BITRATE ALLOCATION FOR MULTIPLE VIDEO STREAMS AT COMPETITIVE EQUILIBRIA

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1 BITRATE ALLCATIN FR MULTIPLE VIDE STREAMS AT CMPETITIVE EQUILIBRIA Mayank Twar, Theodore Groves, and Pamela Cosman Department of Electrcal and Computer Engneerng, Department of Economcs, Unversty of Calforna, San Dego, La Jolla, CA Techncal Area: I-7 (Image and Vdeo Codng) ABSTRACT Methods for multplexng vdeo streams often rely on dentfyng the relatve complexty of the vdeo sequences to mprove the combned overall qualty. In such methods, the qualty of hgh moton vdeos s mproved at the expense of reducton n the qualty of low moton vdeos. In our approach, we use the Edgeworth box soluton for compettve equlbrum to smultaneously mprove the qualty of all the vdeo streams. The proposed method not only uses nformaton about the dfferng complexty of the vdeo streams at every tme-slot but also the dfferng complexty of one stream over tme. All the vdeo sequences do at least as well as ndvdual encodng.. INTRDUCTIN Applcatons where multple compressed vdeo streams are transmtted smultaneously through a shared channel nclude drect broadcast satellte, cable TV, vdeo-on-demand servce, and vdeo survellance. In exstng methods for transmttng multple vdeo streams [ 3], mprovng the overall qualty s the goal. However, not all vdeo streams benet from multplexng processes. Generally, the qualty of hgh complexty vdeos mproves at the expense of reduced qualty of low complexty vdeos. In ths paper, we constran the vdeo multplexng process so that no vdeo stream wll suffer qualty degradaton. All vdeos wll do at least as well as what they acheve by not partcpatng n the multplexng process. The method selects an expected efcent, Pareto optmal (P), allocaton of btrate for multple vdeos. By computng the expected compettve allocaton n the Edgeworth box, a common tool n economcs for equlbrum analyss, we nd a pont where all users perform better or at least as well as what they can acheve ndependently. The rest of the paper s organzed as follows: Secton descrbes the Edgeworth box for solvng for compettve equ- Ths work was supported by the Natonal Scence Foundaton, the Center for Wreless Communcatons at UCSD, and the UC Dscovery Grant program of the State of Calforna. lbra. Secton 3 descrbes the applcaton of the Edgeworth box soluton n btrate allocaton for multple vdeo streams. Results and conclusons are gven n Secton 4.. EDGEWRTH BX FR CMPETITIVE EQUILIBRIUM The Edgeworth box (EWB) [4] s a graphcal tool for exhbtng P allocaton and llustratng a compettve (Walrasan) equlbrum n a pure exchange economy [5], n whch no producton s possble and the commodtes that are ultmately consumed are those that ndvdual users possess as ntal endowments. The users trade these endowments among themselves n a market for mutual advantage. Let there be two users ( =, ) and two goods (j =, ). User 's consumpton vector s x = (x, x ),.e., user 's consumpton of good j s x j 0. Each user s ntally endowed wth an amount c j 0 of good j. The total endowment of good j n the economy s denoted by c j = c j + cj, assumed strctly postve. An allocaton x R 4 + s an assgnment of a non-negatve consumpton vector to each user: x = (x, x ) = ((x, x ), (x, x )). We say that an allocaton s nonwasteful and feasble f x j + xj = cj (the total consumpton of each good s equal to the economy's aggregate endowment of t). In the EWB, user 's quanttes are measured wth the southwest corner as the orgn ( ) as shown n Fgure. User 's quanttes are measured usng the northeast corner as the orgn ( ). For both users, the horzontal dmenson measures quanttes of good and the vertcal dmenson measures quanttes of good. The wdth and heght of the box are c and c, the economy's total endowment of goods and. Any pont n the box represents a dvson of the total endowment between user and. Gven c = (c, c ), user can calculate ts utlty U ( c ). A curve can be drawn for dfferent x whle keepng the utlty at U ( c ). Such curves, known as ndfference curves, can be drawn for dfferent ntal endowments for each user as shown n Fgure. We assume these curves are convex. For any user, the utlty ncreases when we move away from ts orgn. If we draw ndfference

