Research On Key Issues Of Compressed Sensing Measurement Matrix And Reconstruction Algorithms Under Impulsive Noise Environments

On the premise of sparsity of signals, compressed sensing (CS) theory can achieve directinformation sampling to replace the traditional Nyquist theorem, which provides a new way forsampling and compression of signals. For sparse signals with known sparsifying basis, two keyfactors influencing how CS theory is employed to solve practical problems are the construction ofmeasurement matrices and the design of reconstruction algorithms in the presence of noise. Withinthe signal processing system based on CS theory, it is necessary to construct an appropriatemeasurement matrix, especially constructing a special measurement matrix for specific signals, toachieve efficient compression. And then, for the actual application, it is important for us to designreconstruction algorithms in the presence of noise and guarantee the roubustness of algorithms tonoise. The main work and innovation of the dissertation can be generalized as follows.(1) The application of CS system involves robust technologies in different noise environment.However, only a small amount of literatures are dedicated to the CS reconstruction when themeasurements are corrupted by noise at present. And the bounded noise and the additive Gaussianwhite noise are considerd in the traditional framework of reconstruction algorithms in CS theory.And the additive Gaussian white noise is also a kind of bounded noise in the sense of probability.Moreover, performance of the traditional reconstruction algorithms is proportional to the energy ofnoise. There is another common noise called impulsive noise at the real environment. However,the impulsive noise is distinct from the two kinds of noise and the energy of the impulsive noise isalmost unbounded. Thereofore, the traditional reconstruction algorithms cannot achieve effectiverecovery of sparse signals in the presence of impulsive noise. In view of the important issue in theapplication, the performace of support recovery and the reconstruction accuracy of signals for thesubspace pursuit (SP) algorithm are analyzed in the presence of impulsive noise. And the fact thatthe maximum correlation estimate and the least squares estimate in the SP algorithm are not robustto impulsive noise is found. Therefore, the SP algorithm cannot achieve effective reconstructionwhen the measurement vector is corrupted by the impulsive noise. Thus, a new algorithm calledmixed-norm subspace pursuit (MSP) algorithm is proposed and the reconstruction performance ofthe MSP algorithm is proved in theory. This algorithm can take advantage of two kinds ofresiduals to restrict the influence of impulsive noise on reconstruction to guarantee the robustnessof the MSP algorithm to the impulsive noise. (2) Lorentzian iterative hard thresholding (LIHT) algorithm is an efficient algorithm for sparsereconstruction of CS under the impulsive noise environment. However, the study found that theLIHT algorithm is greatly sensitive to the number of impulses and the reconstruction performancedegrades sharply with the increase of impulses. In this case, the lorentzian hard thresholding pursuit(LHTP) algorithm is firstly proposed. In this algorithm, the support of the signal vector is firstlyestimated and then the minimum lorentzian norm optimization problem with respect to this supportis solved. The convergence and reconstruction performance of the LHTP algorithm are proved intheory. Simulation results show that the LHTP algorithm can effectively improve the situation thatthe LIHT algorithm is greatly sensitive to the impulsive noise and the required number ofmeasurement samples is less than that of the LIHT algorithm with the same requirement of thereconstruction accuracy, which means that the compression rate can be higher. In the following, themodified lorentzian iterative hard thresholding (MLIHT) algorithm is proposed. In this algorithm,the1norm is employed as the metric for the selection of measurements that are not corrupted byimpulses and the Barzilai-Borwein method is used to set the stepsize. Simulation results show thatthe MLIHT algorithm is not sensitive to the number of impulses and the required number ofmeasurement samples is also less than that of the LIHT algorithm with the same requirement of thereconstruction accuracy.(3) A new framework based on the Bayesian theory is proposed to achieve effectivereconstruction in the presence of impulsive noise. At first, the Bayesian impulsive noise sparsereconstruction (BINSR) algorithm is proposed for reconstruction of Gaussian sparse signals in thepresence of impulsive noise. In this algorithm, the support of the signal and the measurement setcan be estimated directly from the measurement vector and then the minimum mean square error(MMSE) estimate is employed to estimate the signal vector. On this basis, an adaptive BINSRalgorithm, which is named ABINSR algorithm, is proposed so that the algorithm is no longerdependent on the statistical parameters of the signal and noise. However, the two algorithms aredesigned for Gaussian sparse signals so that neither of them can be used for general signals.Therefore, the Bayesian sparse reconstruction (BSR) algorithm is propsed, which is composed oftwo algorithms including the impulsive noise fast relevance vector machine (INFRVM) algorithmand the bayesian impulse detection (BID) algorithm. Moreover, in the BID algorithm, it is notnecessary for us to discard the measurement samples that are corrupted by impulses so that the adverse effect caused by misoperation can be avoided. Simulation results show that the BSRmethod can achieve effective reconstruction under the impulsive noise environment.(4) The construction of measurement matrix for speech signals is studied in the sixth chapter ofthis dissertation. When the measurements are corrupted by the quantization noise and the impulsivenoise simultaneously, the performance of the BSR algorithm is analyzed. Moreover, thequantization effect of CS on speech signals is primarily analyzed for the independence property ofthe two kinds of noise. And the fact that compared with the fixed quantization, the adaptivequantization can effectively restrict the influence of quantization noise on reconstruction. Moreover,two kinds of matrices are constructed for speech signals including the two block diagonal (TBD)matrix and the approximate truncated circulant autocorrelated matrix and the restricted isometryproperty (RIP) of the two matrices is proved in theory. Furthermore, the TBD matrix, employed asthe measurement matrix, can further restrict the influence of quantization noise on reconstruction.And in the mixed noise environment, the performance of the TBD matrix is also superior to theGaussian random matrix. Moreover, it is proved from the experimental results that the approximatetruncated circulant autocorrelated matrix can achieve far better performance than the Gaussianrandom matrix when the measurements are quantified. It is taken granted that the two measurementmatrices can achieve better compression than general measurement matrices for speech signalswhen the measurement vector is noiseless.