3628 ieee transactions on signal processing,...

3628 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 7, JULY 2012

Symmetric Scalable Multiple DescriptionScalar Quantization

Shahid M. Satti, Student Member, IEEE, Nikos Deligiannis, Member, IEEE, Adrian Munteanu, Member, IEEE,Peter Schelkens, Member, IEEE, and Jan Cornelis, Senior Member, IEEE

Abstract—Real-time data delivery over best-effort error-pronepacket networks has invigorated the study of robust codingschemes, such as scalable multiple description coding (SMDC).In this context, the paper introduces a novel generic symmetricscalable multiple description quantizer (SSMDSQ) which gener-ates perfectly balanced source descriptions. Novel embedded indexassignments are proposed which are used to realize high, as well asmedium-to-low redundancy SSMDSQs. Compared to existing de-signs, it is shown that the proposed quantizer constructions exhibitsuperior distortion-rate (D-R) performance. Moreover, this paperdescribes an innovative extension of the Lloyd-Max algorithm inorder to optimize symmetric and asymmetric scalable multipledescription quantizers. For a family of Generalized Gaussian (GG)source distributions, the proposed optimization algorithm yieldson average a significant D-R performance gain over unoptimizedquantizers. Furthermore, anchored in the designed SSMDSQs,an SMDC framework is established to realize packet-based trans-mission over erasure channels. In this framework, transmissionstrategies are determined for scenarios wherein the average packetloss rate over the transmission link is (a) unknown and (b) can beestimated at the encoder. For both scenarios, SMDC packetizedtransmission is simulated for a family of GG distributions. Exper-imental results confirm that, compared to contemporary schemes,the designed quantizer constructions (with or without optimiza-tion) account for a significant average gain in signal-to-noise ratio(SNR) for a wide range of packet loss rates.

Index Terms—Embedded quantization, error-resilience, mul-tiple description coding, scalable multiple description quantizer.

I. INTRODUCTION

M ULTIPLE DESCRIPTION CODING (MDC) hasemerged as an attractive solution for error-resilient

coding in error-prone packet-switched networks [1]. In contrastto single description coding (SDC), MDC benefits from thetransmission of several correlated source descriptions over

Manuscript received July 18, 2011; revised December 02, 2011 and February20, 2012; accepted February 29, 2012. Date of publication March 20, 2012;date of current version June 12, 2012. The associate editor coordinating the re-view of this manuscript and approving it for publication was Prof. Eduard A.Jorswieck. This work was supported by the government agency for Innovationby Science and Technology (IWT)—Flanders (OptiMMA project) and the post-doctoral mandate of Peter Schelkens and Project G014610N.S. M. Satti, N. Deligiannis, A. Munteanu, and P. Schelkens are with

the Department of Electronics and Informatics (ETRO), Vrije UniversiteitBrussel (VUB), B-1050 Brussels, Belgium. They are also with the In-terdisciplinary Institute for BroadBand Technology (IBBT), Departmentof Future Media and Imaging, B-9050, Ghent, Belgium (e-mail: [email protected]; [email protected]; [email protected];[email protected]).J. Cornelis is with the Department of Electronics and Informatics (ETRO),

Vrije Universiteit Brussel (VUB), B-1050 Brussels, Belgium (e-mail: [email protected]).Digital Object Identifier 10.1109/TSP.2012.2191547

multiple (physical or logical) unreliable channels. Any subsetof these descriptions received at the decoder, can be used toreconstruct the source with a certain fidelity. MDC avoidsretransmission of lost packets, and thus descriptions, if a de-coder’s quality constraint is met upon reception of a subset ofdescriptions. In real networks with a large number of receivers,retransmission of lost packets is not always feasible since itcan lead to network congestion [2]. Moreover, in low delayapplications, such as video conferencing, peer-to-peer videostreaming [3], and real-time delivery of video [4], [5], theuse of retransmission for error resilience is typically not anoption since it introduces a significant structural delay which isprohibitive for such applications. In these cases, MDC formsan appealing alternative.For two-description MDC, the optimal achievable distor-

tion-rate (D-R) regions for generic source probability densityfunctions (pdfs) have been derived by the information theoreticworksof [6] and [7].Later, thefindingsof [8]–[10] have extendedthe theoretically achievableD-R regions to any number of sourcedescriptions. Practical multiple description code designs includepair-wise correlating transforms [11], [12], polyphase trans-forms [13], and multiple description scalar quantization [14],[15]. In the context of the latter, initially, fixed-length multipledescription scalar quantizers (MDSQs) [14] were introduced,while later, significant performance improvements over [14]were reported with their variable-length extensions [15].Existing MDC schemes, e.g., [11]–[15], consider designs

for certain pre-known channel conditions, e.g., availablebandwidth, packet loss rate, etc. However, in real networks,transmission conditions are often variable, meaning that, forlow delay applications, adaptation to varying conditions needsto be performed in real-time. In addition, real-time applicationsare deployed over networks with a high degree of heterogeneityin terms of available bandwidth, user requests and loss char-acteristics, thus simultaneously demanding error-resilience [1]and a scalable representation of the source [16].Scalable MDC (SMDC) combines both scalability and

multiple description based error-resilience in a single codingframework. In SMDC each source description is made scal-able by employing layered coding (LC), hence facilitating itsprogressive encoding and decoding. A scalable MDSQ basedSMDC framework was initially proposed in [17], whereinthe concept of embedded index assignment (EIA) was firstintroduced to achieve separate and combined scalability oftwo source descriptions. The EIA strategy of [17] permits alimited control over redundancy between descriptions. In orderto produce an arbitrary number of scalable descriptions as wellas to permit an enhanced control over the redundancy between

1053-587X/$31.00 © 2012 IEEE

SATTI et al.: SYMMETRIC SCALABLE MDSQ 3629

the quantization levels, embedded MDSQs (EMDSQs) wereintroduced in [18]–[21]. Preliminary EMDSQ instantiations[18], [19] were designed using connected partition cells toyield high redundancy descriptions. In [20], these instantiationswere combined with a minimal-redundancy EIA to achievelow-to-medium overall redundancy. Notice that the main de-sign objective of the EMDSQs of [18]–[21] was the use of adouble dead-zone uniform central quantizer at all quantizationlevels. In scalable SDC, such a quantization strategy is alsotermed as successive approximation quantization (SAQ) [22],which is acknowledged to be near-optimal [23]. This attributesignificantly improved the D-R performance of EMDSQs[18]–[21] over competing designs [17].Utilizing the error-resilience capabilities of MDSQs, a

number of MDSQ-based multimedia data (e.g., image, 3Dmesh) transmission systems [24], [25] have been proposed inthe past. Furthermore, anchored in the scalable MDSQ of [19],an erasure-resilient, multicast video coding architecture wasproposed in [5]. Compared to a data-partitioning-based videocoder [26], the former accounts for an improved compressionversus error-resilience tradeoff for video streaming applicationsover best-effort networks. Other works employing scalableMDSQs for error-resilient transmission of image/video signalsinclude [17], [27], and [28].In packet switched networks, packet losses often occur due

to oversaturated network links or channel degradations. In suchconditions, one can discriminate between channel-unawareand channel-aware encoding methods. For channel-unawareencoding, an SMDC transmission system would optimallysuggest a balanced transmission. In the channel-aware case,the channel knowledge can be exploited to accomplish a moreefficient unbalanced SMDC transmission. In today’s networksboth situations can simultaneously occur. Hence, a singlesystem design is needed to get aligned with the aforementionedtransmission situations.In order to meet these requirements, this paper introduces

a generic two-description symmetric scalable MDSQ (SS-MDSQ), which yields a perfectly symmetric D-R function,i.e., , where the first and the secondargument of denote the rate of the first and the secondsource description, respectively. This is in contrast to existingapproaches [18]–[21], which are characterized by only an ap-proximately symmetric or sometimes even highly asymmetricD-R function. Novel connected-cell and disconnected-cellconstructions are presented which can realize both high andmedium-to-low redundancy SSMDSQs. For GeneralizedGaussian (GG) distributions [29], the proposed SSMDSQs areshown to deliver improved D-R performance versus existingasymmetric quantizers.In addition, this paper presents a generalization of the

Lloyd-Max algorithm [30], [31] to realize locally-optimal,level-constrained and entropy-constrained SSMDSQs. Theproposed algorithm iteratively adjusts quantization thresholdsin order to search for the local minimum of a cost function. It isworth pointing out that the proposed optimization methodologyis not specific to SSMDSQs and can be utilized to optimize anyscalable MDSQ. To the best of the authors’ knowledge, thisis the first work in the related literature which deals with theoptimization of scalable MDSQs.

