Sparse Representation Of Signals Via Wavelet Modulus Maxima And Compressed Sensing Reconstruction

With the rapid advancement of information and technology, the traditional Shannon/Nyquist sampling theorem cannot deal with the problems of the storage, transmission, and processing with the rapid increase of the data, which requires more powerful and high-speed signal processing theory and algorithms, and also needs to enhance the signal processing ability of hardware. In recent years, Donoho, Candes and Tao, who is an Australian Chinese mathematician, discovered a new theory of information acquisition, whose name is Compressed Sensing (CS). The essence of CS is that the algorithm can use the random linear projection of compressible signal to reconstruct the signal exactly.Sparse representation and recovery algorithm is the core problems in CS, therefore, this paper focuses on the two problems. The main topics can be summarized as follows:1. We introduced the basic framework of CS, and gave a detailed analysis about the problems of sparse signal representation, measurement matrices design, and signal reconstruction algorithm. We also expounded preliminary application of CS, and laid the theoretical foundation for further research.2. To solve the problem of sparse signal representation, this dissertation proposed a new algorithm for sparse signal representation and signal reconstruction based on the modulus maxima search. Firstly, the proposed method found the modulus maxima of wavelet coefficients in each layers on the basis of wavelet transformation, then significantly improved the sparsity of the signal by optimizing the coefficients. Secondly, the measurement matrix could be applied to the sparse coefficients to obtain the measurement values. Entropy encoding of the measurement values is used for data compression and transmission. For the decoding step, orthogonal matching pursuit algorithm was utilized to recover the modulus maxima of every level. Lastly, we used the alternating projection algorithm to reconstruct the original signal. Compared with the traditional compressed sensing algorithms, the quality of the signal reconstruction in the proposed algorithm has improved greatly.3. For two-dimension signal reconstruction problems, aimed to the wavelet sub-tree searching of the Tree-based backtracking orthogonal matching pursuit algorithm (TBOMP), this dissertation proposed the method of bottom-up wavelet sub-tree searching. According to the characteristics of wavelet tree structure, our method combined with greedy tree approximation and made further improvements to make the search process more efficient, simple. Then we used the backtracking to optimize the result. Lastly, our algorithm was applied to two-dimensional image reconstruction.