Theoretical Research On Compressed Sensing And Phase Retrieval Via Non-convex Optimization Model

Compressed Sensing(Compressed Sensing,CS)is a new sampling theory proposed by Candes et al.By exploiting the sparsity of signals,the signal can be reconstructed by acquiring the discrete samples of signals by means of random sampling.In recent years,the theory and application of compressed sensing in signal processing,image processing and statistics have attracted the attention of many researchers.The core theory of compressed sensing mainly includes two points.One is the sparse structure of the signal,that is,the signal can be represented with few digital codes under the condition of little information loss.The other is irrelevance.In the process of theoretical proof,it is found that the sampling method of compressed sensing is only related to the signal and a set of determined waveforms,and these waveforms are not related to the sparse space where the signal is located.Based on the framework of compressed sensing,this thesis makes use of the prior knowledge of signals and the non-convex optimization algorithm to complete the research on the sparse signal recovery and the phase recovery of the sparse signal under the framework.The specific research contents of this thesis are as follows:(1)To solve the sparse signal recovery problem,the weighted non-convex optimization model is used to obtain the tk-order RIP conditions for the stable recovery of sparse signals.(2)When the partial support information is known,the sparse signal can be recovered stably under the condition of weak full recovery by using the non-uniform weight weighted lp(0<p?1)minimization model.(3)To solve the phase recovery of sparse signals under the framework,lp(0<p?1)minimization model is used to obtain that if the measurement matrix satisfies the strong restricted isometry property adapted to D(S-DRIP)condition,the model can stably recover the sparse signal under the framework.