A Novel Joint Rate Allocation Scheme of Multiple Streams

Transcription

1 A ovel Jont Rate Allocaton Scheme of Multple Streams Hongfe Fan, Ln Dng, Huzhu Ja*, Xaodong Xe Abstract Encodng multple vdeos n parallel and transmttng them as one jont stream over a lmted bandwdth have become a popular strategy for broadcastng, whch brngs an opportunty to allocate dfferent btrate for each sequence to meet dfferent demands. In ths paper, consderng vsual experence for human bengs, we propose a jont rate allocaton scheme ams to reach an equal vsual qualty among all sequences by mnmzng the dstorton varance of all the sequences (denoted as mnvar problems. Exstng methods assgned bts drectly n proporton to ther complexty measures and we named them as complexty based allocaton scheme (CAS methods. CAS methods rely on the accuracy of the complexty measures whch can hardly be mproved under lmted computng resources. Also complextes may not be drectly related to the dstortons. To address these problems, we present a novel jont rate-dstorton (R-D based allocaton scheme (RDAS n ths paper. Our proposed scheme can ft for dfferent R-D models and n our method we model the R-D relatonshp wth a Hyperbolc functon (RDAS-H. We also derve a closed-form soluton of RDAS-H by a proposed jont R-D relatonshp. We ntegrated the RDAS-H method n HEVC reference software HM6.0. Expermental results demonstrate that our RDAS-H saves 75.29% varance on average over the related CAS-based method n [2], where we apply both LowDelay and RandomAccess confguratons wth four dfferent overall bandwdths for all classes recommended by JCT-VC. Besdes, RDAS-H also saves 36.62% varance on average over our prevous method [7]. The proposed RDAS-H method mproves the performance sgnfcantly whle requrng neglgble computatonal cost. Index Terms Jont rate allocaton, Statstcal multplexng, Rate-Dstorton model, mnvar problems, HEVC W I. ITRODUCTIO ITH the advancement n multmeda technology and dgtal communcatons, the amount of vdeo programs has ncreased n terms of ts applcatons. In ths case a H. Fan, L. Dng, H. Ja, and X. Xe are wth the Insttute of Dgtal Meda, School of Electronc Engneerng and Computer Scence, Pekng Unversty, Bejng 0087, Chna (e-mal: hffan@pku.edu.cn; ln.dng@pku.edu.cn; hzja@pku.edu.cn; donxe@pku.edu.cn. * the correspondng author, H. Ja s wth Pekng Unversty, also wth Cooperatve Medanet Innovaton Center and Beda (Bnha Informaton Research. Ths work s partally supported by the atonal atural Scence Foundaton of Chna ( and 66020, the Major atonal Scentfc Instrument and Equpment Development Project of Chna under contract o. 203YQ030967, atonal Key Research and Development Program of Chna (206YFB and 206YFB Copyrght 208 IEEE. Personal use of ths materal s permtted. However, permsson to use ths materal for any other purposes must be obtaned from the IEEE by sendng an emal to pubs-permssons@eee.org. sngle-program channel s substtuted by broadcastng multple vdeos over a lmted bandwdth, where dfferent vdeos are encoded n parallel and the btrate s multplexed to dfferent vdeos to share the avalable bandwdth. To address such a jont rate allocaton ssue, a smple and drect soluton s to encode each sequence wth equal btrate. evertheless, the dfference n sequence content can be large and ths scheme may lead to a sgnfcant dfference n qualty among the sequences. One sequence wth hgh dstorton wll nevtably lead to poor vsual experence for those who are watchng ths very sequence. Instead, an alternatve soluton s to allocate dfferent btrate for each sequence to meet demands lke gettng smlar vsual experence for the sequences. Specfcally, statstcal multplexng s a way to dynamcally allocate btrate accordng to the dfferent characterstcs of vdeo streams. Typcally, statstcal multplexng methods can be classfed nto two types accordng to dfferent optmzaton targets, whch are mnmum average dstorton (denoted as mnave problem [-5] and mnmum dstorton varance (denoted as mnvar problem [6-7]. The group of frames (GOP of all the sequences encoded n a fxed tme nterval s called a super GOP n ths paper. The btrates of dfferent sequences are allocated by statstcal multplexng methods for each super GOP. The problem of statstcal multplexng can be formulated as to mnmze the average dstorton or the varance of dstorton of all sequences, subject to the constrant that the sum of btrates n one super GOP s lmted by an overall bandwdth, and the channel should be fully utlzed,.e. the total qualty should be maxmzed, mn{ R D k}, mnave problem k mn {, R { k (D k D k 2 }, mnvar problem = s. t. (R c ε R k = R c ( where R k and D k denote the btrate and dstorton for the th stream n the k th super GOP, respectvely; D k s the average dstorton of all sequences n the k th super GOP. s the total number of vdeos n the channel. R c denotes the overall bandwdth for one super GOP. ε s a predefned small value accordng to dfferent applcaton. In ths paper we consder an deal stuaton where ε = 0. The objectve of mnave problems s to acheve maxmum (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 average vsual qualty among the multplexed vdeos. Although mnave methods wll mprove the overall vsual qualty, t may result n dfferent vsual qualtes among multple sequences. In ths case, one sequence wth hgh dstorton wll lead to a poor vsual experence for those who are watchng ths very sequence. Dfferent from mnave methods, mnvar methods am to reach an equal vsual qualty among all sequences, whch result n a better vsual experence. In many practcal applcatons such as broadcastng, mnvar method s a more preferred soluton for a better overall vsual experence. In ths paper, we focus on mnvar problems. In current methods to acheve mnvar, bts are assgned drectly n proporton to ther complexty measures. Snce the btrate allocaton only depends on complexty measures, we classfed them as complexty-based allocaton scheme (CAS methods. Generally n CAS method [6-5], a more accurate complexty measure can lead to a better performance by mnmzng varance of multple programs. Accordng to the expermental results n [2] and [3], the performances of ther proposed complexty measures are better than others. In [2], both the frame actvty and the moton actvty are used to characterze the vdeo complexty. The weghtng factor, whch wll be dynamcally updated usng the data from the prevous GOP, s chosen to balance between frame actvty and moton actvty. In [3], nspred by Structural Smlarty Index (SSIM, a new smlarty ndex (SMI s ntroduced to measure the smlarty between two adjacent frames as temporal complexty. The authors also mentoned a spatal complexty for stll mage. Dfferent from CAS, [6] proposed a scheme by searchng a best dstorton value for the frames to be allocated. An teratve golden-secton search and a refnement search were carred out to reduce the computatonal cost. However, the performance of allocaton wll depend on the search qualty. Typcally, the computng resource for look-ahead approach s lmted whch s far less than that for the formal encodng path. Although the accuracy of complexty measures has been mproved n the state-of-art works, the effcency of allocaton scheme on mnvar problems can hardly be mproved sgnfcantly under such lmted computng constrants. Besdes, there s no mathematcal dervaton to prove that assgnng bts drectly by the proporton of complexty measures can mnmze the varance. In our prevous method [7], we proposed an allocaton scheme by analyze R-D relatonshp and we use an nverse proporton functon as a coarse estmaton of the R-D relatonshp to smplfy the dervaton of the allocaton formula. Results demonstrate that our prevous method performs much better than CAS methods. Though [7] s good, the mscalculaton wll be amplfed when the dfference between the real dstorton fed back by the encoder and the predcted dstorton for the next super GOP s bg. In ths paper, we propose a novel jont rate-dstorton based allocaton scheme. Bandwdth s allocated consderng not only complexty measures but also the R-D relatonshp bult by the nformaton fed back from the encoder. The allocaton scheme s derved to meet the target functon of mnvar problem mathematcally. We can allocate the bandwdth properly f we can buld the R-D relatonshp for all sequences and the proposed scheme can work wth all dfferent R-D models. In the frst step, the btrate and dstorton values of the newly coded super GOPs are obtaned from the encoder. Then, the complexty measures of the next super GOPs are calculated by look-ahead approach. After that, bts are assgned to each sequence accordng to the allocaton formula, consderng the R-D model bult by the feedback btrate, dstorton, and look-ahead complexty measure together. Fnally, dfferent sequences n the current super GOP are coded wth the allocated btrate. The contrbuton of ths work manly les n the followng two aspects: We derve an R-D model based jont rate allocaton scheme (RDAS wth the target functon of mnvar problem, whch s applcable to all knds of R-D models. Furthermore, a Hyperbolc functon based R-D model s appled to the RDAS scheme (RDAS-H to obtan a better allocaton result. 2 We derve a closed-form soluton of RDAS-H nstead of a brute-force soluton to allocate the btrate n O( tme. To acheve ths, we derve a jont R-D model whch descrbes the relatonshp between the average rate and the dstorton for a jont stream of multple sequences based on nformaton theory. The rest of ths paper s organzed as follows. Secton II gves the problem statement of statstcal multplexng. Secton III gves the theoretcal detals of the proposed method. Secton IV shows the expermental results to demonstrate the effectveness of the proposed RDAS-H n comparson wth that of [7] and the related CAS methods n [2]. Ths paper s fnally concluded n secton V. II. JOIT RATE ALLOCATIO PROBLEM AD RELATED WORK In ths secton, we frstly state the mnvar problem n Part A. Then we brefly ntroduce the related work [2] n Part B whch s a representatve CAS method. A. Problem Statement In our system, multple sequences are encoded as a jont stream and btrate s allocated for each sequence at the begnnng of each super GOP. Assume that the k th super GOP s already encoded and we are now allocatng the btrate for the (k + th super GOP. The btrate R k and dstorton D k of the k th super GOP for the th sequence can be obtaned from the encoder. Then the mnvar problem can be rewrtten as follows, mn { (D k+ D k+ 2 = }, s. t. (R c ε R k+ R c (2 R k+ where D k+ s the average dstorton of all sequences n the (k + th super GOP. R k+ s the allocated rate for the th sequence n the (k + th super GOP. In ths paper, dstorton s measured by mean square error (MSE and btrate s measured by bt per pxel (BPP. Accordng to related CAS methods, rates are allocated wth the pre-process of nformaton to meet the target functon. Pre-process nformaton s called complexty measure whch s = (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

