Adaptive error protection coding for wireless transmission of motion JPEG 2000 video
 Giuseppe Baruffa^{1}Email author and
 Fabrizio Frescura^{1}
https://doi.org/10.1186/s136400160111z
© Baruffa and Frescura. 2016
Received: 9 July 2015
Accepted: 25 February 2016
Published: 7 March 2016
Abstract
The delivery of video over wireless, errorprone transmission channels requires careful allocation of channel and source code rates, given the available bandwidth. In this paper, we present a theoretical framework to find an optimal joint channel and source code rate allocation, by considering an intracoded video compression standard such as Motion JPEG 2000 and an errorprone wireless transmission channel. Lagrangian optimization is used to find the optimal code rate allocation, from a PSNR perspective, starting from commonly available source coding outputs, such as intermediate ratedistortion traces. The algorithm is simple and adaptive both on the available bandwidth and on the transmission channel conditions, and it has a low computational complexity. Simulation results, using ReedSolomon (RS) coding, show that the achieved performance, in terms of PSNR and MSSIM, is comparable with that of other methods reported in literature. In addition, a simplified and suboptimal expression for determining the channel code assignment is also provided.
Keywords
1 Introduction
Many multimedia devices are being turned into complete entertainment centers, also by taking profit of wireless transmission. There exist several industrybacked liaisons aimed at transmitting wireless audio/video contents between multimedia home appliances using either the 60GHz band [1–3] or the 2.4–5.0GHz unlicensed spectrum [4, 5], using for this purpose techniques such as UltraWideBand, orthogonal frequency division multiplexing, and multiantenna links.
Moreover, JPEG 2000 [6] is rapidly spreading as a valuable intracoding scheme for video contribution applications [7] due to the high compression efficiency, wide coverage of encoding profiles from lossless to lossy, and the low latency. Recently the International Organization for Standardization (ISO), jointly with the International Electrotechnical Commission (IEC) and the International Telecommunications Union (ITU), added new profiles, to the JPEG 2000 standard, for broadcast video contribution and distribution with an amendment to the JPEG 2000 core coding system [8]. This amendment defines three new profiles, aimed at studio contribution links, specifying encoding parameters and rate limits over seven operating levels for video encoded with JPEG 2000. Even JPEG 2000 over MPEG2 Transport Stream is a recently standardized method suited to this scenario [9]. In this kind of application, wireless cameras may produce a video contribution that has to adapt, in real time, to timevarying transmission channel profiles. In such case, both the available bandwidth and the wireless link bit error rate (BER) may be considered slowly variable with respect to the video frame rate [10].
The streaming of video either directly over the physical layer or using IP packets, is subject to transmission errors. Retransmission of lost/corrupted data or packets is viable, but decreases interactivity and realtime requirements. Thus, forward error correction (FEC) is generally adopted, and the channel code rate may be matched to compressed data error sensitivity, performing unequal error protection (UEP) [11–13].
Many researchers have investigated optimal methods for the protection of intracoded video streams. For instance, JPEG 2000 for wireless (JPWL) [14] has been standardized for this purpose, and several works have shown its good performance either when used on IP networks [15–17] or directly over the physical layer [18–23].
In [24, 25], the authors addressed similar problems showing how ratedistortion optimized (RaDiO) audio/video streaming over packetized networks can be achieved, and they solved this problem using Lagrangian optimization. Cataldi et al. proposed a technique based on raptor codes and sliding windows, where different H.264 [26] code rates are associated to each quality layer [27]. In [28], the authors proposed WynerZiv coding for the protection of a coarse version of the video, where side information is provided by a primary H.26x decoder. Ahmad et al. proposed a UEP system using fountain codes [29]. In general, many of the solutions reported in literature for searching optimal UEP strategies are based on heuristic methods or use optimization algorithms [30–33]. It should be noted, however, that such solutions are based on search strategies characterized by a variable amount of computational complexity, which could prevent their use in realtime and bandwidthadaptive video transmission.
1.1 Review of recent works
In recent years, several researchers have investigated techniques capable to apply differentiated FEC levels to waveletbased image/video compressors, when the multimedia stream is delivered over unreliable or wireless channels.
In [34], a motioncompensated temporal filtering discrete wavelet transform (DWT) video coder is coupled with double binary turbo codes: the joint sourcechannel coding strategy is based on distortion profiling and code statistics. Also, Ho and Tsai [35] used 3Dwavelets, data interleaving, and ReedSolomon (RS) codes for their UEP streaming system. In [36], the authors investigate the performance of MJPEG 2000 and ratecompatible punctured convolutional codes for streaming over a timevarying binary symmetric channel (BSC); in their work, ratedistortion tradeoff of the coding units adapts the error correction code to the bandwidth and error characteristics. Schwartz et al. [37] adopted the DWTbased compression and convolutional coding FEC of CCSDS 122.0B1 and 131.0B1 satellite standards. Their results show that wavelet coefficient UEP outperforms the equal error protection (EEP) method over the simulated AWGN channel.
