Optimal SP frame selection and bit budget allocation for mobile H.264 video streaming
 Stefania Colonnese^{1},
 Stefano Rinauro^{1},
 Lorenzo Rossi^{1} and
 Gaetano Scarano^{1}Email author
https://doi.org/10.1186/16875281201218
© Colonnese et al.; licensee Springer. 2012
Received: 18 April 2012
Accepted: 3 September 2012
Published: 25 September 2012
Abstract
Mobile video streaming services are challenging, as they obey several system constraints, such as random access facilities, efficient server storage, and flexible rate adaptation. Rate adaptation can be performed by means of seamless switching among different encoded bitstreams. The H.264 video coding standard explicitly supports bitstream switching using specific frame coding modes, namely switching pictures (SP). Locations of SP frames affect the overall bit rate and quality of streamed video. In this study, we address the issue of optimal joint selection of the SP frames locations and bit budget allocation at frame layer. The optimization is carried out via a game theoretic approach under assigned system constraints on the overall streaming rate and the maximum random access delay. Numerical simulations show that our frame layer optimal encoding procedure brings advantages in terms of several characteristics of the streamed video, encompassing enhanced ratedistortion, reduced transmission buffer occupancy, equalization of the transmission delays, and more efficient switching.
Keywords
Introduction
Mobile video streaming services are experiencing a boost in cellular networks[1] or in multimedia wireless sensor networks[2] as well as in vehicular applications[3]. In mobile streaming services, the available bandwidth randomly varies, possibly due to changes in network conditions, terminal mobility, and/or handover in heterogeneous networks. The streaming server can react to these variations by streaming content extracted from differently preencoded versions of a given video sequence, i.e. by performing bitstream switching. Bitstream switching can be enabled by encoding INTRA frames coded without reference to any other coded frame. This would result in a significant coding cost, a less efficient bandwidth occupancy, an augmented transmission buffer storage, and worsened frame transmission delay and jitter. One of the new features of H.264 is a new coding mode named switching picture (SP), which allows driftfree bitstream switching[4, 5].
Since their introduction, SP frames gathered the attention of the research community due to their unique characteristics. Applications space from streaming services[6–8], to error control[9]. The optimal selection of SP frame location has recently been addressed in[10]. Within the framework of multiview video coding, the use of SP frames is currently under investigation to allow driftfree switching among different views[11, 12].
Insertion of the socalled primary SP frames into a mobile streamed video offers a set of candidate switching locations. The switch among the differently encoded versions of the video sequence is realized at need via the transmission of a complementary encoded representation, named secondary SP frame. Since both primary and secondary SP frames encompass a motion compensation stage[13], bitstream switching is provided without resorting to the transmission of a dedicated INTRA frame. During the encoding phase, the locations for switching frames are selected, and both primary and secondary SP frames are preencoded and stored at the server side. During the streaming phase, primary or secondary SP frames are transmitted at user convenience, depending on whether a switching is performed or not. As a side effect, SP frames also provide error resilience, which is an important issue in mobile communications. In[4], for instance, SP frames are integrated in a framework where switching is performed within a single compressed stream to achieve both error resilience and rate scalability.
Theoretical and empirical ratedistortion curves of SP frames have been provided in[5]. The rate distortion curve of SP frames unfavorably compares with those of PREDICTED (P) frames, thus limiting the adoption of SP coding mode in mobile video streaming. In this respect, it is clear how the choice of the proper frame coding mode itself significantly affects the overall rate (and quality) of the streamed video. On the other hand, the maximum distance between two consecutive SP frames is usually assigned as a system constraint depending on the desired degree of accessibility. Still, there is a degree of freedom on where to locate the SP frames along the sequence. Large margins of quality improvement—or of bit saving—can be observed by allocating SP frames along the video sequence in accordance with a suitable optimization criterion as well as by optimally allocating the available bit budget among the different frames.
On account of these considerations, in this article, we consider a video streaming framework where different versions of the same sequence are encoded at different qualities and switching among these is realized only by means of SP frames, and we jointly address the problems of SP frame location and bit budget allocation, via a game theoretic approach. The former application of game theory in video coding has been presented in the pioneering work[14], where the authors optimize the perceptual quality of the decoded sequence while guaranteeing fairness in bit allocation among macroblocks via a game theoretic approach. Here, we select the optimal SP frame locations and the optimal bit allocation that maximize the overall quality of the encoded sequence. Specifically, we extend the preliminary results in[15], and we formulate a game to optimize: (i) the bit allocation between different frames of the video sequence and (ii) the frame coding mode selection. Optimization is carried out under highlevel system constraints, such as the temporal distance between successive SP frames and the overall bit budget available to encode the sequence.
