ASYMPTOTIC CLOSED-LOOP DESIGN OF ERROR RESILIENT PREDICTIVE COMPRESSION SYSTEMS. Sina Zamani, Tejaswi Nanjundaswamy, Kenneth Rose

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1 ASYMPTOTIC OSED-LOOP DESIGN OF ERROR RESILIENT PREDICTIVE COMPRESSION SYSTEMS Sia Zamai, Tejaswi Najudaswamy, Keeth Rose Departmet of Electrical ad Computer Egieerig, Uiversity of Califoria Sata Barbara, CA {siazm, tejaswi, ABSTRACT Predictio is used i virtually all compressio systems. Whe such a compressed sigal is trasmitted over ureliable etworks, packet losses ca lead to sigificat error propagatio through the predictio loop. Despite this, the covetioal desig techique completely igores the effect of packet losses, ad estimates the predictio parameters to miimize the mea squared predictio error, ad optimizes the quatizer to miimize the recostructio error at the ecoder. While some desig techiques have bee proposed to accurately estimate ad miimize the ed-to-ed distortio at the decoder that accouts for packet losses, they operate i a closed-loop, which itroduces a mismatch betwee statistics used for desig ad statistics used i operatio, causig a egative impact o covergece ad stability of the desig procedure. Istead, we propose i this paper a effective techique for predictive compressio system desig that accouts for the istability caused by error propagatio due to packet losses, ad ejoys stable statistics durig desig by employig ope-loop iteratios that o covergece mimic closedloop operatio. Simulatio results for a compressio system with a first order liear predictor demostrate the utility of the proposed approach, which offers sigificat performace improvemets over eistig desig techiques. Ide Terms Predictor desig, Asymptotic Closed-Loop desig, Error Resiliece, Liear Predictor. INTRODUCTION Liear predictio is widely used i speech codig, speech sythesis, speech recogitio, audio codig, ad video codig. I compressio systems, the predictio module plays a importat role i eploitig temporal ad spatial redudacies. However, whe such a compressed data is trasmitted over ureliable etworks, errors itroduced due to the ievitable packet losses, propagate through the predictio loop, causig substatial, ad sometimes catastrophic, deterioratio of the received sigal. Despite this, covetioal compressio system desig completely igores the effect of chael loss, ad chooses the predictio parameters to miimize the mea squared predictio error, ad optimizes the quatizer to miimize the recostructio error at the ecoder. This problem ca be alleviated by optimizig the system for the overall ed-to-ed distortio (EED) observed at the decoder, which accouts for the effect of packet loss. A optimal recursive techique to estimate EED at the ecoder was proposed i [], ad utilizig this distortio to optimally select the parameters for motio compesated predictio i video codig was proposed i []. However, desigig optimal predictors ad quatizers, while accoutig for EED is a challegig task, as we eed to This work was supported i part by a gift from Mozilla. work with a stable traiig set that accurately represets the true sigal statistics. The ope-loop (OL) ad closed-loop () approaches were proposed i [3] for predictive vector quatizatio ad have bee widely used sice the. The OL approach uses the origial data as the predictio referece durig desig, but sice the decoder does ot have access to the origial data, the parameters desiged are ot suitable for the statistics see at the decoder. The approach attempts to alleviates the problem of this mismatch by desigig the parameters usig recostructed data obtaied i a closed-loop system as the predictio referece. However, usig these parameters i a closed-loop system geerates ew predictio ad recostructio, which implies they differ from the data the parameters were desiged for. This mismatch i statistics betwee desig ad operatio grows over