The method of projections for finding the common point of convex sets

Gubin, L.G.; Polyak, B. T.; Raik, E.V.

doi:10.1016/0041-5553(67)90113-9

Cited by 647 publications

(351 citation statements)

References 2 publications

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“…As in SSCM, A is divided rowwise into K contiguous blocks as shown in Eq. (11). At iteration t of PSCM, the next point x t+1 is computed as…”

Section: Parallel Surrogate Constraint Methods (Pscm)mentioning

confidence: 99%

“…An important class of iterative methods for the linear feasibility problem is the projection methods developed for the solution of the linear systems by Kacmarz (see [32]), and Cimmino (see [8]). Kacmarz's and Cimmino's works are extended to linear inequalities by Gubin et al [11] and Censor and Elfving [7]. The method in [11] is known as the successive orthogonal projections method.…”

Section: Introductionmentioning

confidence: 99%

“…Kacmarz's and Cimmino's works are extended to linear inequalities by Gubin et al [11] and Censor and Elfving [7]. The method in [11] is known as the successive orthogonal projections method. In this method, an initial guess is successively projected onto hyperplanes corresponding to the boundary of the violated constraints until a feasible point satisfying all inequalities is found.…”

Section: Introductionmentioning

confidence: 99%

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Parallel image restoration using surrogate constraint methods

Uçar

Aykanat

Pınar

et al. 2007

Journal of Parallel and Distributed Computing

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When formulated as a system of linear inequalities, the image restoration problem yields huge, unstructured, sparse matrices even for images of small size. To solve the image restoration problem, we use the surrogate constraint methods that can work efficiently for large problems. Among variants of the surrogate constraint method, we consider a basic method performing a single block projection in each step and a coarse-grain parallel version making simultaneous block projections. Using several state-of-the-art partitioning strategies and adopting different communication models, we develop competing parallel implementations of the two methods. The implementations are evaluated based on the per iteration performance and on the overall performance. The experimental results on a PC cluster reveal that the proposed parallelization schemes are quite beneficial.

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“…As in SSCM, A is divided rowwise into K contiguous blocks as shown in Eq. (11). At iteration t of PSCM, the next point x t+1 is computed as…”

Section: Parallel Surrogate Constraint Methods (Pscm)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Parallel image restoration using surrogate constraint methods

Uçar

Aykanat

Pınar

et al. 2007

Journal of Parallel and Distributed Computing

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“…In general, it will be dependent on the initial guess X 0) . Detailed theoretical discussions of the POCS method can be found in [4,10,15].…”

Section: Set Theoretic Image Coding and Restorationmentioning

confidence: 99%

Fractal Image Coding as Projections Onto Convex Sets

Ebrahimi

Vrscay

2006

Lecture Notes in Computer Science

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Abstract. We show how fractal image coding can be viewed and generalized in terms of the method of projections onto convex sets (POCS). In this approach, the fractal code defines a set of spatial domain similarity constraints. We also show how such a reformulation in terms of POCS allows additional contraints to be imposed during fractal image decoding. Two applications are presented: image construction with an incomplete fractal code and image denoising.

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“…In terms of how the problems have been approached algorithmically, crystallographers are used to dealing with very large numbers of re¯ections and exploit multisolution methods, while the image recovery problem has focused around mathematical techniques using what are called convex sets ' (Sezan, 1992;Combettes, 1996;Combettes & Trussell, 1990;Gubin et al, 1967), and iterative methods whose formal convergence behavior can be analyzed in detail. The mathematics of convex projections is a well established analytical approach to the general problem of image recovery.…”

Section: Introductionmentioning

confidence: 99%

A feasible set approach to the crystallographic phase problem

1999

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The connection between the crystallographic phase problem and the feasible set approach is explored. It is argued that solving the crystallographic phase problem is formally equivalent to a feasible set problem using a statistical operator interpretable via a log-likelihood functional, projection onto the non-convex set of experimental structure factors coupled with a phase-extension constraint and mapping onto atomic positions. In no way does this disagree with or dispute any of the existing statistical relationships available in the literature; instead it expands understanding of how the algorithms work. Making this connection opens the door to the application of a number of well developed mathematical tools in functional analysis. Furthermore, a number of known results in image recovery can be exploited both to optimize existing algorithms and to develop new and improved algorithms.

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The method of projections for finding the common point of convex sets

Cited by 647 publications

References 2 publications

Parallel image restoration using surrogate constraint methods

Parallel image restoration using surrogate constraint methods

Fractal Image Coding as Projections Onto Convex Sets

A feasible set approach to the crystallographic phase problem

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