What is more, based on the designed SSMDSQs, apacket-based SMDC framework is established which canmatch the available channel rate while guaranteeing a fixedaverage distortion at the decoder. In this framework, a balancedtransmission of descriptions is performed when no knowledgeof the channel’s packet loss rate is available at the encoder.Conversely, when the encoder is aware of the channel statistics,a greedy packet scheduling scheme is formulated so as to carryout an unbalanced transmission. In this context, the symmetryof the SSMDSQ-based SMDC’s D-R function reduces the com-plexity of determining optimum packet scheduling comparedto asymmetric contemporary SMDC systems. Experimentalevaluations of the transmission of GG sources over packet losschannels reveal that, within the considered SMDC framework,the proposed quantizer constructions and the optimizationstrategy contribute significant average SNR gains comparedto existing designs. Similar conclusions are obtained when theproposed scheme is utilized for scalable wavelet-based imagetransmission.The remainder of the paper is structured as follows. Scalable

MDSQ and EIA are briefly described in Section II. Section IIIestablishes the sufficient conditions for the design of SS-MDSQs and details novel connected- and disconnected-cellquantizer constructions. Our generalization of the Lloyd-Maxalgorithm to optimize SSMDSQs is presented in Section IV,while Section V evaluates the performance of the consideredquantizer-based SMDC system for transmission over packetloss channels. Finally, Section VI draws the conclusions of thiswork.

II. SCALABLE MULTIPLE DESCRIPTION SCALAR QUANTIZER

A fixed-rate MDSQ [14], [15], [32] quantizes each sourcesample to two or more quantization indices which individu-ally yield a coarse, and jointly produce a fine reconstructionof the source. A scalable MDSQ [17], [18] is a special type ofMDSQ which additionally allows for scalable decoding of pro-duced quantization indices. An -description scalable MDSQcan be fully described by a set of embedded side quantizers

each having embedded levels ,where denotes the finest and denotes the coarsestlevel, respectively. The quantizer partition of is denoted as. For any side quantizer , the partition cells of level are

embedded in the ones of level , i.e., , where. We aim to quantize a zero-mean, independent

and identically distributed (IID) ergodic sourcewith a symmetric pdf . At the encoder,source descriptions—also termed as side descriptions, are

produced using the side quantizers. In particular, for theth side quantizer, a given source sample is first mapped

to an index using the coarsest level partition . At anyfiner level , the source sample is refined to asmaller sub-cell as defined by the refinement symbol . A se-quence of quantization indices, denoted by ,is progressively coded using fixed-length coding (FLC), e.g.,natural binary coding (NBC), or variable-length coding (VLC),e.g., adaptive arithmetic coding [33], thereby producing the thsource description. At the decoder, assuming that all descrip-tions up to a certain level are received, a central quantiza-tion cell , is de-


termined for each source sample. Then, the source sample re-construction is given by the centre-of-mass (centroid) of the de-rived cell. In general, all descriptions need not be received atthe same level, yet one can still determine a joint partition cellfor possibly different quantization levels .In this case, the central quantizer is defined by the corre-sponding central partition , as follows:

(1)

with the convention that consist of a single cell, whichspans the entire source range .

A. Embedded Index Assignment (EIA)

For simplicity, we confine our discussion to the two-de-scription case. Let us consider an IA matrix ,which assigns quantization indices to a side quantizer’s cellpartitions. Starting from single-cell side quantizers, i.e.,

is recursively split times alongeach dimension to create an EIA of levels [18]–[21]. Forinstance, for a particular combination of levelscan be expressed as a block matrix of the form:

(2)

where, a nonempty block of index entries, , defines a spe-cific partition cell of the quantizer . Note that, dependingon the organization of the index entries in , certain blocksmaybe completely or partially empty. We also notice that

and define cells of side quan-tization levels . In general, one observes that

and at levels and, respectively. Hence, using the above notation, side

cells can be referred to as cells at levelsand , respectively.

In order to realize the next level, i.e., , afurther splitting of each of (2), along each dimension, isperformed, and so on. Moreover, notice that at level ,each block contains at most a single index entry.

B. DefinitionsLet denote the side quantization levels

up to which side descriptions are received at the decoder.If denotes the joint reconstruction random variable (RV)then the mean squared error (MSE), which is considered as thedistortion metric, can be formulated as:

(3)

where is the input pdf, denotes the induced centralpartition, is the quantization cell corresponding to the thcentral cell, is the expectation operator, and denotesthe cardinality, i.e., the number of cells in . For any level

, the central distortion and the sides distortionsare defined as

(4)

where, denote RVs defined by the central and the thside quantizer’s codebook, respectively. To compute the entropyof each side quantizer, we define a RV using the probabilitymass function (PMF) of the th level partition of the th sidequantizer, i.e., , where

is the th cell of . The entropy for the th sidequantizer is then expressed as:

(5)

Notice that scalable MDSQ is considered to be balancedif for any level and

.

III. SSMDSQ

In this section, we determine the sufficient conditions for re-alizing an SSMDSQ. Let denote the distortion whena certain combination of levels is received. Anordinary balanced scalable MDSQ cannot ensure the same de-coding distortion for the permutation of . On theother hand, an SSMDSQ always guarantees an equal distortionfor both permutations, i.e., .Symmetric Cells: Let the index entries of be

. The partition cells and are defined asbeing symmetric to each other if for any index in there isan index in . The symmetry between andis denoted as .Symmetric EIA: An EIA is defined as being symmetric if each

partition cell for any combination has a uniquesymmetric partition cell for .In Fig. 1(a), a symmetric EIA is formed by employing a uni-

form four-level splitting, of a certain arrangement, of the entriesin . The symmetric cell pairs at any splitting level are givenbelow:

An example of a symmetric cell pair in corresponding to theIA in Fig. 1(a) is given in Fig. 1(b).Remark 1: In the example of Fig. 1(a), the chosen IA matrixis actually the modified linear (ML) IA used to define the

fixed-rate (nonscalable) MDSQ. For more details on fixed-rateMDSQs, we refer to [14]. In general, all fixed-rate IAs proposed


Fig. 1. (a) Example of a symmetric EIA. “V” denotes vertical split, while “H” denotes the horizontal split of the IA matrix. (b) The symmetric cell pairof the finest central quantizer is highlighted using dashed area under the pdf.

in the literature can be used to create scalable MDSQs by em-ploying bit-slicing of quantization indices. However, in contrastto fixed-rate IAs where the spread is only minimized along thecolumns and the rows of the IA matrix, an explicit design ofEIA must consider the spread minimization along rectangles orsquares cells, similar to the ones shown in Fig. 1(a).Theorem: For a symmetric zero-mean source pdf , a bal-

anced SSMDSQ is always realized if the EIA is symmetric andthe thresholds of are defined using an even function of .

Proof: If the EIA is symmetric and the thresholds ofare defined using an even function of , the particular struc-ture of the index entries within a symmetric cell pair ensure thattheir thresholds have equal absolute values—see the examplein Fig. 1(b). For a symmetric source pdf, this results in equaldistortions of symmetric cells. For a symmetric EIA, whereineach cell of a combination has a unique sym-metric cell in , the total distortion , whichis the sum of distortion contributions from each cell, is alwaysequal to . This confirms the symmetry of the distortionof SSMDSQs. Since the above argument of the same equal ab-solute value thresholds is also true for side symmetric cells,i.e., the cells of permutations, the cells in a side symmetric pair contribute the same dis-

tortion and entropy, namely,and . In other words,

and , are true for any level , therebyproving that a SSMDSQ is always balanced.Remark 2: Assuming EIA to be symmetric, one can construct

symmetric quantizers for zero-mean symmetric pdfs for whichthe thresholds of are not an even function of . The proof isconstructive. With the notations of Fig. 2, one can impose thatthe probability of observing index is equal to the probabilityof observing , i.e., the dashed areas are equal. With this con-dition, one proceeds recursively, starting from given thresholds

to identify . The result is that the entropies onboth sides of the pdf are equal. Subsequently, by imposing equaldistortions in the dashed cells, one determines the reconstruc-tion point in cell for a given reconstruction point (e.g.,the centroid) in cell . Proceeding recursively for each en-sures that the distortions on both sides of the pdf are equal. Ap-plying this procedure on both the central and side quantizers at

Fig. 2. Illustration figure for SSMDSQ construction when thresholds of arenot an even function—see Remark 2.

