Research On Compressive Sensing Signal Reconstruction By Convex Optimization

The signal processing ability of physical systems now faces to the challenge of the rapid development of information technology, the traditional signal processing technology is not suitable for such vast amounts of processing data, thus compressive sensing theory is proposed. Concision information is sensed from the redundant form of signal in compressive sensing theory, which can effectively decrease the amount of processing data of physical systems. Compressive sensing offers a variety of research fields, including signal sparse representation, sensing matrix design, signal reconstruction. Accurately signal reconstruction algoriths play a key role in compressive sensing. Hence, this thesis shows that a few of type algorithms of convex optimization with compressive sensing theory are investigated.Firstly, the basic theory of compressive sensing is reviewed in detail. It is shown shows that sensing matrix should obey restricted isometry property and the reason for signal reconstruction via 1 minimization. Meantime, the simulation results present that traditional way by using 2 minimization to signal reconstruction cannot recovery effectively from compressive sensing signal.Secondly, we study the sparse gradient projection algorithm, iterative shrink-threshold algorithm and homotopy and conclude their basic contents and make a comparison between them. The performance of them is evaluated through simulation. The results show that their convergence rates are not only sensitive to the change in the noise level, but also to the change in the sparsity level or the number of measurements.Finally, iterative weighted gradient projection is proposed which can overcome the shortcomings of them. It makes full use of sparisty of compressive sensing signal and creats the probability vector based on residul to weight the gradient value. The optimize solution is only searched along the gradient directions which are credible. The simulation results show that iterative weighted gradient projection cannot only deal with compressive sensing signal reconstruction and keep low reconstruction error, but also can significantly decrease CPU time and reduce the sensitivity to sparisity and the number of measurements.