Improved Estimation of Transmission Distortion for Error-resilient Video Coding

Transcription

1 1 Improved Estimation of Transmission Distortion for Error-resilient Video Coding Zhifeng Chen 1, Peshala Pahalawatta 2, Alexis Michael Torapis 2, and Dapeng W 1,* 1 Department of Electrical and Compter Engineering, University of Florida, Gainesville, FL Dolby Laboratories, 01 W Alameda Ave, Brban, CA Abstract This paper presents an improved techniqe for estimating the end-to-end distortion, which incldes both qantization error after encoding and random transmission error, after transmission in video commnication systems. The proposed techniqe mainly differs from most existing techniqes in that it taes into accont filtering operations, e.g. interpolation in sbpixel motion compensation, as introdced in advanced video codecs. The distortion estimation for pixels or sbpixels nder filtering operations reqires the comptation of the second moment of a weighted sm of random variables. In this paper, we prove a proposition for calclating the second moment of a weighted sm of correlated random variables withot reqiring nowledge of their probability distribtion. Then, we apply the proposition to extend or previos error-resilient algorithm for prediction mode decision withot significantly increasing complexity. Experimental reslts sing an H.264/AVC codec show that or new algorithm provides an improvement in both rate-distortion performance and sbjective qality over existing algorithms. Or algorithm can also be applied in the pcoming high efficiency video coding (HEVC) standard, where additional filtering techniqes are nder consideration. Index Terms Error-resilient Rate Distortion Optimization (ERRDO), filtering, sbpixel-level distortion estimation, fractional motion estimation, E, mode decision, wireless video I. INTRODUCTION In a typical video encoder, two inds of compression techniqes are sally involved, lossless and lossy compression. Lossless compression can be applied by redcing the redndancy between spatio-temporal neighboring pixels, i.e. throgh intra/inter prediction, and throgh the design and se of better codeword representations for a given probability distribtion, i.e. sing entropy coding techniqes. Lossy compression is sed to frther compress the video by redcing fidelity, e.g. throgh qantization. In traditional video coding or video compression designs, rate-distortion (R-D) theory [1], [2] is proposed to stdy the relationship between bit rate and video distortion indced by the lossy qantization operation. With a R-D fnction, the redndancy of video seqences can be maximally exploited and distortion can be controlled by an acceptable qantization scheme throgh a R-D optimization *Correspondence athor: Prof. Dapeng W, w@ece.fl.ed, This wor was spported in part by the US National Science Fondation nder grant ECCS Copyright (c) 2011 IEEE. Personal se of this material is permitted. However, permission to se this material for any other prposes mst be obtained from the IEEE by sending an to pbs-permissions@ieee.org. (RDO) techniqe. However, in error-prone channels, redcing the redndancy also redces the resilience to random transmission errors which may incr an increase in the end-toend distortion. On the other hand, increasing error resilience, e.g. throgh random intra refreshing, may redce compression efficiency. Error-resilient RDO (ERRDO), or sometimes called loss-aware RDO, is one method that can be sed to alleviate the adverse effects of both bandwidth limitations and random transmission errors. The problem of minimizing the distortion given a bit rate constraint can be formlated as a Lagrangian optimization. De to its discrete characteristics, however, the rate distortion fnction is not garanteed to be convex [3]. Therefore, in this case, the traditional Lagrange mltiplier soltion for continos convex fnction optimization is infeasible. The discrete version of Lagrangian optimization was first introdced in Ref. [4], and then applied in a sorce coding application in Ref. [3]. De to its simplicity and effectiveness, this optimization method is widely sed in traditional video coding applications [5], [6], [7]. In the case of ERRDO, however, the end-to-end distortion is cased by both qantization and pacet transmission errors. While the qantization error is nown to the encoder, the transmission error depends on the particlar channel realization and is therefore nnown to the encoder. Instead, ERRDO can se an estimate of the expected endto-end distortion throgh characterizing the channel behavior, e.g. pacet loss probability (PLP), to help the RDO process. However, predicting end-to-end distortion is particlarly challenging de to 1) the spatio-temporal correlation in the inpt video seqence, that is, a pacet error will degrade not only the video qality of the crrent frame bt also that of the sbseqent frames de to error propagation; 2) the nonlinearity of both the encoder and the decoder, which maes the instantaneos transmission distortion not eqal to the sm of distortions cased by individal error events; and 3) varying PLP in time-varying channels, which maes the distortion process into a non-stationary random process. Some pixel-level end-to-end distortion estimation algorithms have been proposed to assist mode decision as in Ref. [8], [9], [10], [11]. Stochammer et al. [8], [9] proposed a distortion estimation algorithm by simlating K independent decoders at the encoder dring the encoding process and then averaging their simlated distortion. This algorithm, which we will refer to as, is based on the Law of Large Nmbers. That is, the estimated reslt will asymptotically approach the

2 2 expected distortion when K goes to infinity. In the H.264/AVC JM reference software [7], this method is adopted to estimate the end-to-end distortion for mode decision. However, in the algorithm more simlated decoders lead to higher comptational complexity and larger memory reqirements. Also for the same video seqence and the same PLP, different encoders may have different estimated distortions de to the randomly prodced error events at each encoder. In Ref. [10], the Recrsive Optimal Per-pixel Estimate () algorithm is proposed to estimate the pixel-level endto-end distortion by recrsively calclating the first and second moments of the reconstrcted pixel vale. However, nonlinear clipping that contribtes to the transmission distortion is neglected [12]. In addition, enhancing to spport pixel averaging operations, e.g., interpolation filtering, reqires intensive approximation comptation of cross-correlation terms. In Ref. [13], the athors extend by sing the pper bond, obtained from the Cachy-Schwarz approximation, to approximate the cross-correlation terms. However, sch an approximation reqires very high complexity. For example, for an N-tap filter interpolation, each sbpixel reqires N integer mltiplications 1 for calclating the second moment terms; N(N 1)/2 floating-point mltiplications and N(N 1)/2 sqare root operations for calclating the cross-correlation terms; and N(N 1)/2 + N 1 additions and 1 shift for calclating the estimated distortion. On the other