Related studies
Stateoftheart works on SP frames mostly focus on the realization of novel coding techniques aimed at reducing the allocated budget for SP frames. Sun et al.[16] describe a technique to improve the coding efficiency of the SP frames by limiting the mismatch between the prediction reference and the frames to be encoded. In[17], it is shown that by appropriately choosing reference pictures, the size of secondary SP frames can be reduced by up to 40% for randomaccess and up to 2% for rateswitching, without affecting the decoded sequence quality. In recent literature, the problem of coding mode selection has been deeply discussed. In[18], a lowcomplexity procedure to address INTRA mode selection is proposed, while in[19] a ratedistortion approach is employed to derive coding mode assignment procedure for intra, predicted and bidirectional predicted slices. In[10], a scheme to select the best switching points among the encoded bitstream has been introduced in the framework of a specific bandwidth reservation scheme, namely the socalled downstairs reservation scheme. The downstairs reservation scheme is based on reserving the maximum bit rate of the encoded sequence until the frame corresponding to such a maximum is transmitted; then, the reserved rate is reduced to the next highest bit rate and so on. Altaf et al.[10] propose to select as switching points in the bitstream those frames where a change in the reserved bit rate is observed. The resulting SP frame allocation scheme allows the SP frame to be transmitted when the receiver buffer is supposed to be empty, with a minimization of the wasted bits after the bitstream switching[10]. The SP frame selection scheme proposed in[10] is not explicitly related to any optimality criterion on the encoded sequence quality. Besides, in spite of its simplicity, the scheme in[10] is suitable only under a particular reservation scheme and, since the SP frames must coincide with the changes in the reserved bit rate, the degree of accessibility may be severely limited. Thereby, it is worth seeking a procedure for coding mode selection and bit allocation independent of the rate reservation scheme possibly implemented in the streaming system; besides, the procedure should allow the user to choose the desired degree of accessibility.
Organization of the paper
In this study, we introduce an optimization procedure for coding mode selection and bit allocation derived under a game theoretic framework. Specifically, we formulate the optimization problem by representing the frames of a sequence as players whose strategy is the choice of the coding mode and the allocated bits and whose goal is the maximization of the overall sequence quality. Such an encoding optimization procedure is beneficial in different respects, ranging from rate/distortion of the encoded sequences, to networkrelated issues such as equalization of the transmission delay and transmission/playout buffer load.
Optimal SP selection and bit budget allocation
Here, we carry out a frame layer optimization of SP frame selection and bit budget allocation for mobile H.264 video streaming resorting to a game theoretic approach. Since the video encoder controls the resource allocation among different frames in a joint fashion, we recast the problem of coding mode selection and bit allocation in terms of strategy selection in a cooperative game.
Let us then consider a reference streaming framework where the server is equipped with K versions of the same sequence. Each of these flows is encoded at a different quality. The server simultaneously transmits these flows in multicast to the clients. Each client automatically synchronizes to the flow that better matches the experienced channel conditions and the expected video quality. Seamless bitstream switching among the different flows is enabled only via the employment of primary and secondary SP frames.
Setting the maximum temporal distance τ_{max} between two consecutive SP frames results in a system constraint on the random access delay. Thereby, the maximum temporal distance τ_{max} between two consecutive SP frames is chosen so as to cope with the achievable degree of flexibility. The choice of the maximum distance between two consecutive SP frames corresponds to a maximum number of frames between switching points, N_{max} = f_{0}·τ_{max}, f_{0} being the video sequence frame rate. To satisfy this constraint on the maximum number of frames between switching points, the video sequence is partitioned in shorter subsequences of N = floor(N_{max}/2) frames. In each subsequence, exactly one frame shall be coded as a switching one, so as to comply with the choice of τ_{max}. Here, we exploit a game theoretic approach to jointly address the problem of coding mode assignment, i.e. the problem of the selection of the frame where the switching is enabled to occur, and the problem of resource allocation, once the coding modes are correctly assigned.
The game is described as follows:

the players of the game are the N frames within a subsequence;

the player’s strategy is given by its coding mode and by the number of bits allocated for coding;

the player’s utility is its visual quality after decoding.
We wish to encode the N frames in the subsequence at the target bit rate of R[bit/s]. The overall bit budget available for the N frames is B = RN/f_{0}(bit).