2 { Indfference curves for user Drecton of ncreasng preference for user c x Indfference Drecton of ncreasng curves for user preference for user { x c x x Fg.. An Edgeworth box curves for both users n the box, the ponts where the ndfference curves for both users are tangental to each other are P allocatons [5]. The set of all P allocatons s known as the Pareto set. The part of the Pareto set where both users do at least as well as at ther ntal endowments s called the contract curve (Fgure ). Any barganng between the users should result n some pont on the contract curve; these are the only ponts at whch both users do at least as well as at ther ntal endowments and for whch there s no alternatve trade that can make both users better off [5]. Contract Curve c c c Pareto set The budget lne slope = p /p x c B (p) = {x R + : p.x p.c } () A compettve equlbrum for an EWB economy s a prce vector p and an allocaton x = (x, x ) n the EWB such that for =,, U (x ) U (x ) x B (p ) () At an equlbrum, each user 's demanded bundle at prce vector p s x and one user's net demand for a good s exactly matched by the other's net supply. The ntersecton of the budget lne and contract curve, where the budget lne s also tangental to the ndfference curve for both the users on the contract curve, wll result n the compettve equlbrum. At ths equlbrum pont, both users are better off compared to ther ntal endowment. Ths s shown n Fgure. More detals about the EWB and compettve equlbrum can be found n [5]. 3. VIDE MULTIPLEXING USING THE EW BX We extend the concept of the EWB from two users to N vdeo users. The two goods are the bts avalable n two tme-slots (TS). We consder one TS as one GP; however, one can choose TS at any level. The complexty of the EWB ncreases wth box dmenson and there are many TS n each vdeo stream. So, we reduce the problem to two TS for each user. We wll use the terms GP and TS nterchangeably. We generate the rate dstorton (RD) curve for each TS by calculatng the mean squared error (MSE) at dfferent btrates. Note that the complexty of generatng the RD curve can be further reduced by usng the method descrbed n [6]. Suppose that 000 bts are avalable n TS and n TS. Suppose also that user and user each has an ntal endowment of 500 bts n each TS. If the RD curves for the two users are such that gvng user 600 bts n TS and 400 n TS (wth vce versa for user ) produces a more favorable total MSE than the equal ntal endowment, then the EWB approach would favor ths allocaton over the ntal one. Whle ths s the basc dea behnd our approach, often adjacent TS have smlar RD curves. Therefore, lttle benet can be ganed by tradng bts between adjacent TS for two users. ne would lke to trade between the current encodng TS and some other TS wdely separated n tme. Snce we may not know the specc RD curve for some dstant GP, nstead of ths, we wll consder trades between the current encodng TS and some expected or approxmate RD curve for the future. The RD curve for user n TS j s tted by Fg.. The Pareto set and the contract curve n the EWB Suppose users can buy or sell these goods n the market for prces p and p. For any prce system p = (p, p ) and ntal endowments, the budget set for user s: D j (Rj ) = aj + bj R j where R j s the number of bts and Dj s the MSE dstorton for TS j n vdeo stream. a j, bj are the coefcents for generatng ths curve-ttng model and we use the least squares (3)