3 obtaned by the look-ahead approach. The complexty measure of the th vdeo n the k th super GOP s denoted as C k. In ths paper, we also use nformaton of the last GOP fed back from the encoder. Thus, the bts are allocated as, R k+ = R(R k, D k, C k, C k+ (3 where the allocaton scheme R( s derved wth the nformaton meetng the constrants n Eq.2. The framework of jont vdeo codng s llustrated n Fg.. Frstly, frames of ndvdual sequences n one super GOP are pre-processed by look-ahead approach to get complexty values. Then, btrate for each sequence s allocated by the Jont Rate Allocator usng the allocaton scheme n Eq.3. After that, Rate Controller wll calculate QP and λ for ndvdual sequence accordng to the allocated btrate. ext, sequences are encoded separately by the encoder wth the gven QP and λ. At last, the encoded streams are sent to the Jont Buffer. At the same tme, the real btrate, dstorton value and other parameters are sent back to the Jont Rate Allocator on demand for the rate allocaton of the next super GOP. In ths paper, we only focus on the Jont Rate Allocator and assume that all the other parts n the system ncludng Preprocessor, Rate Controller and Encoder are the same as the exstng methods. In Preprocessor, we apply the same complexty measure as [2], and the complexty measure wll be brefly ntroduced n Part B. In Rate Controller, the λ-doman rate control algorthm n [23] s appled, n whch a Hyperbolc R λ model s derved from the Hyperbolc R-D model. Besdes, snce our proposed scheme also use Hyperbolc R-D model, parameters of the model n the Rate Controller wll be reused n the Jont Rate Allocator, whch s denoted as other parameters n Fg.. Many researchers have proposed dfferent methods for complexty measure computaton. In [3] a novel complexty measure s proposed whch outperforms that n [2] accordng to the results of [3]. However, [3] dd not gve all the constant values of ther complexty measure and the performance of [2] and [3] are stll comparable. So we apply the complexty measure n [2] n our system whch s robust wth lttle computatonal cost. The complexty measure n [2] ncludes C Texture and C Moton where C Texture denotes the complexty measure for the frame texture and C Moton denotes that for motons. C Texture s calculated wth, C Texture = H W H W ( lum(, j lum( +, j = j= + lum(, j lum(, j + (5 where H and W represent the heght and the wdth of the frame. lum(, j s the lumnance value of pxel n the poston (, j of the current frame. C Moton s defned as, C Moton = H W M H W lum(, j, k lum(, j, k + k= = j= where lum(, j, k s the lumnance value of pxel at poston (, j of the k th frame and M s the number of frames n a super GOP. Combnng Eq.5 and Eq.6 together, (6 C = θ C Texture + ( θc Moton (7 where θ s a weghtng factor chosen as 0 at the start of encodng. After encodng a super GOP, θ wll be dynamcally updated as the proporton of bts used for codng the I-frame of the whole GOP. Fg.. Framework of jont vdeo codng. B. Related Jont Rate Allocaton Scheme n [2] CAS methods to acheve mnvar can be formulated as, III. THE PROPOSED ALLOCATIO SCHEME In ths secton, we frstly derve an R-D model based allocaton scheme n Part A, whch s applcable to all knds of R-D models. Then, n Part B, Hyperbolc R-D model, whch outperforms other models on HEVC for most cases, s analyzed and appled to our allocaton scheme. Besdes, problems caused by applyng Hyperbolc functon are analyzed. Therefore, n Part C, we solve ths problem and propose a jont R-D model and derve a closed-form soluton for our scheme. And, n Part D, we gve the detaled estmaton of parameters. We fnally analyze the R-D performance when applyng our proposed allocaton scheme n Part E. R k = C k = C k R c (4 Snce complexty measures are often computed wth lmted resource, a more precse complexty measure wth neglgble computatonal costs wll mprove the allocaton effcency. A. Rate-Dstorton Model Based Allocaton Scheme (RDAS To allocate rate meetng the constrant n Eq.2 precsely, we take the Rate-Dstorton relatonshp nto consderaton. Assume the relatonshp between rate and dstorton as R = R(D. The rate of the (k + th super GOP of the th vdeo can be expressed as, (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

