A Perturbation Analysis On Data Separation And Block-sparse Compressed Sensing For Redundant Dictionaries

In recent years, with the advent of age of big data, the technologies of data analysis and data processing have attracted more attention. Based on sparse modeling, an efficient technique to deal with signal data also gets more people’s attention, particularly compressed sensing.So far, researches from various fields such as information theory, image processing, statistics have focused on the different practical aspects of compressed sensing. For different structure forms of signal data, using the existing compressed sensing to analyze signal data has certain limitation. Therefore, it is of great significance to make further study on compressed sensing.In this paper, based on compressed sensing, we further study compressed data separation and block-sparse compressed sensing with redundant dictionary. The main contribution of this thesis is as follows:Chapter one briefly introduces the research background and research significance of compressed sensing. We also analyze research history and present situation of compressed sensing from the aspects of practical application at home and on abroad, and further introduce the present situation of compressed data separation and block-sparse compressed sensing of redundant dictionary. Finally, we give the main works and the full text structure of this thesis.Chapter two mainly presents three basic theories of compressed sensing that are the sparse representation of signals, the design of measurement matrix, and the design of reconstruction algorithms and their relative redundant dictionaries. In this chapter, we also analyze the reconstruction theories and the research achievement of scholars in recent years.Chapter three fist makes a comprehensive introduction to a perturbation analysis on compressed data separation, and then obtains the reconstruction condition and error bound estimation for ?1and ?q(0 < q≤1) minimization. The theoretical result on error bound estimation indicates that the perturbation matrix, the best k-term approximation and the q parameter are the leading influence factors for error control. In this chapter, the theory of nonconvex block-sparse compressed sensing with redundant dictionary is also investigated. Our theoretical results show that under the condition,√22 ≤δ2k|τ< 1, nonconvex?2/?qminimization method with redundant dictionary can stably reconstruct block-sparse signal data. Besides,we also obtain the robust reconstruction condition and error estimation when the block number is no more than four times of the block sparsity of the signal data, this is d≤4k. These studies provide a further theoretical support for compressed sensing.Chapter four presents some simulation experiments and related analysis for the discrete cosine transform and wavelet transform, respectively. Under the additive noise and perturbation, the experiments verify that nonconvex ?q(0 < q≤1) minimization method has strong robustness and stability.Chapter five, the major findings and implications of the study on this thesis will be made clear. In addition, some tentative suggestions will be provided for further study.