Research On The Key Issues Of Compressed Sensing Theory

In the age of big data, the demands for transmission, processing and storage of the huge amount of data resources become more and more important for various complex applications. Digital signal processing technology which is based on Nyquist sampling theorem has faced more potential challenges in many applications such as high-definition images, video, medical imaging, remote sensing image and genomic data. These applications require high Nyquist sampling frequency, however, hardware facilities with high-speed sampling capacity are difficult to make and the costs of sampling are expected to soar. Those are main obstacles for further development of the above applications. The emergence of compressed sensing (CS) theory has broken this deadlock.CS offers a framework for simultaneous sensing and compression of finite dimensional vectors, that relies on linear dimensionality reduction. Quite surprisingly, it predicts that sparse high-dimensional signals can be recovered from highly incomplete measurements by using efficient algorithms. Compared with the traditional digital sampling methods, the signal processing methods based on CS can get the same amount of information with less times of sampling. Compressed sensing is referred to as a random undersampling technique which breaks through the constraints on sampling frequency in Nyquist sampling theory. Compressed sensing technique brings great convenience for magnanimity data storage, transmission and processing. This basic discovery has led to a fundamentally new approach to signal processing, image recovery, and compression algorithms, to name a few areas that have benefited from CS. There are three core issues in CS which are signal sparse representation, measurement matrix design and reconstruction algorithm construction. To apply CS theory in the practical, it is needed to do further research on the three key issues in CS. The focal points of this dissertation includes mainly:(1) A discriminative sparse representation method based on CS.Signals can be sparsely represented is the premise of signals with sensing and compression. Only after the signals are sparsely decomposed or sparsely represented, signals can be compressed and reconstructed. So signals sparse representation is the precondition of CS research. In this paper, we study the fundamental theory of sparse representation and sparse decomposition method, and propose a discriminative sparse representation method based on CS. This method takes the category labels of target signals as signals that are needed sparsely represented, and through repeatedly selection of the sparse representation coefficient to obtain a discriminative sparse representation method with robustness. This method is applied to the pedestrian detection, throught the the screening ability of discriminative sparse representation method, the optimal redundant feature dictionary is obtained. That is the reliable theoretical basis for the feature selection problems in pedestrian detection. Simulations demonstrate that pedestrian detection based on CS framework is feasible and has good performance.(2) An Incoherence Rotated Chaotic measurement matrix based on CS.The design of the measurement matrix plays a very important role in CS. Whether the original signal can be compressed and accurately reconstructed by the receiving end or not is largely depended on the performance of measurement matrix. In this dissertation, we study the condition of the design of the measurement matrix, analyze the structure and performance of commonly used matrices and present a novel simple and efficient measurement matrix named Incoherence Rotated Chaotic (IRC) matrix for high precision reconstruction. We take advantage of the well pseudorandom of chaotic sequence, introduce the concept of rotation, adjust incoherence between measurement matrix and the compressible signal by the incoherence factor, and adopt QR decomposition to obtain the IRC measurement matrix which is suited for sparse reconstruction. The IRC measurement matrix solves two problems which are the limitation of hardware implementation of random measurement matrix and the lack of complete theoretical system of the deterministic measurement matrix, and reveals the unity of determinacy and randomness exactly.(3) An adaptive matching pursuit reconstruction algorithm based on sparsity estimation.The research of reconstruction algorithm has an important significance in CS. CS theory transfer the technology pressure of information acquisition from the sampling end to the receiving end. In order to obtain the high quality reconstruction signal at the receiving end, a more efficient and robust reconstruction algorithm is required. In this dissertation, we propose an adaptive matching pursuit reconstruction algorithm based on sparsity estimation. In this algorithm, the optimal atoms with linear combination are selected after signal sparsity estimation, and the step-size of atom selection can be adaptively adjusted based on the residual error. We optimize the atom selection criterion and obtain more effective atom selection strategy.In conclusion, CS theory which breaks through the constraints on sampling frequency in Nyquist sampling theory for coping with the bottleneck of signal processing technology in the existing large amount of data. The three core issues of CS, sparse representation, measurement matrix design and greed reconstruction algorithm, are further researched in this dissertation. A discriminative sparse representation method with high efficiency and robustness, an IRC measurement matrix with determinate randomness, a reconstruction algorithm based on sparsity estimation with high efficiency and high precision are introduced into CS theory. Based on the above research, we improve signal sparse representation ability, break limitation of hardware implementation and reconstruction accuracy in the existing measurement matrix, and realize efficient and high precision signal reconstruction.