time as the data is fed through the predictio loop i the coder, leadig to istability i both estimatio of predictio parameters, ad desig of quatizers, especially at lower bit rates. Note that this error propagatio ecoutered durig the desig phase due to statistics mismatch, differs from the error propagatio due to packet losses. I this paper we propose to address the challegig problem of tacklig these two types of error propagatio, by desigig the system iteratively, wherei a estimated EED is miimized at each iteratio to accout for packet losses, ad the predictio referece from the previous iteratio is employed i a ope-loop way to esure statistics used for desig ad operatio are matched. Oce the parameters beig desiged coverge, the predictio referece i the curret iteratio will match the referece from the previous iteratio, thus mimickig closed-loop operatio. Hece, we call this the asymptotic closed-loop () approach, which is similar to the approach i [, 5], wherei system desig without accoutig for packet losses is proposed. We specifically describe a framework for rate versus EED optimizatio of a compressio system employig a first order liear predictor. We also propose a ew ecoder architecture, i which the predictio at ecoder is based o the epected decoder recostructios. Eperimetal results substatiate the utility of the proposed approach with sigificat performace improvemets over eistig desig techiques.. PROBLEM SETUP Fig. illustrates a predictive compressio system, wherei iput sigal samples,, 0 < N, are coded by the ecoder to geerate a bitstream, which is trasmitted through a chael to the decoder, where it is decoded to geerate the recostructed samples. The ecoder uses its previous recostructed samples, ˆ e, to geerate predicted samples, e, ad the predictio error, e = e,. This is quatized to geerate ê, ad set over the chael. Whe the decoder receives ê, it adds it to its predicted sample, d, to geerate its recostructed samples, ˆ d,. Note that the recostructed samples //$ IEEE 3 ICASSP 0

2 Ecoder Decoder ˆ d, e ê Q E Chael ˆ (i ) 0 P (i ) ˆ (i ) Σ P (i ) Σ ˆ (i ) e, d, P D Desig a ew Quatizer Q (i) P E ˆ e, Fig. : A predictive compressio system 0 P (i ) Σ Q (i ) e (i) P (i ) ê (i) Σ Q (i) e (i) P (i ) Σ Q (i) ˆ (i) 0 P (i ) Σ Q (i ) Desig a ew Quatizer Q (i) ˆ (i) Desig a ew Predictor P (i) Fig. : Closed-loop traiig approach ê (i) at the ecoder, ˆ e,, ad the decoder, ˆ d,, will differ whe the chael is ureliable ad packets carryig ê are lost. This ucertaity results i ˆ d, beig a radom variable to the ecoder. The problem at had is to desig optimal quatizers ad predictors (P E, Q E ad P D) to miimize the epected EED at the decoder to accout for packet losses. For the mea squared error distortio metric, epected EED at the decoder is, E{D} = = E{( ˆ d, ) } ˆ (i) E{ˆ d, } E{(ˆ d, ) }. () Clearly, to estimate this distortio, first ad secod momets of the decoder recostructios should be accurately estimated at the ecoder. 3. BACKGROUND 3.. Ed to Ed distortio estimatio ad predictio A recursive techique to optimally estimate the epected EED at the ecoder i the presece of packet losses via the first ad secod momets of the decoder recostructios was proposed i [] for video coders. The recursive algorithm optimally estimates the decoder recostructios first momet, E{ˆ j d, }, ad the secod momet, E{(ˆ j d, ) }, for every piel j i frame. These momets are the used to estimate EED at the ecoder usig () to optimally switch betwee iter-frame predictio ad itra-frame predictio, to cotrol the error propagatio through frames. I [], a ew predictio scheme is employed i cojuctio with optimal EED estimatio. Q (i) Q (i) Desig a ew Predictor P (i) Fig. 3: Asymptotic closed-loop traiig approach Covetioal motio compesated predictio employs the ecoder recostructios for predictio, i.e., j e, = ˆ jv e,, where v is the optimal motio vector that miimizes the predictio error. Istead i [], the predictio is based o epected decoder recostructios, i.e., j e, = E{ˆ jv d, }, where v is the optimal motio vector that miimizes the EED of (). This setup plays a importat role i limitig error propagatio durig decoder operatio by appropriately selectig motio vectors to predict from referece blocks that are less likely to be corrupted by error propagatio. 