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Pliable Fraction Repetition Codes for Access-balancing in Distributed Storage
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作者 LI Yueting 《数学进展》 北大核心 2025年第1期73-84,共12页
Fraction repetition(FR)codes are integral in distributed storage systems(DSS)with exact repair-by-transfer,while pliable fraction repetition codes are vital for DSSs in which both the per-node storage and repetition d... Fraction repetition(FR)codes are integral in distributed storage systems(DSS)with exact repair-by-transfer,while pliable fraction repetition codes are vital for DSSs in which both the per-node storage and repetition degree can easily be adjusted simultaneously.This paper introduces a new type of pliable FR codes,called absolute balanced pliable FR(ABPFR)codes,in which the access balancing in DSS is considered.Additionally,the equivalence between pliable FR codes and resolvable transversal packings in combinatorial design theory is presented.Then constructions of pliable FR codes and ABPFR codes based on resolvable transversal packings are presented. 展开更多
关键词 distributed storage system pliable fraction repetition code access balancing resolvable transversal packing
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Coded Distributed Computing for System with Stragglers
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作者 Xu Jiasheng Kang Huquan +5 位作者 Zhang Haonan Fu Luoyi Long Fei Cao Xinde Wang Xinbing Zhou Chenghu 《China Communications》 2025年第8期298-313,共16页
Distributed computing is an important topic in the field of wireless communications and networking,and its high efficiency in handling large amounts of data is particularly noteworthy.Although distributed computing be... Distributed computing is an important topic in the field of wireless communications and networking,and its high efficiency in handling large amounts of data is particularly noteworthy.Although distributed computing benefits from its ability of processing data in parallel,the communication burden between different servers is incurred,thereby the computation process is detained.Recent researches have applied coding in distributed computing to reduce the communication burden,where repetitive computation is utilized to enable multicast opportunities so that the same coded information can be reused across different servers.To handle the computation tasks in practical heterogeneous systems,we propose a novel coding scheme to effectively mitigate the "straggling effect" in distributed computing.We assume that there are two types of servers in the system and the only difference between them is their computational capabilities,the servers with lower computational capabilities are called stragglers.Given any ratio of fast servers to slow servers and any gap of computational capabilities between them,we achieve approximately the same computation time for both fast and slow servers by assigning different amounts of computation tasks to them,thus reducing the overall computation time.Furthermore,we investigate the informationtheoretic lower bound of the inter-communication load and show that the lower bound is within a constant multiplicative gap to the upper bound achieved by our scheme.Various simulations also validate the effectiveness of the proposed scheme. 展开更多
关键词 coded computation communication load distributed computing straggling effect
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Hybrid DF and SIR Forwarding Strategy in Conventional and Distributed Alamouti Space-Time Coded Cooperative Networks
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作者 Slim Chaoui Omar Alruwaili +1 位作者 Faeiz Alserhani Haifa Harrouch 《Computer Modeling in Engineering & Sciences》 2025年第2期1933-1954,共22页
In this paper,we propose a hybrid decode-and-forward and soft information relaying(HDFSIR)strategy to mitigate error propagation in coded cooperative communications.In the HDFSIR approach,the relay operates in decode-... In this paper,we propose a hybrid decode-and-forward and soft information relaying(HDFSIR)strategy to mitigate error propagation in coded cooperative communications.In the HDFSIR approach,the relay operates in decode-and-forward(DF)mode when it successfully decodes the received message;otherwise,it switches to soft information relaying(SIR)mode.The benefits of the DF and SIR forwarding strategies are combined to achieve better performance than deploying the DF or SIR strategy alone.Closed-form expressions for the outage probability and symbol error rate(SER)are derived for coded cooperative communication with HDFSIR and energy-harvesting relays.Additionally,we introduce a novel normalized log-likelihood-ratio based soft estimation symbol(NL-SES)mapping technique,which enhances soft symbol accuracy for higher-order modulation,and propose a model characterizing the relationship between the estimated complex soft symbol and the actual high-order modulated symbol.Further-more,the hybrid DF-SIR strategy is extended to a distributed Alamouti space-time-coded cooperative network.To evaluate the~performance of the proposed HDFSIR strategy,we implement extensive Monte Carlo simulations under varying channel conditions.Results demonstrate significant improvements with the hybrid technique outperforming individual DF and SIR strategies in both conventional and distributed Alamouti space-time coded cooperative networks.Moreover,at a SER of 10^(-3),the proposed NL-SES mapping demonstrated a 3.5 dB performance gain over the conventional averaging one,highlighting its superior accuracy in estimating soft symbols for quadrature phase-shift keying modulation. 展开更多
