Research On Construction Method Of Compressed Sensing Measurement Matrix

At present,the new generation of information technology represented by cloud computing,Internet of Things,mobile Internet,big data and artificial intelligence is booming.Information technology is becoming the core force to promote the global industrial revolution.The rapid development of information technology puts forward new demands for signal sampling technology.Compressed sensing is a novel signal sampling theory which can realize signal sampling and compression at the same time.Once the novel signal sampling theory was proposed,it has challenged the theoretical limitation of the Nyquist-Shannon sampling theorem and plays an important and far-reaching role in the field of information and signal processing.Compressed sensing is regarded as a highly competitive alternative to traditional signal processing operations.The measurement matrix is the decisive factor affecting the reconstruction ability of compressed sensing.Therefore,the construction of measurement matrices is the key problem of compressed sensing theory.In practical application,measurement matrices with good reconstruction performance can reconstruct the signal with higher sparsity in the case of less sampling,while less sampling means lower sampling,storage,transmission,and processing costs.This is the direction of signal sampling technology in the future.The research on the construction of compressed sensing measurement matrices with excellent reconstruction performance,low sampling cost,and easy hardware implementation has great significance to accelerate the application of compressed sensing in the new information field.This thesis focuses on the construction of compressed sensing measurement matrices,and the main research results and innovations are as follows:(1)Aiming at the difficulty in calculating the Spark of measurement matrices,an effective algorithm to calculate the upper bound of the Spark of sparse binary measurement matrices is proposed.For a measurement matrix,the basic idea of the proposed algorithm is that reducing the calculation complexity of the Spark of the measurement matrix by removing a sub-matrix with full column rank from the given measurement matrix to obtain a smaller sub-matrix without full column rank.Taking full advantage of the sparse structure of the binary measurement matrix,a searching method for the full column rank sub-matrix of the binary measurement matrix is proposed.By repeatedly removing the full column rank submatrices of the binary measurement matrix,a new binary matrix with the smallest column dimensions and without full column rank is obtained,and an exhaustive method is used to calculate the Spark of the new small binary matrix.The results of numerical experiments demonstrate that the proposed algorithm can give a smaller upper bound of the Spark of binary measurement matrices.Particularly,comparing with the known lower bound,the Spark of some binary measurement matrices with large dimensions can be accurately calculated by using the proposed algorithm.(2)Aiming at the problem of low applicability of the deterministic binary measurement matrix,a clipping-embedding operation is proposed for binary matrices to generate measurement matrices with more sizes.First,two types of deterministic binary measurement matrices are generated from the incidence matrices of the subspaces over unitary geometry.The coherence and lower bound of the Spark of these unitary geometry deterministic binary measurement matrices are analyzed by using the properties of unitary geometry.Since the unitary group over finite fields is a subgroup of the general linear group,the unitary space over finite fields is a division of the projective space.Therefore,the unitary geometric measurement matrices can be regarded as the sub-matrix of the projective geometric measurement matrices.The construction method of unitary geometry measurement matrices provides a new idea for the research on the construction of the sub-matrices of deterministic measurement matrices.Then,a clipping operation is proposed for deterministic binary measurement matrices which can change the dimensions of binary matrices under the premise of coherence unchanged or increased slowly.The embedding operation that only applies to binary matrices with fixed column weight is generalized to binary matrices with multiple column weights.Based on the clipping operation and the modified embedding operation,a clipping-embedding operation is proposed for binary matrices to generate measurement matrices with more sizes,which can strongly extend the applicability of the deterministic binary matrices in practice.Simulation results show that unitary geometry measurement matrixes and measurement matrices from clipping-embedding operation both have good reconstruction performance for one-dimensional sparse signals and two-dimensional image signals.(3)Aiming at the requirement for the diversity of measurement matrix dimensions under the practical application background,based on the embedding operation and modified progressive edge-growth algorithm,a construction method is proposed to realize the flexible construction of sparse measurement matrices with high applicability and low coherence.First,a novel block sampling model based on the embedding operation is proposed.In the sampling procedure,the proposed model saves a lot of storage space by storing two smaller seed matrices.Meanwhile,the block sampling mechanism makes the proposed model have a lower sampling complexity than Kronecker compressed sensing.Then,in order to construct binary matrices with large column dimension and low coherence,a modified progressive edge growth algorithm is proposed by removing some limited conditions of the traditional progressive edge growth algorithm.Finally,based on the modified progressive edge growth algorithm and the embedding operation,a flexible method is proposed to construct sparse measurement matrices with low coherence and low storage requirement.Simulation results demonstrate that our measurement matrices give better performance than several typical measurement matrices when their storage requirements are in the same level.(4)Aiming at the problem of one-sided objective of existing measurement matrix optimization algorithms,a novel optimization scheme,called progressive coherence minimization scheme,based on Nesterov acceleration gradient is proposed.By analyzing the properties of coherence and spectral norm of the measurement matrices,a progressive optimization scheme of the measurement matrix is proposed which consists of several orthogonal constrained l?-minimization problems.The l?-minimization problem with orthogonal constraints is a non-convex and non-smooth optimization problem.It is relaxed into a smoothed convex problem by penalty function and smoothing technique.Nesterov acceleration gradient is used to solve the smoothed convex approximate optimization problem under the premise of guaranteed convergence.Simulation results demonstrate that the proposed scheme has certain superiority compared with some previous optimization methods,especially in decreasing the spectral norm,and the optimized measurement matrices from the proposed scheme have better sparse signal reconstruction performance.