3.. Closed-Loop versus Asymptotic Closed-Loop Desig I closed-loop iterative desig [], the coder operates i closedloop at each iteratio to geerate predictio errors ad recostructed samples that are used to desig the updated quatizer ad the updated predictor, respectively. At iteratio i, give a quatizer, Q (i ), ad a predictor, P (i ), a traiig set of predictio errors, T (i) : { } N =, for iteratio i is geerated as, where, = P (i ) ( ), () = P (i ) ( ) Q(i ) ( P (i ) ( )). (3) These two equatios are calculated sequetially for all values. The give T (i), we desig a ew quatizer, Q (i). Usig Q (i), a ew set of recostructed samples, ˆ (i), is geerated as per (3) ad based o this, we desig a ew predictor, P (i). These steps are repeated util covergece. Fig. illustrates this closed-loop iterative desig. The major issue with this approach is that whe the updated parameters are employed i closed-loop at a iteratio, ew predictio errors are geerated, which differ from the errors the quatizer was desiged for, ad this implies differet recostructios are geerated, which differ from the referece recostructios the predictor was desiged for. This mismatch i statistics betwee desig ad operatio builds up over time as the data is fed through the predictio loop i the coder, leadig to istability i the iterative desig of both the predictor ad the quatizer, especially at lower bit rates. The desig techique proposed i [, 5], tackles this statistics mismatch issue by desigig the predictor ad the quatizer i a opeloop fashio, while ultimately optimizig the system for closed-loop operatio. Specifically, the predictio is based o recostructios of previous iteratio, i.e.,the predictio errors are geerated as, = P (i ) (ˆ (i ) ). () 3

3 Give the ew predictio errors, we desig a ew quatizer, Q (i). This Q (i) is ow employed to geerate the recostructed samples of et iteratio as, = P (i ) (ˆ (i ) ) Q(i) ( P (i ) (ˆ (i ) )), (5) agai usig the recostructios of previous iteratio for predictio. Give these ew recostructios, we desig a ew predictor, P (i). Note that the equatios () ad (5) are eecuted idepedetly for each sample of the sequece i a ope-loop way. The mai steps of this techique are depicted i Fig. 3. The ope-loop format esures the predictor ad quatizer employ eactly the same recostructed data ad predictio error used for their desig, elimiatig the statistical mismatch issue see i closed-loop desig. O covergece, the predictor ad the quatizer do ot chage, which implies,, i.e., predictig from previous iteratio recostructios is the same as predictig from the curret iteratio recostructios, which is effectively closed-loop operatio. = ˆ(i ). PROPOSED APPROACH We propose a framework to desig a first order predictor ad a quatizer to miimize the EED i (). We first develop the EED estimatio algorithm, the we propose a ecoder architecture i which predictios are based o the epected recostructios at the decoder ad fially we propose the desig approach that accouts for packet loss to improve codig efficiecy ad desig stability... Epected Decoder Distortio ad Recostructios We assume for simplicity of presetatio that each packet cotais oe sample (or alteratively that iterleavig is used). The packet (or sample) loss rate is deoted as p. The predictio model employed at the decoder is a simple first order liear predictor, d, = αˆ d,, () where α is the predictio coefficiet that eeds to be estimated. The quatized predictio error, ê, trasmitted over the chael, may or may ot be