In [15, 38], the authors focus on JPEG 2000, RS coding, and interleaving over wireless channels, simulated by a timevarying BSC with GilbertElliot (GE) model. UEP is performed by variable FEC rates defined by solving a convex optimization problem. Based on interleaving effects, they derive a lower bound for successful image decoding rate in wireless environments. In [39], several UEP schemes are compared and layered JPWL streaming with RTP packetization on wireless channels is studied; the FEC allocation method is comparably faster and less complex than others, although yielding comparable quality.
JPWL has been shown to be both flexible and reactive to variable channel status. In [21, 40], streaming performance is simulated over realistic wireless channels, such as multipleinput multipleoutput or Rayleigh fading ones. In [17], the authors conjugate JPWL with a dynamic bandwidth estimation scheme in order to provide the best layering, scaling, and protection of video streams. Even if source distortion is coarsely estimated, it has been shown that it can be effectively used to find an optimal rate allocation that outperforms EEP [23].
In our previous papers [16, 41], we used JPWL and interleaving over lossy packet networks. The UEP solution, found by means of a recursive, dichotomic search algorithm, was shown to always outperform EEP, and a low complexity interleaving strategy was devised for JPWL implementation on a DSP device. Iqbal et al. [42, 43] devised a family of dynamic programming code allocation methods for FEC protected wireless video streaming. Their protection assignment can provide variable tradeoff between performance and implementation complexity.
A different view was adopted by Bahmani et al. [44]. The method devised by the authors operates mainly at the decoder side, leaving the particular UEP implementation open. By leveraging the error resilience features of JPEG 2000, their method guesses the erased received symbols and improves the error correction capability.
Ouaret et al. [45] compared RS coded JPEG 2000 to the SlepianWolf/WynerZiv distributed video coding (DVC) approach. Their results show that JPEG 2000 results in better quality at high error rates, even if only an EEP scheme was used, whereas DVC performs better at lower packet loss rates. In [46], JPEG 2000 and H.264 streams are protected with UEP and transmitted over lossy packet networks. A performance comparison with multiple description coding shows that UEP achieves better quality. Chen et al. [47], by using progressive digital fountain codes, allowed different users to receive broadcast video at different qualities, depending on the reliability of their UDPbased WiFi link. In [48], the authors describe the emerging MPEG multimedia transport standard for delivering high bit rate video over packetlossy networks, using a lowdensity generatormatrix FEC. They show the effectiveness of their method on the streaming of JPEG 2000 digital cinema.
1.2 Proposed contributions
In this paper, we present a simple mathematical and algorithmic solution for finding an optimal UEP channel code rate allocation strategy. The method used for solving the problem is based on Lagrangian optimization, which is known in the literature and has been applied in several works of other authors [25, 34, 36]. Our proposed solution can be calculated with a closedform expression, directly from the knowledge of the data units ratedistortion and of the transmission channel characteristics.
Differently from other existing strategies, such as those reviewed in Sec. 1.1, our method has a closedform representation for the solution to the UEP problem, and it does not require iterative or dynamic programmingbased strategies. Moreover, the presence of data interleaving enables optimal video quality when channel conditions are timevarying, and data and channeladaptivity can be fulfilled.
The devised method does not rely on a particular channel coding technique, since it can be applied universally to all blockbased FEC schemes. The algorithm itself is also lightweight, as it gives a closedform expression of the optimal channel code allocation strategy, which can be computed in real time and adaptively respond to changes in the available transmission bandwidth and experienced channel BER. This makes the technique suitable for channels with unknown and slowly changing error rate and, even, available bandwidth; the video stream receiver should communicate the experienced error rate to the sender side, which in turn would change the UEP solution accordingly. We also show how the algorithm can be practically applied, using JPEG 2000 source coding and RS channel coding, and present some performance results expressed in terms of either PSNR (peak signaltonoise ratio) or MSSIM (mean structural similarity index metric).
Some mathematical derivations and performance results shown in this paper have already been partially presented in [49]. However, in this paper, we present additional derivations and novel simulation results. First, we describe in detail a method for assigning codewords of a channel code and implementing the designed protection profile. Furthermore, we work out a formula that allows approximating the optimal protection profile without knowledge of the image content, but only by means of a statistical entropy approach. Finally, we also present some results on the simulated transmission of a complete video sequence.
The paper is structured as follows. First, the theoretical framework for optimized unequal error protection is presented in Sec. 2 and a Lagrangian optimization strategy is shown to be able to find a UEP solution, both for particular and general cases. Then, a practical UEP code rate assignment using RS codes is presented in Sec. 3. In Sec. 4, the Monte Carlo simulation setup is described, and the results of several simulated scenarios are presented and discussed. Eventually, conclusions are drawn, followed by an Appendix that contains proofs to assumptions and lemmas.
2 Optimized unequal error protection
Summary of mathematical notation
Symbol  Meaning  Symbol  Meaning 