hand, the pper bond approximation is not accrate for practical video seqences since it assmes that the correlation coefficient is 1, for any two neighboring pixels. In Ref. [14], the athors propose several models to approximate the correlation coefficient of two pixels as fnctions, e.g., an exponentially decaying fnction, of their distance. However, de to the random behavior of individal pixel samples, the statistical model does not prodce an accrate pixel-level distortion estimate. In addition, sch approximations incr extra complexity compared to the Cachy-Schwarz pper bond approximation, since they need additional N(N 1)/2 exponential operations and N(N 1)/2 floating-point mltiplications for each sbpixel. Therefore, the complexity incrred is prohibitively high for real-time video encoders. On the other hand, since both the Cachy- Schwarz pper bond and the correlation coefficient model approximations reqire floating-point mltiplications, additional rond-off errors are navoidable, which frther redce their estimation accracy. In Ref. [12], we proposed a divide-and-conqer method to qantify the effects of for individal terms on transmission distortion, i.e. 1) residal concealment error, 2) Motion Vector (MV) concealment error, 3) propagation error and clipping noise, and 4) correlations between any two of them. Based on or theoretical reslts, we proposed the algorithm in Ref. [11] for error-resilient rate-distortion optimized mode decision with pixel-level end-to-end distortion estimation. In state-of-the-art video codecs, sch as H.264/AVC [15] and HEVC [16], fractional pixel motion compensation with inter- 1 One common method to simplify the mltiplication of an integer variable and a fractional constant is as follows: first scale p the fractional constant by a certain factor; rond it off to an integer; perform integer mltiplication; finally scale down the prodct. polation filtering can have a sbstantial R-D performance gain. However, distortion estimation for pixels or sbpixels nder filtering operations reqires the comptation of the second moment of a weighted sm of random variables. In this paper, we first theoretically derive a proposition for calclating the second moment of a weighted sm of correlated random variables sing a closed-form fnction of the second moments of those individal random variables. This proposition is general in estimating the distortion for any pixels after a filtering operation. Then, we apply the proposition to extend or previos algorithm to sbpixel-level distortion estimation for prediction mode decision withot significantly increasing complexity. This algorithm is referred to as E. The E algorithm only reqires N integer mltiplications, N 1 additions, and 1 shift to calclate the second moment for each sbpixel. Experimental reslts show that, E achieves an average PSNR gain of 0.25dB over the existing algorithm for the mobile seqence when PLP eqals 2%; and E achieves an average PSNR gain of 1.dB over the the algorithm for the foreman seqence when PLP eqals 1%. The rest of this paper is organized as follows. Section II presents the system description and the necessary preliminaries of the algorithm, and serves as a starting point for nderstanding the following sections. In Section III, we first derive the general proposition for the second moment of a weighted sm of correlated random variables, and then apply this proposition to design a low-complexity and highaccracy algorithm for mode decision. Section IV shows the experimental reslts, which demonstrate the advantages of the E algorithm over existing algorithms for H.264/AVC mode decision in error prone environments. Section V concldes the paper. II. SYSTEM DESCRIPTION AND PRELIMINARIES A. Strctre of a Wireless Video Commnication System Fig. 1 shows the strctre of a typical hybrid video coding system. Note that in this system, both the residal and the MV channels are application-layer channels. These channels can be separated for more general applications, e.g. for slice data partitioning nder Uneqal Error Protection (UEP). If the residal and MV pacets are transmitted in the same channel with the same error protection, this becomes a special case where the two channels have the same characteristics and are flly correlated. In Fig. 1, for any pixel in the -th frame, f is the inpt pixel vale at the encoder; ˆf and f are the reconstrcted pixel vales at the encoder and decoder respectively. Sppose the pixel in the -th frame ses the j-th frame as a reference, where j {1, 2,..., J} and J is the nmber of reference frames. If motion estimation is performed before mode decision as in the JM16.0 reference software, the best MV vale mv for each prediction mode and the corresponding j ˆf +mv residal vale e = f for each prediction mode are nown. As a reslt, after the transform, qantization, deqantization, and inverse transform processes at the encoder, the reconstrcted residal ê and the reconstrcted pixel vale

3 3 Inpt - Encoder e T/Q Q -1 /T -1 ˆ + e Channel Residal Channel S(r) 0 1 Decoder Q -1 /T -1 Residal Error Concealment eˆ ( e + e~ Otpt Video captre f fˆ j + mv Motion compensation f ˆ j Clipping fˆ Memory 0 ~ j ' f ~ + mv Motion compensation Clipping ~ f j' Memory ~ f Video display mv Motion estimation MV Channel S(m) 1 m ( v MV Error concealment Fig. 1. System strctre, where T, Q, Q 1, and T 1 denote transform, qantization, inverse qantization, and inverse transform, respectively. ˆf j = Γ( ˆf + ê +mv ) are nown for each prediction mode. Γ( ) is the clipping fnction and is defined by γ L, x < γ L Γ(x) = x, γ L x γ H (1) γ H, x > γ H, where γ L and γ H are low threshold and high threshold of pixel vales. In this paper, we let γ L = 0 and γ H = 255. Let s define f ˆf as qantization error and ζ ˆf f (2) as transmission error. While the qantization error is nown to the encoder after the encoding process, the transmission error is cased from any combinations of three error events, i.e. residal pacet, MV pacet, and propagated errors. In Fig. 1, the residal sed for reconstrction in the decoder, i.e. ẽ, may be either the qantized residal ê or the concealed residal ě depending on the error stats of residal pacet S(r); The MV sed for motion compensation in the decoder, i.e. mv, may be either the tre MV mv or the concealed MV mv ˇ depending on the error stats of MV pacet S(m); as a reslt, the reconstrcted pixel vale in the decoder is f j = Γ( f + ẽ j + mv ) where f is the reference pixel + mv vale for reconstrction at the decoder, which recrsively depends on all error events in the reference trajectory till the intra coded pixel. Depending on whether the MV is correctly received or not, the propagated error can be calclated, according to (2), by ζ j = + mv { ζ j +mv ζ j + ˇ mv = = j ˆf +mv j ˆf +mv f j S(m)=0 +mv, (3) j f S(m)=1, + mv ˇ, where the concealed MV may point to a different reference frame j from the tre reference frame j. Clipping noise is defined as ˆ j ( ˆf + ê +mv ) Γ( + ê ) at the encoder and as ( + ˆf j +mv ẽ ) Γ( f j + mv f j + mv + ẽ ) at the decoder. That is, the clipping noise at the decoder is a fnction of the residal concealment, MV concealment, and