Solution Recovery Of Two Kinds Of Nonconvex Sparse Optimization Models Based On Null Space Properties

When the traditional Nyquist signal sampling technology is difficult to meet the needs of signal processing,the compressed sensing theory emerges.Based on the sparsity of signal,it can realize sampling and compression of signal at the same time,which the efficiency of signal transmission and processing are improved greatly.Traditional compressed sensing requires that the signals are sparse in the sense of orthogonal basis.In fact,a lot of signals in reality are not sparse in this sense,but sparse in the sense of dictionary.Therefore,people extended the traditional compressed sensing model to dictionary-based analysis model.The dictionary-based l1-synthesis model is a relaxation of analysis model and it can recover signal under a highly coherent frame.Since the nonconvex model can not only approximate the original l0problem,but also the number of measurements can be reduced without changing the recovery performance compared with the l1-synthesis model.Based on this,the thesis extends the l1-synthesis model to the nonconvex lq case in the dissertation.At the same time,the nonconvex data separation model in the frame sense is established for the data separation problem.Based on the null space property,the stability of two kinds of model solutions is discussed.The main content is divided into the following three aspects:1.In chapter 1,introduces briefly the background and significance of this dissertation,the important theories mainly involved in the research,such as NSP,RIP,nonconvex model,coherence,framework,data separation and so on.The research progress of compressed sensing based on dictionary signal recovery and data separation model is described and the problems to be solved in this dissertation are presented.2.In chapter 2,a more extensive l_q-synthesis model is established,the conditions of signal recovery without measurement noise and the stable recovery conditions of sparse signal with measurement noise are given.The strong null space property of the lq-synthesis model(q-D-SNSP)in the frame sense is proposed and proves it is the condition for the stable signal recovery of the lq-synthesis model under noisy measurement.Finally,discusses and proves that the stability of the nonconvex synthesis model is equivalent to the dictionary based null space property.3.In the final chapter,discusses mainly the separation of compressible data in the sense of dictionary.Aiming at the practical problem of noise in the process of a actual signal measurement,we combine signal separation problem with coherent dictionary and general noise,establish a dictionary-based data separation model.Meanwhile,this chapter also proposes a modified q-(lp,?)-RNSP that can accurately recover approximate sparse signals under different frame representations,and discusses the stability of model restoration by means of this property.