Studies On Reconstruction Algorithms Of Compressed Sensing And Applications In Image Reconstruction

Compressed sensing(CS)is a novel sampling theory utilizing the sparsity of sig-nals,implementing sampling and compressing simultaneously,and reconstructing sig-nals from very low sampling rate.In recent years CS has attracted extensive attention and found many applications,including medical imaging,information theory,radar,and pattern recognition.CS studies the reconstruction algorithm of sparse signals which can be seen as solving an underdetermined linear equations.If the coefficient matrix(measurement matrix)satisfies some properties,the signals can be reconstructed by a feasible algorith-m.The reconstruction algorithm is the important part of CS relating to the effectiveness of CS application.This thesis studies a greedy algorithm and a thresholding algorithm based on Laplace quasi-norm of CS theory,a weighted method improving the proper-ty of measurement matrix,and the reconstruction algorithms for magnetic resonance imaging(MRI)and phase retrieval.The main contributions and innovations are listed as follows.1.The least square technique is the important part of greedy algorithms.Com-bining least square technique and quasi-Newton iterative projection(QNIP)algorithm,a new greedy algorithm is proposed.The convergence is analysed,under a certain condition on the restricted isometry property(RIP)of the measurement matrix.The numerical results show the effectiveness of the proposed algorithm.2.Based on studying the properties of Laplace quasi-norm,we derive the thresh-olding point of Laplace regularization model and a quasi-analytic thresholding repre-sentation for the solution,and then a thresholding algorithm for the Laplace regulariza-tion is proposed.The numerical results show that the proposed algorithm has similar computation time and higher reconstruction rate than fp(0?<? 1)thresholding algo-rithms.3.The reconstruction algorithms of CS rely on the small restricted isometry con-stant(RIC)of measurement matrix,while in some imaging systems,the measurement matrices do not have small RIC.Hence,based on singular value decomposition,we pro-pose a weighted approach for measurement matrix,and prove that in some situations the proposed method improves RIC which extends the applications of CS.Furthermore,this weighted approach is preliminarily applied in computed tomography(CT).Simu-lations results show that the proposed method improves the reconstruction quality.4.The application of CS in magnetic resonance imaging(MRI)is studied.Com-bining Hessian Schatten norm and adaptive dictionary,a novel regularization model is proposed.Using split Bregman iteration,the model is split into several subproblems,and then a corresponding algorithm is also proposed.Numerical experiments are con-ducted on several test images with a variety of sampling patterns and ratios both in noiseless and noise scenarios.The simulations results demonstrate the superior perfor-mance of the proposed algorithm.5.In phase retrieval,the phase information is lost.It can be viewed as nonlinear CS,so we extend the typical orthogonal matching pursuit algorithm into phase retrieval.A novel algorithm for sparse phase retrieval and its modified version which has high recovery rate are proposed.Moreover,the recovery condition is given and the numerical results show the effectiveness of the proposed algorithm.