3 approach to nd these coefcents. Note that the above functon s convex. ther curve-ttng models are avalable n the lterature [7]. Let the utlty for user be U (x, x ) = (a + b x + a + b x ) (4) that s, the negatve sum of the MSE n both TS. The utlty s a convex functon. Let the ntal endowment for user be c. Then the ndfference curve through the ntal endowment for user can be derved as (a + b x + a + b x ) = (a + b c + a + b c ) (5) for dfferent combnatons of x. A compettve equlbrum s found by solvng max U (x x, x ) s.t. p x +p x = p c +p c, = to N,x (6) and N N x j = c j j =, (7) = = The Lagrangan expresson for user s L = U (x, x ) + λ (p c + p c p x p x ) (8) By dfferentatng L wth respect to x, x, and λ, equatng the results to 0, we get N b j p c + p c N p j p b + = c j p b j =, (9) = = To determne the compettve equlbrum, we need to nd the slope p /p. Therefore, we assume p = and solve Eq. 9 numercally for p. Wth p, we nd x j whch s the soluton for compettve equlbrum. Frst, we consder the constant bt allocaton for each TS (EQL TS). Here each vdeo n every TS receves an equal number of bts to encode ts vdeo. Ths rate control method s appled n many vdeo standards such as H.64/AVC [8]. We now compare the compettve equlbrum bt allocaton for varous vdeo streams. The rst TS s always consdered to be the current TS that we are encodng. If we assume that we have some nformaton about the future, lke the average RD curves for future TS, then we can use such nformaton for tradng bts for the current TS wth the average of remanng TS (REM TS). Ths s an ex ante approxmaton model where we assume some nformaton about the future. If we have no future nformaton then we predct the future by lookng at the prevous TS (ex post) wth the assumpton that the average RD curve of prevous TS wll be smlar to that of the future. We trade bts for the current TS wth the average of prevous TS (PRE TS). The performance of PRE TS depends on how exactly the past TS represents the future TS. Both REM TS and PRE TS are solved for compettve equlbrum. For comparson, f each user has full nformaton about ts RD curves n all TS, then t can dvde the bts among all the TS based on ther relatve complexty. All vdeo streams use ths crtera for bt allocaton for ther TS ndependently. Snce the total number of bts s constant for each TS, we normalze the number of bts produced n each vdeo stream by the total avalable bts for a TS (FUL TS). In ths paper, these four bt allocaton methods are compared for vdeo multplexng. 4. RESULTS The smulaton was performed usng the baselne prole of H.64/AVC [9] reference software JM.0 [0]. The - second test vdeos contanng varyng types of scenes and moton were taken from a 7 mnute travel documentary at a resoluton of 76 0 pxels and frames per second. The GP sze s 5 frames (I-P-P-P). The frames nsde a TS are encoded usng H.64 rate control [8]. The codng parameters such as resoluton or GP sze can be changed for any approprate applcaton and the results are expected to be smlar. Fgure 3 shows the result of multplexng four vdeo streams. The four curves n each plot represent the multplexng methods descrbed prevously. Each plot shows PSNR versus btrate (rangng from 5-35 (50-70 kbps)). We calculate the MSE of each frame and average across all frames of a vdeo then convert to PSNR. The performance of EQL TS s worst n all vdeos, and ths s the method used n most vdeo standards for GP level rate control. For archved vdeo we know RD curves for all TS and we see that FUL TS performs the best. The PSNR gan over EQL TS vares from db for g to.-.5 db for g9. However, ths method cannot be used for real-tme vdeo multplexng. If a user knows the average RD curve for future TS, then ths s sufcent to mprove the vdeo qualty as shown by REM TS. Ths method nds the compettve equlbrum pont for the current TS when compared to ts average of remanng TS. Ths method mproves the qualty of each vdeo stream from db for g to db for g9 over the EQL TS method. Fnally, we assume that we have no pror knowledge about the vdeo and we predct the future RD curves by lookng at the prevous TS. Agan we compute the compettve equlbrum for the current TS when compared to the average of the prevous TS (PRE TS curve n the gure). Ths method mproves the PSNR from db for g to db for g9. Smlarly, Fgure 4 shows the results for two vdeos (the two most extreme cases) when sx vdeo streams are multplexed together usng the methods descrbed above. The PSNR of all the sx vdeos mproves from these multplexng methods for a wde range of btrates. g9 vdeo agan performs the best whle the performance for REM TS, PRE TS, and FUL TS are the same n g5 vdeo. We note that the largest PSNR gan s acheved by ndng the compettve equlbrum when there s a lot of uctuaton

