Restricted Isometry Property Based Signal Reconstruction

Compressive sensing?CS?breaks through the limitation of Nyquist-Shannon sampling theorem and becomes a new type of sampling theory which guides information acquisition.CS combines the sampling process with the compression process,which effectively reduces the amount of data needed to signal reconstruction and the storage space.Generally speak-ing,CS has three important layers:?1?Sparse domain of the signal,?2?Selecting sensing matrix,?3?Designing reconstruction algorithm.In this paper,we are mainly studying recon-struction algorithms.The reconstruction algorithms mainly include optimization algorithm,greedy algorithm,iterative thresholding algorithm.The l₁-minimization is a convex op-timization algorithm with wide applications and relatively mature research.Partial hard thresholding?PHT?pursuit is a greedy algorithm.Partial hard thresholding pursuit needs less time than l₁-minimization.In Chapter 1,we introduce the research background and research development of CS,the notations and definitions about CS.In Chapter 2,we consider the stable recovery of sparse or approximated sparse signals x?Rn from highly corrupted linear measurements b = Ax+f +e,where f ? Rm is a sparse error vector whose nonzero entries may be arbitrarily large and e ? Rm is a stochastic noise.We propose an extended Dantzig selector model which considers sparsity of both x and f.We establish sufficient conditions under the generalized restricted isometry property,which guarantee the stable signal recovery from extended Dantzig selector model and extended Lasso model respectively.In Chapter 3,we study partial hard thresholding pursuit method.We use a new al-gorithm which we called normalised partial hard thresholding pursuit algorithm.The new algorithm is insensitive to matrix scaling and is guaranteed to be stable.