Lossy Compression via Sparse Linear Regression: Computationally Efficient Encoding and Decoding

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources under the squared-error distortion criterion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off as a function of the block length n is the following. With computational resource (space or time) per source sample of O((n/\log n)^2), for a fixed distortion-level above the Gaussian distortion-rate function, the probability of excess distortion decays exponentially in n. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has good empirical performance, especially at low and moderate rates.Comment: 14 pages, to appear in IEEE Transactions on Information Theor

Quantization as Histogram Segmentation: Optimal Scalar Quantizer Design in Network Systems

An algorithm for scalar quantizer design on discrete-alphabet sources is proposed. The proposed algorithm can be used to design fixed-rate and entropy-constrained conventional scalar quantizers, multiresolution scalar quantizers, multiple description scalar quantizers, and Wyner–Ziv scalar quantizers. The algorithm guarantees globally optimal solutions for conventional fixed-rate scalar quantizers and entropy-constrained scalar quantizers. For the other coding scenarios, the algorithm yields the best code among all codes that meet a given convexity constraint. In all cases, the algorithm run-time is polynomial in the size of the source alphabet. The algorithm derivation arises from a demonstration of the connection between scalar quantization, histogram segmentation, and the shortest path problem in a certain directed acyclic graph

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Hypergraph-based Source Codes for Function Computation Under Maximal Distortion

This work investigates functional source coding problems with maximal distortion, motivated by approximate function computation in many modern applications. The maximal distortion treats imprecise reconstruction of a function value as good as perfect computation if it deviates less than a tolerance level, while treating reconstruction that differs by more than that level as a failure. Using a geometric understanding of the maximal distortion, we propose a hypergraph-based source coding scheme for function computation that is constructive in the sense that it gives an explicit procedure for defining auxiliary random variables. Moreover, we find that the hypergraph-based coding scheme achieves the optimal rate-distortion function in the setting of coding for computing with side information and the Berger-Tung sum-rate inner bound in the setting of distributed source coding for computing. It also achieves the El Gamal-Cover inner bound for multiple description coding for computing and is optimal for successive refinement and cascade multiple description problems for computing. Lastly, the benefit of complexity reduction of finding a forward test channel is shown for a class of Markov sources

Improving the Rate-Distortion Performance in Distributed Video Coding

Distributed video coding is a coding paradigm, which allows encoding of video frames at a complexity that is substantially lower than that in conventional video coding schemes. This feature makes it suitable for some emerging applications such as wireless surveillance video and mobile camera phones. In distributed video coding, a subset of frames in the video sequence, known as the key frames, are encoded using a conventional intra-frame encoder, such as H264/AVC in the intra mode, and then transmitted to the decoder. The remaining frames, known as the Wyner-Ziv frames, are encoded based on the Wyner-Ziv principle by using the channel codes, such as LDPC codes. In the transform-domain distributed video coding, each Wyner-Ziv frame undergoes a 4x4 block DCT transform and the resulting DCT coefficients are grouped into DCT bands. The bitplaines corresponding to each DCT band are encoded by a channel encoder, for example an LDPCA encoder, one after another. The resulting error-correcting bits are retained in a buffer at the encoder and transmitted incrementally as needed by the decoder. At the decoder, the key frames are first decoded. The decoded key frames are then used to generate a side information frame as an initial estimate of the corresponding Wyner-Ziv frame, usually by employing an interpolation method. The difference between the DCT band in the side information frame and the corresponding one in the Wyner-Ziv frame, referred to as the correlation noise, is often modeled by Laplacian distribution. A soft-input information for each bit in the bitplane is obtained using this correlation noise model and the corresponding DCT band of the side information frame. The channel decoder then uses this soft-input information along with some error-correcting bits sent by the encoder to decode the bitplanes of each DCT band in each of the Wyner-Ziv frames. Hence, an accurate estimation of the correlation noise model parameter(s) and generation of high-quality side information are required for reliable soft-input information for the bitplanes in the decoder, which in turn leads to a more efficient decoding. Consequently, less error-correcting bits need to be transmitted from the encoder to the decoder to decode the bitplanes, leading to a better compression efficiency and rate-distortion performance. The correlation noise is not stationary and its statistics vary within each Wyner-Ziv frame and within its corresponding DCT bands. Hence, it is difficult to find an accurate model for the correlation noise and estimate its parameters precisely at the decoder. Moreover, in existing schemes the parameters of the correlation noise for each DCT band are estimated before the decoder starts to decode the bitplanes of that DCT band and they are not modified and kept unchanged during decoding process of the bitplanes. Another problem of concern is that, since side information frame is generated in the decoder using the temporal interpolation between the previously decoded frames, the quality of the side information frames is generally poor when the motions between the frames are non-linear. Hence, generating a high-quality side information is a challenging problem. This thesis is concerned with the study of accurate estimation of correlation noise model parameters and increasing in the quality of the side information from the standpoint of improving the rate-distortion performance in distributed video coding. A new scheme is proposed for the estimation of the correlation noise parameters wherein the decoder decodes simultaneously all the bitplanes of a DCT band in a Wyner-Ziv frame and then refines the parameters of the correlation noise model of the band in an iterative manner. This process is carried out on an augmented factor graph using a new recursive message passing algorithm, with the side information generated and kept unchanged during the decoding of the Wyner-Ziv frame. Extensive simulations are carried out showing that the proposed decoder leads to an improved rate-distortion performance in comparison to the original DISCOVER codec and in another DVC codec employing side information frame refinement, particularly for video sequences with high motion content. In the second part of this work, a new algorithm for the generation of the side information is proposed to refine the initial side information frame using the additional information obtained after decoding the previous DCT bands of a Wyner-Ziv frame. The simulations are carried out demonstrating that the proposed algorithm provides a performance superior to that of schemes employing the other side information refinement mechanisms. Finally, it is shown that incorporating the proposed algorithm for refining the side information into the decoder proposed in the first part of the thesis leads to a further improvement in the rate-distortion performance of the DVC codec