4 R k+ = R k+ (D k+ (8 the dstorton value changes n a small range. When the k th super GOP s encoded, we have, where D k+ s the dstorton correspondng to R k+. R k+ ( s the R-D relatonshp of all frames n the th vdeo n the (k + th super GOP. Consderng the target functon of mnvar problem gven n Eq.2, the deal allocaton for the (k + th super GOP s as follows, [, ], D k+ = D k+, s. t. R k+ = = R c (9 whch means the dstorton values of every sequence n the (k + th super GOP are the same. Thus, we denote D k+ as the correspondng dstorton for an deal allocaton. We sum up R k+ of dfferent sequences n the (k + th super GOP together, Assume that R k+ R c = R k+ = R k+ = R k+ (D k+ = > 0, we dvde Eq.8 by Eq.0 and get, = R k+ (D k+ = R k+ (D k+ (0 R c ( As we can see, Eq. s an R-D model based jont allocaton scheme derved from the target functon of mnvar problem. The allocaton scheme s applcable for all knds of R-D models. Gven the R-D relatonshp andd k+, the allocaton btrate for the (k + th super GOP can be calculated drectly. B. The Proposed RDAS-H In ths part, we frstly revew our prevous work [7] n whch we proposed a RDAS scheme whch used an nverse proporton functon for a coarse estmaton of the R-D relatonshp (RDAS-I. To further mprove the effcency of RDAS-I, we propose a refned RDAS scheme n whch the R-D relatonshp s modeled by the Hyperbolc functon (RDAS-H. In RDAS-I, a quadratc R-D relatonshp s modeled by expandng rate-dstorton functon nto a Taylor seres n [8]. Therefore, the nverse proporton functon based R-D model s a coarse estmaton of the quadratc model whch s derved n [7] and [34] as, γ C2 D = R (2 where γ s a constant parameter whose value reflects the characterstc of a sequence. C s the complexty measure of the sequence whch can be calculated wth Mean Absolute Dfference (MAD between the pxels of the current frame and that of the reference frame accordng to the dervaton n [8]. In [7], the model n Eq.2 was verfed to be acceptable when R k = γ k C k 2 D k (3 We assume that the frame types are the same n each super GOP wth both LowDelay and RandomAccess confguratons and the characterstcs between adjacent super GOPs of a sequence are smlar. So, γ k+ γ k = D k R k C k 2 (4 where γ k+ s the estmated value of γ k+. Combnng Eq., Eq.3 and Eq.4 together, the allocaton btrate can be derved as, R k+ = whch can be rewrtten as, where, R k+ γ k+ j= 2 C k+ D k+ j γ k+ j 2 C k+ D k+ = X k+ j X k+ j= X k+ R c R c (5 = D k R 2 k C k+ C 2 (6 k Expermental results show that RDAS-I outperforms CAS n [2]. The advantage by applyng the nverse proporton model s that we can remove D k+ n Eq.5. Otherwse, D k+ must be estmated to calculate R k+. However, the nverse proporton based R-D model s not precous enough to descrbe the relatonshp between R and D, wth whch the allocaton error may be amplfed when scene changes. Consderng the sde effect when applyng the nverse proporton model, a more precse R-D model s preferred to conduct a better allocaton scheme. In ths paper, we propose a refned RDAS n whch the R-D relatonshp s modeled by the Hyperbolc functon whch outperforms other models on HEVC for most cases [22][23]. The Hyperbolc functon based R-D model s one of the hghly effectve R-D models to represent the characterstcs of sequences [20][2] as, D = α R β (7 where α and β are parameters related to vdeo contents. Consderng R-D model n Eq.7, we use the Hyperbolc functon based R-D model for each sequence n a super GOP, then we can have the R-D relatonshp for each sequence n the (k + th super GOP as, (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

5 R k+ = α k+ β k+ D k+ (8 where α k+ and β k+ wll be dscussed n part D. Combnng Eq. and Eq.8 together, the allocaton btrate can be derved as, the average dstorton of the communcaton system and D s a constant value. Here we use squared-error dstorton. We denote the mnmum average nteractve nformaton wth the constrant D D as R I (D, whch can be expressed as R I (D = mn E(X X 2 D I(X; X (23 R k+ = α k+ = α k+ β k+ D k+ β k+ D k+ R c (9 In Eq.9, we fnd that we need to estmate α k+, β k+ and D k+ to obtan R k+ wth a gven R c. Therefore, we wll dscuss the estmaton of D k+ n part C and that of (α k+, β k+ n part D. C. A Closed-Form Soluton for D k+ wth the Proposed Jont R-D Model In Eq.9, one problem caused by applyng Hyperbolc model s that D k+ must be estmated. A smple way s to brute-force all possble dstortons, D k+ = D, mn R {D [0,MAX 2 c ]} α k+ = D β k+ (20 where MAX s the maxmum possble pxel value of the mage. We can accelerate Eq.20 wth the bnary search algorthm or other algorthms. However, a closed-form soluton s more preferred. To derve a closed-form soluton for D k+, we defne a jont R-D model R(D to descrbe the relatonshp between the average rate and dstorton for a jont stream of multple sequences. ow, we dscuss the jont R-D relatonshp n one super GOP wth the constrant of the deal allocaton n Eq.9, R(D = R (D = (2 where the subscrpt denotes the th sequence. D s measured by MSE and R(D represents the average BPP of all sequences n one super GOP. Snce the MSE of dfferent sequences s the same n the mentoned deal allocaton, R(D and R (D have the same nput D. To derve the jont model R(D, we analyze the dfference of the R-D model between one sngle sequence and a jont stream. Laplace dstrbuton s commonly used for modelng the dstrbuton of transformed coeffcents of natural mages [28]. Therefore, we assume that one sngle transformed coeffcent s a zero mean Laplace source X~Laplace(0, b and the probablty densty of X s, p(x = x 2b e b (22 Accordng to the nformaton theory [25], we can always fnd one nformaton channel whch can mnmze the average mutual nformaton value wth the constrant D D where D s where X s the output value of the channel. E(X X 2 s the average dstorton D. I(X; X s the average nteractve nformaton. R I (D s consdered as the lower bound of the rate to encode source X wth the constrant D D. Accordng to work [25], we have, I(X; X = h(x h(x X = h(x h(x X X (24 h(x h(x X (25 h(x h ( (0, E(X X 2 (26 = h(x 2 ln 2πeD (27 where h(x s the entropy of source X. h(x X s the condtonal entropy. (0, E(X X 2 s a zero mean normal dstrbuton wth varance E(X X 2. Here Eq.25 follows from the fact that condtonng reduces entropy (Theorem n [25] and Eq.26 follows from the fact that the normal dstrbuton maxmzes the entropy for a gven second moment (Theorem n [25]. Accordng to the defnton of entropy, we have, h(x = p(x ln p(x = ln(2be (28 where p(x s the probablty densty functon n Eq.22. Combne Eq.24-28, we can rewrte Eq.23 as, R I (D = ln(2be 2 ln 2πeD = 2e ln ( 2 πd b2 (29 ext, we dscuss the calculaton of the mnmum average nteractve nformaton of all transformed coeffcents of one stream. In the current vdeo codng system, ntra and nter predcton modes are used to elmnate the redundancy among pxels before transformng, and we can consder the transformed coeffcents as ndependent values. Besdes, these coeffcents may have dfferent dstrbuton parameter b n Eq.22. For example, the dstrbutons of pxels of ntra blocks and nter blocks are dfferent. Thus, the problem can be consdered as calculatng the mnmum average nteractve nformaton of a parallel Laplace source X j ~Laplace(0, b j, j =,2,, P where P s the number of coeffcents n one super GOP. Here b j mght be equal for coeffcents wth the same mode. The relatonshp between R I and D of a parallel Laplace source X j s, P R I (D = 2e ln ( 2 πd b j 2 j= = P 2 2 P P 2e ln πd ( b j (30 j= (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

