Research On Observation Matrix In Compression Perception

Traditional signal sampling compression process is based on the classic Nyquist sampling theorem, under this theoretical frame-work, since the amount of sampling data increases, the cost of sampling is so high, and there is even a waste of resources and other issues. In recent years, compressed sensing (CS) theory arisen in the signal processing field has caught extensive attention of scholars. CS theory indicates that:As long as a signal is sparse or compressible, we can sample this signal at a rate much lower than the Nyquist rate, and recover this signal accurately. This theory has broken through the limitations of the Nyquist sampling theorem, has a great scientific theoretical significance, and provides a new way for the signal acquisition and sensor designing, has a very broad application prospect and great industrial value, so it is worthy of further study.Since construction of measurement matrices is the core issue of CS theory, and it plays a decisive role in the application of CS theory, how to construct a proper measurement matrix is a very important research direction. At present, although scholars have achieved some valuable results in this direction, existing CS models are not perfect in theory, there is a series of problems demanding prompt solutions. This thesis focuses on the key issues of measurement matrices in CS theory.This thesis begins with introducing the application background and current research of CS theory, and focuses on analyzing the problems in measurement matrices, points out the value of studying measurement matrices. We describe the main contents of CS theory and related key theories, including some basic mathematical concepts, provide a theoretical basis for the research in the subsequent chapters. Based on analyzing existing theories and methods of measurement matrices, the research of this thesis focuses on four aspects as follows:First, based on CS theory, we study sparse random matrices with fixed column sparsity or fixed row sparsity and general sparse random matrices respectively. When these sparse random matrices satisfy the restricted isometry property (RIP), we deduce the lower bound conditions that the number of measurements should satisfy; we propose the definition of the sparse ratio of sparse random matrices, when the first two sparse random matrices satisfy the RIP, we deduce the upper and lower bounds conditions that the sparse ratio should satisfy; and we analyze the performance of three matrices.Second, since the performance of binary sparse measurement matrices is bad when the signal is not so sparse in binary signal recovery, we propose to construct a non-binary sparse measurement matrix. The novel measurement matrix enables us to design a suboptimal and effective recovery algorithm by fully exploiting the structural features. Moreover, we analyze and estimate the un-recovery probability based on the tree structure to evaluate the recovery performance.Third, we study Toeplitz matrices which are close related to channel estimation applications, optimize the structure of existing Toeplitz matrices, reduce the required number of random quantities in generating matrices, and prove that optimized Toeplitz matrices still satisfy the RIP; we propose the exact conditions of constructing Toeplitz matrices to be used as measurement matrices.Fourth, since existing measurement matrices suffer from high computational burden, based on the structure of circulant matrices (Toeplitz matrices are a special case), combining the advantages of sparse measurement matrices, we propose to construct a sparse block circulant matrix to reduce the computational burden. The RIP of the proposed sparse block circulant matrix is also guaranteed with overwhelming probability.Finally, this thesis summarizes the research contents. Furthermore, to make CS more efficient, this thesis points out the direction and problems of further study.