To elaborate, let us denote by c_{ i } the coding mode assigned to the frame i = 0,…,N−1, and by c = [c_{0},…,c_{N−1}] the coding mode Ntuple corresponding to the entire subsequence. The coding mode c_{ i } takes a value in a finite set$\mathcal{L}$ of cardinality L representing the coding modes provided by the video encoder. The generic Ntuple c takes a value in a finite set$\mathcal{M}$ of cardinality M, i.e.$\mathcal{M}=\{{\mathbf{c}}^{\left(0\right)},\dots ,{\mathbf{c}}^{(M1)}\}$. Due to coding constraints, generally$\mathcal{M}\subseteq {\mathcal{L}}^{N}$, so that M ≤ L^{ N }. Here, we consider the case where c_{ i }represents a binary choice between P and SP coding mode.^{a} Let us remark here that once the number of allowed frames for each coding mode in each subsequence has been set, the Ntuples$\mathbf{c}\in \mathcal{M}$ are different permutations of the same values.^{b}
Let r_{ i } be the number of bits allocated to the i th frame and let u_{ i }= u_{ i }(c_{ i },r_{ i }) denote the utility of the i th player, i.e. the visual quality of the i th frame, i = 0,…,N − 1. Each player is characterized by the initial utility${u}_{i}^{0}$, which measures the minimal visual quality that must be guaranteed, and by the corresponding number of allocated bits${r}_{i}^{0}$ required to achieve the quality${u}_{i}^{0}$. In assigning the minimal quality that must be guaranteed to each frame${u}_{i}^{0},i=0,\dots ,N\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}1$, different priors can be adopted.
Hence, the optimal allocated bits are obtained as${r}_{i}^{\left({m}_{\mathrm{opt}}\right)},i=0,\dots ,N1$.
In order to find m_{opt}, let us recall that, due to coding constraints, all the Ntuples${\mathbf{c}}^{\left(m\right)}\in \mathcal{M}$ are permutations of the same values, let us say${c}_{\left(m\right)}^{i}$,i = 0,…,N − 1. On account of this observation, the set${\mathcal{K}}_{m}=\left\{{k}_{i}^{\left(m\right)}\stackrel{\text{def}}{=}K\right({c}_{i}^{\left(m\right)}\left)\right\},i=0,\dots ,N1$, collecting the values of the weight function K(·) over the elements of the Ntuple c^{(m)}is affected by different choices of c^{(m)}only in the form of a permutation of its elements. As a consequence, it is easily seen how the denominator in (6) takes the same values for all the possible choices of c^{(m)}, and hence has then no effect in (7).
Under the hypothesis of uniform minimal quality all over the sequence, i.e.${u}_{i}^{0}={u}_{min},i=0,\dots ,N\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}1$, the solution provided by (9) corresponds to the choice of progressively assigning the less efficient coding mode (higher values of the coding cost k_{ i }) to the frame with lower amounts of innovation (smaller values of g(σ_{ i })).
The condition (9) directly comes from the following Proposition, whose proof is reported in Appendix.
Proposition 1
To recap, our optimal allocation and coding mode selection leads to the following criteria for maximization of the overall quality of the decoded sequence (1):
Coding mode selection criterion: the coding mode assignment used in the subsequence of N frames is performed by coding with the less efficient coding modes the frames with the smallest amount of innovation.
Bit budget allocation criterion: the bit allocation is performed in two steps: at first an initial allocation is performed in order to satisfy the minimum quality constraints, then the remaining bit budget after this first assignment is fairly redistributed among the frames in the subsequence of N frames.
The optimal coding procedure optimized according to these criteria is summarized in Appendix, where also a few implementation details are discussed.
The abovesummarized criteria resulting from the maximization of the objective function (1) basically lead to smoother instantaneous fluctuations of the video bit rate. The smooth traffic behavior is a result of the cooperative game underlying the optimized procedure of the coding mode selection and frame bit allocation. In fact, the constraint on the initial quality${u}_{i}^{0}$ is satisfied with minimal initial budget$\sum _{i=0}^{N1}{r}_{i}^{0}$ when the less efficient SP coding mode is assigned to the frame with the minimum amount of innovation. This assignment reduces the unbalance of the initial bit budgets${r}_{i}^{0}$ required to guarantee the desired initial quality of the different frames. Besides, after the initial allocation, the remaining bit budget is fairly allocated among the frames. Hence, the fair allocation resulting from the joint quality improvement pursued by optimizing the objective function (1) in a cooperative fashion results in a smoother behavior of the video traffic.