6 Accordng to Eq.30, the R I -D relatonshp for coeffcents of the same sequence n one super GOP s equvalent to that of a parallel Laplace source X j ~Laplace(0, b, j =,2,, P P P. where b = ( j= b j Then, smlar wth the transformed coeffcents of one stream, transformed coeffcents of a jont stream n one super GOP can also be consdered as a parallel Laplace source X,j ~Laplace(0, b,j, =,2,,, j =,2,, P and (, j denote the j th coeffcent n the th sequence. Therefore, the R I (D of a jont stream X,j n one super GOP s, P 2 R I (D = 2e ln ( 2 πd b,j = j= = P 2 2 ln 2e P P πd ( b,j (3 = j= Smlarly, from Eq.3, the R I -D relatonshp for coeffcents of the jont stream n one super GOP s consstent wth that of a parallel Laplace source X j ~(0, b, j =,2,, P P where b = ( = j= b,j P, that s the R-D relatonshp of a jont stream can be equvalent to that of one sngle sequence. The Hyperbolc R-D model n [20] s derved wth three assumptons whch are satsfed by most natural sequences. If the jont stream stll satsfes these assumptons, we can also model the jont R-D relatonshp wth the Hyperbolc functon. To verfy these assumptons, we defne h[x] as the normalzed dscrete hstogram of the P transformed coeffcents a[j], j =,2,, P of one sequence n a super GOP. x s the value of transformed coeffcents so that x h[x] =. Here P s consdered to be suffcently large and the hstogram s suffcently regular. The values of ths hstogram are nterpolated to defne a functon p(x 0 for all + x R such that p(x dx =. Ths p(x s the probablty densty of a random varable X. The frst assumpton s that p(x s symmetrc, (a Class B (b Class D Fg.3. The log 2 z-log 2 s (z curves for Class B and Class D. s ( k P = a[j k ], k =,2,, P (33 Snce 0 < k, we defne a functon s (z for any z P [0,] nterpolated by the values s ( k. P The next two assumptons are, d log 2 s (z < 0, z [0,] (34 d log 2 z d 2 log 2 s (z 0, z (0, (35 (d log 2 z 2 The three assumptons can be verfed by the followng analyss. Assumpton. In Fg.2, we gve the x-p(x dstrbutons for the jont streams of Class B and Class D n one super GOP. The dstrbutons are obtaned under LowDelay confguraton wth QP 27 and the coeffcents n skpped blocks are all consdered as zeros. Moreover, snce IDCT s appled n HEVC, coeffcents of dfferent TU szes have been multpled by dfferent constant values. So, we dvde coeffcents of TU sze 32 32, 6 6, 8 8, and 4 4 by 4, 8, 6, and 32, respectvely. The symmetry of the probablty densty assumpton n Eq.32 can be clearly observed and therefore t s verfed. 2 Assumpton 2. In Fg.3, the log 2 z-log 2 s (z curve for all the sequences of Class B and Class D and ther jont streams are shown. As a result, the property of the jont stream curve s smlar wth that of normal sequences. We can fnd that log 2 s (z strctly decreases when log 2 z ncreases. So the assumpton n Eq.34 s satsfed. 3 Assumpton 3. Moreover, the curve s obvously concave whch means Eq.35 s satsfed. p(x = p( x (32 We sort the P transformed coeffcents a[j] by ther ampltudes. The ampltude of the k th coeffcent s wrtten as a[j k ] and a[j k ] a[j k+ ], k =,2,, P. We defne, (a Class B - LD (b Class B - RA (a Class B (b Class D Fg.2. The x- p(x dstrbutons of jont stream for Class B and Class D. (c Class D - LD (d Class D - RA Fg.4. The R-D curves for the jont stream and the correspondng hyperbolc fttng curve of Class B and Class D under both LowDelay and RandomAccess confguratons (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

7 So, accordng to the above analyss, the concluson s that a jont stream of coded streams can also be modeled usng a Hyperbolc R-D model and R(D n Eq.2 can be modeled as, R(D = R (D = α D β (36 = We verfy our jont R-D model n Eq.36 wth some experments. In Fg.4, we gve the R-D curves of one super GOP of Class B and Class D under both LowDelay and RandomAccess confguratons. The length of super GOP s set as 6. The horzontal axs and vertcal axs denote MSE and BPP, respectvely. The dot lne s the R-D curve of sngle sequences and the sold lne s the curve of the correspondng jont stream. Besdes, we also gve the fttng curve of jont stream usng Hyperbolc functon wth dash lne and the coeffcent of determnaton R 2 s also gven n the fgure. As a result, the dash lne well fts the sold lne whch means our proposed Hyperbolc jont model can model the jont R-D relatonshp well. Thus, for the k th super GOP, the parameter of jont R-D relatonshp can be calculated as, α k D k β k = (α k D k β k = (37 where D Real s the real dstorton value and R Real s the real rate value of the th stream. α and β are calculated by Eq.39, n whch α k and β k are estmated by Least Mean Square (LMS method wth R-D values calculated when QPs are 22, 27, 32, and 37. D k s the average dstorton of all sequences obtaned wth QP 22, 27, 32, and 37. Expermental results of E(D Real are calculated on sequences recommended by JCT-VC and the results are gven n Table I. LowDelay confguraton s appled n HM frames are tested wth QP 22, 27, 32, and 37. The R-D relatonshp of each sequence s ftted separately wth Hyperbolc functon and the correspondng parameters are gven n α and β column. R( column denotes the btrate correspondng to the gven dstorton (. D s the average MSE of each Class coded wth QP 22, 27, 32, and 37. D equals to 8, 30, 33, 35, 9, and for Class A to Class F. Lne Real R of Jont Stream gves the sum of the real BPP for each class. Lne Estmate R of Jont Stream gves the BPP estmated wth the proposed jont R-D relatonshp. Lne E(D Real gves the error defned n Eq.40. ote that the errors of R( 2 3 D and R( 4 3 D always equal to 0 because of Eq.38. As shown n the expermental results, the error E(D Real can almost be neglected. Snce we have derved the jont R-D relatonshp, consderng that the channel should be fully utlzed for each super GOP, we have, where α k and β k are the parameters related to ndvdual vdeo s content n one super GOP. Consderng practcal applcatons, we propose a fast algorthm to calculate α k and β k wthn O( tme as, So, we can calculate D k+ as, R c = R k (D k = α k D kβ k (4 (α k β d k k = { (α k (2d k β k = = α k d k β k = α k (2d k β k (38 where d k = 2 3 D k, and D ks the average value of D k. α k and β k are the estmated values of α k and β k. Eq.37 can be rewrtten as, (α k (2d k β k = β k = log 2 (α β k d k = k (39 α k = (α β k d k k β k { d k = To further verfy the error of the proposed jont R-D relatonshp R k (D k = α k D kβ k wth real R-D values, we defne the error functon as, E(D Real = α D β Real = R Real = RReal (40 D k+ = ( R c R k+ n Eq.9 s fnally derved as, R k+ = R c = α k+ β k+ (42 α k+ [α k+ β k+ β k+ α k+ ( R c β k+ β k+ ] α k+ ( R c (43 In our proposed RDAS-H, btrate s allocated accordng to Eq.43. Besdes, our model can be drectly used under dfferent codng structures or dfferent rate control schemes such as [23] or [33]. D. Parameter Estmaton for RDAS-H Snce there s no closed-form expresson derved for α and β n the Hyperbolc model, we need to estmate α k+ and β k+ by the obtaned value of the k th super GOP to allocate btrate for the (k + th super GOP n Eq (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