propagated errors. Denote { r, m} as the error event that both residal and MV are correctly received at the decoder for pixel ; the clipping noise nder this event will be j { r, m} = ( f + ê j +mv ) Γ( f + ê +mv ). Some important notations sed in this paper are listed in Table I. Throghot this paper, we addˆonto the reconstrcted variables at the encoder; addˇonto the concealed variables at the decoder; and add onto the variables, which are sbject to the random channel error at the decoder. B. Preliminaries abot the algorithm In Ref. [12], we tae a divide-and-conqer approach to derive the second moment of ζ, i.e. E[( ζ ) 2 ]. We first divide the transmission reconstrcted error into for components: three random errors (residal concealment, MV concealment, and propagated errors) de to their different physical cases, and clipping noise, which is a non-linear fnction of these three random errors. This error decomposition allows s to qantify the effects of below for terms on transmission distortion: (1) residal concealment error (R), (2) MV concealment error (M), (3) propagated error pls clipping noise (P), and (4) correlations between any two of the error sorces (C). Based on this decomposition, we developed a practical algorithm, called algorithm, to estimate the pixel-level end-to-end distortion (PEED) for mode decision in Ref. [11]. In Ref. [11], the end-to-end distortion fnction for each pixel in the -th frame is defined as D,ET E E[(f f ) 2 ], and can be derived by D,ET E = E[(f f ) 2 ] = E[(f ˆf + ζ ) 2 ] = (f ˆf ) 2 + E[( ζ ) 2 ] + 2(f ˆf ) E[ ζ ]. Define ε ê ě as the residal concealment error when the residal pacet is lost; define ξ j j ˆf ˆf +mv + mv ˇ as the MV concealment error when the MV pacet is lost; and denote P as the PLP for pixel in the -th frame. Under the assmptions that S(r) is independent of e and (4)

4 4 TABLE I NOTATIONS f : Vale of the pixel with position in the -th frame, i.e. pixel. e : Residal vale of the pixel. mv : MV of the pixel. : Clipping noise of the pixel. { r, m}: Clipping noise of the pixel nder the condition that both residal and MV are correctly received at the decoder. ζ : Transmission error of the pixel, defined in (2). ε : Residal concealment error of the pixel. ξ : MV concealment error of the pixel. ζ j +mv : 1) propagated error of the pixel whose reference pixel is pointed to the j-th frame by the tre MV mv ; : 2) transmission error of the pixel with position + mv in the j-th frame according to (2). ζ j + ˇ mv : 1) propagated error of the pixel whose reference pixel is pointed to the j -th frame by the concealed MV mv ˇ ; : 2) transmission error of the pixel with position + mv ˇ in the j -th frame according to (2). S(m) is independent of ξ, i.e. the pacet transmission error is independent from the vales of residal and MV, we proved in Refs. [12], [11] that withot slice data partitioning, E[ ζ ] and E[( ζ ) 2 ] can be calclated by (5) and (6) 2, E[ ζ ] = P (ε + ξ + E[ ζ j + mv ˇ ]) + (1 P ) E[ ζ j + +mv { r, m}], E[( ζ ) 2 ] = P ((ε + ξ) 2 + 2(ε + ξ) E[ ζ j + mv ˇ ] + E[( ζ j ) 2 ]) + (1 P + mv ˇ ) E[( ζ j + +mv { r, m}) 2 ], (6) where E[ ζ j ] and E[( ζ j + mv ˇ + mv ˇ ) 2 ] in the j -th frame have been calclated by (5) and (6) and stored in memory dring encoding of the j -th frame; E[ ζ j +mv+ { r, m}] and E[( ζ j +mv and (8), where E[ ζ j (5) + { r, m}) 2 ] can be calclated by (7) +mv ] and E[( ζ j +mv ) 2 ] have been calclated and stored in memory dring encoding of the jth frame. Note that in this paper, for simplicity, we se E( ) to also represent the estimate of E( ), i.e. Ê( ), in Ref. [11]. Existing pixel-level algorithms, e.g., the algorithm, are based on the integer pixel MV assmption to derive an estimate of D,ET E. In order words, in (5), (6), (7) and (8), mv is with integer-pixel accracy. Therefore, their application in state-of-the-art encoders is limited de to the possible se of fractional motion compensation. For sbpixel motion compensation, E[ ζ j ] and E[( ζ j +mv +mv) 2 ] need to be estimated based on the interpolated pixel vales, i.e. a weighted sm of several neighboring pixels. That is, E[( ζ j +mv ) 2 ] reqires the comptation of the second moment of a weighted sm of correlated random variables and therefore the comptation of several cross-correlation terms. This is also tre for distortion estimation for pixels or sbpixels nder other filtering operations. In Section III, we will extend algorithm to solve this problem with low complexity. III. THE EXTENDED ALGORITHM FOR MODE DECISION In this section, we first derive a general proposition for any second moment of a weighted sm of correlated random 2 In Ref. [12], [11], we assme that if S(m) = 1, the reference pixel, pointed by the concealed MV mv ˇ, comes from the 1-th frame; in this paper, we denote the j -th frame for reference pixel pointed by the concealed MV mv ˇ withot sch an assmption. variables; then we apply it to extend to estimate the end-to-end distortion for sbpixel motion compensation and design the algorithm for mode decision. A. Sbpixel-level Distortion Estimation for Error Resilient Video Encoding Typically, the rate distortion optimized mode decision consists of two steps. First, the R-D cost is calclated by J(ω m ) = D (ω m ) + λ R(ω m ), (9) where D = 1 V V l D; V l l is the set of pixels in the l-th MB (or sb-mb, i.e. any bloc size for a given prediction mode) of the -th frame; ω m is the prediction mode [17]; R(ω m ) is the encoded bit rate for mode ω m, ω m Ω m and Ω m is the mode set for mode decision; λ is the preset Lagrange mltiplier. Then, the optimal prediction mode that minimizes the rate-distortion (R-D) cost is fond by ˆω m = arg min {J(ω m )}. (10) ω m Ω m For the algorithms, if the MV of one bloc for encoding is fractional, the MV has to be ronded to the nearest integer. This bloc will se the reference bloc pointed to by the ronded MV as a reference. However, in state-ofthe-art codecs, sch as H.264/AVC and HEVC, interpolation filters are sed to interpolate a reference bloc. Therefore, the distortion of nearest neighbor approximation is not optimal for sch an encoder. In order to optimally estimate the distortion for pixels with interpolation filtering, or any other filtering in general, we need to extend the existing algorithm. In H.264/AVC, motion compensation accracy is in nits of one qarter of the distance between lma samples. 3 The prediction vales at half-sample positions are obtained by applying a one-dimensional 6-tap Finite Implse Response (FIR) filter horizontally and vertically. The prediction vales at qarter-sample positions are generated by averaging samples at integer- and half-sample positions [17]. Tae in ζ j +mv (5), (6), (7) and (8) for example. Denote v j = + mv and s in a sbpixel position in the j-th frame. All neighboring pixels in the integer position, sed to interpolate the reconstrcted pixel vale at v, are denoted by and with a weight w i, i 1, 2,..., N, where N = 6 for the half-sample 3 Note that considering the chroma distortion does not always improve the R-D performance bt indces more complexity. Therefore, we only consider lma components in this paper.