关键词 Cooperative communication soft information relaying soft symbols modeling cooperative diversity gain distributed Alamouti space-time code
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Low complexity DCT-based distributed source coding with Gray code for hyperspectral images 被引量:1
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作者 Rongke Liu Jianrong Wang Xuzhou Pan 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2010年第6期927-933,共7页
To compress hyperspectral images, a low complexity discrete cosine transform (DCT)-based distributed source coding (DSC) scheme with Gray code is proposed. Unlike most of the existing DSC schemes, which utilize tr... To compress hyperspectral images, a low complexity discrete cosine transform (DCT)-based distributed source coding (DSC) scheme with Gray code is proposed. Unlike most of the existing DSC schemes, which utilize transform in spatial domain, the proposed algorithm applies transform in spectral domain. Set-partitioning-based approach is applied to reorganize DCT coefficients into waveletlike tree structure and extract the sign, refinement, and significance bitplanes. The extracted refinement bits are Gray encoded. Because of the dependency along the line dimension of hyperspectral images, low density paritycheck-(LDPC)-based Slepian-Wolf coder is adopted to implement the DSC strategy. Experimental results on airborne visible/infrared imaging spectrometer (AVIRIS) dataset show that the proposed paradigm achieves up to 6 dB improvement over DSC-based coders which apply transform in spatial domain, with significantly reduced computational complexity and memory storage. 展开更多
关键词 image compression hyperspectral images distributed source coding (DSC) discrete cosine transform (DCT) Gray code band-interleaved-by-pixel (BIP).
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Recurrent neural network decoding of rotated surface codes based on distributed strategy
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作者 李帆 李熬庆 +1 位作者 甘启迪 马鸿洋 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第4期322-330,共9页
Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error corre... Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models.In this paper,we use a distributed decoding strategy,which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases.Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder.The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy.Then we test the decoding performance of our distributed strategy decoder,recurrent neural network decoder,and the classic minimum weight perfect matching(MWPM)decoder for rotated surface codes with different code distances under the circuit noise model,the thresholds of these three decoders are about 0.0052,0.0051,and 0.0049,respectively.Our results demonstrate that the distributed strategy decoder outperforms the other two decoders,achieving approximately a 5%improvement in decoding efficiency compared to the MWPM decoder and approximately a 2%improvement compared to the recurrent neural network decoder. 展开更多
关键词 quantum error correction rotated surface code recurrent neural network distributed strategy
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Distributed Video Coding Using LDPC Codes for Wireless Video
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作者 P. APARNA Sivaprakash REDDY Sumam DAVID 《Wireless Sensor Network》 2009年第4期334-339,共6页
Popular video coding standards like H.264 and MPEG working on the principle of motion-compensated pre-dictive coding demand much of the computational resources at the encoder increasing its complexity. Such bulky enco... Popular video coding standards like H.264 and MPEG working on the principle of motion-compensated pre-dictive coding demand much of the computational resources at the encoder increasing its complexity. Such bulky encoders are not suitable for applications like wireless low power surveillance, multimedia sensor networks, wireless PC cameras, mobile camera phones etc. New video coding scheme based on the principle of distributed source coding is looked upon in this paper. This scheme supports a low complexity encoder, at the same time trying to achieve the rate distortion performance of conventional video codecs. Current im-plementation uses LDPC codes for syndrome coding. 展开更多
关键词 SYNDROME coding Cosets distributed Source coding distributed VIDEO coding (DVC).