8 ClassA ClassB ClassC ClassD ClassE ClassF TABLE I THE VERIFICATIO OF THE ESTIMATIO I EQ.20 OF SEQUECES RECOMMEDED BY JCT-VC. Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 PeopleOnStreet Traffc Real R of Jont Stream Estmated R of Jont Stream E(D Real 0.22% 0.00% 0.08% 0.09% 0.06% 0.00% 0.07% Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 BasketballDrve BQTerrace Kmono ParkScene Cactus Real R of Jont Stream Estmated R of Jont Stream E(D Real 2.00% 0.00% 0.69% 0.75% 0.46% 0.00% 0.56% Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 BasketballDrll BQMall PartyScene RaceHorses Real R of Jont Stream Estmated R of Jont Stream E(D Real 0.07% 0.00% 0.03% 0.03% 0.02% 0.00% 0.02% Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 BasketballPass BlowngBubbles BQSquare RaceHorses Real R of Jont Stream Estmated R of Jont Stream E(D Real 0.7% 0.00% 0.06% 0.07% 0.04% 0.00% 0.06% Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 FourPeople Johnny KrstenAndSara Real R of Jont Stream Estmated R of Jont Stream E(D Real 0.97% 0.00% 0.35% 0.38% 0.24% 0.00% 0.30% Sequence α β R( 2 R( 2 3 R( 5 6 R(D R( 7 6 R( 4 3 R( 3 2 ChnaSpeed SldeEdtng SldeShow Real R of Jont Stream Estmated R of Jont Stream E(D Real 0.52% 0.00% 0.9% 0.22% 0.4% 0.00% 0.8% As we have mentoned before, the rate control algorthm n [23] s appled n our system, n whch a Hyperbolc model s also used. Thus, the result of the Rate Controller can be reused. In [23], the method to estmate parameters of the Hyperbolc model s proposed, whle the estmaton method s further mproved n the recent work [27] and we use the parameter estmaton method of [27] nstead of ths part of [23] n our Rate Controller. The Hyperbolc model used n [23][27] s, λ = Α R Β (44 where Α and B are parameters related to vdeo contents. And, (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

9 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 9 λ = dd dr (45 After the encodng of the k th super GOP, we can obtan A and B from the Rate Controller. Combnng Eq and the model n Eq.7, we can have the followng relatonshp of α, β and Α, B as, α { β = = ( A B + β β (46 where s the sequence ndex, k s the ndex of the super GOP, and p denotes the p th frame n one super GOP. In case that Hyperbolc model s not used n the Rate Controller and we cannot obtan A and B, we gve another soluton to calculate α, β wth R, D by a smlar algorthm n [27]. Consderng the defnton of the Lagrange multpler and the Hyperbolc R-D relatonshp, we have two equatons, look-ahead approach. Therefore, we try to detect the relatonshp of α, β and the complexty measure C n Eq.5 by some expermental results. We test all sequences from Class A to F under both RandomAccess and LowDelay confguratons and the results are all shown n Fg.5. The horzontal axs denotes C and vertcal axs denotes α or β. Each pont represents C, α and β of one super GOP. The length of one super GOP s set as 6 and each ntra-frame s nserted as the frst frame of every super GOP. We calculated α k and β k by Eq.49 and complexty measure C by Eq.7 for each super GOP. Snce the parameter estmaton s not our man work n ths paper, we use a smple and effcent lnear functon to express the relatonshp between α and C accordng to the results shown n Fg.5. What s more, the relatonshp of β and C looks ndependent to some extent. Therefore, we estmate α k+ and by, β k+ = C k+ { α k+ β k+ C k = β k α k (50 { λ = dd = dr R = α α β β D D β (47 Snce λ k, R and D can always be obtaned after the frame s encoded, we calculate α and β as, where α k+ and β k+ are the estmated value of α k+ and β k+. Accordng to our expermental results n the next Secton, we fnd ths smple estmaton performs better than drectly estmatng α k+, β k+ wth α k, β k. { = D λ R = R β α β D (48 (a α C (b β C Fg.5. The relatonshp between the parameter of Hyperbolc R-D model and the complexty measure. As gven n Eq or Eq.46, we can obtan α and β for each frame. However, how to calculate α k and β k as the parameters of the R-D relatonshp for all the frames n one super GOP stll remans unsolved. We cannot calculate α k and β k by the average value of the whole super GOP because λ s dfferent for each frame. Therefore, we calculate α k and β k wth a smlar way as that we have mentoned n Eq.39, { M β k = log 2 M p= α k = (α R k R k M D k β k (2D β (49 where M s the number of frames n a super GOP.R k, D k are the average BPP and MSE for the whole super GOP. To further estmate α k+ and β k+, the only nformaton we can obtan s the complexty measure n Eq.5-7 calculated wth E. Rate-Dstorton Performance Analyss for RDAS-H Although the proposed scheme focuses on mnmzng the varance among multple sequences, we are stll nterested n the R-D performance of RDAS-H. To analyze the R-D performance, we assume the parameters for the Hyperbolc R-D model are accurate and the model can precsely express the R-D relatonshp. To compare wth RDAS-H, we allocate btrate for dfferent sequences equally and set ths allocaton method as the anchor named AVG. Then we have, { R C = α D β (5 R c = α D β (52 where Eq.5 and Eq.52 descrbe the R-D relatonshp of applyng AVG and RDAS-H as allocaton scheme, respectvely. In Eq.5, the subscrpt denotes the sequence ndex and α, β are the parameters of each sequence. Eq.5 means that each sequence has a dfferent dstorton but the same btrate. In (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

10 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 0 Eq.52, α, β are the parameters for a jont R-D relatonshp. Eq.52 means that each sequence has the same dstorton and a dfferent btrate, whle the overall btrate s R c. Snce the overall btrate of two schemes are the same, we only need to compare the dstorton, D D 0, = D { D 0, = RDAS H s better AVG s better (53 So, we need to fnd out whether D D = s postve or negatve. D D = can also be expressed as, D D = ( R c α β (R c α = = β (54 Accordng to Eq.39, α, β can be calculated wth α and β, so the R-D performance wll totally depend on α, β and R c. Let us take the parameters of the hyperbolc R-D model for dfferent sequences and the jont stream n Table I as an example to calculate D, and we gve the results n = D Table II. Dfferent R c values are tested and column R denotes the btrate R(D gven n Table I. We fnd that n most stuatons D D = s negatve, whch means RDAS-H s better than AVG. Snce Eq.5 s not always negatve, the R-D performance s affected by the characterstcs of the sequences and the overall bandwdth. However, as a result shown n Table II, the R-D performance of our proposed RDAS-H wll be better than AVG n most cases (the usually usable range, and we can also draw a concluson that our RDAS-H wll not have a bad R-D performance compared to AVG n most cases (and also the typcal stuatons. TABLE II THE THEORETICAL R-D PERFORMACE OF SEQUECES RECOMMEDED BY JCT-VC. Sequence /4R /3R /2R R 2R 3R 4R ClassA ClassB ClassC ClassD ClassE ClassF IV. EXPERIMETAL RESULTS We ntegrate the proposed RDAS-H nto HEVC reference software HM 6.0. Both LowDelay and RandomAccess confguratons are used. The λ-doman rate control algorthm n [23] s appled and only frame-level rate control s turned on for all the methods n ths paper (ncludng the proposed scheme and related schemes. Test sequences recommended by JCT-VC contanng 6 classes are tested. The length of a super GOP s set as 6. In our experments, we jontly encode all sequences n one class each tme and the sequence numbers for Class A to Class F are 2, 5, 4, 4, 3, and frames are tested for Class C, Class D, Class E, and Class F. 240 frames are tested for Class B and 44 frames for Class A. Intra Perod s set as 6. In each experment, we encode sequences wth fxed QPs n advance, and then we calculate R c by summng up the btrate of each sequence wth the correspondng fxed QPs. Varance measures how far a set of numbers are spread out from ther mean. Accordng to the target functon descrbed n Eq.2, a smaller varance of dstorton represents a better allocaton model. Thus, the evaluaton ndex for the mnvar problem s defned as, Varance k = (PSR k k PSR 2 = (55 where Varance k represents the varance of Peak Sgnal to ose Rato (PSR of the k th super GOPs of all the sequences. PSR k represents the PSR value of the k th super GOP n the th vdeo and ks PSR the mean value of PSRs of dfferent sequences n the k th super GOP. To verfy the performance of our proposed scheme, consderng the target of our multplexng problem s to mnmze the varance of dstorton, we defne the varance savng rato (VSR between our proposed scheme and the anchor as, VSR anchor = Varance anchor Varance proposed Varance anchor (56 where Varance anchor and Varance proposed are the average varance values of all super GOPs for the anchor and the proposed scheme, respectvely. A larger VSR anchor represents a better performance of the proposed scheme compared wth the anchor. The man objectves of our experments are two-folds: ( to explore the effectveness of our proposed methods descrbed n Part C and Part D n Secton III, and (2 to evaluate the performance of our soluton (RDAS-H compared wth the related scheme n [2] and that n our prevous work [7]. Proposed methods verfcaton: In ths set of experments, the man objectve s to verfy the estmaton accuracy of α k+, β k+ and D k+ n Eq.9. Verfcaton of D k+ estmaton. We frst evaluate our proposed closed-form soluton of D k+, whch s derved n Eq.42. We compare our estmaton wth the brute-force method descrbed n Eq.20 and the brute-force method can acheve a better performance theoretcally. These two methods are both ntegrated nto our RDAS-H and the other confguratons are set the same. QP value s set as 32, and the comparson results are gven n Table III. Our proposed estmaton s denoted as Ours, whle the brute-force method s denoted as Brute-force. We also gve the VSR performance n the column VSR where Brute-force s set as the anchor. A (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