4 (a) g vdeo stream (b) g9 vdeo stream (c) g0 vdeo stream (d) g vdeo stream Fg. 3. PSNR varaton wth btrate for four multplexed vdeo streams n the vdeo moton, for example g9. Conversely, the PSNR gan s low f the moton uctuaton n a vdeo stream s low, for example g. Most of the vdeo streams have a lot of moton uctuaton and scene change, so multplexng them by computng the compettve equlbrum wll mprove the qualty. The performance of PRE TS depends on how accurate s the representaton of future TS from past TS. As can be seen from Fgure 3, all the vdeo streams gan from the multplexng process. The PSNR gan vares from one vdeo to another, dependng on the content. The multplexng method usng the compettve equlbrum borrows bts from a low moton TS of a vdeo and gves these bts to another vdeo n the same TS wth the promse of takng t back when the need arses. So, the multplexng method exchanges bts between vdeo streams as well as across the TS. Ths leads to another observaton that the qualty uctuaton for each vdeo stream s reduced because the hgh moton TS gets more bts than the low moton TS nstead of gettng the same number of bts for all TS. Fgure 5 shows the PSNR uctuaton for g9 for all the multplexng methods. The EQL TS method has the hghest uctuaton and FUL TS method has the lowest. In the end, all the vdeos receve equal numbers of bts n the multplexng method unlke prevous methods for vdeo multplexng where some vdeos get more bts than the other vdeos. By changng the encodng technque nsde a GP (e.g., usng multple reference frame predcton or usng herarchcal B- frames), along wth these multplexng methods, the overall vdeo qualty can be expected to further mprove. In concluson, we dscussed four methods for multplexng vdeo streams. We proposed two novel methods of multplexng vdeo streams usng the EWB soluton for ndng compettve equlbrum. The results show PSNR mprove-

5 (a) g9 vdeo stream Tme Slot Fg. 5. Varaton of PSNR wth TS for g9 vdeo at 50 kbps (b) g5 vdeo stream Fg. 4. PSNR varaton wth btrate for sx multplexed vdeo streams ment for all vdeo streams unlke prevous methods [ 3] where the qualty of some vdeos s mproved whle deteroratng the qualty of other vdeos. The PSNR gan s greater for vdeos wth hgher moton uctuaton. In future work, we would lke to examne better estmaton of future frames from the prevous frames. 5. REFERENCES [] L. Wang and A. Vncent, Bt allocaton and constrants for jont codng of multple vdeo programs, IEEE Trans. on Crcuts and Systems for Vdeo Tech., vol. 9, no. 6, pp , Sept [] G. Su and M. Wu, Efcent bandwdth resource allocaton for low-delay multuser vdeo streamng, IEEE Trans. on Cr. and Sys. for Vdeo Tech., vol. 5, no. 9, pp. 4 37, Sept [3] M. Twar, T. Groves, and P. Cosman, Multplexng vdeo streams usng dual-frame vdeo codng, Proc. IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng, Mar [4] F. Edgeworth, Mathematcal psychcs: An essay on the applcaton of mathematcs to the moral scences, London: C.Kegan Paul and Co., 88. [5] A. Mas-Colell, M. Whnston, and J. Green, Mcroeconomc theory, xford Unversty Press Inc., 995. [6] L. Ln and A. rtega, Bt-rare control usng pecewse approxmated rate-dstorton characterstcs, IEEE Trans. on Cr. and Sys. for Vdeo Tech., vol. 8, no. 4, pp , Aug [7] K. Stuhlmüller, N. Färber, M. Lnk, and B. Grod, Analyss of vdeo transmsson over lossy channels, IEEE Journal on Selected Areas n Comm., vol. 8, no. 6, pp. 0 03, June 000. [8] K.-P. Lm, G. Sullvan, and T. Wegand, Text descrpton of jont model reference encodng methods and decodng concealment methods, Jont Vdeo Team of IS/IEC MPEG and ITU-T VCEG, JVT-K049, Mar [9] T. Wegand, G. Sullvan, G. Bjontegaard, and A. Luthra, vervew of the H.64/AVC vdeo codng standard, IEEE Transactons on Crcuts and Systems for Vdeo Tech., vol. 3, no. 7, pp , July 003. [0] H.64/AVC ref. software,