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Tree-Structured Parallel Regeneration for Multiple Data Losses in Distributed Storage Systems Based on Erasure Codes 被引量:4
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作者 孙伟东 王意洁 裴晓强 《China Communications》 SCIE CSCD 2013年第4期113-125,共13页
To reduce the time required to complete the regeneration process of erasure codes, we propose a Tree-structured Parallel Regeneration (TPR) scheme for multiple data losses in distributed storage systems. Under the sch... To reduce the time required to complete the regeneration process of erasure codes, we propose a Tree-structured Parallel Regeneration (TPR) scheme for multiple data losses in distributed storage systems. Under the scheme, two algorithms are proposed for the construction of multiple regeneration trees, namely the edge-disjoint algorithm and edge-sharing algorithm. The edge-disjoint algorithm constructs multiple independent trees, and is simple and appropriate for environments where newcomers and their providers are distributed over a large area and have few intersections. The edge-sharing algorithm constructs multiple trees that compete to utilize the bandwidth, and make a better utilization of the bandwidth, although it needs to measure the available band-width and deal with the bandwidth changes; it is therefore difficult to implement in practical systems. The parallel regeneration for multiple data losses of TPR primarily includes two optimizations: firstly, transferring the data through the bandwidth optimized-paths in a pipe-line manner; secondly, executing data regeneration over multiple trees in parallel. To evaluate the proposal, we implement an event-based simulator and make a detailed comparison with some popular regeneration methods. The quantitative comparison results show that the use of TPR employing either the edge-disjoint algorithm or edge-sharing algorithm reduces the regeneration time significantly. 展开更多
关键词 distributed storage system erasure code REPLICATION regeneration tree
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Locally Minimum Storage Regenerating Codes in Distributed Cloud Storage Systems 被引量:2
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作者 Jing Wang Wei Luo +2 位作者 Wei Liang Xiangyang Liu Xiaodai Dong 《China Communications》 SCIE CSCD 2017年第11期82-91,共10页
In distributed cloud storage systems, inevitably there exist multiple node failures at the same time. The existing methods of regenerating codes, including minimum storage regenerating(MSR) codes and minimum bandwidth... In distributed cloud storage systems, inevitably there exist multiple node failures at the same time. The existing methods of regenerating codes, including minimum storage regenerating(MSR) codes and minimum bandwidth regenerating(MBR) codes, are mainly to repair one single or several failed nodes, unable to meet the repair need of distributed cloud storage systems. In this paper, we present locally minimum storage regenerating(LMSR) codes to recover multiple failed nodes at the same time. Specifically, the nodes in distributed cloud storage systems are divided into multiple local groups, and in each local group(4, 2) or(5, 3) MSR codes are constructed. Moreover, the grouping method of storage nodes and the repairing process of failed nodes in local groups are studied. Theoretical analysis shows that LMSR codes can achieve the same storage overhead as MSR codes. Furthermore, we verify by means of simulation that, compared with MSR codes, LMSR codes can reduce the repair bandwidth and disk I/O overhead effectively. 展开更多
关键词 distributed cloud storage systems minimum storage regenerating(MSR) codes locally repairable codes(LRC) repair bandwidth overhead disk I/O overhead
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DESIGN OF EXACT REGENERATING HIERARCHICAL CODE FOR DISTRIBUTED STORAGE SYSTEM 被引量:1
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作者 Hao Jie Lu Yanbo +1 位作者 Liu Xinji Xia Shutao 《Journal of Electronics(China)》 2013年第3期290-298,共9页
Erasure code is widely used as the redundancy scheme in distributed storage system. When a storage node fails, the repair process often requires to transfer a large amount of data. Regenerating code and hierarchical c... Erasure code is widely used as the redundancy scheme in distributed storage system. When a storage node fails, the repair process often requires to transfer a large amount of data. Regenerating code and hierarchical code are two classes of codes proposed to reduce the repair bandwidth cost. Regenerating codes reduce the amount of data transferred by each helping node, while hierarchical codes reduce the number of nodes participating in the repair process. In this paper, we propose a "sub-code nesting framework" to combine them together. The resulting regenerating hierarchical code has low repair degree as hierarchical code and lower repair cost than hierarchical code. Our code can achieve exact regeneration of the failed node, and has the additional property of low updating complexity. 展开更多