Fig. 3. Illustration of the D-R surface of a two-description SSMDSQ.

all quantization levels leads to balanced SSMDSQs, with sym-metric EIA, for which the thresholds of are not necessarilyan even function of . However, in our work we do not followthe design methodology sketched above. First, nothing can besaid about global optimality, and the thresholds depend on theinput pdf. Furthermore, the thresholds and reconstruction pointsneed to be determined recursively for each cell in order to ensuresymmetry. This corresponds to a computationally expensive re-cursive placement of quantizers’ thresholds which is unattrac-tive from a practical point of view.Remark 3: In this paper, it is considered that the descriptions’

bit-streams are transmitted using packets of fixed size. At first,the bit-planes resulting from the quantization process per de-scription are coded with NBC in case of FLC or with adaptivearithmetic coding in case of VLC. Per description, the coded


TABLE IDISTORTION PERFORMANCE OF THE SCALABLE MDSQS OF FIG. 4 FOR A UNIT-VARIANCE GAUSSIAN SOURCE. MEAN DENOTES THE MEAN OF TWO SIDE

DISTORTION READINGS, I.E., . = DOUBLE DEAD-ZONE. DIV

Fig. 4. EIAs of connected-cell scalable MDSQs. (a) The asymmetric EIA of[19], (b) the equivalent symmetric EIA.

bit-planes are put together in the order of their significance tocreate a single scalable bit-stream. Each such bit-stream per de-scription is thereafter split into a number of fixed size packets.This way each packet per description yields a progressive in-crement in the description rate, say bits per source sample(bpss). In other words, if we receive packets from the thdescription, , then we can decode the description at

bpss. For any combination of packets receivedfrom the two sides, the unique D-R symmetry behavior of SS-MDSQs yields .From here onward, we assume that thresholds for are al-

ways defined as an even function of , allowing to interchange-ably refer to a certain scalable MDSQ or its EIA.

A. Connected-Cell Scalable MDSQs

The most attractive feature of connected-cell scalableMDSQs is that their central and side quantizers have connectedpartition cells at any level. Although largely reducing the im-plementation complexity of such quantizers, this characteristicmay lead to a sub-optimal D-R performance—see Theorem 5in [34]. An additional drawback of maintaining partition cellsconnected is that the resulting quantizers can only generatehigh-redundancy side descriptions.The EMDSQs of [18], [19] are representative examples of

connected-cell scalable MDSQs. In Fig. 4(a), the two level EIAof [19] is depicted. Notice that, since connected-cell side andcentral partition cells are realized at each level, entries alonga row (or a column) are consecutive. This implies that

, where . Upon a close look at the EIAof [19]—see Fig. 4(a), one observes that its index arrangementis highly asymmetric. Table I reports the distortion performanceof the resulting EMDSQ [19]. We note that the side distortionvalues in Table I deviate by a large magnitude from their mean.The same behavior can be also observed for side entropies.A straightforward way to produce a connected-cell sym-

metric scalable MDSQ is by sequentially populating the IAmatrix, such that the sufficient requirements of the symmetric

EIA definition are fulfilled. Namely, starting from the top-leftcorner let an entry is placed at position in the IA matrix.Then, symmetry demands that an entry is placed atposition . In this way a symmetricIA matrix is assembled and uniform splitting is carried out toderive a symmetric EIA, as in Fig. 4(b). The latter has the sametotal number of index entries with the one of Fig. 4(a), thereforethe two EIAs are equivalent. The experimental validation ofthe designed SSMDSQ—see Table I, demonstrates that for agiven number of received bit-planes the symmetric EIA ex-hibits lower central and mean side distortions compared to theasymmetric EIA. For both cases, the finest central quantizersare the same, thus giving equal distortion .

B. Disconnected-Cell Scalable MDSQs

Quantizers belonging to this family are composed of discon-nected partition cells. This trait increases their implementationcomplexity, however, contrary to connected-cell quantizers, itenables them to generate medium-to-low redundancy source de-scriptions [20]. Generally, their side quantizers do not employa dead-zone cell and their reconstruction value is given by thecentroid. The symmetric ML IA of [14] and the asymmetricEIA of [17] are the most representative examples of discon-nected-cell scalable MDSQs.Table II, depicts a D-R comparison of the ML(8,2) IA of

[14] against the equivalent EIA of [17] for a zero-mean, unit-variance Gaussian source. The granular distortion [35], [36],which is the product of the side and the central distortion, isreported for several side rates. For a fair comparison betweensymmetric and asymmetric EIAs, the overall side distortion istaken as the mean of distortions of the two sides. Thus the re-ported granular distortion is

.FromTable II we notice that, in contrast to the EIA of [17], the

symmetric ML(8,2) IA always produces perfectly balanced sidedescriptions. The second row of Table II includes the percentagedeviation from the mean (%Div) of and , for theEIA of [17]. Interestingly, a large deviation between the twoside distortions is observed, demonstrating that the EIA of [17]realizes highly unbalanced descriptions. Furthermore, Table IIclearly shows that the EIA of [17] outperforms the ML IA of[14] in terms of granular distortion. The obtained gains are at-tributed to the connected central partition EIA of [17] and to thefact that the side descriptions are largely unbalanced, yieldingsignificantly lower central distortion values. However, recallthat, in case the encoder is unaware of the channel conditions,an SMDC system has to employ a balanced transmission. Thiscannot be achieved using asymmetric EIAs, e.g., [17], due totheir highly unbalanced nature.To resolve this drawback, we propose a novel symmetric

disconnected-cell EIA strategy which combines the improved


TABLE IIGRANULAR DISTORTION OF THREE DISCONNECTED-CELL SCALABLE MDSQS FOR A UNIT-VARIANCE GAUSSIAN SOURCE (EXACT VARIANCE 1.0028).IS CONSIDERED UNIFORM. EACH QUANTIZER CONSISTS OF THREE EMBEDDED LEVELS (I.E., ). FOR THE ASYMMETRIC EIA OF [17], %DIV OF THE

SIDE DISTORTION FROM THEIR MEAN VALUE IS GIVEN IN PARENTHESIS

Fig. 5. The index scans which are employed in order to construct a symmetric EIA. (a) scan. (b) scan. (c) scan. (d) scan. (e) scan.

Fig. 6. Constructive design of a disconnected-cell symmetric EIA with : (left) ; (middle) ; (right) . Embedded levels can be differentiatedusing different line styles (solid line: ), (dashed line: ) and (dotted line: ).