Remarkably, the smoothness of the traffic of the encoded video resulting from such cooperative optimization is expected to be beneficial in a realistic network scenario, in terms both of transmission delay jitter of the video traffic and of load of network buffers. These benefits will be quantitatively assessed in “Experimental results” section.
Final remarks
it can easily be proved that such a model results in exactly the same optimal criteria obtained with the quality model in (3).
Let us finally observe that once the optimal bit budget per frame is allocated, any rate control scheme can be employed to encode the sequence; for instance, the algorithm described in[14] can further be applied at a macroblock layer for individual frame coding.
To sum up, while recent literature works refer to withinframe optimization techniques, the scope of our optimization includes different encoded frames, jointly taking into account system constraints and ratedistortion aspects. Besides, we have demonstrated that the optimization problem phrased in (1) can separately be solved w.r.t. the two tuples c_{0},…,c_{N−1} and r_{0},…,r_{N−1}. Moreover, although the finite and discrete nature of the tuple c_{0},…,c_{N−1}would allow to find the optimal c_{ i }’s using an exhaustive search, we have provided a closed form solution.
The so found novel resource allocation procedure extends to framelevel the macroblockoriented resource allocation procedure found in[14], also accounting for the unbalance between coding mode efficiency.
Experimental results
In this section, we show some experimental results of the herein analyzed optimized method, obtained using the H.264 codec[25] on different test sequences in QCIF and CIF format at the reference frame rates of f_{0} = 10 and 30 fps.
The optimization is performed by first partitioning the sequence in groups of N frames with N = f_{0}·1, corresponding to at least one SP frame every 2 s, so as to achieve a good compromise between accessibility and compression efficiency. For every group of N frames, the switching coding mode c_{ i }= SP (SP frame) is assigned to the frame with minimum RMSV of the innovation process; the remaining frames in the subsequence are coded as c_{ i }= P (P frame). The optimization algorithm is then applied by evaluating the bit budget for each frame to be encoded. Following the guidelines in Step IV of “Optimal coding procedure” in Appendix, we compute the initial frame bit budget${r}_{i}^{0},i=0,\dots ,N1$ by applying a coarse quantization coding stage with quantization parameters fixed so as to assure the desired average initial quality set to PSNR_{0} = 20dB.^{e} The optimal frame bit budgets r_{ i },i = 0,…,N−1 are then evaluated by fairly redistributing the remaining bit budget according to (5). Finally, the sequence is encoded under the per frame bit budget constraints r_{ i }. A coarse rate control procedure is implemented using a constant within frame QP; such strategy can be further refined using a spatially varying QP as described in[14].
For comparison, we consider also a suboptimal coding approach with fixed SP periodicity (one SP frame each N frames), using quantization parameters chosen according to the analysis in[5].
In all these cases, maximization of the objective function (1) leads to smoother instantaneous bit rate fluctuations; the effect is even more noticeable on the sequence in CIF format, showing that the higher the objective bit rate, the more effective the optimal allocation scheme is.
Average bit rate and PSNR measurements for the optimal and suboptimal (periodical SP allocation) strategies
Sequence  Format  fps  Bit rate R(kb/s)  PSNR (dB)  Bit rate R^{PER} (kb/s)  PSNR^{PER}(dB) 

Foreman  QCIF  10  89  35.0  89  34.8 
Coastguard  QCIF  10  88  31.9  91  31.9 
Mother and Daughter  CIF  10  799  46.5  829  46.5 
Foreman  QCIF  30  93  28.3  103  28.1 
Movie  CIF  30  469  41.6  480  41.3 
Let us now refer to the same scenario as in Figure6, when a specific bandwidth reservation scheme, namely the Downstairs Reservation (DR) scheme is employed. As variable bit rate (VBR) video data are likely to exhibit severe bit rate fluctuations on both short and large scales, suitable smoothing procedure are designed to realize the transmission of VBR by means of a series of constant bit rate (CBR) segments. Video server is then required to reserve the correct amount of bandwidth to effectively transmit each segment. Several techniques to achieve such piecewise CBR reservations have been proposed in recent literature. Among others, the DR scheme exhibits the desired property of avoiding upwards bandwidth reallocations, that is, every CBR segment is characterized by a bit rate equal or less than the previous segments. Such a characteristic deeply simplifies the network admission control procedures.
and the largest values among the A_{ l }’s is employed for the reservation of the following segment. The procedure is iterated for the entire sequence.
