Sparse Signal Reconstruction Theory And Algorithm Via Prior Information

With the advent of the information age,data is gradually applied in many field.How to effectively store,collect and transmit,these exponentially increasing data has been a hot attention in academia.As a effective high dimensional data processed and novel theory,Compressed sensing can reconstruct a signal with high probability by utilizing the sparsity and compressibility of signal data.In several years,Compressed sensing has been applied many field such as medical imaging,image processing,model selection.Based on the theory of Compressed sensing,this paper makes a study on the reconstruction theory and algorithm of sparse signal under incorporating prior information.The main contents of this paper are the following:The first chapter briefly introduces the background and significance of the study of Com-pressed sensing and introduces the research progress of Compressed sensing.Then,we sum-marize the main work and the overall structure of the paper.The second chapter mainly introduces the three aspects of Compressed sensing:sparse representation of signal,the design of measurement matrix,reconstruction algorithms.In the third chapter,we study the problem of l₁-l₁or l₁-l₂minimization with prior information to reconstruct sparse signal.According to null space property?NSP?,we give the necessary and sufficient conditions for accurate reconstruction of l₁-l₁minimization.Then,we proposed a strong null space property?SNSP?for l₁-l₂minimization to ensure that the sparse signal can be recovered under incorporating prior information.Finally,the numerical experiments show the l₁-l₁and l₁-l₂minimization with prior information can exactly recover the sparse signal.Furthermore,l₁-l₁minimization with prior information has a greater advantage on success rate and convergence compared other models.In the fourth chapter,we introduce the problem of sparse signal recovery with prior information by iterative reweighted least squares algorithm?IRLS?.We modify the IRLS al-gorithm and establish a theoretical analysis of the IRLS algorithm by incorporating prior information?PI-IRLS algorithm?,including the error estimate and convergence result.Our results show that the error bound depends on the best s+k term approximation and the regularization parameter,and convergence result depends the regularization parameter.A series of numerical experiments are carried out to demonstrate that PI-IRLS algorithm can accurately reconstruct the original signal with partially known support.The fifth chapter summarizes and analyzes of the further valuable research on Compressed sensing and prior information.