D-R advantage of [17] with the balanced side distortion char-acteristics of the ML IA of [14]. In particular, consider thedesign of an EIA for , with sidecells using a certain number of diagonals , where

. The proposed EIA starts by ini-tializing the level IA using the symmetric ML IA withparameters —see [14]. Each index of the resultingIA is subsequently split into 2 2 sub-indices to realize thefinest level. Depending on whether is evenor odd, predefined index scans, at level , are carefullyperformed to conform to the definition of a symmetric EIA.Five index scans are employed which are shown in Fig. 5.The complete algorithm is given in the form of pseudo codein the Appendix. In general, one can split each index entry ofthe ML IA at into a block of sub-entries, where

. To yield a symmetric final IA, when derivingthe indices of level , the index scanning of the th andthe corresponding th block, should be a reversemirror of each other, where denotes the number of entriesat level . Note that, although any can beused to realize a symmetric EIA, in our realization we choose

for simplicity. In detail, for , the reversed mirroredand scanning patterns are applied to all entries belonging

to diagonals —when is odd—and to all entriesbelonging to diagonals —when is even. In theformer case, the lower diagonal and the upper diagonal

scans are applied to all elements of the diagonals indexedby 1 and , respectively. Notice that for these elements,one can still simply use the and scans instead of theand scans; however, this would notably increase the totalnumber of entries of the proposed EIA compared to those ofthe ML IA [14] or the EIA of [17], for a given odd value of

. Finally, the scan, which is the reverse mirror of itself, isonly used to split the specific index of the ML IAat level.Fig. 6 depicts the resulting symmetric EIAs for different

number of diagonals. We notice that, unlike the ML IA of [14]and the EIA of [17], the proposed EIA strategy may includepartially filled diagonals. However, for a given and , thetotal number of index entries of the proposed and the existingIAs would roughly be the same. The last row of Table II reportsthe granular distortion of the proposed symmetric EIAbased SSMDSQ. Clearly, the proposed EIA strategy yieldsthe lowest values, while still maintaining perfectlybalanced side distortions.Moreover, Fig. 7 plots the side versus the central distortion

for the three families of EIAs with respect to the theoreticalbound given by [35]. Different points for agiven family are obtained by varying the number of diagonals inthe IA matrix. As a general multiple description rule, at a givenside rate, the side distortion should increase when lowering theredundancy (i.e., by increasing the number of diagonals)—seethe curves in Fig. 7(a)–(c) corresponding to the ML IA [14]and the proposed symmetric EIA. For the asymmetric EIA of[17], although the second side description obeys this rule—seeFig. 7(b), the same is not true for the first side description, asshown in Fig. 7(a). In particular, for the first side of the EIAof [17], we notice that . This causesthe mean side distortion for to be lower than the meanside distortion for —see Fig. 7(c). Such a behavior isspecifically explained by the asymmetric nature of EIA of [17].In contrast to [17], the proposed EIA strategy ensures a smoothdecay of side distortion, similar to ML IA. Moreover, Fig. 7corroborates that the proposed EIA strategy accounts for a better


Fig. 7. Central versus side distortion comparison for an IID Gaussian source with unit-variance at [individual (a)–(b), mean (c)]. Three pointsare plotted for a given EIA family, each corresponding to a certain number of diagonals in the IA matrix. The left-most points correspond to , the middlepoints to , while the right-most points correspond to .

side versus central distortion tradeoff for a certain achievableredundancy point.

IV. QUANTIZER OPTIMIZATION

In this section, a novel algorithm to optimize an SSMDSQ,driven by the pdf of the source to be coded, is presented. Ourapproach is generic since it can optimize different types of SS-MDSQs with different D-R characteristics. The proposed opti-mization algorithm is a novel extension of the Lloyd-Max algo-rithm [30], [31] to the multiple description embedded setting,wherein both level-constrained and entropy-constrained opti-mization can be performed using the same algorithm.Considering a two-description SSMDSQ, we define the ag-

gregate side and aggregate central distortions as a weighted sumof per-level quantities, given by (4), as follows:

(6)

Similarly, the aggregate side entropies are defined as

(7)

and in (6)and (7) denote central and side weighting functions, respec-tively. These weighting functions can be used to weight the cen-tral and the side quantities at each refinement level according totheir relative importance in a given application. Additionally,we impose that the sum of weighting factors should be equal to1, i.e., .We aim to jointly optimize the side and central quantizers.

The optimization is formulated as a minimization problem asfollows:

(8)

(and ) and (and ) in the above equation denotethe target value of aggregate distortion and aggregate rate of thefirst (and the second) side, respectively.

A. Lagrangian Formulation

The Lagrangian functional for the multi-constraint optimiza-tion problem of (8) can be formulated using the Lagrangian con-stants and , as follows:

(9)

Since an SSMDSQ is balanced, one can takeand , as the two side quantities are equally im-portant from an optimization point of view. Let be suchthat it minimizes over all for a fixed , whileminimizes over all for a given , for any

. Then satisfies the first order necessary con-ditions [37] subject to the constraints

and . The parame-ters and are varied until the constraints on side distortions

and side entropies , given in (8) are fulfilled.1) Optimal Decoder: For a given encoder the optimal de-

coder is the one that reconstructs all source samples withina given quantization cell using the centroid. Hence, at any level, for the th cell of the th side, the optimum side decoder isdefined by the following relation:

(10)

Using a training set , the reconstructionpoints can be experimentally approximated as

(11)

where denotes the number of training samples fallingin the side partition cell . Similarly, the reconstruction pointsat level for the central cell , which is formed by the cellof the first side quantizer and by the cell of the second side

quantizer, can be determined as

(12)


2) Optimum Encoder: The optimum encoder for a givendecoder is given by the minimum of the scalar cost functionof (9), that is [see (13) at the bottom of the page], where, theexpectation in the central distortion is defined by the side quan-tization indices , which correspond to nonempty entriesin the EIA. The cells denote the coarser th level side cellscorresponding to the finest level and side cells,respectively. Bearing in mind that the side cells are composedof one or more central cells, one can take the expectation oper-ator in (13) leading to

(14)

The minimum in (14) is achieved when the expression inside theexpectation operator is individually minimized for each sampleof the training set , i.e., by optimally mapping each sampleto a side cell pair that results in the smallest value of

the aforementioned expression. In fact, this is equivalent to cre-ating an optimal finest central partition; since the finest centralpartition together with the EIA defines side cells at any level. Using the training set , the expression inside the expec-tation operator can be minimized by defining the finest centralpartition as [see (15) at the bottom of the page], where

. Defining

(16)

and

(17)

expression (15), which refers to the finest central partition cells, reduces to

(18)

The finest central partition , as given by (18), is solvedusing the extreme-point algorithm of [14].

B. Optimization Algorithm

For a given value of and , the optimization algorithm is asfollows:

ALGORITHM:

1. SET iteration counter . Initialize .INITIALIZE the encoder-decoder pair using a twodescription symmetric SSMDSQ (i.e., symmetric EIAuniform finest central quantizer).SET the stopping threshold to an appropriate small value.

2. Determine the optimum decoderfor , using the per-level decoder

for the center, , and the sides, , as given by (11),(12), respectively.

3. Determine the optimum encoder for the decoderfound in the last step by solving (18) using the

extreme-point algorithm of [14].4. Compute the cost function using the encoder-decoderpair .

5. IF THEN STOP. ELSE andGO TO step 2.

(13)

(15)


The complexity of the optimization increases exponentiallywith due to the exponential increase in the total number ofquantization cells. The appropriate value of and are anyset of values that fulfill the constraints on aggregate rate anddistortions, as given in (8). For FLC of quantization indices, theentropy constraints in (8) become meaningless and one can set

. However, for VLC, both and need to be carefullysearched so as to find the appropriate local minima which fulfillsall the constraints of (8). One way to choose proper and isby iteratively narrowing the possible range of values betweentwo extreme points using the bisection method [23].Another important aspect is the appropriate selection of the

weighting functions . We found experimentally that, athigh rates and should be strictly increasing functions ofthe rate—or otherwise, strictly decreasing functions of level .However, at low rates, it is better to use the reverse weightingstrategy, wherein, and are strictly increasing functions oflevel . In this paper, the selected side and central weightingfactors for any level are taken as

and

where . Although the proposed optimization algo-rithm is presented for two-side SSMDSQs, it can be straight-forwardly extended to any scalable MDSQ with more than two(say ) sides, by defining the following cost function:

(19)

where the aggregate central and side quantities are given as

Unlike SSMDSQ, for an asymmetric scalable MDSQ we haveand , where . More-

over, in contrast with SSMDSQs where one weighting functionfor all side quantizers is defined, an

asymmetric SSMDSQ optimization needs a separate weightingfunction for each side . Sincethe number of terms in (19) grows linearly with , we can saythat for a fixed , the complexity of the proposed optimizationalgorithm would increase linearly with . However, it is worthpointing out that designing an efficient multidimensional IA is asignificantly complex problem by itself. A solution foris given in [32]. Alternatively, a less complex way of generating

, number of descriptions is described in [38].