B  Size of the compressed bitstream, in bits  m _{ i }  \({m_{i}} = \Sigma _{j = i}^{{N_{k}}  1}{M_{j}}\), complementary cumulative distortion (CCD) 
N _{ k }  Number of pieces the bitstream is split into  \(\hat m\)  \(\hat m = {(\Pi _{i = 0}^{{N_{k}}  1}{m_{i}})^{1/{N_{k}}}}\), geometric mean of the CCD 
k  Size of each piece (with 2byte CRC) before channel coding, in bits  \(\bar {r}_{\text {min}}\), \(\bar {n}_{\text {min}}\)  Minimum average channel code rate and piece length after channel coding, in bits or bytes 
n _{ i }  Size of the ith piece after channel coding, in bits  σ ^{2}  Variance of the compressed frame coefficients 
p _{ i }  p _{ i }(n _{0},n _{1},…,n _{ i }), probability that no transmission errors occur up to piece i  Δ ρ  MSE profile sampling step width, in bits/pixel 
M _{ i }  MSE distortion of reconstructed frame using the first i pieces  H  Differential entropy of the frame 
Q  Number of available transmission alphabet symbols in the symmetric channel  \(\tilde N\)  Number of mother RS codes 
P _{ S }  Symbol error rate of the Qary symmetric channel  \(\tilde {r}_{l}\)  \(\tilde {k}_{l} / \tilde {n}_{l}\), channel code rate of the lth RS code 
\(\tilde {P}_{S}\)  Qary symbol error rate estimated at the receiver  w _{ i }  Index of the RS code used in the ith piece 
P _{ b }  Effective bit error rate of the Qary symmetric channel  α _{ i }, β _{ i }  Fractional number of codewords, in piece i, with RS channel code rate \(\tilde {r}_{w_{i}}\) and \(\tilde {r}_{w_{i} + 1}\), respectively 
γ _{ b }  Average signaltonoise ratio of the AWGN or Rayleigh fading channels  α i′, β i′  Integer number of codewords, in piece i, with RS channel code rate \(\tilde {r}_{w_{i}}\) and \(\tilde {r}_{w_{i} + 1}\), respectively 
h(n _{ l })  Probability that a piece of n _{ l } bits is channeldecoded without residual errors  \(\delta _{\alpha '_{i}}\)  Amount of rate \(\tilde {r}_{w_{i}}\) codewords normalized to the total number of codewords 
C, s, d  Amplitude, decay, and offset constants of the loglinear model for the combined transmission channel/channel code performance  N _{row}, N _{col}  Dimensions (rows and columns) of the block interleaver 
f _{ D }  Doppler spread of the Rayleigh fading channel  λ  Lagrange multiplier used for the optimization step 
r _{ s }  Source code rate, in bits/pixel  \(\bar {r}\)  Channel code rate 
R  Combined source and channel rate, in bits/pixel  N _{ F }  Number of frames in the YUV video sequence 
R _{ b }  Transmitted bit rate, after source and channel coding  \(\epsilon _{i}, \bar {\epsilon }\)  Perframe and average MSE of the received video sequence 
\(\bar {n}\)  Average piece length after channel coding, in bits or bytes  \(\Gamma _{i}, \bar {\Gamma }\)  Perframe and average PSNR (in dB) of the received video sequence 
\(\iota _{i}, \bar {\iota }\)  Perframe and average MSSIM of the received video sequence  F _{ E }(ε)  a posteriori cumulative distribution function of the perframe MSE ε _{ i } 
Lemma 1.
where \(\bar {n}=k/\bar {r}\) is the average protected piece length, m _{ i } is a complementary cumulative distortion (CCD), \({m_{i}} =\! \Sigma _{j = i}^{{N_{k}}  1}{M_{j}}\), and \(\hat {m}\) is the CCD geometric mean \(\hat {m} = {\left (\Pi _{i = 0}^{{N_{k}}  1}{m_{i}}\right)^{1/{N_{k}}}}\).