5 E[ ζ j +mv + { r, m}] = E[( ζ j +mv + { r, m}) 2 ] = interpolation, and N = 2 for the qarter-sample interpolation in H.264/AVC. Therefore, the interpolated reconstrcted pixel vale at the encoder is and at the decoder From (2), we have ζ v j = w i = Since E[ ζ j E[ ζ j v ˆf j v = f j v = ˆf j w i w i w i j j w i ( ˆf f ) = ˆf j, (11) f j. (12) f j w i ζ j. ˆf 255, E[ ζ j < +mv] ˆf 255 ˆf, E[ ζ j > +mv] ˆf E[ ζ j ], ˆf +mv 255 E[ ζ j +mv] ˆf, ( ˆf 255) 2, E[ ζ j < +mv] ˆf 255 ( ˆf ) 2, E[ ζ j > +mv] ˆf E[( ζ j ) 2 ], ˆf 255 E[ ζ j +mv] ˆf, (13) ] has been calclated by the algorithm, ] can be very easily calclated by E[ ζ j v ] = However, calclating E[( ζ j v E[( ζ j v w i E[ ζ j ]. (14) ) 2 ] is not straightforward since ) 2 ] = E[( w i ζ j ) 2 ] (15) is in fact the second moment of a weighted sm of correlated random variables. B. A Proposition for Calclating the Second Moment of a Weighted Sm of Correlated Random Variables The Moment Generating Fnction (MGF) can be sed to calclate the second moment for random variables [18]. However, to estimate the second moment of a weighted sm of random variables, the traditional moment generating fnction sally reqires nowing their probability distribtion and assmes they are independent. However, in a practical video codec, most random variables, e.g. those involved in the averaging operations, are not independent and their probability distribtions are nnown. Therefore, some approximations, sch as the Cachy-Schwarz pper bond approximation [13] or the correlation coefficient model approximation [14], are sally adopted. However, those approximations are of high complexity. For example, for each sbpixel, with the N-tap filter interpolation, the Cachy-Schwarz pper bond approximation reqires N integer mltiplications for calclating the second moment terms, N(N 1)/2 floating-point mltiplications and N(N 1)/2 sqare root operations for calclating the cross-correlation terms, and N(N 1)/2 + N 1 +mv additions and 1 shift for calclating the estimated distortion. The correlation coefficient model reqires an additional N(N 1)/2 exponential operations and N(N 1)/2 floatingpoint mltiplications when compared to the Cachy-Schwarz pper bond approximation. In a wireless video commnication system, the comptational capability of the real-time encoder is sally very limited, and floating-point processing is ndesirable. Therefore, it is desirable to design a new algorithm for the calclation of the second moment in (15) sing only integer operations. Proposition 1 is a reslt of this motivation. Proposition 1: For any N correlated random variables {X 1, X 2,..., X N } and w i R, i {1, 2,..., N}, the second moment of the weighted sm of these random variables is given by (16). E[( w i X i ) 2 ] = w i [w j E(Xj 2 )] N 1 =1 l=+1 j=1 [w w l E(X X l ) 2 ] 5 (7) (8) (16) The proof of Proposition 1 is provided below. Note that in H.264/AVC, most averaging operations, e.g., interpolation, deblocing, and bi-prediction, are special cases of Proposition 1 in that N w i = 1. Therefore, we can extend the algorithm throgh the consideration of Proposition 1. In (16), since E(Xj 2 ) has been estimated by the algorithm, the only nnown is N 1 N =1 l=+1 [w w l E(X X l ) 2 ]. However, we will see that this nnown can be assmed to be negligible for the prposes of mode decision. C. The Extended Algorithm for Mode Decision 1) Algorithm design: Replacing X and X l in (16) by ζ i and ζ j, and from (2) we have E[(X X l ) 2 ] = E[( ζ i ζ j ) 2 ] = E[ ˆf i f i ( ˆf j f j )] 2 = E[( ˆf i ˆf j ) ( f i f j )] 2. (17) In (17), both ˆf i ˆf j and f i f j depend on the spatial correlation of the reconstrcted pixel vales in position i and j. When i and j are located in the same neighborhood, they are very liely to be transmitted in the same pacet. In other words, either both f i and f j se the tre MV and residal for reconstrction, or both f i and f j se the concealed MV and residal for reconstrction. Therefore, f i f j will not change too mch from ˆf i ˆf j, and hence E[( ζ i ζ j ) 2 ]

6 6 Proof: E[( w i X i ) 2 ] = E[ (wj 2 Xj 2 ) + = E[ = = j=1 j =1 w j N w i j=1 N w i j=1 j=1 =1 (w w l X X l )] l=1 (l ) (w j Xj 2 ) j=1 (w j w j Xj 2 ) + j =1 (j j) N 1 [w j E(Xj 2 )] E{ [w j E(X 2 j )] N 1 =1 l=+1 =1 l=+1 =1 (w w l X X l )] l=1 (l ) N 1 [w w l (X 2 + Xl 2 )] + [w w l E(X X l ) 2 ]. =1 l=+1 (2 w w l X X l )} is mch smaller than E[( ζ i ) 2 ] and E[( ζ j ) 2 ] in (16). On the other hand, distortion is estimated for one MB or one sb-mb as in (9) for mode decision. When the cardinality Vl is large, N 1 N v Vl j=i+1 [w i w j E( ζ i ζ j ) 2 ] converges to a constant for all modes with high probability de to the smmation over the same samples in each mode. For simplicity, we will call it negligible term in the following sections. Therefore, in (16) only the first term on the righthand side needs to be calclated withot too mch loss in precision. Since N w i = 1, we calclate E[( ζ v) 2 ] for mode decision by E[( ζ v) 2 ] = [w i E( ζ i ) 2 ]. (18) With the N-tap filter interpolation, the complexity in (18) is dramatically redced to only N integer mltiplications, N 1 additions, and 1 shift. Here, we propose the following algorithm to extend the algorithm for mode decision. Algorithm 1: Rate distortion optimized mode decision for an MB in the -th frame ( >= 1). 