dis2i;1

@Di;1

qi;1

produced by the quantization step q; can beapproximated by the entropy such that

R XN

mN

Prmlog2 Prm;

where Z mqq=2

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1;m

*qa;q

1

Imag1 rN @qi;1 si;1

l1ai@

@qi;1Ri;1

qi;1

si;1

0: 18

Simplifying (18) and performing similar calculusfor @qi;2 permit to obtain the system (19) (forjAf1; 2g for a two-channels scheme.

@Di;j@Ri;j

qi;j

si;j

ljai

Dis2i;jCi;j1rN

mjEj ; 19a

X#SBi1

aiRi;jqi;j

si;j

Rl 0: 19b

Resolution of system (19) which has #SB 1equations and #SB 1 unknowns gives us theoptimal sets of quantization steps fqi;1g; fqi;2g:

The proposed algorithm is based on modeling ofR and D functions as we will show in next section.

4. Model-based R and D

In each subband the probability density func-tion of the wavelet coefcients can be approxi-

@Di;j@Ri;j

*q

PNm1

2@f@

pa;1 *q=22

ln f0;0a; *q l1ai@

@qi;1Ri;1

qi;1

si;1

Then, Eq. (14) becomes

@Jfqi;1; qi;2g@qi;1

C @ q

M. Pereira et al. / Signal Processing:932mated with generalized Gaussian. Therefore,Prm mqq=2

pa;sx dx

is the probability of the quantization level m:According to [37], the best decoding value, when

using the mean squared error (MSE) as distortionmeasure, for the quantization level m; is

#x

Rmqq=2mqq=2 xpa;sx dx

Prm:

Then, the MSE can be expressed as

D XN

mN

Z mqq=2mqq=2

x #x2pa;sx dx:

Setting *q qi;j=si;j ; it has been shown in [24,22]that @Di;j=@Ri;j *q can then be calculated as

with

@fn;m@ *q

a; *q m 1

2

n1pa;1 m *q

*q

2

"

m 1

2

n1pa;1 m *q

*q

2

*qn:

*f1;ma; *qf0;ma; *qf1;ma; *q2@f0;m

@ *q a; *q

f0;ma; *q2

PN

m1@f0;m@ *q a; *qln f0;ma; *q 1

ln 2 20we have

pa;sx aejbxja

with

b 1

s

G3=aG1=a

sand a

ba2G1=a

:

For a given subband, the coder output bit rate R

e Communication 18 (2003) 92594521

5. Proposed algorithm

5.1. Ci;j parameter

The Ci;j parameter dened in Eq. (19) dependson s2i;1Di;1 and s

2i;2Di;2: One problem is that the

distortions involved are unknown before system(19) is solved due to their dependence on thequantization steps fqi;1g; fqi;2g: We propose analgorithm that initializes the Ci;j parameters to 1and iteratively modies their values according tothe current s2i;1Di;1 and s

2i;2Di;2: More precisely, if

we dene S as the set of all possible subbands ofdescriptor 1 and 2, we search the subband in Swith the highest distortion. Then we set thecorrespondent Ci;j value to rN : Note that in eachiteration the subbands of both descriptors suchthat Ci;j was already set are excluded from thesearch set S: It is easily veried that #SBiterations are always performed. This algorithmis detailed in Fig. 2.

5.2. @D=@R; R and D functions

The functions in (19) are not easily invertible.However, for generalized Gaussian model, it hasbeen shown in [22] that @D=@R; R and D functionscan be tabulated in order to simplify the inversion.The tables for these functions can be found in[22,24] for different generalized Gaussian shapeparameters a sampled in ]0, 2].