Average bit rate and PSNR measurements for the optimal presented allocation scheme, the suboptimal strategy (periodical SP allocation) and the work in [[10]] (“Foreman” test sequence, 30 fps, QCIF format)
Allocation strategy  Bit rate (Kb/s)  PSNR (dB) 

Optimal  93  28.3 
Suboptimal (fixed SP periodicity)  103  28.1 
Work in[10]  103  28.1 
Endtoend frame loss rate for different transmission and playout buffer under DR scheme (“Foreman”, QCIF, 30 fps, 100 Kb/s)
SP frames Allocation  tx buffer (Kbit)  

scheme with DR  Playout buffer (Kbit)  
10.2  10.2  15  15  
460  500  390  430  
Optimal  0.02  0.01  0.09  0.02 
Periodical SP allocation  0.06  0.03  0.17  0.09 
Work in[10]  0.05  0.03  0.22  0.12 
Average bit rate and PSNR measurements for the optimal presented allocation scheme, the suboptimal strategy (periodical SP allocation) and the work in [[10]] (“Sport” test sequence, 30 fps, CIF format)
Allocation strategy  Bit rate (Kb/s)  PSNR (dB) 

Optimal  437  36.9 
Suboptimal (fixed SP periodicity)  489  36.8 
Work in[10]  491  36.8 
Endtoend frame loss rate for different transmission and playout buffer under DR scheme (“Sport”, CIF, 30 fps, 450 Kb/s)
SP frames  tx buffer (Kbit)  

Allocation scheme  Playout buffer (Mbit)  
20.3  21.2  22.1  23  
2.57  2.37  2.27  2.47  
Optimal  0.01  0.01  0.03  0.01 
Periodical SP allocation  0.21  0.20  0.17  0.16 
Work in[10]  0.20  0.19  0.17  0.17 
Endtoend frame loss rate for different transmission and playout buffer (“Foreman”, QCIF, 30 fps, 100 Kb/s)
SP frames  tx buffer (Kbit)  

Allocation scheme  Playout buffer (Kbit)  
4.2  7.8  10.2  10.2  
420  390  390  418  
Optimal  0.07  0.08  0.07  0.05 
Periodical SP allocation  0.21  0.17  0.17  0.11 
Work in[10]  0.20  0.17  0.17  0.11 
Endtoend frame loss rate for different transmission and playout buffer (“Sport”, CIF, 30 fps, 450 Kb/s)
SP frames  tx buffer (Kbit)  

Allocation scheme  Playout buffer (Kbit)  
19.4  33  36.4  43.2  
2.11  2.27  2.35  2.11  
Optimal  0.09  0.02  0.01  0.05 
Periodical SP allocation  0.22  0.11  0.10  0.12 
Work in[10]  0.22  0.12  0.09  0.11 
All the presented results clearly highlight the impact of the optimal criteria for coding mode assignment and bit allocation with respect to stateoftheart approaches.
Until now, we have considered optimization of primary SP frames, i.e. the random access frames of the encoded video sequence. When a switching is requested during a streaming session, the server sends a different version of the access frame, namely the secondary SP frame, for decoder buffer synchronization purposes. Since also secondary SP frames are encoded by motion compensation, optimization of primary SP allocation is beneficial for secondary SP bit allocation too. Numerical simulations have shown a variable gain of the optimal allocation scheme over the suboptimal one; in the case of bitstream switching between 70 and 100 Kb/s version of the QCIF sequence “Foreman”, we have observed a reduction up to 20%, with an average value of 10%, of the bits allocated to the SP secondary frames.
Conclusion
In this study, we have presented a procedure for optimal framelevel coding mode selection and bit budget allocation, with application to mobile H.264 video streaming. The optimization procedure is here derived via a game theoretic approach. The cooperative game underlying the optimized procedure of the coding mode selection and frame bit allocation basically leads to smoother instantaneous fluctuations of the video bit rate. Numerical simulation results show that the encoding optimization procedure is beneficial in different respects, ranging from rate/distortion of the encoded sequences, to networkrelated issues such as equalization of the transmission delay, and transmission, playout buffer load.
Appendix
Proof of Proposition 1
Let us consider the finite set${\mathcal{A}}^{\left(n\right)}=\{{a}_{1},\dots ,{a}_{n}\}$ with a_{ i }≥ 0,i = 1,…,n, sorted in descending order, i.e. a_{ i }≥ a_{i−1}, and the finite set${\mathcal{P}}^{\left(n\right)}=\{{p}_{1},\dots ,{p}_{n}\}$ with p_{ i }≥ 0,i = 1,…,n, sorted in ascending order, i.e. p_{ i }≤ p_{i−1}. Moreover, let us denote by F^{(n)} the set of all the possible permutations f:{1,…,n}→{1,…,n} of the first n integers.
