Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

Davidi, Ran; Herman, Gabor T.; Censor, Yair

doi:10.1111/j.1475-3995.2009.00695.x

Cited by 71 publications

(114 citation statements)

References 32 publications

Supporting

Mentioning

113

Contrasting

Order By: Relevance

“…In the problems discussed in [45], the number of unknowns was 59,049. In the examples given in [59] (a paper devoted to radiation therapy planning), problems of the form (1) were considered with the number N of unknowns only 515 but the number of pairs of constraints M = 128,688.…”

Section: Scientific Publicationsmentioning

confidence: 99%

“…According to them, that code uses the algorithm they refer to as TwIST [14] with split Bregman [50] as the substep. For the projection method we used a block-iterative algorithm with superiorization [24,45]. Superiorization uses perturbations to steer the iterative process of a projection method towards a minimizer of the given convex function φ.…”

Section: Examples From Computerized Tomographymentioning

confidence: 99%

See 1 more Smart Citation

On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

et al. 2011

View full text Add to dashboard Cite

International audienceThe effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many real-world applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints)

show abstract

Section: Scientific Publicationsmentioning

confidence: 99%

Section: Examples From Computerized Tomographymentioning

confidence: 99%

On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

et al. 2011

View full text Add to dashboard Cite

show abstract

“…Projection methods may have different algorithmic structures, such as block-iterative projections (BIP), see, e.g., [22, 24] and references therein, or string-averaging projections (SAP), see, e.g., [17] and references therein, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavior patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems.…”

Section: Linear Superiorizationmentioning

confidence: 99%

Can linear superiorization be useful for linear optimization problems?

Censor¹

2017

Inverse Problems

View full text Add to dashboard Cite

Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are (i) Does linear superiorization provide a feasible point whose linear target function value is lower than that obtained by running the same feasibility-seeking algorithm without superiorization under identical conditions? and (ii) How does linear superiorization fare in comparison with the Simplex method for solving linear programming problems? Based on our computational experiments presented here, the answers to these two questions are: “yes” and “very well”, respectively.

show abstract

“…Instead of seeking an optimal solution for the optimization problem based on regularization, it should obtain a superior solution with respect to a given merit function, which is computationallyefficient and circumvents the need to select a regularization parameter. The potential usefulness of superiorization has been demonstrated in several iterative algorithms for inverse problems in image reconstruction which generally belong to the class of convex feasibility problems (CFP) [14][15] [16] [17]. The fundamental principle, mathematical formulations and a general framework for the superiorization methodology are summarized in [18].…”

Section: Introductionmentioning

confidence: 99%

A Heuristic Superiorization-Like Approach to Bioluminescence Tomography

2013

View full text Add to dashboard Cite

Abstract-Bioluminescence tomography (BLT) is a powerful molecular imaging technology designed for the localization and quantification of bioluminescent sources in vivo. With the forward process modeled by the diffusion approximation equation, BLT is the inverse problem to reconstruct the distribution of internal bioluminescent sources subject to Cauchy data. Due to the non-uniqueness of BLT in general, adequate prior information such as nonnegativity and source support constraints must be utilized to obtain a physically favorable BLT solution. Iterative algorithms such as the well-known Expectation Maximization (EM) algorithm and the Landweber algorithm which are suitable for incorporating these knowledge-based constraints are widely used in practice. In the current work, we investigate the application of a superiorization-like approach to BLT. A superiorization-like version of prototypical iterative algorithms for BLT in a general framework, denoted by S-BLT, is presented. For the EM algorithm as the underlying iterative algorithms for BLT and S-BLT, superiorized by the total variation (TV) merit function, preliminary simulation results for a heterogeneous phantom are reported to demonstrate the viability of the approach and evaluate the performance of the proposed algorithm. It is found that total variation superiorization of BLT can significantly improve the visualization effect of the reconstruction with the sources set as a particular case of radial basis functions.

show abstract

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

Cited by 71 publications

References 32 publications

On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

Can linear superiorization be useful for linear optimization problems?

A Heuristic Superiorization-Like Approach to Bioluminescence Tomography

Contact Info

Product

Resources

About