11 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < negatve VSR ndcates a better performance of the brute-force method than our estmaton for D k+. Our estmaton has a neglgble performance loss compared wth the theoretcal best performance, whch proves the effcency of our method. Verfcaton of α k+ and β k+. Then, we verfy the parameters estmaton for the Hyperbolc model. In Part D Secton III, we derve the relatonshp between parameter (α, β wth complexty measure C by some expermental results and then estmate α k+ and β k+ n Eq.50. We also compare our estmaton method wth a drect estmaton method whch estmates (α k+, β k+ wth (α k, β k n Eq.49. The results are shown n Table IV where our proposed estmaton method s denoted as Ours whle the drect estmaton method s denoted as Drect estmaton. The expermental envronment s the same as that n Table III and Drect estmaton s set as the anchor when calculatng VSR. Our estmaton n Eq.50 defntely mproves the performance by savng 4.60% and 9.75% varance on average under LowDelay and RandomAccess confguratons respectvely. 2 Entre soluton (RDAS-H verfcaton: In Table V, we verfy the performance of the proposed RDAS-H method compared wth three other methods. Four dfferent total btrates are tested and btrate for each class s gven n column Rate. In column Scheme, AVG represents that each stream s allocated wth the same btrate. CAS[2] s the method gven n [2] whch s brefly descrbed n Eq.4. RDAS-H s our proposed method and RDAS-I[7] s our prevous method n [7] applyng the same complexty measure wth that n CAS[2]. We apply the rate control algorthm [23] n all these methods. Besdes, VSR AVG/CAS/RDAS-I gves the VSR between RDAS-H and AVG/CAS/RDAS-I, respectvely. Snce the frst super GOP s allocated wth the same btrate among all three methods, the varance of the frst super GOP s not consdered for the performance evaluaton. To analyze the results, we frst focus on VSR n lne VSR AVG and VSR CAS[2]. Our proposed method sgnfcantly outperforms AVG and CAS [2] wth VSR 82.63% and 75.29% on average under dfferent btrates. Moreover, comparng wth our prevous method RDAS-I [7], the result s stll consderable wth VSR 36.62%. As the overall btrate decreasng, VSR for the Hyperbolc ncreases because the nverse proporton R-D model s not precse enough. Because when the overall btrate s low, dstorton wll change more whle btrate changes a lttle and Hyperbolc R-D model descrbes ths relatonshp better. Therefore, the more precse Hyperbolc R-D model wll be more effectve at low bandwdth. To clearly llustrate the results n Table V, we gve the varance curves of AVG, CAS[2], RDAS-I[7], and RDAS-H n Fg.6. The curves show the average varance value of each frame correspondng to the results of the thrd group n Table V. The horzontal and vertcal axes denote the frame number and the varance, respectvely. In the ttle, LD and RA denote the LowDelay and RandomAccess confguratons. As shown n Fg.6, the proposed RDAS-H outperforms RDAS-I [7] method and the related CAS [2] method n almost all super GOPs. Besdes QP values commonly used, our proposed RDAS-H can also work wth a lower or hgher QP. In Fg.7, the VSR values between RDAS-H and AVG/CAS/RDAS-I are gven wth QP 2, 7, 22, 27, 32, 37, 42, and 47. Accordng to our results, our scheme can work under dfferent QPs. The effcency of our proposed scheme reles on the accuracy of R-D model, and the accuracy of R-D model depends on the estmaton of parameters (e.g. α, β n the Hyperbolc model n Rate Controller. Based on the authors experence, α, β are dffcult to estmate when QP s very low or very hgh whch may affect our allocaton effcency to some extent. The estmaton of α, β s not our man purpose n ths paper, and we wll focus on ths part n our future work. Accordng to our target functon, our scheme only focuses on mnmzng the varance among sequences. However, we stll want to know how the proposed RDAS-H affects the R-D performance. We evaluate our soluton wth BD-BR to further prove the effectveness of our soluton. Therefore, we gve the BD-BR performance of CAS [2], RDAS-I [7], and RDAS-H n Table VI, where AVG s set as the anchor. The frst TABLE III THE ACCURACY OF D k+ ESTIMATIO OF OUR METHOD AD BRUTE-FORCE. Confguraton Sequence Varance Brute-force Ours VSR ClassA % ClassB % ClassC % LowDelay ClassD % ClassE % ClassF % Average % ClassA % RandomAccess ClassB % ClassC % ClassD % ClassE % ClassF % Average % TABLE IV THE ACCURACY OF α k+ AD β k+ ESTIMATIO OF OUR METHOD AD DIRECT ESTIMATIO. Confguraton Sequence Varance Drect estmaton Ours VSR ClassA % ClassB % ClassC % LowDelay ClassD % ClassE % ClassF % Average % ClassA % RandomAccess ClassB % ClassC % ClassD % ClassE % ClassF % Average % (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