received by the decoder. If the curret packet is received (with probability p), the decoder uses it to geerate the recostructed sample as, ˆ d, = d, ê. (7) Whe the packet is lost, a simple cocealmet of settig residue to zero is employed, which gives the recostructed sample as, ˆ d, = d,. () Thus the first ad secod momet of the decoder recostructed samples, required to estimate EED give i (), are calculated recursively at the ecoder as, E{ˆ d, } = ( p)e{ê αˆ d, } pe{αˆ d, } = ( p)ê αe{ˆ d, } (9) E{(ˆ d, ) } = ( p)e{(ê αˆ d, ) } pe{(αˆ d, ) } = ( p)(ê αê E{ˆ d, }) α E{(ˆ d, ) } (0) e ê Σ Q Chael e, = αe{ˆ d, } Ecoder P ( p)ê E{ˆ d, } D d, = αˆ d, Fig. : Architecture of the proposed coder Decoder.. Predictio Based o the Epected Decoder Recostructios Packet losses cause the recostructios at the ecoder ad the decoder to differ. Thus to close the gap betwee predictio at the ecoder ad the decoder, we employ the epected decoder recostructios for predictio at the ecoder, i.e., P ˆ d, e, = αe{ˆ d, }. () The predictio error, e = e,, is the quatized to geerate, ê. The overall proposed architecture is show i Fig.. We desig the predictio coefficiet α to miimize the EED, by solvig for α i the equatio give by settig the partial derivative of EED with respect to α to 0. The EED i () is depedet o α through equatios (9) ad (0). The equatio to be solved is, E{D} α = E{ˆ d, } ( p)ê E{ˆ d, } αe{(ˆ d, ) } = 0, () which gives us the solutio as, α = E{ˆ d, }( ( p)ê ). (3) E{(ˆ d, ) } Note that although ê is depedet o α, we assume that the modificatios i α across our desig iteratios are small eough to ot chage the quatizatio itervals..3. Asymptotic Closed-Loop Desig We employ the approach for a stable system desig by elimiatig the statistical mismatch issue of closed-loop desig. This is achieved by operatig i a ope-loop way, wherei we employ previous iteratio s first ad secod momets of the decoder recostructios, to estimate curret iteratio s momets ad predictio. Give a set of decoder recostructios first momets, E{ˆ d } (i ), ad secod momets, E{(ˆ d ) } (i ), of iteratio i, the predictor ad quatizer are iteratively desiged i a ier loop. I a subiteratio s of the ier loop, give a set of quatized predictio errors, ê (i,s ), the optimal predictio coefficiet is estimated as, α (i,s) = E{ˆ d, } (i ) ( ( p)ê (i,s ) ) E{(ˆ d, ) } (i ). () D 33

4 -ER -ER -ER (a) (b) (c) Fig. 5: Average decoder distortio (i db) of, ad the proposed -ER desig approaches, for a first order predictive coder with etropy costraied scalar quatizer, at various bit rates for packet loss rates of (a)5% (b)0% (c)0%. The ew predictio errors are ow geerated i a ope-loop fashio as, e (i,s) = α (i,s) E{ˆ d, } (i ). (5) A optimal quatizer, Q (i,s), is ow desiged for this set of ew predictio errors, which is used to geerate a ew set of quatized predictio errors, ê (i,s) = Q (i,s) (e (i,s) ). These subiteratios are repeated util covergece to obtai curret subiteratios fial quatizer, Q (i), fial predictio coefficiet, α (i), ad fial set of quatized predictio errors,. The first ad secod momets of the decoder recostructios are ow updated i the outer loop i a opeloop way as, E{ˆ d, } (i) E{(ˆ d, ) } (i) = ( p) α (i) E{ˆ d, } (i ) () = ( p)(( ) α (i) E{ˆ d, } (i ) ) (α (i) ) E{(ˆ d, ) } (i ). (7) These momets are ow used i the et iteratios ier loop to update the predictor ad quatizer. Iteratios are repeated util covergece. Note that although the etire desig is i ope-loop, o covergece it emulates closed-loop operatio. This is achieved as o covergece the quatizer ad predictor do ot chage, i.e., Q (i) = Q (i ) ad α (i) = α (i ), which implies E{ˆ d, } (i) = E{ˆ d, } (i ), thus employig previous iteratio s momets is the same as estimatig curret iteratio s momets recursively ad employig them for predictio i a closed-loop way. 5. EXPERIMENTAL RESULTS To validate our proposed method, we