C. Performance Comparison

This section evaluates the performance of the proposed op-timization algorithm for the family of GG distributions [29].GG distributions are commonly used to model histograms ofwavelet coefficients in images [39]. A zero-mean, GG distribu-tion of variance is expressed by

(20)

denotes the Gamma function, is the shape (ordecay) control parameter, and is the scaling factor[29]. We consider three instances of the GG distribution of (20),defined by (Laplacian pdf), and(Gaussian pdf). First, a training sequence of floating pointsamples, drawn from each of the considered source distribu-tions, is used to carry out the quantizer optimization, as ex-plained in Section IV. The convergence rate of the optimiza-tion algorithm depends on the selected value of the termina-tion threshold . It has been experimentally observed that for

the algorithm converges in between 45 to 60 iter-ations; on average, the optimum is attained after 50 iterationsfor all three source pdfs. The optimized quantizers are thereafterused to quantize the actual source samples. The experiments arecarried out using sources of 25344 samples and each D-R pointis evaluated as a mean over 100 trials. To compare the D-R per-formance of different scalable MDSQs, the signal-to-noise ratio(SNR) metric (dBs) is employed.Table III reports the SNR improvements brought by the

proposed optimization algorithm for the connected-cell scal-able MDSQs of Fig. 4 at various side rates and for a Gaussiansource pdf. Both level-constrained (using FLC) and entropy-constrained (using VLC) cases are considered. Since the op-timization is done for the average quantities, overembedded levels, it cannot be generally ensured that the distor-tion performance at each rate point is also improved. For thelevel-constrained case the optimization always results in SNRimprovements at all rate points with respect to the unoptimizedquantizers. However, the same is not true for the entropy-con-strained case. Negative sign values in Table III indicate instanceswhen the optimized distortion is higher than the unoptimizeddistortion. However, for the majority of rate points, the opti-mized distortion is far below the unoptimized one, leading to anaverage improvement of central and side SNRs—see Table III.To demonstrate that the proposed algorithm improves theoverall D-R, Table III also reports the accumulated gain :

(21)

where , i.e., the source SNR is evaluated using a total ofrate points. Table III clearly points out that the overall D-R

improvement in case of VLC is approximately 0.5 dB, while inFLC it grows larger than 2.5 dB, for both the symmetric andasymmetric connected-cell quantizers.Similar to the connected-cell case, the optimization algo-

rithm provides an important average gain for disconnected-cellquantizers as well—see Table IV. In this experimental setup,three source pdfs are quantized using both level-constrained andentropy-constrained optimized quantizers. Again, the obtainedSNR gains in the level-constrained case are higher comparedto the entropy-constrained optimization case. Moreover, oneobserves that larger optimization gains are obtained for MLIA based quantizers while lesser but still significant gains areachieved for the asymmetric EIA of [17]. The smallest opti-mization gains are obtained for the proposed symmetric EIAbased SSMDSQs. This is to be expected, since the proposed


TABLE IIISNR DIFFERENCE (IN DBS) BETWEEN THE OPTIMIZED AND THE UNOPTIMIZED SCALABLE MDSQS OF

TABLE I, FOR A UNIT-VARIANCE GAUSSIAN SOURCE.

TABLE IVSNR DIFFERENCE (IN DBS) BETWEEN THE OPTIMIZED AND THE UN-OPTIMIZED SCALABLE MDSQS OF TABLE II, FOR THREE UNIT-VARIANCE SOURCE PDFS

symmetric EIA serves as an initialization and the optimizationalgorithm converges to a local minimum.To verify their performance for real multimedia data, the

proposed SSMDSQs are utilized for SMDC of wavelet-decom-posed images. Table V reports the reconstructed image qualityversus rate when subbands resulting from the 6-level CDF9/7 wavelet transform are quantized using the five diagonalSSMDSQ and the equivalent asymmetric scalable MDSQ of[17]. The LL subband is losslessly coded and the ensuinginformation is duplicated in both descriptions. The bit-planesof the other subbands are coded using adaptive arithmeticcoding in quality scalable manner. For the optimized case, thesubband histograms are first modeled using GG distributions.For each wavelet subband, the estimated shape-parameteris mapped to the closest value in the set andthe SSMDSQ optimized for the source pdf corresponding tothe selected value of is then employedto quantize the subband. Since the original optimization isdone for zero-mean, unit-variance sources, the quantizer’sthresholds and reconstruction alphabets are adjusted basedon the estimated mean and the standard deviationof each subband’s coefficients. The results in Table V showthat the proposed SSMDSQ (with and without optimization)provide improved performance over the scalable MDSQ of[17]. We highlight that for the case of joint decoding of thedescriptions, the improvement in the PSNR of the reconstructedimage can be up to 4–5 dBs. Furthermore, visual comparisonsof joint descriptions decoding at level , shown

in Fig. 8, demonstrate that the reconstructed image qualityprovided by the proposed scheme is significantly improvedcompared to that of [17]. Additionally, one may notice that,unlike the asymmetric scalable MDSQ of [17], the side PSNRsfor the SSMDSQ are nearly equal at different side rates, therebyconfirming its fairly balanced description structure.The coding system used for computing the results of Table V

is a basic one, wherein, the generated bit-planes were pro-gressively coded using adaptive arithmetic entropy coding. Ingeneral, scalable image codecs, e.g., [23], [40], also employbit-plane coding schemes such as run-length coding, quad-tree(QT) coding, to further exploit the spatial correlation within agiven bit-plane. In Table VI, a performance comparison of theproposed SSMDSQ coupled with the QT-L codec of [40] isgiven against duplicated JPEG2000 [23] (duplicating the singledescription coding of the input image). For comparison pur-poses, we also include the single description coding results forthe JPEG2000 [23] and the QT-L codec in [40], showing thatthey perform similarly. The central reconstruction values showthat the SSMDSQ based image codec outperforms duplicatedJPEG2000.

V. TRANSMISSION VIA PACKET LOSS CHANNEL

We consider transmission via a single link which is dividedinto multiple logical channels, namely, packets representingdifferent source descriptions are transmitted using time-divi-sion multiplexing. Let denote the average probabilityof packet loss on the transmission link. For a two-description


TABLE VPERFORMANCE EVALUATION OF THE PROPOSED SSMDSQ FOR WAVELET-BASED SMDC OF LENA (512 512) AND BARBARA (512 512) IMAGES.

(RESP. ) DENOTE THE CENTRAL AND THE SIDES PEAK-SNR IN DBS [RESP. MEAN SIDE RATE IN BITS PER PIXEL (BPP)]

Fig. 8. Visual comparison of images reconstructed using the central quantizer at level . From left, the first and the third image are the reconstruction usingthe asymmetric EIA [17]. Reconstructions using the proposed optimized SSMDQ are shown by the second and the fourth image. The reported rate is the sum rateof two sides, i.e., .

TABLE VITHE UNOPTIMIZED SSMDSQ-BASED QT IMAGE CODEC VERSUS THE EQUIVALENT SINGLE AND MULTIPLE (DUPLICATED) DESCRIPTION IMAGE CODING

SYSTEMS. FOR THE SSMDSQ-BASED SYSTEM, THE CENTRAL RECONSTRUCTION QUALITY IS REPORTED

SMDC, the average expected distortion at the decoder, whensending packets for the th description, , isgiven by

(22)

where

(23)

In (23), represents the discrete time Heaviside (unit-step)function.For a symmetric D-R surface, , of which a

schematic representation is illustrated in Fig. 3, and for a given, (22) yields a symmetric average distortion surface—see

Fig. 9. Notice that in this work we consider IID packet erasuresover the transmission link. In general, transmission channelsmay well encounter burst packet losses. In such scenarios, thesimplification resulting from the assumption of IID losses, i.e.,

(23) can still be employed, provided that bursts of losses can bespread across independent packets by interleaving [5].

