The Research On Algorithms Of Distributed Compressed Sensing And Their Applications

Compressed sensing (CS) has received an increased interest recently for high demand for fast, efficient algorithms and in-expensive devices in signal processing. Compared with traditional Nyquist paradigm, the compressive sensing framework, relying on obtaining sparse solutions to underdetermined linear systems, can recover the signals from far fewer measurements than is possible employing Nyquist sampling rate. The issue of limited number of samples can be found in multiple scenarios, e.g. when the number of data capturing devices are limited, measurements are very expensive or slow to acquire such as in radiology and imaging techniques via neutron scattering. In such scenarios, a promising solution is given by CS. CS utilizes sparsity of signals in certain transform domain and the incoherency of the measurements with the original domain. Essentially, the sampling and compression are combined into one step by measuring minimum samples which contain maximum information of the signal. This process removes the requirement to obtain and store large number of worthless samples. CS has been applied in various fields such as image processing and gathering geophysics data. These applications are feasible for the inherent sparsity of many real world signals like sound, image, video etc. Recently, the major applications occur in communication and network domain.Unfortunately, CS only considers recovery of single signal. In case of multi-signals, CS only exploits the intra-signal correlations, without taking the correlations of the multi-signals into account. The correlations are helpful to promote the precision and efficiency of recovery. That means CS has broad prospects for development. For exploiting the correlations of the multi-signals, the theory of distributed compressed sensing (DCS) arises in recent years. DCS is seen as the combination of distributed source coding (DSC) and CS. DCS compresses multi-signals independently but recoveries them jointly. The basic framework of DCS contains the concepts of sparse representation, measurement and joint reconstruction. In this framework, if the multi-signals can be represented sparsely by a basis, each of them can be observed and encoded by using another incoherent basis and acquire the measurements which are far less than the signal length. The small fewer measurements are transmitted to the decoder. Under some right conditions, collected data can be joint reconstructed precisely at central decoder.During encoding, DCS compresses each signal independently. By taking advantage of the inter-and intra-signal correlations, DCS can decrease a large number of measurements, especially when the common part of the signals dominates. On the other hand, during decoding, DCS transfers the complexity from encoder to joint decoder rather than reducing the complexity of the whole process. This is very appropriate for some distributed applications that need low computational complexity at the encoder, such as wireless sensor networks and video coding. Since DCS is proposed, it is widely used in fields such as video coding, image fusion, multi-input multi-output (MIMO) channel estimation, object recognition etc.In diverse distributed scenarios, there are various intra- and inter-signal correlations that are presented by jointly sparse models (JSM). Compared with CS, algorithms for DCS are related to the mode of correlation. That means for different jointly sparse models, the DCS algorithms are different. The core of JSM is describing sparse or compressible signal as the sum of two parts: sparse common part and sparse innovation part. The common part means the indexes of the nonzero components are supported on the same set. The innovation part is the remaining of the signal except the common part. In existing literatures, the major JSM are:(1) common signal model, (2) common support-set model, (3) mixed signal model and (4) mixed support-set model, where mixed support-set model can be seen as a generalization over the others. In this paper, the proposed DCS algorithms are related to mixed support-set model.The major researches of this paper are listed below:1. Underdetermined blind source separation based on DCSBlind source separation (BSS) is the process that separates the source signals from observations without priori information about the sources or the model. It is called underdetermined blind source separation (UBSS) if the mixing matrix is underdetermined. Generally, UBSS is composed of two stages:estimating the mixing matrix and separating the sources. DCS exploits the intra- and inter-signal correlations to reconstruct signals jointly from few measurements. The similarities between UBSS and DCS are:(1) Both of them apply linear transformation to the original signals (the former is linear mixing, the latter is linear measuring).(2) Either the mixing matrix or the measurement matrix is underdetermined.(3) Sparsity is required for both of them.Based on the comparison between UBSS model and DCS model, UBSS model is decomposed and restructured in this paper. After being decomposed, the source matrix is transformed to a signal ensemble, where each column is seen as a "signal" to be measured. Meanwhile, the mixing matrix is decomposed and transformed to be seen as a measurement matrix. That means the UBSS model has been transformed to a DCS model, where an existing DCS algorithm is used to recover the signal ensemble. A series of simulations demonstrate the performance of the proposed method. Relations between the performance and sparsity of the sources are investigated in the simulations. Additionally, simulations demonstrate the complexity of the proposed method is lower than that of traditional UBSS methods.2. Underdetermined blind source separation based on stagewise DCSFor the source matrix of UBSS model,i-th row represents the samples of i-th source and the number of rows is the number of source. On the other hand, for the signal ensemble of DCS model,i-th column represents the measurements of i-th signal while the number of columns is the number of signals. It is proposed in this paper that the UBSS model