The protection rate for piece i depends on the average protection rate \(\bar {n}\) plus a modification term.

The modification term logarithmically weights the CCD at piece i, normalized by the geometric mean of the CCD.

If the channel code has higherror correction performance and/or the channel conditions are good, the parameter s is large, which gives a small modification term.

The protection profile depends on the equivalent transmission channel conditions only by means of the parameter s, not C and d.

Since m _{ i } is monotonically not increasing and ln(·) is a monotonic function, the modification term is monotonically not increasing, i.e., pieces at the beginning of the bitstream are more protected than those at the end.

The protection level at piece i depends on the cumulative amount of distortion of all following pieces.

The shape of the protection profile is determined by the CCD. Ordinate extrema are defined only by the channel/code combined performance.
With respect to other similar solutions presented in the literature, and described in Sec. 1.1, the main advantage of our method is that a closedform solution to the UEP problem is readily available, without needing iterative or dynamic programmingbased solving strategies. The proposed solution is data and channel adaptive; regular, low bit rate feedback from the receiver lets the transmitter modify the UEP strategy, which, considering also the presence of data interleaving, enables optimal decompressed video quality when channel conditions change with time. Moreover, the side information produced at no cost during the compression process allows implementing a wellcrafted protection profile, which minimizes the expected amount of distortion due to missing or corrupted data at the receiver. Eventually, we also want to outline that when stringent realtime requirements are needed, and the wireless channel is timevarying, deep interleaving matrices are necessary to intersperse the symbol losses occurring in the channel far away (for high Doppler spread f _{ D }), and the decoding delay increases correspondingly.
Upon looking carefully at the solution (3) proposed in Lemma 1, it can be noticed that an additional condition to be satisfied is that n _{ i }≥k, 0≤i<N _{ k }, meaning that we cannot overprotect the pieces at the beginning, since there would be not enough bits to allocate for the last pieces, not even the source coding bits; especially at high symbol error probabilities, the protection profile, given the total bit budget, could be extremely unbalanced and might provide values lower than k.
Lemma 2.
for a given equivalent channel error performance s.
Lemma 2 can be proved after some work on (3) and supposing that \(\ln \left (m_{i} / \hat {m} \right) < 0\), for large i close to N _{ k }.
In all cases where the exact ratedistortion curve of the compressed image is not known or cannot be calculated exactly, an approximation of (3) can be done, if the source coding process is expected to generate an ideal progressive codestream with a typical ratedistortion curve.
Lemma 3.
for N _{ k } large, where Δ ρ is the bit rate sampling step of the MSE profile.
Proof of this lemma is given in Appendix A.3 Proof of Lemma 3.
Our proposed UEP method has been devised in order to be as general as possible, with application scenarios that can extend also to other source and channel coding methods. For instance, considerations on the distortion reduction carried by data packets may be similarly done also for the network abstraction layer (NAL) units used in H.264 or H.265. In this case, NAL units naturally segment the video stream in pieces for which there is a correspondence with the pieces used in Fig. 1. The computation of the distortion profile can be achieved in several ways, if temporal and spatial frames intra/interdependency is maintained or not; layering methods for postcompression reordering of NAL units, similar to those adopted by the scalable video coding (SVC) extension of H.264 are then possible [52].
3 UEP profile generation using RS coding
In the scenario adopted in this paper, when RS coding is used, a strategy must be devised for the practical implementation of the optimal UEP profile found with (3). First, we consider a list of \(\tilde {N}\) mother code rates \(\{\tilde {r}_{l}\} = \{(\tilde {k}_{l} / \tilde {n}_{l}) \}\), \(l = 0, 1, \ldots, \tilde {N}  1\), ordered by decreasing code rate and not containing repeated code values, i.e., \( \tilde {r}_{l} > \tilde {r}_{l + 1}\), \(j = 0, 1, \ldots, \tilde {N}  2\).
4 System simulation and performance
4.1 Simulation setup
For purposes of assessing the performance of the technique presented in this paper, we have prepared a 512frame video composed by the first 32 frames of each of the following 16 clips with CIF resolution (352 × 288, 30 frames/s) and YUV 4:2:0 format, combined in sequence: akiyo, bus, coastguard, crew, flower, football, foreman, harbor, husky, ice, news, soccer, stefan, tempete, tennis, and waterfall [53]. Only the luminance (Y) component of the video frames has been used to perform the optimization strategy and the transmission and reception over a simulated channel.