关键词 distributed storage Network coding Regenerating code Hierarchical code Repair cost
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Code Design and Latency Analysis of Distributed Matrix Multiplication with Straggling Servers in Fading Channels 被引量:2
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作者 Ning Liu Kuikui Li Meixia Tao 《China Communications》 SCIE CSCD 2021年第10期15-29,共15页
This paper exploits coding to speed up computation offloading in a multi-server mobile edge computing(MEC)network with straggling servers and channel fading.The specific task we consider is to compute the product betw... This paper exploits coding to speed up computation offloading in a multi-server mobile edge computing(MEC)network with straggling servers and channel fading.The specific task we consider is to compute the product between a user-generated input data matrix and a large-scale model matrix that is stored distributively across the multiple edge nodes.The key idea of coding is to introduce computation redundancy to improve robustness against straggling servers and to create communication redundancy to improve reliability against channel fading.We utilize the hybrid design of maximum distance separable(MDS)coding and repetition coding.Based on the hybrid coding scheme,we conduct theoretical analysis on the average task uploading time,average edge computing time,and average output downloading time,respectively and then obtain the end-to-end task execution time.Numerical results demonstrate that when the task uploading phase or the edge computing phase is the performance bottleneck,the hybrid coding reduces to MDS coding;when the downlink transmission is the bottleneck,the hybrid coding reduces to repetition coding.The hybrid coding also outperforms the entangled polynomial coding that causes higher uplink and downlink communication loads. 展开更多
关键词 mobile edge computing distributed matrix multiplication coded computing cooperative transmission
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LDPC based differential unitary space-frequency coding for MIMO-OFDM systems
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作者 姜海宁 罗汉文 +2 位作者 田继锋 宋文涛 刘兴钊 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2005年第6期294-298,共5页
This paper proposes a novel LDPC based differential unitary space-frequency coding (DUSFC) scheme for MIMO-OFDM systems when neither the transmitter nor the receiver has access to the channel state information (CSI). ... This paper proposes a novel LDPC based differential unitary space-frequency coding (DUSFC) scheme for MIMO-OFDM systems when neither the transmitter nor the receiver has access to the channel state information (CSI). The new DUSFC strategy basically consists of coding across transmit antennas and OFDM tones simultaneously as well as differential modulation in the time-domain. It can fully exploit the inherent advantages provided by the multipath fading channels, resulting in a high degree of diversity. The state-of-the-art low-density parity-check (LDPC) codes are concatenated with our DUSFC as channel coding to improve the bit error rate (BER) performance considerably. Owing to the maximum multipath diversity and large coding advantages, LDPC-DUSFC strongly outperforms the differential unitary space-time coded OFDM techniques re- cently proposed in literature. The corresponding iterative decoding algorithm without channel estimation is finally provided to offer significant performance gain. Simulation results illustrate the merits of the proposed scheme. 展开更多
关键词 Differential modulation Unitary space-time codes space-frequency coding MIMO OFDM LDPC
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Improved scheme with cooperative diversity based on distributed space-time block coding
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作者 Du Na Xu Dazhuan 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2009年第3期463-469,共7页
An improved scheme with cooperative diversity based on distributed space-time block coding (WCD- DSTBC) is proposed, which effectively achieves diversity gains and improves the performance of the system by sharing s... An improved scheme with cooperative diversity based on distributed space-time block coding (WCD- DSTBC) is proposed, which effectively achieves diversity gains and improves the performance of the system by sharing some single-antenna users' antennas to form a virtual antenna array and combining with distributed spacetime block coding (DSTBC) mode. Then the relation between the system BER and the interuser BER for WCDDSTBC scheme is theoretically derived and the closed-form expression of BER for WCD-DSTBC system is obtained. The simulation results show that the proposed WCD-DSTBC scheme achieves distinct gains over the non-cooperative multi-carrier CDMA (MC-CDMA) system. When system BER is le-3 and interuser BER is le-3, about 2.5 dB gain can be gotten. When interuser channel state information (CSI) outgoes the users' individual CSI, about 3 dB gain is also achieved. 展开更多
关键词 multiple input multiple output cooperative diversity distributed space-time block coding multicarrier CDMA.
