Numerical Studies of the Generalized \u3cem\u3el\u3c/em\u3e₁ Greedy Algorithm for Sparse Signals

The generalized l1 greedy algorithm was recently introduced and used to reconstruct medical images in computerized tomography in the compressed sensing framework via total variation minimization. Experimental results showed that this algorithm is superior to the reweighted l1-minimization and l1 greedy algorithms in reconstructing these medical images. In this paper the effectiveness of the generalized l1 greedy algorithm in finding random sparse signals from underdetermined linear systems is investigated. A series of numerical experiments demonstrate that the generalized l1 greedy algorithm is superior to the reweighted l1-minimization and l1 greedy algorithms in the successful recovery of randomly generated Gaussian sparse signals from data generated by Gaussian random matrices. In particular, the generalized l1 greedy algorithm performs extraordinarily well in recovering random sparse signals with nonzero small entries. The stability of the generalized l1 greedy algorithm with respect to its parameters and the impact of noise on the recovery of Gaussian sparse signals are also studied

The Convergence Rate of the Chebyshev SIM under a Perturbation of Foci of an Elliptic Domain

The Chebyshev semiiterative method (CHSIM) is a powerful method for finding the iterative solution of a nonsymmetric real linear system Ax = b if an ellipse excluding the origin well fits the spectrum of A. The asymptotic rate of convergence of the CHSIM for solving the above system under a perturbation of the foci of the optimal ellipse is studied. Several formulae to approximate the asymptotic rates of convergence, up to the first order of a perturbation, are derived. These generalize the results about the sensitivity of the asymptotic rate of convergence to a perturbation of a real-line segment spectrum by Hageman and Young, and by the first author. A numerical example is given to illustrate the theoretical results

The Convergence of Block Cyclic Projection with Underrelaxation Parameters for Compressed Sensing Based Tomography

The block cyclic projection method in the compressed sensing framework (BCPCS) was introduced for image reconstruction in computed tomography and its convergence had been proven in the case of unity relaxation (λ=1). In this paper, we prove its convergence with underrelaxation parameters λ∈(0,1). As a result, the convergence of compressed sensing based block component averaging algorithm (BCAVCS) and block diagonally-relaxed orthogonal projection algorithm (BDROPCS) with underrelaxation parameters under a certain condition are derived. Experiments are given to illustrate the convergence behavior of these algorithms with selected parameters

The Convergence of Block Cyclic Projection with Over-Relaxation Parameters for Compressed Sensing Based Tomography

The convergence of the block cyclic projection for compressed sensing based tomography (BCPCS) algorithm had been proven recently in the case of underrelaxation parameter λ∈(0,1]. In this paper, we prove its convergence with overrelaxation parameter λ∈(1,2). As a result, the convergence of the other two algorithms (BCAVCS and BDROPCS) with overrelaxation parameter λ∈(1,2) in a special case is derived. Experiments are given to demonstrate the convergence behavior of the BCPCS algorithm with different values of λ

Error Analysis of Reweighted \u3cem\u3el\u3c/em\u3e₁ Greedy Algorithm for Noisy Reconstruction

Sparse solutions for an underdetermined system of linear equations Φx=u can be found more accurately by l1-minimization type algorithms, such as the reweighted l1-minimization and l1 greedy algorithms, than with analytical methods, in particular in the presence of noisy data. Recently, a generalized l1 greedy algorithm was introduced and applied to signal and image recovery. Numerical experiments have demonstrated the convergence of the new algorithm and the superiority of the algorithm over the reweighted l1-minimization and l1 greedy algorithms although the convergence has not yet been proven theoretically. In this paper, we provide an error bound for the reweightedl1 greedy algorithm, a type of the generalized l1 greedy algorithm, in the noisy case and show its improvement over the reweighted l1-minimization