1) Inpt: QP, PLP. 2) Initialization of E[ ζ 0 ] and E[( ζ 0 ) 2 ] for all pixel. 3) Loop for all available modes for each MB. 3a) estimate E[ ζ j +mv ] via (14) and E[( ζ j +mv ) 2 ] via (18) for all pixels in the MB, 3b) estimate E[ ζ j +mv + { r, m}] via (7) and E[( ζ j +mv + { r, m}) 2 ] via (8) for all pixels in the MB, 3c) estimate E[ ζ ] via (5) and E[( ζ ) 2 ] via (6) for all pixels in the MB, 3d) estimate D via (4) for all pixels in the MB, 3e) estimate R-D cost for the MB via (9), End 4) Via (10), select the best mode with minimm R-D cost for the MB. 5) Otpt: the best mode for the MB. Algorithm 1 is referred to as the Extended (ERPMC). Note that if an MV pacet is lost, in order to redce both the MV concealment and distortion estimation complexity, the concealed MV mv ˇ does not necessary se fractional accracy. That is, the E algorithm conceals the MV with integer accracy. Therefore, E[ ζ j and + mv ˇ ] E[( ζ j + mv ˇ ) 2 ] in (5) and (6) do not reqire (18). 2) Complexity analysis: In Ref. [11], we compare the complexity of to that of and in theory. In this section, we first compare the complexity of E to in theory. Then, we test the complexity of E// in JM16.0 and compare them to the defalt ERRDO algorithm in JM16.0, i.e. ; we also test the RDO withot error-resilient algorithms as a benchmar. comptational complexity is calclated from (4), (5), (6), (7) and (8), where the first moment and the second moment of the reconstrcted error of the best mode shold be stored after the mode decision (Note that the reconstrcted error in previos frames cold be regarded as the propagated error in the crrent frame, recrsively). Therefore, 2 nits of memory are reqired to store those two moments for each pixel. Note that the first moment E[ ζ j taes vales +mv] in { 255, 254,..., 255}, i.e., 8 bits pls 1 sign bit per pixel, and the second moment E[( ζ j +mv) 2 ] taes vales in {0, 1,..., }, i.e., 16 bits per pixel. From the discssion above, we now that the difference, in comptational complexity, of E compared to is the estimation of E[ ζ j ] in (7) and E[( ζ j +mv +mv) 2 ] in (8). To be more specific, in E E[ ζ j ] and E[( ζ j +mv +mv) 2 ] inclde a fractional MV while in E[ ζ j +mv ] and E[( ζ j +mv ) 2 ] inclde only an integer MV. Therefore, the additional complexity of E compared to is the calclation of (14) and (18). Note that (14) and (18) are not needed in E if: 1) the prediction mode is intra; 2) the MV is an integer MV. For MVs pointing to half-pixel positions, the vales of {w i } are {1, 5, 20, 20, 5, 1} in (14) and (18); i.e., there are N = 6 integer mltiplications, N 1 = 5 additions, and 1 right shift with 5 bits to mae sre N w i = 1. By sing the same simplification of the JM interpolation implementation, (18) can be calclated by E( ζ 3.5 ) 2 = {20 [E( ζ 3 ) 2 + E( ζ 4 ) 2 ] 5 [E( ζ 2 ) 2 + E( ζ 5 ) 2 ] + [E( ζ 1 ) 2 + E( ζ 6 ) 2 ] + 16} >> 5. In fact, we may frther simplify the implementation of (18) by replacing the integer mltiplication operations by additions

7 7 and shifts as E( ζ 3.5 ) 2 = {[E( ζ 3 ) 2 + E( ζ 4 ) 2 ] 4 + [E( ζ 3 ) 2 + E( ζ 4 ) 2 ] 2 [E( ζ 2 ) 2 + E( ζ 5 ) 2 ] 2 [E( ζ 2 ) 2 + E( ζ 5 ) 2 ] + [E( ζ 1 ) 2 + E( ζ 6 ) 2 ] + 16} 5. (19) Note that in (19), E( ζ 3 ) 2 + E( ζ 4 ) 2 and E( ζ 2 ) 2 + E( ζ 5 ) 2 are invoed two times, bt are conted only once since the temporary reslt is saved in the CPU register. As a reslt, the half-pixel position needs 8 additions (ADDs) and 4 shifts only. For qarter-pixel positions additional comptations will need to be performed. We observe that there are two practical implementations for ERMP, each one having different comptational complexity and memory reqirements. In the first, we can store the first and second moments of the transmission error for half-pixel positions, which will reqire 3 times the memory compared to storing only the integer pixel moments. However, this will redce the complexity of calclating the moments for qarter-pixel positions. In the second, the moments for all fractional positions are calclated on the fly. This is also the method sed in the algorithm available in the JM. The complexity of this method is analyzed below. Note that the comptational complexity of calclating (14) and (18) for different sbpixel positions is different. In this paper we only show the complexity analysis for those half-pixel positions