A. Balanced Transmission of Descriptions

In today’s packet switched networks, nodes randomly droppackets to match link rates. In point-to-point communicationemploying layered coding, packets arrive at a network node’squeue in the order of their significance. Therefore, it is morebeneficial to drop packets which arrived later rather than packetswhich arrived earlier. Assuming such a packet drop strategy, wefirst consider scalable multiple description transmission whenthe encoder is unaware of the packet loss probability on thechannel. For a balanced transmission, packets from the two de-scriptions are alternatively placed over the transmission link andarrive in the same order at the network node. If the network nodeis forced to drop packets, the packet dropping strategy will al-ways be one of the following:


Fig. 9. Average experimental D-R at as a function of transmittedpackets , for a unit-variance Gaussian source. The best and the com-plementary best TPs are illustrated using solid and dashed lines, respectively.

Fig. 10. profiles for symmetric and asymmetric connected-cell scalableMDSQs at bpss, for a unit-variance Gaussian source. Dashedlines correspond to unoptimized (unopt), while solid lines are used for optimized(opt) quantizers.

where the first and the second term in denote the numberof packets dropped from the first and the second description,respectively. Thus, the transmission is still approximatelybalanced for any number of packets dropped by the networknode. This is not true for unbalanced transmission, for whichpackets belonging to the two descriptions are not alternatelyplaced on the transmission link. In this case, the network nodemay drop all packets belonging to a single description, thusenabling only source reconstruction with side distortion, whichis usually much higher than the central distortion. As a conse-quence, for an unbalanced transmission, a certain loss patternmay lead to a much higher decoding distortion compared toother loss patterns for the same total number of lost packets.Hence, one concludes that a balanced transmission of descrip-tions is always a better choice if the amount of packet loss isunknown at the encoder. The performance of a scalable MDSQin the considered packet-based SMDC system is reported usingaverage SNR .Connected-Cell Quantizers: The profile of the con-

nected-cell EMDSQ of [19] and the proposed SSMDSQ withthree levels , for a unit-variance Gaussian source,over a wide range of packet loss probability , is depicted in

Fig. 11. profiles for symmetric and asymmetric disconnected-cellscalable MDSQs, at , for two unit-variance source pdfs.(a) Gaussian. (b) Laplacian.

Fig. 10. Both FLC and VLC results are reported. As expected,due to rate savings, VLC outperforms FLC for a given packetloss probability . The results demonstrate that the proposedunoptimized SSMDSQ outperforms the unoptimized EMDSQof [19] for all values greater than 1% and 5% when con-sidering FLC and VLC, respectively. In addition, when opti-mization is applied—see Section IV, the proposed optimizedSSMDSQ outperforms the optimized EMDSQ of [19] forvalues greater than 3% for FLC. Alternatively, in case of VLC,both optimized quantizers operate on par for values above7%. The obtained performance improvements for high packetloss probabilities, show the benefit of balanced transmission, asachieved using the proposed SSMDSQ compared to the unbal-anced EMDSQ of [19]. For very low packet loss probability,the proposed SSMDSQ falls behind the EMDSQ of [19], whichreveals the penalty for providing a perfectly symmetric D-Rfunction. Furthermore, in SMDC-based transmission, the pro-posed optimization algorithm brings notable gains for boththe proposed (SSMDSQ) and the contemporary (EMDSQ [19])connected-cell quantizers, as shown in Fig. 10.Disconnected-Cell Quantizers: For disconnected-cell quan-

tizers, profiles for unit-varaince Gaussian and Laplaciansource pdfs are depicted in Fig. 11. We notice that the optimized


TABLE VIIAVERAGE DECODED PSNR (IN DBS) FOR THE LENA IMAGE AT

BPP USING BALANCED TRANSMISSION OF CODEDDESCRIPTIONS. THE EXPERIMENTAL SETTINGS OF TABLE V ARE REUSED TO

COMPUTE THE RESULT IN THIS TABLE

quantizers significantly outperform their unoptimized equiva-lents, for both pdfs. One observes that the improvements at-tained by the proposed optimization algorithm are higher for theML IA [14] based quantizer, which comprises disconnected sideand central cells, than for quantizers which feature connectedcentral and disconnected side cells, e.g., the asymmetric EIA[17] and the proposed symmetric EIA based quantizers. This isbecause in the former, a reduction in the cost for a central cellwill always result in a decrease of the cost for the correspondingside cells, and vice versa. However, this is not always true forquantizers belonging to the second category, as the optimizationof the cost for the central and the side cells may have conflictingrequirements. In essence, the proposed symmetric EIA basedquantizer consistently outperforms the existing ML IA [14] andEIA [17] based quantizers, with and without optimization.Table VII reports the average decoded PSNR, for the SS-

MDSQ and the equivalent scalable MDSQ of [17], when mul-tiple description transmission of the Lena image is carried outover a packet loss channel. The experimental setup of Table Vis reused to compute average PSNR for the decoded image, forfour packet loss rates. In this context, the proposed SSMDSQwithout optimization improve the quality of the reconstructedimage by 2–3 dBs over that of [17]. On top of this, the proposedoptimization approach yields an additional PSNR gain of up to0.5 dB on average.

B. Unbalanced Transmission of Descriptions

In case the packet loss rate can be known at the encoder, anefficient transmission strategy is established by an unbalancedtransmission of descriptions. Depending on , we formulate agreedy packet scheduling strategy which minimizes —givenby (22), for a certain total of packets sent over the trans-mission channel. For a given , an appropriate combinationof packets can be found as

(24)

By sequentially increasing the total number of transmittedpackets, exhaustive search solutions of (24) define a so-calledbest transmission path (TP). Specifically, the encoder regulatespacket contributions from both sides, as suggested by thebest TP, to minimize at the decoder. We notice that when

, then , and. Therefore, for , the best TP corre-

sponds to traversing the source D-R at the minimum sourcedistortion.

Fig. 12. Performance comparison of the proposed disconnected-cell SSMDSQin an SMDC system, for an IID unit-variance Gaussian. Packet transmission issimulated for a packet loss channel with .

Notice that in the aforesaid packet scheduling, for a given, only packet combinations which agree with the sched-

uling of packets are searched. In general, an optimumcombination for may not lead to an optimum combi-nation for . Therefore, since the packet scheduling strategyis considered greedy, it may lead to reduced performance of anSMDC system for large . A better approach would be todetermine an overall best TP by considering all possible com-binations for the highest and accordingly scheduling thetransmission for smaller rates. However, such an approach re-quires all packets of all descriptions to be available beforehandwhich brings an unreasonable structural delay in many real-timeapplications, e.g., multimedia transmission.For an SSMDSQ-based SMDC system, which guarantees a

symmetric D-R surface, a permutation of the solution of (24) isalso a valid solution. Therefore, this leads to the existence of twocomplementary best TPs as demonstrated in Fig. 9. Since boththe best and its complementary TPs result in the same averagedistortion , the determination of the best TP in a SSMDSQbased SMDC scenario requires searching only half of thespace, as defined by (or ). This is in contrastto SMDC systems based on contemporary asymmetric scalableMDSQs, e.g., [17], in which one needs to perform an exhaustivesearch for any pair, for which .Fig. 12 shows the versus the total number, , of

packets transmitted from two descriptions for the three evalu-ated quantizers. As explained above, greedy packet schedulingis employed to carry-out an unbalanced transmission of de-scriptions for a unit-variance Gaussian source. Notice that fora fair comparison, the mean performance between the two bestTPs, corresponding to and , is reportedfor the asymmetric EIA based quantizer of [17]. The resultsdemonstrate that the proposed SSMDSQ quantizer performsbetter compared to the other two assessed quantizers, that is theML IA [14] and EIA [17] based quantizers. Similar results havebeen observed for a GG and for a Laplacian sourcepdf. Also, the results confirm that the proposed optimizationalgorithm contributes notable gains in all three quantizers.Fig. 13 plots the with respect to the number of trans-

mitted packets for the best TP (as defined for unbalanced trans-mission) and for the TP corresponding to the balanced trans-mission, for the unoptimized SSMDSQ-based SMDC system.


Fig. 13. Balanced versus unbalanced packet transmission using an entropy-coded SMDC system equipped with the proposed disconnected-cell SSMDSQ.(a) Zero-loss channel. (b) Lossy channel with .