5.3. Global bit allocation procedure

As can be seen in Fig. 3, we compute Ri;j usingthe given parameters Ci;j ; lj ;mj ; rN ; Eq. (19a) andtabulated @D=@R functions. If the debit constraint(19b) is not veried, we recompute the Ri;j using

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M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945 933Fig. 2. Computation of Ci;j parameter. Fig. 3. Global bit allocation procedure.

new lj : If it is veried we compute: qi;j using Ri;jand the tabulated R functions; Di;j using tabulatedD functions; Dj using Di;j and (9); and nally Ci;jusing Di;j :The above steps are performed #SB times. After

#SB iterations all Ci;j are computed and thealgorithm can proceed.The last step of bit allocation is the verication

of the penalty (9). If this is not veried, we reiteratefrom the beginning with a new m parameter. If the

6. Channel model and redundancy

The two main elements which describe a channelare the transmission rate and the channel capacity.The transmission rate was dened by Shannon asR Hx Hyx for discrete or continuous chan-nels. Where Hx is the entropy of the input andHyx the conditional entropy or the equivocation.The channel capacity C is dened as the maximum ofR when the input varies over all possible collectionpx: This channel capacity was dened by Shannon

ARTICLE IN PRESS

on

Q

Q

M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945934penalty is veried, the algorithm stops and theoutput of bit allocation gives the optimal quanti-zation steps qi;j :

5.4. Bit allocation complexity

The complexity of our algorithm is the complex-ity of the EBWIC bit allocation presented in[25,24]. The complexity of this coder is presentedin [24] and detailed in [21]. In this work, theauthors conclude that the highest cost of model-based allocation method corresponds to thecomputation of generalized Gaussian distributionparameters for each subband. They need fouroperations (two additions and two multiplications)for wavelet coefcient to compute s and aparameters. Assuming that, complexity of theremaining part of the algorithm is lower than 1operation for each image pixel. The authorsconclude that the complexity of such allocationmethod is less than ve simple arithmetic opera-tions for each image pixel.We can then conclude that our bit allocation

complexity is less than ve operations for eachimage pixel.

Rl( Dl ),

BitAllocati

coefficient

Wavelet

subbandBERFig. 4. General Mfor some special cases. The BSC and the Gaussiancases. However, in most mobile radio systems, thechannel exhibits Rayleigh fading, aggravated bytypically log-normally distributed shadowing or slowfading, resulting in a time variant channel capacity.Lee [16] derived an estimate of the channel capacity inRayleigh fading environments.In this paper we focus on binary symmetric channel

(BSC), Gaussian and Rayleigh channel models.

6.1. Binary symmetric channel

For the BSC case we have two possible symbolseach with a probability p of coming throughundisturbed, and 1 p of being changed into theother of the pair. The capacity can be written asC 1 p log2 p 1 plog21 p bits/symbol.maxHx 1 for this channel model. Thus theredundancy parameter presented in Section 3.3can be estimated by

rN maxpxHX C

maxpxHx

p log2 p 1 plog2 1 p: 22

R1( D1),

R2( D2),

Scalaruantization

EntropyCoder

EntropyCoder

ScalaruantizationDC scheme.

memory only until these coefcients have beenencoded [26]. No motion compensation is per-formed.Fig. 1 shows the complete scan-based MDC

scheme. The bit allocation procedure, presented inSection 3, is followed by a simple scalar quantiza-tion and the encoding of each subband uses acontext based arithmetic coder [22,39] as presentedin Fig. 4.

7.1.1. Smart arithmetic coding

In order to provide synchronization and mini-

ARTICLE IN PRESS

Fig. 5. Side PSNR vs. central PSNR

Fig. 6. BSC channel at BER 0:001: Image coded at 0:5 bpp:PSNR 29:41 dB:

Imag6.2. Additive white Gaussian noise channel

For a band-limited Additive White GaussianNoise (AWGN), the channel capacity can beexpressed as C=B log21 g bits/symbol, whereB is the channel bandwidth in symbol/s and g is thesignal to noise ratio (SNR). The SNR g is denedas g S=N ; where S is the received signal powerand N is the AWGN power within the channelbandwidth. For this channel model, maxHxdepends on the modulation. For instance, if weconsider a QPSK modulation, maxHx 2 andin this case the redundancy parameter in Section3.3 is estimated by

rN maxpxHX C

maxpxHx2 log21 S=N

2

1log21 S=N

2: 23

6.3. Rayleigh channel

In the case of Rayleigh models, an upper boundapproximation for the normalized channel capa-city was introduced by Lees [16] asC=BElog2 e e

