A Preconditioned Landweber Scheme And Its Application To Limited-Angle Reconstruction And Compressed Sensing

CT image reconstruction,reconstructing the internal information of object from the projection data of X-ray passing through the interior of object,is summed theoretically down to the inversion of Radon transform.The projection data related to angle variable is sometimes limited,resulting in the limited-angle image reconstruction problem.Limited-angle image reconstruction is summed down to the inversion of the limited-angle Radon transformation.Algebraic reconstruction is to solve discrete linear system of Radon transformation.When the known angle range is small,the theoretical ill-posedness of limited-angle Radon transform causes that its discrete linear system is also ill-posed.Thus,the limited-angle image reconstruction is a very challenging problem.The illposedness of linear system is caused by the big condition number of coefficient matrix.This paper studies a preconditioned Landweber scheme and its application to the limited-angle reconstruction and compressed sensing.Main contributions and innovations are as follows:1.For the linear system with a big condition number,we propose a reweighted method,left-multiplying the normal equation of the linear system a positivedefine symmetric matrix related to its coefficient matrix and repeating this process many times,and prove that the condition number decreases monotonically to 1 as the weighting times approaches infinity.Correspondingly,we give a preconditioned Landweber iteration scheme and prove that the numerical accuracy of the iteration solution is improve with the appropriate large weighting times.Numerical experiments show that our method is valid.2.We give the selection method of the weighted parameters and the estimations of the upper bounds of the maximum eigenvalues used in the reweighting process,analyze the convergence of the preconditioned Landweber iteration scheme,give an optimized reconstruction model and the corresponding RwAEDS algorithm by combining the preconditioned Landweber iteration scheme and AEDS algorithm,and apply it for the limited-angle image reconstruction.Experimental results on simulated and real projection data show that our algorithm is valid for the limited-angle CT image reconstruction with a known appropriate small angle range.3.In compressed sensing,a small enough restricted isometry constant(RIC)of the sensing matrix satisfying the restricted isometry property(RIP)is the powerful guarantee on the precise reconstruction of a sparse discrete signal.Under a certain condition,the smaller condition number of sensing matrix,the smaller its RIC is.We use the proposed reweighted method to weight the original reconstruction equation,then derive a new equivalent reconstruction equation with a smaller RIC and apply it for sparse signal reconstructions.Numerical experiments based on different measurement matrices and recovery algorithms show that the results of the obtained reconstruction equation outperform those of original reconstruction equation.