can be seen as a DCS model if the source matrix is sparse enough and the number of sources is large. That means each column of the source matrix is seen as a " signal" and the mixing matrix is seen as the measurement matrix. Then a DCS algorithm is used to recover the "signal ensemble". In order to ensure the reconstruction precision, the "signal ensemble" is recovered stage by stage according to the order of sampling. In this paper, it is proved theoretically that the complexity of the propose method is lower than that of other UBSS methods. A series of simulations investigates the relations between the performance of the proposed method and sparsity of the sources, the number of the sources and the number of the stages, respectively.3. A DCS algorithm based on threshold and tailoringLike CS, the core of DCS is to find the support sets of the measured signals. In this paper an DCS algorithm based on greedy pursuit is proposed to find the support set iteratively. At each iteration some coordinates are selected via a threshold, where the threshold is determined by the correlations between the columns of the measurement matrix and the residual. The selected coordinates are incorporated into the estimate of support set obtained at last iteration. Then the estimate of support set is checked by a tailoring mechanism. Simulations demonstrate that performance of the proposed algorithm is more robust against the number of measurements, signals and sparsity, compared with other existing DCS algorithms. In noisy case, the proposed algorithm can also recover signals with acceptable precision. Additionally, the feasibility of the recovery of the proposed algorithm is analyzed theoretically in this paper.4. Generalized orthogonal matching pursuit for DCSOrthogonal matching pursuit (OMP) is a classic algorithm in CS. OMP inherits the principle of selecting atoms of matching pursuit. Moreover, OMP ensures optimality via orthogonalizing the selected atoms iteratively so that the number of iterations is reduced. Based on OMP, generalized orthogonal matching pursuit (GOMP) is proposed recently. Based on GOMP, generalized orthogonal matching pursuit for DCS (DCS-GOMP) is proposed in this paper to jointly recover signals that conform to mixed support-set model. For promoting the recovery efficiency, more than one coordinates are selected and added into the estimate of support set at each iteration where the number of coordinates is determined by the correlations between the columns of the measurement matrix and the residual. Recovery conditions based on restricted isometry property (RIP) of DCS-GOMP are proposed in this paper. Base on these conditions, feasibility of recovery strategy of DCS-GOMP is proved. After obtaining the estimate of support set, reconstruction is acquired via a least squares solution. In this paper, a series of simulations demonstrate the relations between performance of DCS-GOMP and the number of measurements and sparstiy of original signals, compared with other existing DCS algorithms.5. Applications of the proposed DCS algorithms for recovery of speech signals and imagesIn this paper, algorithm based on threshold and tailoring and DCS-GOMP are applied to jointly recover speech signals and image signals. Taking into account that speech signals are sparse in frequency domain, both of the proposed DCS algorithms in this paper are used to recover three high-dimensional speech signals from low-dimensional measurements. Simulations demonstrate the recovery errors are acceptable. Taking into account that image signals are approximately sparse in wavelet domain, the proposed DCS algorithms in this paper are used to jointly recover the areas of test images. Note that each area can be represented by a few wavelet coefficients. By jointly recovering coefficients of the areas, the proposed algorithms can recover the whole image well.The major innovations of this paper are:1. Existing CS-based UBSS methods mainly exploit the relations between CS and UBSS models. By transforming UBSS model to CS model, sources are separated by CS algorithms. The problem of these methods is, all sources are concatenated to be a " signal" whose length is equal to the sum of lengths of all sources. This increases recovery complexity significantly. The proposed method in this paper transforms UBSS model to DCS model but not concatenating the sources. By comparing the performed time with other algorithms for estimating the sources, the proposed method in this paper is less computational demanding.In another case, if the sources are sparse enough and the number of sources is large, traditional UBSS algorithms are computational demanding. In this case, the UBSS model is seen as a DCS model without transforming. Then the sources are recovered by DCS algorithms stage by stage. This procedure is less computational demanding, compared with other algorithms for estimating the sources.2. In this paper a DCS algorithm based on threshold and tailoring is proposed. By backtracking mechanism, the coordinates selected at current iteration are incorporated into the estimate of support set. Then the estimate is checked to determine the indexes that should be removed. Different with other algorithms based on backtracking, tailoring mechanism is used to remove incorrect coordinates.Additionally, after a few iterations, remaining correct coordinates that should be found are few. In this case, selecting coordinates with fixed number at following iterations is computational demanding or even incorrect. In this paper, the number of selected coordinates at each iteration is controlled by a threshold that determined by the correlation between columns of the measurement matrix and the residual. The flexible threshold is conducive to recovering exactly.3. Different with many existing DCS algorithms, neither of the DCS algorithms proposed in this paper requires sparsity as the priori knowledge for recovery. Furthermore, the proposed algorithms require few measurements for exact recovery and are robust against the changed sparsity. When applied in recovery of speech signals and images, the errors are acceptable. These demonstrate the application value of the research in this paper.