1=ge ln g 1=g bits/symbol.For this channel model, maxHx depends alsoon modulation. For instance, if we consider aQPSK, maxHx 2 and in this case theredundancy parameter in Section 3.3 is written as

rN maxpxHX C

maxpxHx

1log2 e e

N=Se ln S=N N=S2

: 24

7. Results

7.1. Scan-based DWT video coder

For spatial decomposition our coder uses 9-7biorthogonal lter [1] and three level of decom-position. For temporal decomposition it uses the(2,2) lter and performs two level of decomposi-tion. The frames of the video sequence areacquired and processed on the y to generate the

M. Pereira et al. / Signal Processing:3D wavelet coefcients and the data are stored ine Communication 18 (2003) 925945 935mize the error propagation in the case of channel

ARTICLE IN PRESS

for le

Imagerrors, each spatio-temporal subband is dividedinto blocks. Then, arithmetic coding is performedon each block independently. Since the blockdivision requires side information, the block-sizemust be related to the channel BER such that highBER implies small block-sizes. For error detec-tion, when the number of coded coefcients isknown, it is possible to verify if the arithmeticcoder stops correctly. In case an error occurs andthe arithmetic coder is misplaced in the bitstream,we synchronize the decoder to start at the

Fig. 7. Lena image coded at 1 bpp: BSC channel at BER 0:001right image PSNR is 24:99 dB:

M. Pereira et al. / Signal Processing:936beginning of the next block.It is well known that this usual method of error

detection is largely insufcient. It is not rare tohave the correct number of coefcients decodedand the arithmetic coder well placed, but wrongcoefcients decoded. This problem is more difcultto solve. Some methods try to use the informationof the neighbor blocks, but their effectiveness isnot optimal. Here, we propose to use someinternal information included in the arithmeticcode. The arithmetic coder works with two mainregisters: the code string that represents the base ofthe interval (lower bound) and the interval. Witheach binary decision the current probabilityinterval is subdivided into two subintervals, andthe registers are modied in accordance. The statesof registers depend only on the probability of theindividual events. In our scheme, we include the nlast bits of the interval register inside the nalbitstream (in our experiments n 8 was sufcientfor BERX0:01: The decoder simply veries if itsnal interval register correspond to the oneincluded by the coder inside the bitstream. If it isnot the case, it means an error occurs in the blockand then, it is discarded.

7.2. Simulations

For simulations, we use the 512 512 pixelsLena image, 3 s of QCIF silent video and 3 s of

ft image and 0.01 for right image. Left image PSNR is 33:89 dB;

e Communication 18 (2003) 925945QCIF akiyo video. For Lena image, the total bitrate was set to 1 bpp (i.e. Rl 0:5 bpp) or 0:5 bpp(i.e. Rl 0:25 bpp). The silent video was com-pressed to 200 kbits=s (30 frames/s) and the akiyovideo was compressed to 300 kbits=s (30 frames/s).All channel simulations were performed 10 times.Note that mean PSNR values are computed byaveraging decoded MSE values and then convert-ing the mean MSE to the corresponding PSNRvalues. Visual results of video presents always theframes 1, 11, 21, 31, 41, 51, 61, 71, 81 and 91 of thevideo.

7.2.1. Central PSNR vs. side PSNR for still image

For 1 bpp central bit rate and Lena image,central PSNR vs. side PSNR is plotted in Fig. 5 forvalues of rN between 0 and 1. Our results arecompared to the referenced methods: [19]Ref.[13]Ref. [31]Ref. 3. We present two different

results. In the rst one (New 1) bit allocationprocedure was performed with Cj f1; rN ; 1;rN ;yg; resulting in a very simple MDC scheme.In the second one (New 2), we found the best set ofCj using the algorithm of Section 5. The New 2method presented here provides the best results.