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Trifocal tensor based side information generation for multi-view distributed video code
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作者 Lin Xin Liu Haitao Wei Jianming 《High Technology Letters》 EI CAS 2010年第3期268-273,共6页
Distributed video coding (DVC) is a new video coding approach based on Wyner-Ziv theorem. The novel uplink-friendly DVC, which offers low-complexity, low-power consuming, and low-cost video encoding, has aroused mor... Distributed video coding (DVC) is a new video coding approach based on Wyner-Ziv theorem. The novel uplink-friendly DVC, which offers low-complexity, low-power consuming, and low-cost video encoding, has aroused more and more research interests. In this paper a new method based on multiple view geometry is presented for spatial side information generation of uncalibrated video sensor network. Trifocal tensor encapsulates all the geometric relations among three views that are independent of scene structure; it can be computed from image correspondences alone without requiring knowledge of the motion or calibration. Simulation results show that trifocal tensor-based spatial side information improves the rate-distortion performance over motion compensation based interpolation side information by a maximum gap of around 2dB. Then fusion merges the different side information (temporal and spatial) in order to improve the quality of the final one. Simulation results show that the rate-distortion gains about 0.4 dB. 展开更多
关键词 MULTI-VIEW distributed video coding (DVC) camera sensor networks trifocal tensor side information
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A rate control approach to distributed source coding for interferential multispectral image compression
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作者 宋娟 Li Yunsong Wu Chengke Wang Yangli 《High Technology Letters》 EI CAS 2010年第2期133-137,共5页
Distributed source coding (DSC) is applied to interferential multispectral image compression owing to strong correlation among the image frames. Many DSC systems in the literature use feedback channel (FC) to cont... Distributed source coding (DSC) is applied to interferential multispectral image compression owing to strong correlation among the image frames. Many DSC systems in the literature use feedback channel (FC) to control rate at the decoder, which limits the application of DSC. Upon an analysis of the image data, a rate control approach is proposed to avoid FC. Low-complexity motion compensation is applied first to estimate side information at the encoder. Using a polynomial fitting method, a new mathematical model is then derived to estimate rate based on the correlation between the source and side information. The experimental results show that our estimated rate is a good approximation to the actual rate required by FC while incurring a little bit-rate overhead. Our compression scheme performs comparable with the FC based DSC system and outperforms JPEG2000 significantly. 展开更多
关键词 interferential multispectral image distributed source coding (DSC) rate control motion compensation polynomial fitting
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Log Likelihood Ratio-Based Relaying for Distributed Turbo Codes
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作者 Peiyao Yang Hai Li Shujuan Hou 《Journal of Beijing Institute of Technology》 EI CAS 2018年第2期176-181,共6页
A distributed turbo codes( DTC) scheme with log likelihood ratio( LLR)-based threshold at the relay for a two-hop relay networks is proposed. Different from traditional DTC schemes,the retransmission scheme at the... A distributed turbo codes( DTC) scheme with log likelihood ratio( LLR)-based threshold at the relay for a two-hop relay networks is proposed. Different from traditional DTC schemes,the retransmission scheme at the relay,where imperfect decoding occurs,is considered in the proposed scheme. By employing a LLR-based threshold at the relay in the proposed scheme,the reliability of decoder-LLRs can be measured. As a result,only reliable symbols will be forwarded to the destination and a maximum ratio combiner( MRC) is used to combine signals received from both the source and the relay. In order to obtain the optimal threshold at the relay,an equivalent model of decoderLLRs is investigated,so as to derive the expression of the bit error probability( BEP) of the proposed scheme under binary phase shift keying( BPSK) modulation. Simulation results demonstrate that the proposed scheme can effectively mitigate error propagation at the relay and also outperforms other existing methods. 展开更多
关键词 cooperative communication relaying scheme distributed turbo codes log likelihood ratio(LLR)
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Side information refinement in distributed video coding
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作者 刘鸿彬 Zhao Debin +1 位作者 Ma Siwei Gao Wen 《High Technology Letters》 EI CAS 2012年第4期423-427,共5页