First, the partial distortions M _{ i } for each frame in the video sequence have been calculated. To this purpose, JPEG 2000 compression has been performed using Kakadu 6.0 [54], with default parameters, no visual weighting, and the ‘rate’ option on every frame. The portion of each JPEG 2000 codestream located after the startofdata (SOD) marker has been split into multiple pieces, each one with a size of (k−2) bytes (after CRC insertion, the piece will be of k bytes). Then, a new codestream has been constructed using the original header data, with an amended startoftile (SOT) marker to account for the new codestream length, a number i of codestream pieces, and the endofcodestream (EOC) marker. The obtained codestream has been decompressed using Kakadu 6.0, and M _{ i } has been calculated as the distortion of the reconstructed frame. Although this process of determining M _{ i } is cumbersome, it should be said that the JPEG 2000 encoding process is able, per se, to provide such values; during the encoding procedure of JPEG 2000, an accurate ratedistortion estimation of the compressed frame is calculated, since the distortion values are gathered for the selection of the compressed wavelet coefficients with embedded block coding optimized truncation (EBCOT) of the bitstream [55]. In this work, we have favored a direct calculation of the distortion values, in order to achieve more precise results.
When the piece boundaries are not coincident with the codestream interruption points decided by the JPEG 2000 compressor, we adopt a continuum hypothesis, i.e., we assume that the intermediate distortions at the piece boundaries can be calculated using linear interpolation from the nearest known, available distortions. This assumption is generally valid, since JPEG 2000 is a positionprogressive encoder, and distortions are related to the way wavelet coefficients are truncated by EBCOT, in order to best satisfy the qualityrate constraints imposed on the compression process.
Moreover, in order to make a fair comparison among different channel code rates, we have kept fixed the total amount of data sent on the channel, i.e., the combined source and channel code rate R.
The transmission of the compressed stream has been simulated, using MATLAB, on three different types of channels. The first type is a Qary symmetric channel (Q=256), characterized by symbol error rates P _{ S } ranging from 10^{−3} to 10^{−1}. In this type of channel, errors occur at a symbollevel; since the bit errors are equiprobable and uniformly distributed over the symbol bits (log2Q=8 bits/symbol), there is a simple relationship between bit and symbol error rates when the number of bits per symbol is large, i.e., P _{ b }≅P _{ S }/2 ([56] Section 4.41).
where Q(·) is the Gaussian tail function, and the Rayleigh channel BER is calculated for the maximum uncorrelated Doppler spread [56].
The UEP profile has been generated using (3), given the distortion profile M _{ i } and the channel parameter s. Then, the codestream has been split into pieces that have been protected according to the determined UEP profile, using RS coding with \(\tilde {N} = 24\) mother code rates \(\left \lbrace \tilde {r}_{l} \right \rbrace = \left \lbrace 32/36, 32/38, 32/40, \ldots, 32/80 \right \rbrace \), and adopting the codeword allocation strategy given by (7). The effect of the channel is simulated by randomly changing the transmitted bytes according to the simulation symbol error rate P _{ S }. The erroraffected codestream has been recomposed by terminating it at the last errorfree received piece (thanks to the CRC codeword). In this way, any image reconstruction artifact due to wrong/erased codestream bytes has been eliminated, and the reconstructed image MSE is that used by the UEP allocation strategy. The JPEG 2000 header (about 300 bytes) has been considered as transmitted on a reliable channel, since it represents the most critical section of the codestream. At the receiving side, the JPEG 2000 header has been prepended to the JPEG 2000 bitstream bytes, and only the portion of the header carrying information on the bitstream size (Psot field of the SOT marker) has been changed accordingly. Performance has been evaluated as objective visual quality, and YPSNR has been used as objective quality indicator. In addition, we used MSSIM to faithfully represent the subjective evaluation by a human observer. The overall performance has been calculated by averaging the PSNR and MSSIM values calculated at each frame of the video sequence. The performance of the UEP method has been directly compared with that of an EEP method.
Additionally, comparisons with existing techniques in literature have been done using a static reference image, the 512 × 512 pixel grayscale version of lena, compressed at a total bit rate (joint source and channel code rate) of R=0.5 bits/pixel. For each simulated channel error rate, at least 100 independent transmissions of the image have been repeated, and the results averaged. Since both the video sequence and the static image cases cover a standard definition application scenario, we have also used a static image frame from the highdefinition 1920 × 1080 pixels crowdrun RGB sequence [53], in order to show some properties of the calculated UEP profiles.