7.2.2. Results for still image

Figs. 69 show some visual results of transmis-sion of Lena image at 1 and 0:5 bpp over BSC andGaussian channels with BER 0:001 and 0.01.The mean PSNR we obtained in the case of0:5 bpp and BER 0:001 was 28:62 dB and for aBER 0:01; it was 24:79 dB:As mentioned in Section 1.1, we can nd great

amount of works dedicated to MDC for onoffchannels but very reduced amount of work

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Fig. 8. Gaussian channel at BER 0:001: Image coded at0:5 bpp: PSNR 31:27 dB:

Fig. 9. Lena image coded at 1 bpp: Gaussian channel at BER 0:0034:90 dB; right image PSNR is 25:49 dB:

Table 1

Mean PSNR (dB) results of QCIF silent video, coded at

200 kbits=s: Channel transmission at BER 0:01; same rN forthe two cases (with and without noise)

Channel Y U V

BER BER BER

0 102 0 102 0 102

UMTSIndoor 31.47 25.57 39.06 38.56 40.06 39.76

UMTSPedestrian 31.47 28.02 39.06 38.67 40.06 39.89

UMTSVehicular 31.47 26.26 39.06 38.58 40.06 39.68

M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945 9371 for left image and 0.01 for right image. Left image PSNR is

Table 2

Mean PSNR (dB) results of QCIF silent video, coded at

200 kbits=s: Channel transmission at BER 0:001; same rN forthe two cases (with and without noise)

Channel Y U V

BER BER BER

0 103 0 103 0 103

Gaussian 33.17 31.50 40.72 40.00 41.51 41.15

UMTSIndoor 31.54 30.08 39.11 39.07 40.09 40.09

UMTSPedestrian 31.54 31.45 39.11 38.99 40.09 40.04

UMTSVehicular 31.54 31.45 39.11 39.04 40.09 40.060

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Y

Image Communication 18 (2003) 92594510

15

20

25

30

35PS

NR

M. Pereira et al. / Signal Processing:938dedicated to MDC for channels as BSC orGaussian. In [36] the authors constructed anMDC for networks with packet lost and/or biterrors. They provide a mechanism for bit alloca-tion between the redundancy in terms of FECs andredundancy that is meant to correct for packetloss. In this work, they present same results withLena image compressed at 0:5 bpp: When BSCtransmission for a BER 0:001; the PSNR is27:5 dB and for a BER 0:01 the PSNR obtainedis 20:2 dB: Showing that our MDC outperformsthis one. Moreover, the results of [36] were

0

5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49Fram

Fig. 10. Mean PSNRs of each frame for Y component for Gaussian

MDC; yellow: SDC; light blue: SDC TC:

U

0

5

10

15

20

25

30

35

40

45

PSNR

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 4Fra

Fig. 11. Mean PSNRs of each frame for U component for Gaussian

MDC; yellow: SDC; light blue: SDC TC:obtained with seven levels of decomposition andassuming the LL band uncorrupted.

7.2.3. Results for video

For video, we present results for Gaussian andUMTS channels. UMTS channel2 presents threedifferent models: indoor, pedestrian, and vehicu-lar. They are dened in [7].

52 55 58 61 64 67 70 73 76 79 82 85 88 91 94e

channel at BER 0:001; pink: no noise; dark blue: proposed

9 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94me

channel at BER 0:001; pink: no noise; dark blue: proposed

2The authors wish to thank France Telecom for providing a

UMTS simulator.

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V

Imag20

25

30

35

40

45PS

NR

M. Pereira et al. / Signal Processing:To compare with, we present results obtainedwith the proposed MDC when subject to noise ornot. We present also some results obtained whentransmission of a singular description coder (SDC)with a similar codec. In this case the video iscompressed at 200 kbits=s: We also use an SDCwith a Turbo Coder SDC TC with a bit rate of200 kbits=s including the channel rate.In Tables 1 and 2 are presented the mean PSNR

for BER 0:001 and 0.01 for the proposed MDC.