Side information (SI) is one of the key issues in distributed video coding (DVC) and affects the compression performance of DVC largely. This paper proposes an SI refinement algorithm, in which the Wyner-Ziv (WZ... Side information (SI) is one of the key issues in distributed video coding (DVC) and affects the compression performance of DVC largely. This paper proposes an SI refinement algorithm, in which the Wyner-Ziv (WZ) frame is split into two parts based on checkerboard pattern, and the two parts are encoded independently but decoded sequentially. In the decoding process, the part 1 is first decoded with the initial SI and partially decoded part (PDP) 1 is used to improve the motion vectors (MVs) and SI of both parts. At the next stage, the part 2 is decoded with the improved SI and PDP 2 is used to further refine MVs of the part 2. Then, SI of both parts are further refined. Simulation results show that the proposed algorithm can improve the peak signal to noise ratio (PSNR) by up to 1.43 dB when compared with traditional DVC codec. 展开更多
关键词 distributed video coding (DVC) side information (SI) checkerboard pattern
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Design of the Quantization Matrix in the Distributed Compressed Sensing Video Coding*
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作者 Yueyue Dai Xinhua Rui Xuanyu Zhao 《Journal of Computer and Communications》 2016年第5期16-23,共8页
In the frame of compressed sensing distributed video coding, the design of the quantization matrix directly affects the reconstruction quality of the receiving terminal of the video. In this article, we present a new ... In the frame of compressed sensing distributed video coding, the design of the quantization matrix directly affects the reconstruction quality of the receiving terminal of the video. In this article, we present a new design method of the Gaussian quantization matrix adapting to the compressed sensing coding, for that the distribution of the parameters of the image is featured of the characteristic of approximately normal distribution after measured by compressive sensing. By this way, the parameters of a certain quantity of the image frames depending on the video sequences generated by the Gaussian quantization matrix possess certain adaptive capacity. By comparison with the plan of the traditional quantization, the quantization matrix presented in this article would improve the reconstruction quality of the video. 展开更多
关键词 Compressed Sensing distributed Video coding Gaussian Quantization Matrix
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MAP based blocking artifact reduction for DCT domain distributed video coding
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作者 Liu Hongbin Zhao Debin +1 位作者 Ma Siwei Gao Wen 《High Technology Letters》 EI CAS 2012年第1期77-83,共7页
This paper proposes a maximum a posteriori (MAP) based blocking artifact reduction algorithm for discrete cosine transform (DCT) domain distributed video coding, in which the SI and the initial reconstructed Wyner... This paper proposes a maximum a posteriori (MAP) based blocking artifact reduction algorithm for discrete cosine transform (DCT) domain distributed video coding, in which the SI and the initial reconstructed Wyner-Ziv (WZ) frame are utilized to further estimate the original WZ frame. Though the MAP estimate improves quality of the artifact region, it also leads to over-smoothness and decreases quality of the non-artifact region. To overcome this problem, a criterion is presented to discriminate the artifact and the non-artifact region in the initial reconstructed WZ frame, and only the artifact region is updated with the MAP estimate. Simulation results show that the proposed algorithm provides obvious improvement in terms of both objective and subjective evaluations. 展开更多
关键词 distributed video coding(DVC) blocking artifact reduction maximum a posteriori(MAP)
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Bitrate Control of Distributed Source Coding by Frame Prediction Method
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作者 Kin Honn Chiam Mohd Fadzli Mohd Salleh 《通讯和计算机(中英文版)》 2016年第1期36-41,共6页
关键词 信源编码 控制帧 分布式 预测 码率 编码模型 解码器 编码器
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PERIOD DISTRIBUTION OF CYCLIC CODES OVER F_q + uF_q +···+u^(m-1)F_q 被引量:2
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作者 Li Ping Zhong Jiawei Zhu Shixin 《Journal of Electronics(China)》 2014年第6期547-551,共5页
In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period... In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period of cyclic codes over R is given. The period distributions of cyclic codes over R and their dual codes are determined by employing generator polynomial. The counting formulas of the period distributions of cyclic codes over R and their dual codes are obtained. 展开更多
关键词 Cyclic codes Dual codes PERIOD Period distribution
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