As anticipated, due to the symmetry of the D-R function in-curred by the proposed SSMDSQ, the two complementary bestTPs, for unbalanced transmission, lie on top of each other. Wenotice that, for the zero packet loss channel state, a certain im-provement in is provided by unbalanced versus balancedtransmission. However, this performance difference diminishesas increases—see Fig. 13(b). This implies that, for large ,the best unbalanced TP is approximated by the TP determinedby balanced transmission. This in turn motivates the employ-ment of the proposed SSMDSQs, which can produce a scalableand perfectly balanced transmission of descriptions.

VI. CONCLUSION

In contrast with contemporary asymmetric scalable MDSQdesigns, this paper introduced a novel SSMDSQ which can beemployed to attain perfectly balanced packetized transmissionof descriptions. In particular, we determined the necessaryconditions and we proposed novel EIA constructions to realizepractical SSMDSQs. The proposed SSMDSQs outperformexisting asymmetric scalable MDSQ designs in terms of D-Rperformance. In addition, we propose a novel generalizationof the Lloyd-Max algorithm to carry out pdf specific iterativeoptimization of scalable MDSQs. Based on the optimizedquantizers, a packet-based SMDC framework is established

for transmission via packet erasure channels. In this context,it is shown that the symmetry of the D-R surface, incurred bythe proposed SSMDSQs, can facilitate balanced transmissionwhen the packet loss rate is unknown at the transmitter. Addi-tionally, when an unbalanced transmission is considered, theproposed quantizers simplify the problem of defining the besttransmission path with respect to asymmetric quantizers. ForSMDC-based communication, simulation results for packe-tized transmission of GG and image data over erasure channelsclearly show the advantage of the proposed SSMDSQs andverify the improvements brought by the proposed optimizationstrategy.

APPENDIX

The pseudo code to realize a disconnected-cell symmetricEIA is given in the following table.

REFERENCES[1] V. K. Goyal, “Multiple description coding: Compression meets the net-

work,” IEEE Signal Process. Mag., vol. 18, no. 5, pp. 74–93, Sep. 2001.[2] M. Alasti, K. Sayrafian-Pour, A. Ephremides, and N. Farvardin, “Mul-

tiple description coding in networks with congestion problem,” IEEETrans. Inf. Theory, vol. 47, no. 3, pp. 891–902, Mar. 2001.

[3] V. N. Padmanabhan, H. J. Wang, and P. A. Chou, “Resilient peer-to-peer streaming,” in Proc. IEEE Int. Conf. Netw. Protocols, Redmond,WA, Nov. 2003, pp. 16–27.

[4] N. V. Boulgouris, K. E. Zachariadis, A. Kanlis, and M. G. Strintzis,“Multiple description wavelet coding of layered video using optimalredundancy allocation,” EURASIP J. Appl. Signal Process., vol. 2006,pp. 1–19.

[5] F. Verdicchio, A. Munteanu, A. Gavrilescu, J. Cornelis, and P.Schelkens, “Embedded multiple description coding of video,” IEEETrans. Image Process., vol. 15, no. 10, pp. 3114–3130, Sep. 2006.

[6] L. Ozarow, “On a source coding problem with two channels and threereceivers,” Bell Syst. Tech. J., vol. 59, pp. 1909–1921, Dec. 1980.

[7] A. A. El Gamal and T. M. Cover, “Achievable rates for multiple de-scriptions,” IEEE Trans. Inf. Theory, vol. 28, no. 6, pp. 851–857, Nov.1982.

[8] R. Venkataramani, G. Kramer, and V. K. Goyal, “Multiple descriptioncoding with many channels,” IEEE Trans. Inf. Theory, vol. 49, no. 9,pp. 2106–2114, Sep. 2003.

[9] R. Puri, S. S. Pradhan, and K. Ramchandran, “N-channel symmetricmultiple descriptions—Part II: An achievable rate region,” IEEE Trans.Inf. Theory, vol. 51, no. 4, pp. 1377–1392, Apr. 2005.

[10] C. Tian and J. Chen, “New coding schemes for the symmetricdescription problem,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp.5344–5365, Oct. 2010.


[11] Y. Wang, M. T. Orchard, V. Vaishampayan, and A. R. Reibman, “Mul-tiple description coding using pairwise correlating transforms,” IEEETrans. Image Process., vol. 10, no. 3, pp. 351–366, Mar. 2001.

[12] V. K. Goyal and J. Kovacevic, “Generalized multiple descriptioncoding with correlating transforms,” IEEE Trans. Inf. Theory, vol. 47,no. 6, pp. 2199–2224, Sep. 2001.

[13] W. Jiang and A. Ortega, “Multiple description coding via polyphasetransform and selective quantization,” in Proc. SPIE Int. Conf. VisualCommun. Image Process., San Jose, CA, 1999, pp. 998–1008.

[14] V. A. Vaishampayan, “Design of multiple description scalar quan-tizers,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 821–834, May1993.

[15] V. A. Vaishampayan and J. Domaszewicz, “Design of entropy-con-strained multiple description scalar quantizers,” IEEE Trans. Inf.Theory, vol. 40, no. 1, pp. 245–250, Jan. 1994.

[16] W. Equitz and T. Cover, “Successive refinement of information,” IEEETrans. Inf. Theory, vol. 37, no. 2, pp. 269–275, Mar. 1991.

[17] T. Guionnet, C. Guillemot, and S. Pateux, “Embedded multipledescription coding for progressive image transmission over unreliablechannels,” in Proc. IEEE Int. Conf. Image Process., Thessaloniki,Greece, Oct. 2001, vol. 1, pp. 94–97.

[18] A. I. Gavrilescu, A. Munteanu, P. Schelkens, and J. Cornelis, “Em-beddedmultiple description scalar quantizers,” IEE Electron. Lett., vol.39, no. 13, pp. 979–980, Jun. 2003.

[19] A. I. Gavrilescu, A. Munteanu, P. Schelkens, and J. Cornelis, “High-redundancy embeddedmultiple-description scalar quantizers for robustcommunication over unreliable channels,” in Proc. Int. Workshop onImage Anal. Multimedia Interact. Serv., Lisboa, Portugal, Apr. 2004,vol. CD-version..

[20] A. I. Gavrilescu, A. Munteanu, P. Schelkens, and J. Cornelis, “A newfamily of embedded multiple description scalar quantizers,” in Proc.IEEE Int. Conf. Image Process., Singapore, Oct. 2004, vol. 1, pp.159–162.

[21] A. I. Gavrilescu, A. Munteanu, P. Schelkens, and J. Cornelis, “Gen-eralization of embedded multiple description scalar quantizers,” IEEElectron. Lett., vol. 41, no. 2, pp. 63–65, Jan. 2005.

[22] J. M. Shapiro, “Embedded image coding using zerotrees of waveletcoefficients,” IEEE Trans. Signal Process., vol. 41, no. 12, pp.3445–3462, Dec. 1993.

[23] D. S. Taubman and M. W. Marcellin, JPEG2000: Image CompressionFundamental, Standards and Practice. Dordrecht, Germany: KluwerAcademic, 2001.

[24] S. D. Servetto, K. Ramachandran, V. A. Vaishampayan, and K.Nahrstedt, “Multiple description wavelet based image coding,” IEEETrans. Image Process., no. 9, pp. 813–826, May 2000.

[25] M. O. Bici and G. B. Akar, “3Dmesh coding using multiple descriptionscalar quantizers,” in Proc. Int. Conf. Image Process., Atlanta, GA,2006, pp. 1–4.

[26] I. V. Bajic and J. W. Woods, “Domain-based multiple descriptioncoding of images and video,” IEEE Trans. Image Process., vol. 12,no. 10, pp. 1211–1225, Oct. 2003.

[27] A. I. Gavrilescu, F. Verdicchio, A. Munteanu, I. Moerman, J. Cornelis,and P. Schelkens, “Scalable multiple-description image coding basedon embedded quantization,” EURASIP J. Image Video Process., vol.2007, pp. 1–11, Jan. 2007, 81813.

[28] O. Crave, B. Pesquet-Popescu, and C. Guillemot, “Robust video codingbased on multiple description scalar quantization with side informa-tion,” IEEE Trans. Circuits Syst. Video Technol., vol. 20, no. 6, pp.769–779, Jun. 2010.