0

5

10

15

F

Fig. 12. Mean PSNRs of each frame for V component for Gaussian

MDC; yellow: SDC; light blue: SDC TC:

Y

0

5

10

15

20

25

30

35

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

Fra

PSNR

Fig. 13. Mean PSNRs of each frame for Y component for UMTS I

MDC; yellow: SDC case.e Communication 18 (2003) 925945 939We compare with the case without noise. We cansee that the performances of the proposed MDC inthe presence of noise are slightly the same thanwithout noise, especially for the U and Vcomponents. For the Gaussian case and BER 0:001; same simulations with the SDC TCsimulator gives the following PSNR: Y PSNR 28:66 dB; UPSNR 36:50 dB and VPSNR 38:12 dB: We can see that the proposed MDCprovides a gain of 3 dB over a standard method.

rame

channel at BER 0:001; pink: no noise; dark blue: proposed49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94

me

ndoor channel at BER 0:01; pink: no noise; blue: proposed

ARTICLE IN PRESS

Y

Imag15

20

25

30

35PS

NR

M. Pereira et al. / Signal Processing:940Figs. 10, 11 and 12 present the mean PSNR ofdifferent frames for Y, U, and V, respectively,when transmission over Gaussian channel with aBER 0:001: We can conclude that the proposedMDC and the SDC TC are able to recover fromchannel losses. However, the MDC presents betterPSNRs since the SDC TC is designed for aworst case transmission. In these gures, we cansee that the Y component is the most sensitive to

0

5

10

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46Fra

Fig. 14. Mean PSNRs of each frame for Y component for UMTS Pe

MDC; yellow: SDC case.

Y

0

5

10

15

20

25

30

35

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

PSNR

Fig. 15. Mean PSNRs of each frame for Y component for UMTS V

MDC; yellow: SDC case.e Communication 18 (2003) 925945noise. In the rest of the paper, we present onlyresults for this component.Figs. 1315 present the mean PSNR of different

frames for Y when transmission over UMTSchannel for the three different models, respec-tively. The BER in this case is 0.01.Finally, we show some visual results in Figs. 16

and 17 for Gaussian transmission for BER 0:001:These images allow to compare the proposed MDC

49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94

me

destrian channel at BER 0:01; pink: no noise; blue: proposed49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94

Frame

ehicular channel at BER 0:01; pink: no noise; blue: proposed

with the SDC TC: Fig. 18 represents silent videofor UMTS transmission for BER 0:01:

8. Conclusion

In this paper, we proposed an MDC schemefor noisy time-varying channels. It includes

scan-based DWT and an efcient bit allocationprocedure that dispatches source redundancybetween the different channels. The amount ofredundancy is estimated based on the channel stateand the a priori channel model. The scan-basedDWT allows the development of a stripe-basedMDC that takes into account channel changes intime.

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M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945 941Fig. 16. Gaussian channel at BER 0:001: Two left columns using the proposed MDC and two right columns with the SDC TC:

ARTICLE IN PRESS

ImagM. Pereira et al. / Signal Processing:942We show that the proposed MDC is asimple alternative for real-time video transmissionwhere methods that use error control schemessuch as FEC or ARQ are not suitable for delayreasons.With MDC, a long burst error or even the loss

of an entire description does not have a cata-strophic effect, as long as not all bitstreamsexperience failures simultaneously. Thus, one

Fig. 17. Gaussian channel at BER 0:001: Two left columns using te Communication 18 (2003) 925945could even use fewer error control bits for eachbitstream.

Acknowledgements

The authors want to acknowledge the anon-ymous reviewers for their advice which improvedthe quality of the paper.

he proposed MDC and two right columns with the SDC TC:

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M. Pereira et al. / Signal Processing: Image Communication 18 (2003) 925945 945

Multiple description image and video coding for wireless channelsIntroductionError control techniques or multiple description coding (MDC)?Prior work on MDCMain contributions of the paper

Problem statementProposed bit allocation for MDCProposed scheme for DWTCentral distortion modelingRedundancy parameterConstraintsSolution of the problem

Model-based R and DProposed algorithmCi,j parameterpartD/partR, R and D functionsGlobal bit allocation procedureBit allocation complexity

Channel model and redundancyBinary symmetric channelAdditive white Gaussian noise channelRayleigh channel

ResultsScan-based DWT video coderSmart arithmetic coding

SimulationsCentral PSNR vs. side PSNR for still imageResults for still imageResults for video

ConclusionAcknowledgementsReferences