Research On Inverse Problem Based On Sparse Representation

Inverse problems mainly include linear compressed sensing problems and nonlinear phase retrieval problems.Compressed sensing breaks through the limitations of the traditional Nyquist sampling theorem,and achieves the purpose of compression while sampling the signal,reducing the cost of signal acquisition and storage.The multiple measurement vector algorithm can effectively reconstruct the original signal,and the algorithm has a lower complexity.Recovering only the original signal from the signal’s amplitude or intensity information is called phase retrieval.In the optimization algorithm of phase retrieval,the gradient descent method is usually used to iteratively update the initial estimate,but this type of algorithm converges slowly and is easy to fall into the local optimal solution.The stochastic gradient method has a fast convergence speed,and is combined with a better search direction to find the global optimal solution.Therefore,it is of great significance to design three effective algorithms for solving inverse problems.The specific contents of this study are as follows:(1)The multiple reweight linear program(MRLP)algorithm is designed.First,multiple measurement vectors are transformed the single measurement vector equation.Second,the equations are solved by the linear program.Then,an effective weight update strategy is selected to reconstruct the original signal with a high probability.The MRLP algorithm is compared with the classical multiple measurement vector algorithm.The recovery performance of the algorithm under the correlation parameters and observation numbers between different source signals are studied.Compared with other algorithms,MRLP algorithm can recover the original signal with a high probability,and its failure rate is low,and it has better recovery performance.(2)The phase retrieval algorithm based on the Polak-Ribiere-Polyak(PRP)stochastic gradient descent wirtinger flow(PSGD-WF)algorithm is studied.First,a mathematical model for solving the phase retrieval problem is established,and the convergence speed of the algorithm is accelerated by using the stochastic gradient descent method.At the same time,the PRP search direction is used as the optimal descent direction in the process of stochastic gradient descent,and the global optimal solution can be obtained.Under different test images,the PSGD-WF algorithm is compared with different phase recovery algorithms.Experimental results show that the PSGD-WF algorithm has a high peak signal to noise ratio(PSNR)value,can reconstruct the original image with high quality,and has certain robustness to Gaussian white noise.(3)The sparse wirtinger flow(SWF)phase retrieval algorithm is explored.The SWF algorithm uses the PRP search direction as the optimal descent direction in the process of gradient descent.The SWF algorithm is compared with Th WF algorithm 、 Alt Min Sparse algorithm and SPARTA algorithm,and the results show that the success rate of SWF algorithm is higher than that of Th WF algorithm and Alt Min Sparse algorithm,but lower than that of SPARTA algorithm,and the SWF algorithm for sparse signal processing is worth further studying.