evaluated it for a compressio system with first order liear predictio ad a etropy costraied scalar quatizer. The Geeralized Lloyd Algorithm (GLA) was used to desig the etropy costraied scalar quatizer. We used the speech files available i the EBU SQAM database [7] as our dataset, as liear predictio is commoly employed i speech codig. However, ote that the proposed approach is applicable to predictive compressio of ay sigal with temporal correlatios. The first half of the speech files were used as traiig set (resultig i more millio samples) ad the secod half as test data. The predictio coefficiet was iitialized to zero. We evaluated the followig three differet desig techiques:. The closed-loop desig procedure discussed i Sectio 3., which completely igores the packet losses (referred as ).. The algorithm discussed i [5], which also igores the packet losses ad ca be obtaied by settig p = 0 i the update formulas of Sectio.3 (referred as ). 3. Our proposed method (referred as -ER). The system performace is evaluated by plottig the decoder recostructio error s sigal to oise ratio () averaged over te differet loss patters versus average bitrate, as show i Fig. 5 for differet packet loss rates. Clearly, the proposed approach cosistetly outperforms both ad uder all testig scearios, with gais of up to 7 db over ad gais of up to.5 db over. The proposed approach provides higher performace improvemets as the packet loss rate icreases, sice accoutig for error propagatio due to packet losses becomes critical i these cases. Compared to, our method provides larger gais at higher bitrates, as relies more o the high quality previous recostructios for predictio, uaware of the fact that these recostructios at the decoder will be corrupted as a result of error propagatio due to packet losses. These results substatiate the sigificat utility of the proposed approach.. CONUSION I this paper we proposed a effective ad robust desig techique for a predictive compressio system. We accout for the presece of a ureliable chael by desigig the system to optimize the EED at the decoder. We the elimiate the statistical mismatch issue suffered by covetioal closed-loop approaches, by employig a stable iterative desig approach that operates i a ope-loop way ad o covergece mimics closed-loop operatio. By carefully desigig the system parameters, error propagatio at the decoder is effectively cotaied. Sigificat performace improvemets see i eperimetal evaluatio results demostrate the utility of the proposed approach. Future research directios iclude, etedig the proposed desig techique to higher order predictors, ad employig a powerful optimizatio techique for desig of the etropy costraied scalar quatizer to accout for packet losses. 7. REFERENCES [] R. Zhag, S. L. Reguatha, ad K. Rose, Video codig with optimal iter/itra-mode switchig for packet loss resiliece, 3

5 IEEE Joural o Selected Areas i Commuicatios, vol., o., pp. 9 97, 000. [] H. Yag ad K. Rose, Source-chael predictio i error resiliet video codig, i IEEE Iteratioal Coferece o Multimedia ad Epo (ICME), 003, vol., pp. II 33. [3] V. Cuperma ad A. Gersho, Vector predictive codig of speech at kbits/s, IEEE Trasactios o Commuicatios, vol. 33, o. 7, pp. 5 9, 95. [] H. Khalil, K. Rose, ad S. L. Reguatha, The asymptotic closed-loop approach to predictive vector quatizer desig with applicatio i video codig, IEEE Trasactios o Image Processig, vol. 0, o., pp. 5 3, 00. [5] H. Khalil ad K. Rose, Robust predictive vector quatizer desig, i Data Compressio Coferece (DCC), 00, pp. 33. [] P.-C Chag ad R. M. Gray, Gradiet algorithms for desigig predictive vector quatizers, IEEE Trasactios o Acoustics, Speech ad Sigal Processig, vol. 3, o., pp , 9. [7] G Waters, Soud quality assessmet materialrecordigs for subjective tests: Users hadbook for the ebu sqam compact disk, Europea Broadcastig Uio (EBU), Tech. Rep, 9. 35