On Optimization Of Sensing Matrices For Compressed Sensing Systems

Compressed sensing is a new theory for signal processing and sampling, which has at-tracted extensive attention. It takes full advantage of sparsity (or compressibility) of signal, and converts the traditional signal sampling into a random measurement process. Compressed sensing combines the compressing and sampling together. At last, the original signal can be reconstructed by a reconstruction algorithm in the receiving end. This method can save the storage and transmission space compared to the Nyquist sampling theorem, and at the same time the sampling frequency can be reduced. In the frame of compressed sensing theory, the choice of measurement matrix is the key problem, which not only is closely related with the sampling and compressing effect, but also can influence the reconstruction result.This thesis introduces the basic theoretical framework of CS and analyzes the existing measurement optimal algorithm, then proposes a new algorithm. The main research contents and contributions are given as follows:1. The framework of CS theory is systematically presented in terms of the sparse represen-tation of signals, measurement matrices, signal reconstruction algorithms. Analyze the key-points of measurement matrices, such as the principles for measurement matrices design, and the common matrices.2. Two methods for optimizing the measurement matrices are analyzed, which are pro-posed by Michael Elad and Duarte-Carvajalino separately. The former represents the iterative algorithm while the latter is non-iterative. Compare their performance and re-construction result through simulation.3. A new design criterion for optimal measurement matrices is proposed, which is defined in terms of three popularly used target Grams, including the identity-based Gram, the frame-based Gram and dictionary-based Gram. What’s more, according to the matrix decomposition theory, an algorithm is derived for optimizing the sensing matrix based on the newly proposed criterion. 4. Based on the existing research on gradient descent methods, a new design measure of optimal measurement matrices is proposed, which is defined as the sum of the weighted l1-norm-based coherence factors. A gradient-based algorithm is derived for solving this problem. Extensive experiments have been carried out and the simulation results have demonstratedthe feasibility and effectiveness of the new proposed methods.