[29] S. Nadarajah, “A generalized normal distribution,” J. Appl. Statist., vol.3, no. 7, pp. 685–694, Sep. 2005.

[30] S. P. Lloyd, “Least square quantization in PCM,” IEEE Trans. Inf.Theory, vol. 28, no. 2, pp. 129–137, Mar. 1982, also unpublished mem-orandum, Bell Lab., 1957.

[31] J. Max, “Quantizing for minimum distortion,” IRE Trans. Inf. Theory,vol. 6, no. 1, pp. 7–12, Mar. 1960.

[32] C. Tian and S. S. Hemami, “Sequential design of multiple descriptionscalar quantizers,” in Proc. Data Compression Conf. (DCC’04), Aug.2004, pp. 32–41.

[33] I. H. Witten, R. M. Neal, and J. G. Cleary, “Arithmetic coding for datacompression,” Commun. ACM, vol. 3, no. 6, pp. 520–540, Jun. 1987.

[34] D. Muresan and M. Effros, “Quantization as histogram segmentation:Optimal scalar quantizer design in network systems,” IEEE Trans. Inf.Theory, vol. 54, no. 1, pp. 344–366, Jan. 2008.

[35] J. C. Batllo and V. A. Vaishampayan, “Asymptotic performance ofmultiple description transform codes,” IEEE Trans. Inf. Theory, vol.43, no. 2, pp. 703–707, Mar. 1997.

[36] C. Tian and S. S. Hemami, “Universal multiple description scalar quan-tization: Analysis and design,” IEEE Trans. Inf. Theory, vol. 50, no. 9,pp. 2089–2102, Sep. 2004.

[37] D. G. Luenberger, Linear and Non-Linear Programming, 2nd ed.Reading, MA: Addison-Wesley, 1984.

[38] J. Cardinal, “Multistage index assignments for m-description coding,”in Proc. IEEE Int. Conf. Image Process., Barcelona, Spain, 2003, vol.3, pp. 249–252.

[39] S. Mallat, “A theory for multiresolution signal decomposition: Thewavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol.11, no. 7, pp. 674–693, Jul. 1989.

[40] P. Schelkens, A. Munteanu, J. Barbarien, M. Galca, X. Giro-Nieto, andJ. Cornelis, “Wavelet coding of volumetric medical datasets,” IEEETrans. Med. Imag., Special Issue on “Wavelets in Med. Imag.”, vol.22, no. 3, pp. 441–458, Mar. 2003.

Shahid Mahmood Satti (S’09) was born inRawalpindi, Pakistan, in 1980. He received the B.Sc.degree in electrical engineering from the Universityof Engineering and Technology Taxila, Pakistan,in 2003, and the M.Sc. degree in communicationsystems from the Technical University of Munich,Germany, in 2007.Since 2008, he has been pursuing the Ph.D. degree

at the Department of Electronics and Informatics(ETRO), Vrije Universiteit Brussel (VUB). Hismain research interests include multiresolution

analysis, wavelet-based compression, scalable single and multiple descriptionquantization, and scalable joint source channel coding.

Nikos Deligiannis (S’08–M’10) received theM.Eng.degree in electrical and computer engineering fromthe University of Patras, Patras, Greece, in 2006, andthe Ph.D. degree in applied sciences (awarded withthe highest distinction and congratulations of the jurymembers) from the Vrije Universiteit Brussel (VUB),Brussels, Belgium, in 2012.From December 2006 to September 2007, he was

with the Wireless Telecommunications Laboratory,University of Patras. He joined the Departmentof Electronics and Infomatics, Vrije Universiteit

Brussel, in October 2007. His current research interests include statisticalchannel modeling, multimedia coding, distributed source coding, multipledescription coding, wireless cellular networks, and location positioning.Dr. Deligiannis was the corecipient of the 2011 ACM/IEEE International

Conference on Distributed Smart Cameras Best Paper Award.

Adrian Munteanu (M’07) received the M.Sc.degree in electronics from Politehnica University ofBucharest, Romania, in 1994, the M.Sc. degree inbiomedical engineering from University of Patras,Greece, in 1996, and the Doctorate degree in appliedsciences (awarded with the highest distinction andcongratulations of the jury members) from VrijeUniversiteit Brussel (VUB), Belgium, in 2003.During 2004–2010, he was a Postdoctoral

Fellow with the Fund for Scientific Re-search—Flanders (FWO), Belgium, and since

2007, he has been a Professor at VUB. He is also research leader of the4Media group at the Interdisciplinary Institute of Broadband Technology(IBBT), Belgium. His research interests include scalable image and videocoding, distributed video coding, scalable coding of 3D graphics, 3D videocoding, error-resilient coding, multiresolution image and video analysis,video segmentation and indexing, multimedia transmission over networks,and statistical modeling. He is the author or coauthor of more than 200journal and conference publications, book chapters, patent applications, andcontributions to standards.Dr. Munteanu is the recipient of the 2004 BARCO-FWO prize for his Ph.D.

work.


Peter Schelkens (M’97) received the degree inelectronic engineering in VLSI-design from theIndustriële Hogeschool Antwerpen-Mechelen(IHAM), Campus Mechelen. He then receivedthe M.Sc. degree in applied physics, a biomedicalengineering degree (medical physics), and the Ph.D.degree in applied sciences from the Vrije UniversiteitBrussel (VUB).He currently holds a professorship with the Depart-

ment of Electronics and Informatics (ETRO), VUB.He is a member of the scientific staff of the Interdisci-

plinary Institute for Broadband Technology, Belgium. Additionally, since 1995,he has also been affiliated to the Interuniversity Microelectronics Institute, Bel-gium, as scientific collaborator. His research interests are situated in the field ofmultidimensional signal processing encompassing the representation, commu-nication, security, and rendering of these signals while especially focusing oncross-disciplinary research. He is a coeditor of the books The JPEG 2000 Suite(NewYork:Wiley, 2009) andOptical and Digital Image Processing (NewYork:Wiley, 2011). He has published more than 200 papers in journals and conferenceproceedings, and holds several patents, as well as contributed to several stan-dardization processes.Dr. Schelkens is the Belgian head of delegation for the ISO/IEC JPEG stan-

dardization committee, editor/chair of part 10 of JPEG 2000: “Extensions forThree-Dimensional Data” and PR Chair of the JPEG committee. Since 2010,he has been a member of the Board of Councilors of the Interuniversity Micro-electronics Institute. He is a member of SPIE and ACM, and is currently theBelgian EURASIP Liaison Officer. In 2011, he acted as General (co-)Chair ofIEEE International Conference on Image Processing (ICIP) and the Workshopon Quality of Multimedia Experience (QoMEX).

Jan Cornelis (SM’80) received the M.S. degree inelectronics and electromechanics from the Vrije Uni-versiteit Brussel (VUB) in 1973, and the Ph.D. degreein applied sciences in 1980.He started his university career in 1973, and

cocreated the new Department of Electronics(VUB-ETRO). His major research interest as a post-doctoral researcher in the Cardiology Department ofthe academic hospital (UZ Jette) were biomedicalsignal processing. Since 1983, he was a full-timeprofessor, with various teaching duties in design of

electronic circuits, digital image processing, medical imaging, and securitystatistics. He was head of the Department of Electronics and Informatics(ETRO), VUB, until September 2008. He is now Coordinator of the researchgroup IRIS (computer vision, image processing) and Academic Coordinatorfor Technology Transfer, VUB. His current research interest is in image andvideo compression and medical imaging. He is also active in R&D-policyand -management: cofounder of “ NV”—a university incubation fund,and ICAB NV—Incubation Centre of VUB. During 2001–2008, he was ViceRector for research at VUB. He is currently setting up an interuniversityearly seed fund. He is an author or coauthor of more than 450 internationalpublications in journals and conference proceedings, promoter of 26 Ph.D.degrees and (co)inventor of four patents.Dr. Cornelis became Deputy Head of Cabinet in the Ministry of Science

Policy and Innovation for 6 months in 2009. He is a Board Member of the In-stitute of Broad Band Technology (IBBT) and represents the Flemish Govern-ment on the Board of Directors of the Interuniversity Microelectronics Centre(IMEC).

3628 ieee transactions on signal processing,...

Documents