4.2 Simulated performance results
4.2.1 Performance for static images on BSC
We first report the performance obtained with static images. Both images (lena and crowdrun) were compressed to a total rate of R=2.5 bits/pixel, comprising both the source and the channel code bits.
Clearly, (1−δ _{ α,i }) is the normalized amount of codewords with rate \(\tilde {r}_{w_{i} + 1}\).
4.2.2 Performance for video sequence on BSC
The PSNR and MSSIM histories Γ _{ i } and ι _{ i } plotted in Fig. 7 are obtained on a simulated channel with an error rate of P _{ S }=7×10^{−2}, equivalent to a bit error rate of P _{ b }=3.5×10^{−2}. The red line depicts the quality of the received UEP frames after decompression, with additional artifacts due to the errors introduced by the loss of pieces during transmission on the channel. For comparison purposes, we have also reported the quality of an EEP profile (blue line), having the same total rate R. The average performance of the UEP method results in a PSNR of \(\bar {\Gamma }=25.3\) dB, while the PSNR of the EEP method is of \(\bar {\Gamma }=19.6\) dB. Similarly, we have an MSSIM of \(\bar {\iota }=0.84\) and \(\bar {\iota }=0.66\), for the UEP and EEP methods, respectively.
4.2.3 Performance on AWGN and Rayleigh channels
4.3 Computational complexity
The optimization problem requires the knowledge of the distortion profile of the image. Using JPEG 2000 compression, the partial distortions M _{ i } (and the CCD m _{ i }) can be easily obtained during the rate allocation step of the JPEG 2000 bitstream preparation [55]; thus, these values can be obtained virtually at no cost.
For the calculation of the C, s, and d coefficients, a lookup table (LUT) can be used to store the parameters, for different values of the packet size N _{ k }, of the channel bit/symbol error rate P _{ b }/ P _{ S }, and possibly even for different channel coding algorithms (e.g., convolutional, binary RS, lowdensity parity check codes). The LUT can then be accessed to provide the parameters that will be used in the protection profile generation, with a large saving with respect to storing the entire UEP profile, for each combination of the three variables. As for generating the coefficients stored in the LUT, they can be calculated offline and smoothly interpolated to provide all the intermediate values that could be requested by the system.
in which case it takes N _{ k } logarithms, (N _{ k }−1) additions, 1 division, and 1 exponentiation to be computed. Then, we need (N _{ k }−1) additions for the computation of the CCD function, N _{ k } multiplications for the logarithm operand (one division to obtain the inverse of the geometric mean), N _{ k } logarithm operations, N _{ k } multiplications for logarithm result scaling (one division to obtain the inverse of s, if not already saved in this form in the LUT), and N _{ k } additions. In summary, to implement (3), a total of (3N _{ k }−2) additions, 2N _{ k } multiplications, 2N _{ k } logarithms, 3 divisions, 1 exponentiation is needed. Assuming that natural logarithms and powers of e can be implemented by means of another LUT, with a sufficient precision once the dynamic ranges of the operands have been characterized, the asymptotic complexity becomes \(\mathcal {O}\left (N_{k}\right)\). Differently, the solutions presented in [16] or in [58] require multiple evaluations of expressions similar to (2), which are more cumbersome to calculate from the computational viewpoint.
5 Conclusions
The transmission of video over errorprone wireless channels is a problem that can be solved by using an adequate error protection layer added to the streams, once the characteristics of the channel are known. In this work, we have presented a UEP strategy devised to allocate channel code bits, using an optimization algorithm that is computationally light during the search of the UEP profile. The general formulation of the problem has been solved using a Lagrangian minimization method. The discovered closedform UEP expression requires already available data, such as the image ratedistortion curve, the average error protection code rate, the typical allowed packet size for transmission, and the channel error model (represented by one parameter). In addition, we have also presented a practical method for implementing the discovered UEP profile using RS codes. The simulated performance of the proposed UEP strategy shows that the results outperform those obtained using an EEP strategy, that they are comparable with the UEP performance results of other methods presented in similar works, yet having a lower computational complexity, and that this UEP method can be used to effectively counteract the impairments introduced by an errorprone transmission channel.
6 Appendix A
6.1 A.1 Loglinear approximation of residual BER curves
Fitting parameters C, s, and d found for the curves in Fig. 13
k  