12 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 2 : TABLE V COMPARISOS OF THE VARIACES OF CAS[2], RDAS-I[7], AD RDAS-H. QP Confguraton Scheme ClassA ClassB ClassC ClassD ClassE ClassF Average Low Delay Random Access Low Delay Random Access Low Delay Random Access Low Delay Random Access AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 83.39% 90.36% 88.79% 89.55% 70.98% 77.37% 83.4% VSR CAS 65.99% 86.6% 78.60% 72.83% 60.88% 70.2% 72.43% VSR RDAS-I 6.2% 58.45% 42.99% 30.04% 23.4% 3.0% 33.64% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 73.44% 87.03% 92.38% 90.78% 77.45% 76.50% 82.93% VSR CAS 50.35% 87.70% 83.50% 8.62% 67.% 64.77% 72.5% VSR RDAS-I 20.90% 36.35% 45.34% 46.84% 9.09% 6.3% 30.80% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 80.28% 83.26% 90.0% 80.42% 80.35% 75.53% 8.66% VSR CAS 67.00% 78.3% 84.35% 77.56% 72.6% 69.0% 74.82% VSR RDAS-I 7.28% 42.0% 43.5% 38.66% 23.45% 28.72% 32.2% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 66.27% 78.26% 89.5% 85.75% 88.77% 76.22% 80.80% VSR CAS 46.9% 76.00% 82.74% 74.53% 83.72% 67.34% 7.75% VSR RDAS-I 20.67% 30.2% 39.05% 36.28% 7.78% 24.54% 28.09% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 85.88% 88.29% 92.65% 90.40% 85.46% 73.66% 86.06% VSR CAS 79.0% 84.67% 90.35% 8.54% 79.99% 68.30% 80.66% VSR RDAS-I 4.3% 64.37% 60.3% 46.08% 5.0% 28.9% 42.65% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 6.55% 8.26% 8.59% 83.72% 92.0% 76.38% 79.42% VSR CAS 5.43% 8.46% 77.4% 75.33% 89.38% 69.03% 74.0% VSR RDAS-I 2.84% 45.34% 45.9% 47.60% 26.46% 30.09% 36.09% AVG CAS[2] RDAS-I[7] RDAS-H VSR AVG 94.23% 75.88% 93.33% 90.05% 9.33% 70.76% 85.93% VSR CAS 9.92% 72.34% 9.50% 80.54% 88.42% 64.6% 8.56% VSR RDAS-I 67.94% 53.32% 62.30% 67.24% 25.94% 2.78% 49.75% AVG CAS[2] RDAS-I RDAS-H VSR AVG 65.50% 84.69% 82.33% 84.62% 9.64% 76.30% 80.85% VSR CAS 54.80% 78.98% 8.5% 74.83% 89.3% 68.07% 74.58% VSR RDAS-I 26.05% 50.66% 46.30% 49.80% 3.90% 33.69% 39.73% (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

13 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 3 Fg.6. Comparsons of varance curves of AVG, CAS[2], RDAS-I[7], and RDAS-H for Class A-F under both LowDelay and RandomAccess confguratons. TABLE VI COMPARISOS OF BD-BR OF CAS[2], RDAS-I[7], AD RDAS-H. Confguraton Scheme ClassA ClassB ClassC ClassD ClassE ClassF Average CAS[2] -2.23% -4.34% -.7%.37% -0.59% -.85% -.56% LowDelay RDAS-I[7] -2.7% -.78% -0.28% 0.% -0.20% -2.33% -.% RDAS-H -2.29% -2.32% -0.94% 0.43% 0.35% -2.29% -.8% CAS[2] -3.80% -0.90% -2.66%.56% -0.50% -2.54% -.47% RandomAccess RDAS-I[7] -4.92% -2.9% -2.38%.09% 0.63% -.22% -.50% RDAS-H -5.7% -.70% -3.23%.40% 0.87% -.02% -.48% VSR 00% 80% 60% 40% 20% 0% AVG - LD CAS - LD RDAS-I - LD AVG - RA CAS - RA RDAS-I - RA QP Fg.7. The VSR values wth dfferent QPs of the proposed RDAS-H compared wth AVG, CAS[2], and RDAS-I[7]. super GOP s excluded from the results because the frst super GOP s allocated wth the same btrate for all the methods. The expermental R-D performance of our proposed RDAS-H s better than that of AVG n most cases and the BD-BRs of CAS[2], RDAS-I[7], and our proposed method are comparable. As a result, our method gets a better mn-var performance wthout RD performance reducton. Snce the computatonal complexty of the HEVC encoder can be hgh, many researchers have devoted ther efforts on reducng the computatonal complexty of the HEVC encoder[30-32]. Therefore, we also analyze the computatonal complexty of our proposed scheme because we don t want to spend too much complexty on the allocaton part. Accordng to our allocaton method n Eq.43, the tme cost s only related to the number of sequences but the resolutons of the sequences. In table VII, we gve the tme cost of our allocaton scheme and that of the whole HEVC encoder. The tme cost for encodng each sequence s gven n the Encoder (s column and the tme cost for our proposed allocaton scheme s gven n the RDAS-H (μs column. We can fnd that the encoder wll take more tme to process a larger resoluton sequence but our allocaton scheme doesn t. And the tme cost of our proposed scheme s only related to the number of sequences. Besdes, the tme for the allocaton s 0 7 tmes less than that for encodng. TABLE VII THE TIME COST OF OUR ALLOCATIO METHOD AD THAT OF THE WHOLE HEVC ECODER. Sequence Resoluton umber of RDAS-H HEVC Sequences (μs Encoder(s Class A Class B Class C Class D Class E Class F / V. COCLUSIO In ths paper, we proposed a novel jont rate-dstorton based allocaton scheme derved by an deal allocaton stuaton to solve the mnvar problem. The proposed RDAS can work wth dfferent R-D models. Furthermore, a Hyperbolc functon based R-D model s appled to the RDAS scheme (RDAS-H to obtan a better allocaton result. We derve a closed-form soluton of RDAS-H nstead of a brute-force soluton by dervng a jont R-D model based on nformaton theory. Compared wth the related CAS n [2], the proposed RDAS-H mproves the performance by savng varance 75.29% on (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

