Research On Convex Optimization Algorithm Based On Compressed Sensing

With the rapid development of information technology, the amounts of processing data increase quickly. The traditional signal processing mode has not been able to adapt to this situation. The signal processing ability experience great challenges. Compressed sensing theory is proposed. This theory can reduce the amount of signal processing data effectively by extract the concision information from the redundant raw signals. There are several research fields in the compressed sensing area, such as signal sparse representation, selection and design of observation matrix and signal reconstruction. The key point of the compressed sensing is to make signal reconstruction accurately with small amount of observed values. In this dissertation, we make a research on several convex optimization algorithms with compressed sensing theory to process the signal reconstruction.In the first place, we make a detailed introduction on the basic theory about compressed sensing. The compression process and reconstruction process is described particularly. It is essential for the basic theory description to describe the sparse decomposition process and the characteristic of observation matrix. In addition, we analyze the specialty and difference between the convex relaxation method and greedy algorithm.In the next place, we make an analysis on the difference between the l1minimization in signal reconstruction and the l0and l2minimization in traditional way. In order to make farther research, we introduce the basic theory of the sparse gradient projection algorithm and Interior-Point method. Then we make a conclusion about the characteristics of these methods theoretically. In the next, the simulation experiments are made to test the performance of these methods. The experimental results show that their convergence rates relay on the algorithm scale, the sparsity level of signal and the noise level.The last but not the least, in allusion to the phenomenon which the strategy stated above cannot converge stably. We proposed the gradient projection with quasi-Newton method for sparse reconstruction and the better stopping criterion for convex optimization algorithms. It makes full use of the correction mechanism and superlinear convergence of quasi-Newton method. The approximation of the objective function decreases the computational complexity. The correction mechanism makes up the error on search direction so as to find the optimal solution in the possible gradient directions. The stopping criterion can makes a good balance between the accuracy of algorithm and the CPU time. The experimental results show that gradient projection with quasi-Newton method cannot only deal with the compressed sensing signal reconstruction with low reconstruction error, but also can significantly decrease CPU time and improve the stability and the convergence of the algorithm.