512  1024  1600  
P _{ S }  C  s  d  C  s  d  C  s  d 
1E3  0.0085  0.1893  544  0.0180  0.0783  1088  0.0282  0.0461  1700 
3E3  0.0000  0.0984  623  0.0009  0.0512  1186  0.0074  0.0314  1806 
5E3  0.0000  0.0886  660  0.0008  0.0453  1228  0.2666  0.0345  1747 
1E2  0.0000  0.0801  684  0.0008  0.0364  1298  0.0761  0.0255  1854 
3E2  0.0002  0.0451  772  0.0026  0.0248  1451  0.0237  0.0163  2148 
5E2  0.0003  0.0404  849  0.0027  0.0178  1628  0.1258  0.0127  2281 
1E1  0.0011  0.0174  1119  0.0246  0.0077  1996  0.2450  0.0054  2707 
In Fig. 13, solid lines represent the results of simulations, whereas the dashed lines are obtained by evaluating (1) with the bestfit model parameters of Table 3, for every channel symbol error rate.

The amplitude C is generally much lower than 1.

The decay s increases as the channel/channel code performance improves.

The offset d is the point where the error rate curve becomes linear, and is higher than 10^{−2}.

d is greater than the values of k that we have used.
For other types of transmission methods, such as, for instance, BPSK on AWGN or Rayleigh fading channels, if a bit/byte interleaver interspersing consecutive errors is present, then the results and comments discussed above are still valid.
6.2 A.2 Proof of Lemma 1
Proof.
since products of h(n _{ l }) terms can be neglected. Supposing that all h(n _{ l }) have the same value, it can be found that when h(n _{ l })<2×10^{−2}, the approximation (14) is valid with an error lower than 10 %.
6.3 A.3 Proof of Lemma 3
Proof.
0≤i<N _{ k }, where the lower bound expresses the differential entropy H of the actual source, and the upper bound is calculated considering the hypothesis of Gaussian source (with encoded image coefficients that are Gaussian distributed with variance σ ^{2}), respectively.
Declarations
Acknowledgements
The authors thank Paolo Micanti and Barbara Villarini for the help on theoretical aspects and simulations carried out for this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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