 (i)
(10) is true for n = 2;
 (ii)
if (10) is true for n = m − 1, then it is true also for n = m.
Given the ordering of the elements of the sets${\mathcal{A}}^{\left(2\right)}$ and${\mathcal{P}}^{\left(2\right)}$, the term (a_{1}−a_{2})(p_{2}−p_{1}) is always nonnegative, and hence (11) proves (i).
which proves (ii).
Optimal coding procedure
Here, we summarize the coding algorithm steps, optimized according to the criteria exposed in “Optimal SP selection and bit budget allocation” section.
Step I: Sequence Partitioning— The coding optimization algorithm is applied by first partitioning the overall sequence in subsequences of equal length N. In each and every subsequence exactly one SP frame shall be introduced.
Step II: Innovation process RMSV estimation— According to the guidelines provided by Proposition 1, the RMSVs σ_{ i }i = 0,…,N − 1 of the innovation process of the N frames in each subsequence are estimated as the RMSV of the motion–compensation residuals and are sorted in ascending order. We observe that, during the coding process, the motion compensation residuals are generated with respect to the decoded reference frame. Here, we estimate the RMSV of the motion–compensation residual with respect to the original reference frame. This design choice is well suited to be implemented in streaming systems, since it leads to allocate the primary SP frames at the same time index in all the encoded bitstreams. This circumstance enables streaming server rate adaptation by seamless switching among preencoded bitstreams.
Step III: Coding Mode Assignment— Once the RMSV has been evaluated, the SP coding mode is assigned to the frame with the minimum RMSV of the innovation process. The values for K(c_{ i }) can be directly derived from the rate distortion curves at a typical distortion value, or assigned through an a priori criterion. In the specific case of only two possible coding modes (P or SP), it is sufficient to establish an ordering between K(c_{ i }= SP) and K(c_{ i }= P), according to the hypothesis of Proposition 1, regardless of their numerical values.
Step IV: Rate Evaluation— After the choice of the coding mode of each frame, the preliminary assignment of the initial rates${r}_{i}^{0}$ is performed, based on the assignment of the qualities${u}_{i}^{0},i=1,\dots ,N1$. Recent investigations on the theoretical and experimental ratedistortion performance of SP and P frames have highlighted that a given level of distortion is achieved by higher rate for SP frames than for P frames[5]. Hence, to avoid initial quality fluctuations, a larger initial bit budget${r}_{i}^{0}$ is assigned to the SP frame. The r_{ i }i = 0,…,N−1, are then straightforwardly evaluated using (5).
Step V: Frame Coding— Once the bit budget per frame r_{ i } has been assigned, the subsequence is ready to be encoded. The optimal frame coding under an assigned bit budget per frame can be performed according to different rate control techniques. For instance the optimal approach presented in[14] can be applied; according to this algorithm, the quantization parameter is properly chosen for each macroblock, in order to meet the fairest bit allocation among macroblocks satisfying the bit budget constraint. If the whole frame is encoded by a single quantization parameter, this latter shall be chosen equal to the minimum value compatible with the assigned value r_{ i }.
Endnotes
^{a} Extension to the case where B frames are also considered is straightforward, provided the number of allowed B frames in the subsequence is fixed.
^{b} For instance, if one and only one out of the N frames in each subsequence is allowed to be an SP frame and the other frames are set as P frames, then it results that M = N and all the Ntuples c will exhibit the form c = [PP⋯SP⋯P], thus differing one from the other only in the location assigned to the SP frame.
^{c} As in[14], here we set$g\left({\sigma}_{i}\right)={\sigma}_{i}^{\alpha}$ with α = 0.8.
^{d} As the optimal allocated bits evaluated according to (5) are real values, they must be quantized to provide an input to the encoder. For instance, the closest integer to${r}_{\left(m\right)}^{i}$ can be considered as the assigned rate. The quantization loss thus introduced by this approximation comprises the effect of a single bit on the whole frame and it is therefore negligible.
^{e} Let us remark that, when the objective function is maximized, all the N frames overcome this limit. The minimum initial quality can be instead regarded as a parameter that allows to determine initial bit budget, which is assigned in an unbalanced way and, by complement, the residual bit budget, which is assigned on a fair basis.
^{f} Innetwork losses are neglected in this test.
Declarations
Authors’ Affiliations
References
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