interpolated by integer pixels. It is very easy to extend this analysis for all fractional positions. Since (14) and (18) are calclated on the fly, there is no additional memory reqirement for E. Note that in most CPUs, a shift can be integrated into an ADD, therefore not impacting complexity. As a reslt, the complexity of E can be fond in Table II 4. For, half-pixel position motion compensation reqires 8 ADDs and 4 shifts more than with integer-pixel position motion compensation for each simlated decoder; that is is 8N d ADDs more than the complexity of that in Ref. [11], where N d means the nmber of simlated decoders at the encoder; the defalt vale in JM is N d = 30. We also cite the complexity of and with integer MV accracy from Ref. [11] in Table II for reference. Since the theoretical complexity comparison only acconts for half-pixel positions, it wold be beneficial to evalate complexity in a real encoding environment. In reality, MVs cold point to any position, integer or fractional. The nearest integer MV is sed to approximate the fractional MV in and. NO ERRDO means the normal RDO mode decision process, withot any error resiliency considerations available in the JM. Table III shows the reslts for the mobile seqence at CIF resoltion with P LP = 5%. The experimental setp is the same as those in Section IV. A system based on an AMD Opteron(tm) 2356 processor at 2.29GHz was sed. It can be seen that the exectation time of E// is only slightly higher of that of ERRDO. However, reqires considerable more exection time than these schemes. Other seqences and channel conditions show similar reslts. 4 Note that in H.264/AVC seven inter prediction modes are spported, i.e., 16 16, 16 8, 8 16, 8 8, 8 4, 4 8, and 4 4. Nine intra 4 4 and 8 8 modes, as well as for modes for lma intra prediction are spported. Total complexity is calclated for all prediction modes. D. Merits and Limitations of E Algorithm 1) Merits: Since both the Cachy-Schwarz pper bond approximation [13] and the correlation coefficient model approximation [14] indce floating-point mltiplications, rondoff error is navoidable. The algorithm by Yang et al. [14] needs extra complexity to mitigate the effect of rond-off error dring distortion estimation. In contrast, one of the merits of Proposition 1 is that it only needs integer mltiplications and additions. In H.264/AVC and HEVC, w i (and w i w j ) can be scaled to an integer vale withot any rond-off error for all coding modes. As a reslt, rond-off error can be avoided in the E algorithm. In Ref. [19], the athors prove that a low-pass interpolation filter will decrease the frame-level propagated error nder some assmptions. In fact, it is easy to prove that when N w i = 1 and Vl is large, the negligible term is larger than or eqal to zero. Even at the MB-level, the negligible term is larger than or eqal to zero with very high probability. From (16), we see that the bloc-level distortion decreases, with very high probability, after the interpolation filtering. One additional benefit of (16) is to gide the design of the interpolation filter. Traditional interpolation filter design aims to minimize the prediction error. With (16), we may design an interpolation filter by maximizing N N =1 l=+1 [w w l E(X X l ) 2 ] nder the constraint of N j=1 [w j E(Xj 2)]. 2) Limitations: In Algorithm 1, the second moment of propagated error E[( ζ j +mv) 2 ] is estimated by ignoring the negligible term to redce the complexity. A more accrate alternative method is to estimate E( ζ i ζ j ) 2 ] recrsively by storing the vale in memory. This will be considered in or ftre wor. IV. EXPERIMENTAL RESULTS In this section, we compare the R-D performance and sbjective performance of the E algorithm with that of the and the algorithms for mode decision in H.264/AVC. Since the original does not spport the interpolation filtering operation and its extensions [13], [14] indce many floating-point operations and rond-off errors, we only se the same nearest integer MV approximation to show how its R-D performance differs from E,, and. To compare all algorithms nder mlti-reference pictre motion compensated prediction, we also enhance the original algorithm [10] with mlti-reference capability. A. Experimental Setp The JM encoder and decoder were sed in the experiments. The first 100 frames from several CIF resoltion, 30fps test video seqences were tested nder different PLP settings from 0.5% to 5%. The co-located pixel copy from the previos frame method was sed for error concealment in all algorithms. The first frame is assmed to be correctly received. The High profile of H.264/AVC, sing CABAC for entropy coding bt withot B slices, with 3 slices per pictre and 3 reference frames, was sed. Constrained intra prediction was also enabled. In the algorithm, the nmber of simlated decoders is 30.

