Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Eggermont, P. P. B.; Herman, Gabor T.; Lent, Arnold

doi:10.1016/0024-3795(81)90139-7

Cited by 239 publications

(112 citation statements)

References 15 publications

Supporting

Mentioning

109

Contrasting

Unclassified

Order By: Relevance

“…It is also a generalization of the block-iterative projections (BIP) method of Aharoni and Censor [1] (see [16,Algorithm 5.6.1]) which uses orthogonal (least Euclidean distance) projections onto convex sets. It also generalizes the work of Eggermont et al [20] and the work of Elfving [21]. For an extensive survey of linear least squares algorithms, including projection methods, see Björck [7].…”

Section: Introductionmentioning

confidence: 60%

Iterative algorithms with seminorm‐induced oblique projections

Censor

Elfving

2003

Abstract and Applied Analysis

View full text Add to dashboard Cite

A definition of oblique projections onto closed convex sets that use seminorms induced by diagonal matrices which may have zeros on the diagonal is introduced. Existence and uniqueness of such projections are secured via directional affinity of the sets with respect to the diagonal matrices involved. A blockiterative algorithmic scheme for solving the convex feasibility problem, employing seminorm-induced oblique projections, is constructed and its convergence for the consistent case is established. The fully simultaneous algorithm converges also in the inconsistent case to the minimum of a certain proximity function.

show abstract

Section: Introductionmentioning

confidence: 60%

Iterative algorithms with seminorm‐induced oblique projections

Censor

Elfving

2003

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…To solve system (4), we use a Landweber-Kaczmarz method (cf., e.g., [3,11]), which reads in our context as follows: for a given initial guess Φ (0) ∈ V for the turbulences, the iterates are given by…”

Section: Solution With the Landweber-kaczmarz Methodsmentioning

confidence: 99%

“…The Landweber-Kaczmarz method is an iterative method that operates only on sub-domains of the turbulent layers in every iteration, which makes it computationally very competitive. It is particularly well suited for high dimensional ill-posed problems (cf, e.g., [3]). Note that we will always assume that the wavefronts have already been reconstructed from wavefront sensor measurements, using, e.g., the CuRe (Cumulative reconstructor) [23,18].…”

Section: Introductionmentioning

confidence: 99%

An H¹ -Kaczmarz reconstructor for atmospheric tomography

Eslitzbichler

Pechstein

Ramlau

2013

Journal of Inverse and Ill-Posed Problems

View full text Add to dashboard Cite

Atmospheric tomography is a prerequisite step before wavefront perturbations can be corrected in Multi Conjucate Adaptive Optics (MCAO) systems. Recently, the use of a Landweber-Kaczmarz iteration has been proposed to solve the tomography problem. However, due to the geometric partitioning of the update steps in this method, discontinuities in the solution appear that are physically implausible. We investigate augmenting the Landweber-Kaczmarz iteration with a smoothing step based on solving an elliptic partial differential equation to alleviate these discontinuities.

show abstract

“…The behaviour of this algorithm for inconsistent systems, when relaxation parameters are allowed, is studied by Censor et al (1983) and Eggermont et al (1981). Atkinson and Soria (2007) compare the ART method to other algebraic methods (e.g.…”

Section: Artmentioning

confidence: 99%

Simultaneous multiplicative column-normalized method (SMART) for 3-D ionosphere tomography in comparison to other algebraic methods

Gerzen

Minkwitz

2016

Ann. Geophys.

View full text Add to dashboard Cite

Abstract. The accuracy and availability of satellite-based applications like GNSS positioning and remote sensing crucially depends on the knowledge of the ionospheric electron density distribution. The tomography of the ionosphere is one of the major tools to provide link specific ionospheric corrections as well as to study and monitor physical processes in the ionosphere.In this paper, we introduce a simultaneous multiplicative column-normalized method (SMART) for electron density reconstruction. Further, SMART+ is developed by combining SMART with a successive correction method. In this way, a balancing between the measurements of intersected and not intersected voxels is realised. The methods are compared with the well-known algebraic reconstruction techniques ART and SART. All the four methods are applied to reconstruct the 3-D electron density distribution by ingestion of ground-based GNSS TEC data into the NeQuick model.The comparative case study is implemented over Europe during two periods of the year 2011 covering quiet to disturbed ionospheric conditions. In particular, the performance of the methods is compared in terms of the convergence behaviour and the capability to reproduce sTEC and electron density profiles. For this purpose, independent sTEC data of four IGS stations and electron density profiles of four ionosonde stations are taken as reference. The results indicate that SMART significantly reduces the number of iterations necessary to achieve a predefined accuracy level. Further, SMART+ decreases the median of the absolute sTEC error up to 15, 22, 46 and 67 % compared to SMART, SART, ART and NeQuick respectively.

show abstract

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Cited by 239 publications

References 15 publications

Iterative algorithms with seminorm‐induced oblique projections

Iterative algorithms with seminorm‐induced oblique projections

An H¹ -Kaczmarz reconstructor for atmospheric tomography

Simultaneous multiplicative column-normalized method (SMART) for 3-D ionosphere tomography in comparison to other algebraic methods

Contact Info

Product

Resources

About

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Cited by 239 publications

References 15 publications

Iterative algorithms with seminorm‐induced oblique projections

Iterative algorithms with seminorm‐induced oblique projections

An H1 -Kaczmarz reconstructor for atmospheric tomography

Simultaneous multiplicative column-normalized method (SMART) for 3-D ionosphere tomography in comparison to other algebraic methods

Contact Info

Product

Resources

About

An H¹ -Kaczmarz reconstructor for atmospheric tomography