Recovery algorithms play a key role in compressive sampling (CS). Most of current CS recovery algorithms are originally designed for one-dimensional (1D) signal, while many practical signals are two-dimensional (2D). By utilizing 2D separable samplin
g, 2D signal recovery problem can be converted into 1D signal recovery problem so that ordinary 1D recovery algorithms, e.g. orthogonal matching pursuit (OMP), can be applied directly. However, even with 2D separable sampling, the memory usage and complexity at the decoder is still high. This paper develops a novel recovery algorithm called 2D-OMP, which is an extension of 1D-OMP. In the 2D-OMP, each atom in the dictionary is a matrix. At each iteration, the decoder projects the sample matrix onto 2D atoms to select the best matched atom, and then renews the weights for all the already selected atoms via the least squares. We show that 2D-OMP is in fact equivalent to 1D-OMP, but it reduces recovery complexity and memory usage significantly. Whats more important, by utilizing the same methodology used in this paper, one can even obtain higher dimensional OMP (say 3D-OMP, etc.) with ease.
Distributed Arithmetic Coding (DAC) proves to be an effective implementation of Slepian-Wolf Coding (SWC), especially for short data blocks. To study the property of DAC codewords, the author has proposed the concept of DAC codeword spectrum. For equ
iprobable binary sources, the problem was formatted as solving a system of functional equations. Then, to calculate DAC codeword spectrum in general cases, three approximation methods have been proposed. In this paper, the author makes use of DAC codeword spectrum as a tool to answer an important question: how many (including proper and wrong) paths will be created during the DAC decoding, if no path is pruned? The author introduces the concept of another kind of DAC codeword spectrum, i.e. time spectrum, while the originally-proposed DAC codeword spectrum is called path spectrum from now on. To measure how fast the number of decoding paths increases, the author introduces the concept of expansion factor which is defined as the ratio of path numbers between two consecutive decoding stages. The author reveals the relation between expansion factor and path/time spectrum, and proves that the number of decoding paths of any DAC codeword increases exponentially as the decoding proceeds. Specifically, when symbols `0 and `1 are mapped onto intervals [0, q) and [(1-q), 1), where 0.5<q<1, the author proves that expansion factor converges to 2q as the decoding proceeds.
Distributed Arithmetic Coding (DAC) is an effective implementation of Slepian-Wolf coding, especially for short data blocks. To research its properties, the concept of DAC codeword distribution along proper and wrong decoding paths has been introduce
d. For DAC codeword distribution of equiprobable binary sources along proper decoding paths, the problem was formatted as solving a system of functional equations. However, up to now, only one closed form was obtained at rate 0.5, while in general cases, to find the closed form of DAC codeword distribution still remains a very difficult task. This paper proposes three kinds of approximation methods for DAC codeword distribution of equiprobable binary sources along proper decoding paths: numeric approximation, polynomial approximation, and Gaussian approximation. Firstly, as a general approach, a numeric method is iterated to find the approximation to DAC codeword distribution. Secondly, at rates lower than 0.5, DAC codeword distribution can be well approximated by a polynomial. Thirdly, at very low rates, a Gaussian function centered at 0.5 is proved to be a good and simple approximation to DAC codeword distribution. A simple way to estimate the variance of Gaussian function is also proposed. Plenty of simulation results are given to verify theoretical analyses.