8 8 B. R-D Performance TABLE II COMPLEXITY COMPARISON IN THEORY Algorithms comptational complexity memory reqirement inter mode 25 ADDs, 8 MULs E (half-pixel) intra mode 7 ADDs, 6 MULs 25 bits/pixel total complexity 266 ADDs, 1 MULs inter mode 9 ADDs, 8 MULs intra mode 7 ADDs, 6 MULs 25 bits/pixel total complexity 154 ADDs, 1 MULs inter mode 10N d ADDs, N d MULs (half-pixel) intra mode N d ADDs, N d MULs 8N d bits/pixel total complexity 83N d ADDs, 20N d MULs inter mode 2N d ADDs, N d MULs (integer pixel) intra mode N d ADDs, N d MULs 8N d bits/pixel total complexity 27N d ADDs, 20N d MULs inter mode 7 ADDs, 8 MULs intra mode 4 ADDs, 7 MULs 24 bits/pixel total complexity 101 ADDs, 147 MULs TABLE III COMPLEXITY COMPARISON IN EXPERIMENT Algorithm E NO ERRDO Time in second De to space limitations, we only show the plots of PSNR vs. bit rate for video seqences foreman and mobile nder P LP = 0.5% and P LP = 2% in Figs. 2 and 3 respectively. The experimental reslts show that E achieves the best R-D performance; achieves the second best R- D performance; achieves better performance than in some cases sch as at high rate in Fig. 2, bt worse performance than in other cases sch as in Fig. 3 and at the low rate in Fig. 2. It is interesting to see that for some seqences and channel conditions, E achieves a notable PSNR gain over. This is, for example, evident in mobile and foreman. For some other cases, however, E only achieves a marginal PSNR gain over (e.g., coastgard and football ). From the analysis in Section III-A, we now that the only difference between and E is the j ζ +mv estimate of the propagated error in (7) and (8). Therefore, the performance gain of E over only comes from inter modes, since they both se exactly the same estimates for intra modes. Ths, the higher percentage of intra modes in coastgard and football may reslt in a marginal PSNR gain of E over. For most seqences and channel conditions, we observe that in most cases the higher the bit rate for encoding, the more the PSNR gain of E over, sch as in Fig. 2 and Fig. 3(a). In (4), the end-to-end distortion consists of both qantization and transmission distortion. The E algorithm gives a more accrate estimation of propagated error in transmission distortion than the algorithm. When the bit rate for sorce encoding is very low, with rate control the controlled Qantization parameter (QP) is large, and hence the qantization distortion becomes the dominant part in the end-to-end distortion. Therefore, the PSNR gain of E over is marginal. On the contrary, when the bit rate for sorce encoding is high, the transmission distortion becomes the dominant part in the end-to-end distortion. Therefore, the PSNR gain of E over is notable. However, this is not always tre as observed in Fig. 3(b). The reason is as follows. In the JM, the Lagrange mltiplier in (9) is a fnction of QP. A higher bit rate or smaller QP also cases a smaller Lagrange mltiplier; therefore, the rate cost in (9) becomes smaller, which may prodce a higher percentage of intra modes. In sch a case, the PSNR gain of E over decreases. As a reslt, different seqences give different reslts depending on whether the effect of increase of intra modes dominates over the effect of decrease of qantization distortion. has poorer R-D performance than E. This may be since 30 simlated decoders are still not enogh to achieve a reliable distortion estimate. Meanwhile, complexity increase is considerable compared to E. It is also interesting to see that the integer MV approximation for is only valid for some seqences, sch as foreman, while this approximation gives poor R-D performance for some other seqences, sch as mobile. However, the nearest neighbor approximation for in all seqences achieves good performance. This is becase approximates the first and second moments of j the propogated error ζ by the ronded MV, while +mv approximates the first and second moments of the reference j f +mv pixel vale by the ronded MV. Since the propagated errors are mch smaller and more stable than the reference pixel vales, shows better and more stable performance sing integer MV approximation. Table IV shows the average PSNR gain (in db) of E over RPMC,, and for different video seqences and different PLP. The average PSNR gain is obtained sing the BD-PSNR method in Ref. [20], which measres the average distance (in PSNR) between two R-D crves. From Table IV, we see that E achieves an average PSNR gain of 0.25dB over for the seqence mobile nder P LP = 2%; it achieves an average PSNR gain of 1.dB over for the foreman seqence nder P LP = 1%; and it achieves an average PSNR gain of 3.18dB over for the mobile seqence nder P LP = 0.5%. C. Sbjective Performance Since PSNR may not be as meaningfl for error concealment, sbjective performance is also evalated. Fig. 4 shows the sbjective qality of the 84-th frame and the 99-th frame of foreman seqence nder a PLP of 1% and a bit rate of

9 E E Fig Bit rate (b/s) (a) PSNR vs. bit rate for foreman : (a) PLP=0.5%, (b) PLP=2% Bit rate (b/s) (b) E E Bit rate (b/s) (a) Bit rate (b/s) Fig. 3. PSNR vs. bit rate for mobile : (a) PLP=0.5%, (b) PLP=2%. TABLE IV AVERAGE PSNR GAIN (IN DB) OF E OVER, AND Seqence coastgard football foreman mobile PLP 5% 2% 1% 0.5% 5% 2% 1% 0.5% 5% 2% 1% 0.5% 5% 2% 1% 0.5% E vs E vs E vs (b) 250bps. These reslts sggest a similar performance as those presented in Section IV-B. We can ths conclde that E achieves the best performance. D. Discssion 1) Effect of clipping noise on the mode decision: Since does not consider the effect of clipping noise on the transmission distortion, it over-estimates the end-to-end distortion for inter modes. Hence, wold tend to select intra modes more often than E,, and, which will lead to higher encoding bit rates. To verify this conjectre, we tested all seqences nder the same QP settings, from 20 to, withot rate control. We observed that the algorithm always prodced a higher bit rate than other schemes as shown in Fig. 5 and Fig. 6. 2) Effect of transmission errors on mode decision: One can observe three characteristics for E/// algorithms vs NO ERRDO. 1) The nmber of intra MBs increases since the transmission error is acconted for dring mode decision; 2) The nmber of sip mode MBs also increases, since the transmission error will increase the transmission distortion in all other modes except for this mode; 3) if we allow the first frame to be erroneos, the second frame will have high percentage of intra MBs. This is becase only the vale of 128 can be sed to conceal the reconstrcted pixel vales if the first frame is lost, while if other frames are lost the collocated pixel in the previos frame can be sed to conceal the reconstrcted pixel vales. Therefore, the propagated error from the first frame will be mch higher than the error from other frames. As a reslt, if the first frame is allowed to be lost with a certain probability, the second frame will contain a high percentage of intra MBs de to ERRDO.

