Research On The Sparse Representation And Measurement Matrix In Compressed Sensing

In 2006,the theory of compressed sensing has been proposed by Donho,et al.The compressed sensing has been proposed as a solution to relieve the pressure.The theory shows that if the signal is sparse or compressible in the transform domain,it could be projected from high dimensional to low dimensional in the transform domain with measurement matrix that nearly has no coherence with the sparse basis.The original signal can reconstructed accurately with high probability by solving optimization problem.The signal sparse is one of the main factor,which is the primary condition of the compressed sensing.Measurement matrix is the key point to compressed sensing because it guarantees that measurements in low dimensional include enough information of the original signal.In this paper,we research on sparse representation and the optimization of measurement matrix,the main work is as follows:1.This paper introduces the mathematical model and conditions of the theory of compressed sensing,discusses the theory of sparse representation and the principle of measurement matrix.The commonly used complete dictionaries and observation matrices have been introduced.2.An optimized objective function of the dictionary learning algorithm base on mask matrix has been introduced.The principles and steps of the compressed sampling matching pursuit algorithm and the coefficient reuse orthogonal matching pursuit algorithm are introduced in detail.In the process of solving the sparse coefficient,we optimize the operation memory by combining the optimization method of the mask matrix.The dictionary learning algorithm of K-SVD is analyzed.In this paper,an improved dictionary learning algorithm is proposed by using the singular value decomposition and the prior information and the construction matrix of the last iterative process.The simulation experiment proves the effectiveness of the algorithm.3.The Gram matrix algorithm and the gradient-based projection algorithm by reducing the mutual coherence of observation matrix and transform bases are studied.According to the relation between the minimum singular value of matrix and the independence of matrix vector,three matrix transformation algorithms QR decomposition,SVD decomposition and row vector orthogonal are introduced.Simulation experiment shows that the matrix transformation algorithm can optimize the minimum singular value of matrix.In the process of solving the measurement matrix,the adaptive step is used to replace the fixed step.The method of reducing the minimum singular value of the measurement matrix is reduced by the matrix transformation algorithm,and the independence of the column vector is optimized.Experiments demonstrate the improved matrix optimization algorithm based on matrix transformation can effectively improve the reconstruction effect of the measurement matrix.4.Sparse random observation matrix and sparse binary random observation matrix are introduced.Toeplitz measurement matrix and rotation measurement matrix are studied.By introducing the construction of deterministic binary block diagonal matrix,a construction method of the measurement matrix of sparse deterministic measurement matrix is proposed and the validity of the construction method is verified through the coherence,the time complexity and the space complexity.Experiments show that the construction method of measurement matrix effectively improves the quality of reconstructed images.