Research On Coding Matrix And Recovery Algorithm Of Compressed Sensing

Compressed sensing can greatly reduce the number of measurements required to restore a sparse signal.This theory breaks through the limitations of the Shannon-Nyquist sampling theorem,brings new changes to the theory of signal sampling,and has been widely used in many fields,such as image compression,signal recovery,nuclear magnetic resonance imaging.Under the impetus of a group of mathematicians,coding experts and signal processing experts,compressed sensing has become a hot research topic.In general the study of compressed sensing can be divided into two directions:signal recovery algorithm and coding matrix?also known as the perceptual matrix or the measurement matrix?.Orthogonal Matching Pursuit?OMP?algorithm is a classic greedy algorithm among the known recovery algorithms of sparse signals.Many sufficient conditions in terms of the well-known restricted isometry property were reported in the literature to ensure OMP algorithm recover sparse signals or their support.One objective of this thesis is to develop a new such condition.Specifically,we prove that,under the context of l? noise,if the restricted isometry property of a coding matrix satisfies ?_K+1<1/?K+1?^1/2,then OMP algorithm can successfully recovers support of any K-sparse signal ? provided that min_i?supp???|?i|is no less than some certain constant.Our condition is better than the known ones in the literature in the sense that it means that more sparse signals can be recovered by OMP.Another objective of this thesis is to derive new sufficient condition for OMP recovering the sparse signal under Gauss noise which is a familiar noise in signal processing field.More precisely,we show that,under Gauss noise,OMP algorithm can restore the original sparse signal with high probability as long as the coding matrix and sparse signal satisfy some conditions.We also obtain such a probability.Another core problem of compressed sensing is the design of the coding matrix.The coding matrices can be generally divided into two categories:deterministic coding matrix and random ones.The deterministic coding matrix has several advantages over random ones including small storage space,convenient hardware implementation.Most importantly,the deterministic coding matrix can reconstruct sparse signals at one hundred percent probability.Therefore,how to construct a good deterministic coding matrix is very important in compressed sensing.The third objective of this thesis is to construct a series of deterministic coding matrix from pseudo-random sequences with asymptotically optimal correlation.The goodness of the correlation of pseudo random sequence ensures the low coherence of the generated coding matrices.Compared with the Gauss random matrix,Bernoulli matrix,discrete Fourier matrix and the coding matrices based on BCH codes,the numerical results show that our coding matrices obtained in this thesis have better performance to restore the sparse signal.