10 10 (a) (b) (c) (d) (e) (f) (g) (h) Fig. 4. (a) E at the 84-th frame, (b) at the 84-th frame, (c) at the 84-th frame, (d) at the 84-th frame, (e) E at the 99-th frame, (f) at the 99-th, (g) at the 99-th frame, (h) at the 99-th frame E Bit rate (b/s) (a) Fig E Bit rate (b/s) (b) PSNR vs. bit rate for foreman : (a) PLP=0.5%, (b) PLP=2%. V. C ONCLUSION In this paper, we proved a new proposition for calclating the second moment of a weighted sm of correlated random variables withot reqiring nowledge of the random variable probability distribtions. Then, we apply this proposition to extend or previos algorithm in estimating the fractional-level end-to-end distortion for prediction mode decision withot significantly increasing complexity. Experimental reslts show that E achieves on average a PSNR gain of 0.25dB over the existing algorithm for the mobile seqence when PLP eqals 2%; E achieves an average PSNR gain of 1.dB over the the algorithm for the foreman seqence when PLP eqals 1%. Experimental reslts also show that sbjective qality was also improved. R EFERENCES [1] C. E. Shannon, Coding theorems for a discrete sorce with a fidelity criterion, IRE Nat. Conv. Rec. Part, vol. 4, pp , [2] T. Berger, Rate distortion theory: A mathematical basis for data compression. Prentice-Hall, Englewood Cliffs, NJ, [3] Y. Shoham and A. Gersho, Efficient bit allocation for an arbitrary set of qantizers. IEEE Trans. Acost. Speech Signal Process., vol., no. 9, pp , [4] H. Everett III, Generalized Lagrange mltiplier method for solving problems of optimm allocation of resorces, Operations Research, vol. 11, no. 3, pp , [5] A. Ortega and K. Ramchandran, Rate-distortion methods for image and video compression, IEEE Signal Processing Magazine, vol. 15, no. 6, pp , [6] G. Sllivan and T. Wiegand, Rate-distortion optimization for video compression, IEEE Signal Processing Magazine, vol. 15, no. 6, pp , [7] H.264/AVC reference software JM16.0, Jl [Online]. Available: [8] T. Stochammer, M. Hannsela, and T. Wiegand, H. 264/AVC in wireless environments, IEEE Transactions on Circits and Systems for Video Technology, vol. 13, no. 7, pp , [9] T. Stochammer, T. Wiegand, and S. Wenger, Optimized transmission of h.26l/jvt coded video over pacet-lossy networs, in IEEE ICIP, [10] R. Zhang, S. L. Regnathan, and K. Rose, Video coding with optimal inter/intra-mode switching for pacet loss resilience, IEEE Jornal on Selected Areas in Commnications, vol. 18, no. 6, pp , Jn [11] Z. Chen and D. W, Prediction of Transmission Distortion for Wireless Video Commnication: Algorithm and Application, Jornal of Visal Commnication and Image Representation, vol. 21, no. 8, pp , 2010.

11 E E Fig Bit rate (b/s) (a) PSNR vs. bit rate for mobile : (a) PLP=0.5%, (b) PLP=2% Bit rate (b/s) (b) [12], Prediction of Transmission Distortion for Wireless Video Commnication: Part I: Analysis, IEEE Transactions on Image Processing, 2011, accepted. [13] A. Leontaris and P. Cosman, Video compression for lossy pacet networs with mode switching and a dal-frame bffer, IEEE Transactions on Image Processing, vol. 13, no. 7, pp , [14] H. Yang and K. Rose, Advances in recrsive per-pixel end-to-end distortion estimation for robst video coding in H. 264/AVC, IEEE Transactions on Circits and Systems for Video Technology, vol. 17, no. 7, p. 845, [15] ITU-T Series H: Adiovidal and Mltimedia Systems, Advanced video coding for generic adiovisal services, Nov [16] T. Wiegand, W.-J. Han, B. Bross, J.-R. Ohm, and G. J. Sllivan, WD1: Woring Draft 1 of High-Efficiency Video Coding, Gangzho, Oct. 2010, JCTVC-C403, 3rd JCT-VC Meeting. [17] T. Wiegand, G. J. Sllivan, G. Bjontegaard, and A. Lthra, Overview of the h.264/avc video coding standard, IEEE Transactions on Circits and Systems for Video Technology, vol. 13, no. 7, pp , Jl [18] G. Casella and R. L. Berger, Statistical Inference, 2nd ed. Dxbry Press, [19] K. Sthlmller, N. Farber, M. Lin, and B. Girod, Analysis of video transmission over lossy channels, IEEE Jornal on Selected Areas in Commnications, vol. 18, pp , Jn [20] G. Bjontegaard, Calclation of average PSNR differences between RDcrves, 13th VCEG-M33 Meeting, Astin, USA, Alexis M. Torapis received the Diploma degree in Electrical and Compter Engineering from the National Technical University of Athens (NTUA), Greece, in 1995 and the Ph.D. degree in Electrical and Electronic Engineering from the Hong Kong University of Science & Technology, HK, in Alexis has held in the past varios research and development positions with companies sch as Microsoft, Thomson, DoCoMo Labs USA, and Dolby Laboratories. He is crrently with Magnm Semicondctor Inc. as a Senior Director of Video Algorithm Engineering focsing on the development of next generation video processing and compression hardware system designs. Alexis is a senior member of the IEEE, and a member of the ACM, SPIE, and SMPTE. In 2000 he received the IEEE HK section best postgradate stdent paper award for his wor, and in 2006 he was acnowledged as one of 10 most otstanding reviewers by the IEEE Transactions on Image Processing. Alexis crrently holds 7 US patents and has more than 90 US and international patents pending. Alexis has made several contribtions to several video coding standards, and in particlar to H.264/MPEG-4 AVC, on a variety of topics, sch as motion estimation and compensation, rate distortion optimization, rate control and others, and crrently serves as a co-chair of the development activity on the H.264 Joint Model (JM) reference software. Zhifeng Chen received Ph.D. degree in Electrical and Compter Engineering from the University of Florida, Gainesville, Florida, in From 2002 to 2003, he was an engineer in EPSON (China), and from 2003 to 2006, he was a senior engineer in Philips (China), both woring in mobile phone system soltion design. He joined Interdigital Inc. in 2010, where he is crrently a staff engineer woring on video coding research. Dapeng W (S 98 M 04 SM 6) received Ph.D. in Electrical and Compter Engineering from Carnegie Mellon University, Pittsbrgh, PA, in Crrently, he is a professor of Electrical and Compter Engineering Department at University of Florida, Gainesville, FL. Peshala V. Pahalawatta received his Ph.D. degree in electrical engineering from Northwestern University, Evanston, IL, in He is crrently a Staff Engineer with the Image Technology grop at Dolby Laboratories Inc., Brban, CA. His research interests inclde image and video compression and transmission, image and video qality evalation, and compter vision.