14 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 4 average under LowDelay and RandomAccess confguratons wth QP 22, 27, 32, and 37. Besdes, the proposed method saves 36.62% varance over our prevous method RDAS-I n [7], n whch the R-D model s bult by an nverse proporton functon. Besdes, the proposed RDAS-H can also obtan a hgher R-D performance compared wth the methods whch allocate btrate for dfferent sequences equally n most cases. In ths work we emphasze on reachng an equal vsual qualty among all sequences. However, t wll be a better dea f we encode jont vdeos accordng to the users heterogeneous characters and how to transmt jont vdeos to meet every user s demand s another dffcult problem whch s part of our future work. REFERECES [] Z. He; S. K. Mtra, Optmum bt allocaton and accurate rate control for vdeo codng va ρ-doman source modelng, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 2, no. 0, pp , Oct [2] M. Twar; T. Groves; P. C. Cosman, Compettve Equlbrum Btrate Allocaton for Multple Vdeo Streams, IEEE Transactons on Image Processng, vol. 9, no. 4, pp , Aprl 200. [3] C. Pang; O. C. Au; F. Zou; J. Da, Model-based optmal dependent jont bt allocaton of H.264/AVC statstcal multplexng, IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng, 202, pp , Mar [4] C. Pang; O. C. Au; J. Da; F. Zou, Dependent Jont Bt Allocaton for H.264AVC Statstcal Multplexng Usng Convex Relaxaton, IEEE, vol. 23, no. 8, pp , Aug [5] F. Shao; G. Jang; W. Ln; M. Yu; Q. Da, Jont Bt Allocaton and Rate Control for Codng Mult-Vew Vdeo Plus Depth Based 3D Vdeo, IEEE Transactons on Multmeda, vol. 5, no. 8, pp , Dec [6] L. Wang; A. Vncent, Jont rate control for mult-program vdeo codng, Internatonal Conference on Consumer Electroncs, Dgest of Techncal Papers, pp. 40-4, June 996. [7] L. Boroczky; A. Y. ga; E. F. Westermann, Statstcal multplexng usng MPEG-2 vdeo encoders, IBM Journal of Research and Development, vol. 43, no. 4, pp , July 999. [8] L. Boroczky; A. Y. ga; E. F. Westermann, Jont rate control wth look-ahead for mult-program vdeo codng, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 0, no. 7, pp , Oct [9] Z. G. L; C. Zhu; F. Pan; G. Feng; X. Yang; S. Wu;. Lng, A novel jont rate control scheme for the codng of multple real tme vdeo programs, Internatonal Conference on Dstrbuted Computng Systems Workshops, pp , [0] J. Yang; X. Fang; H. Xong, A jont rate control scheme for H.264 encodng of multple vdeo sequences, IEEE Transactons on Consumer Electroncs, vol. 5, no. 2, pp , May [] M. Jacobs; J. Barbaren; S. Tondeur; R. V. Walle; T. Pardaens; P. Schelkens, Statstcal multplexng usng SVC, IEEE Internatonal Symposum on Broadband Multmeda Systems and Broadcastng, pp. -6, Aprl [2] Y. Wang; L. Chau; K. Yap, Jont Rate Allocaton for Multprogram Vdeo Codng Usng FGS, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 20, no. 6, pp , June 200. [3] W. Yao; L. Chau; S. Rahardja, Jont Rate Allocaton for Statstcal Multplexng n Vdeo Broadcast Applcatons, IEEE Transactons on Broadcastng, vol. 58, no. 3, pp , Sept [4] W. Yao; L. Chau; S. Rahardja, Jont Rate Allocaton of Stereoscopc 3D Vdeos n ext-generaton Broadcast Applcatons, IEEE Transactons on Broadcastng, vol. 59, no. 3, pp , Sept [5] Z. He; D. O. Wu, Lnear Rate Control and Optmum Statstcal Multplexng for H.264 Vdeo Broadcast, IEEE Transactons on Multmeda, vol. 0, no. 7, pp , ov [6] W. Yao; L. Chau; S. Rahardja, Jont Rate Allocaton For Statstcal Multplexng of SVC, IEEE Internatonal Conference on Image Processng, pp , Sept [7] H. Fan; L. Dng; X. Xe; H. Ja; W. Gao, Jont Rate Allocaton wth Both Look-ahead And Feedback Model For Hgh Effcency Vdeo Codng, IEEE Internatonal Conference on Multmeda and Expo, 206. [8] T. Chang; Y. Zhang, A new rate control scheme usng quadratc rate dstorton model, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 7, no., pp , Feb [9] G. J. Sullvan; T. Wegand, Rate-dstorton optmzaton for vdeo compresson, IEEE Sgnal Processng Magazne, vol. 5, no. 6, pp , ov [20] S. Mallat; F. Falzon, Analyss of low bt rate mage transform, IEEE Transactons on Sgnal Processng, vol. 46, no. 4, pp , Aprl 998. [2] M. Da; D. Logunov; H. Radha, Rate-dstorton modelng of scalable vdeo coders, Internatonal Conference on Image Processng, vol. 2, pp , Oct [22] M. R. Ardestan; A. A. B. Shraz; M. R. Hashem, Rate-dstorton modelng for scalable vdeo codng, IEEE Internatonal Conference on Telecommuncatons, pp , Aprl 200. [23] B. L; H. L; L. L; J. Zhang, Lambda Doman Rate Control Algorthm for Hgh Effcency Vdeo Codng, IEEE Transactons on Image Processng, vol. 23, no. 9, pp , Sept [24] T. Berger, Rate Dstorton Theory. Englewood Clffs, J: Prentce-Hall, 97. [25] T. M. Cover; J. A. Thomas, Elements of Informaton Theory, Tsnghua Unversty Press, 99: [26] W. Szepansk, Δ-entropy and Rate Dstorton Bounds for Generalsed Gaussan Informaton Sources and Ther Applcatons to Image Sgnals, Electroncs Letters, vol. 6, no. 3, pp:09-, 980 [27] S. L ; M. Xu ; Z. Wang ; X. Sun, "Optmal Bt Allocaton for CTU Level Rate Control n HEVC", IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. PP, no. 99, pp. -, July 206. [28] F. Muller, "Dstrbuton shape of two-dmensonal DCT coeffcents of natural mages", Electroncs Letters, vol. 29, no. 22, pp , Oct [29] S. Ma; J. S; S. Wang, "A study on the rate dstorton modelng for hgh effcency vdeo codng", IEEE Internatonal Conference on Image Processng, pp. 8-84, 202. [30] Z. Pan; J. Le, Y. Zhang; X. Sun; S. Kwong, "Fast moton estmaton based on content property for low-complexty H.265/HEVC encoder," IEEE Transactons on Broadcastng, vol. 62, no. 3, pp , Sept [3] Z. Pan; Y. Zhang; S. Kwong, "Effcent moton and dsparty estmaton optmzaton for low complexty multvew vdeo codng," IEEE Transactons on Broadcastng, vol. 6, no. 2, pp , 205. [32] Z. Pan; P. Jn; J. Le; Y. Zhang; X. Sun; S. Kwong, "Fast Reference Frame Selecton Based on Content Smlarty for Low Complexty HEVC Encoder", Journal of Vsual Communcaton and Image Representaton, vol. 40, Part B, pp , Oct [33] L. Xu; D. Zhao; X. J; L. Deng; S. Kwong; W. Gao. Wndow level rate control for smooth vsual qualty and smooth buffer occupancy, IEEE Transactons on Imagng Processng, Vol. 20, no. 3, pp , 20. [34] Y. Lu; Q. Huang; S. Ma; D. Zhao; W. Gao; S. C; H. Tang, "A ovel Rate Control Technque for Multvew Vdeo Plus Depth Based 3D Vdeo Codng", IEEE Transactons on Broadcastng, vol. 57, no. 2, pp , July 20. Hongfe Fan receved the B.S. degree n software from Shangha Jao Tong Unversty, Shangha, Chna, n 203. From Sep. 203, he s workng toward the Ph.D. degree wth the atonal Engneerng Laboratory for Vdeo Technology and the Cooperatve Medanet Innovaton Center, School of Electroncs Engneerng and Computer Scence, Pekng Unversty, Bejng, Chna. Hs research nterests nclude vdeo codng and mage processng (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

15 > REPLACE THIS LIE WITH YOUR PAPER IDETIFICATIO UMBER (DOUBLE-CLICK HERE TO EDIT < 5 Ln Dng receved the B.S. degree n computer scence from Unversty of Electronc Scence and Technology of Chna, Chengdu, Chna, n 203. From Sep. 203, she s workng toward the Ph.D. degree wth the atonal Engneerng Laboratory for Vdeo Technology and the Cooperatve Medanet Innovaton Center, School of Electroncs Engneerng and Computer Scence, Pekng Unversty, Bejng, Chna. Her research nterests nclude vdeo codng and mage processng. Huzhu Ja receved hs Ph.D. degree n electrcal engneerng from the Chnese Academy of Scences, Bejng, Chna, n He s currently wth the atonal Engneerng Laboratory for Vdeo Technology, Pekng Unversty, Bejng. Hs research nterests nclude: ASIC desgn, mage processng, multmeda data compresson and VR. Xaodong Xe receved hs Ph.D. degree n electrcal engneerng, Unversty of Rochester, USA. He was a Senor Scentst at Eastman Kodak Company, ew York, USA, from 994 to 997; a Prncpal Scentst at Broadcom Corporaton, Calforna, USA, from 997 to He s currently a Professor wth the Department of Electrcal Engneerng, Pekng Unversty, Bejng, Chna. Hs research nterests nclude multmeda SoC desgn, embedded system. He holds 24 U.S. Patents (c 208 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.