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block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2002年第1期24卷 40-58页

作者： Censor, Y Elfving, T Univ Haifa Dept Math IL-31905 Haifa Israel Linkoping Univ Dept Math SE-58183 Linkoping Sweden

We formulate a block- iterative algorithmic scheme for the solution of systems of linear inequalities and/ or equations and analyze its convergence. This study provides as special cases proofs of convergence of ( i) the recently proposed component averaging ( CAV) method of Censor, Gordon, and Gordon [ Parallel Comput., 27 ( 2001), pp. 777 808], ( ii) the recently proposed block- iterative CAV ( BICAV) method of the same authors [ IEEE Trans. Medical Imaging, 20 ( 2001), pp. 1050 1060], and ( iii) the simultaneous algebraic reconstruction technique ( SART) of Andersen and Kak [ Ultrasonic Imaging, 6 ( 1984), pp. 81 94] and generalizes them to linear inequalities. The first two algorithms are projection algorithms which use certain generalized oblique projections and diagonal weighting matrices which reflect the sparsity of the underlying matrix of the linear system. The previously reported experimental acceleration of the initial behavior of CAV and BICAV is thus complemented here by a mathematical study of the convergence of the algorithms.

关键词： block-iterative algorithms component averaging (CAV) block-iterative CAV simultaneous algebraic reconstruction technique oblique projections linear feasibility problem

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block-iterative algorithms with underrelaxed Bregman projections

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SIAM JOURNAL ON OPTIMIZATION 2002年第1期13卷 283-297页

作者： Censor, Y Herman, GT Univ Haifa Dept Math IL-31905 Haifa Israel CUNY Grad Ctr Dept Comp Sci New York NY 10016 USA Linkoping Univ Dept Math S-58183 Linkoping Sweden

The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes, and the question of how to relax Bregman projections onto convex sets that are not linear (i.e., not hyperplanes or half-spaces) has remained open. A definition of the underrelaxation of Bregman projections onto general convex sets is given here, which includes as special cases the underrelaxed orthogonal projections and the underrelaxed Bregman projections onto linear sets as given by De Pierro and Iusem [ J. Optim. Theory Appl., 51 ( 1986), pp. 421 440]. With this new definition, we construct a block-iterative projection algorithmic scheme and prove its convergence to a solution of the convex feasibility problem. The practical importance of relaxation parameters in the application of such projection algorithms to real-world problems is demonstrated on a problem of image reconstruction from projections.

关键词： convex feasibility projection algorithms Bregman functions block-iterative algorithms underrelaxation

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Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems

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INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH 2009年第4期16卷 465-479页

作者： Byrne, Charles Univ Massachusetts Lowell Dept Math Sci Lowell MA 01854 USA

We obtain the following upper bounds for the eigenvalues of the matrix A(dagger)A. For any a in the interval [0, 2] let c(aj) = Sigma(T)(i=1) vertical bar A(ij)vertical bar(a), r(ai) = Sigma(J)(j=1) vertical bar A(ij)vertical bar(2-a), and c(a) and r(a) the maxima of the c(aj) and r(ai), respectively. Then no eigenvalue of the matrix A(dagger)A exceeds the maximum of Sigma(J)(j-1) c(aj)vertical bar A(ij)vertical bar(2-a), over all i, nor the maximum of Sigma(T)(i=1) r(ai)vertical bar A(ij)vertical bar(a), over all j. Therefore, no eigenvalue of A(dagger)A exceeds c(a)r(a). Using these bounds, it follows that, for the matrix G with entries Gij = A(ij)root alpha(i)root beta(j), no eigenvalue of G(dagger)G exceeds one, provided that, for some a in the interval [0, 2], we have alpha(i) <= r(ai)(-1), and beta(j) <= c(aj)(-1). Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax = b, including the CAV, BICAV, CARP1, SART, SIRT, and the block-iterative DROP and SART methods.

关键词： block-iterative algorithms singular-value bounds

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Choosing parameters in block-iterative or ordered subset reconstruction algorithms

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2005年第3期14卷 321-327页

作者： Byrne, C Univ Massachusetts Lowell Dept Math Sci Lowell MA 01854 USA

Viewed abstractly, all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y = Px, where y is a vector with positive entries and P a matrix whose entries are nonnegative and with no purely zero columns. The expectation maximization maximum likelihood method, as it occurs in emission tomography, and the simultaneous multiplicative algebraic reconstruction technique are slow to converge on large data sets;accelerating convergence through the use of block-iterative or ordered subset versions of these algorithms is a topic of considerable interest. These block-iterative versions involve relaxation and normalization parameters, the correct selection of which may not be obvious to all users. The algorithms are not faster merely by virtue of being block-iterative;the correct choice of the parameters is crucial. Through a detailed discussion of the theoretical foundations of these methods, we come to a better understanding of the precise roles these parameters play.

关键词： block-iterative algorithms image reconstruction ordered subsets

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PARALLEL COMPUTING WITH block-iterative IMAGE-RECONSTRUCTION algorithms

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APPLIED NUMERICAL MATHEMATICS 1991年第5期7卷 399-415页

作者： ZENIOS, SA CENSOR, Y HAIFA UNIV DEPT MATH & COMP SCIIL-31905 HAIFAISRAEL

Fully discretized models for image reconstruction from projections give rise to huge and sparse nonlinear optimization problems. We study the use of parallel and vector supercomputers for the iterative reconstruction of medical images. A block-iterative version of the Multiplicative Algebraic Reconstruction Technique (MART) is implemented on a CRAY X-MP/48. The implementation exploits the block structure of the algorithm which allows us to take advantage both of the vector architecture of the computer and of the multiple processors for parallel computations. Results indicate that block-iterative algorithms are suitable for parallel and vector implementations and can reconstruct with relative efficiency highly discretized images which give rise to very large optimization problems.

关键词： COMPUTERIZED TOMOGRAPHY IMAGE RECONSTRUCTION block-iterative algorithms PARALLEL COMPUTING ENTROPY MAXIMIZATION VECTORIZATION MULTITASKING

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Choosing parameters in block-iterative or ordered-subset reconstruction algorithms

Choosing parameters in block-iterative or ordered-subset rec...

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Nuclear Science Symposium/Medical Imaging Conference

作者： Byrne, C Univ Massachusetts Lowell MA 01854 USA

ISBN: (纸本)0780387007

Viewed abstractly, all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y = Px, where y is a vector with positive entries and P a matrix whose entries are nonnegative and with no purely zero columns. The expectation maximization maximum likelihood (EMML) method as it occurs in emission tomography and the simultaneous multiplicative algebraic reconstruction technique (SMART) are slow to converge on large data sets;accelerating convergence through the use of block-iterative or ordered subset versions of these algorithms is a topic of considerable interest. These block-iterative versions involve relaxation and normalization parameters the correct selection of which may not be obvious to all users. The algorithms are not faster merely by virtue of being block-iterative;the correct choice of the parameters is crucial. Through a detailed discussion of the theoretical foundations of these methods we come to a better understanding of the precise roles these parameters play.

关键词： block-iterative algorithms image reconstruction ordered-subset methods expectation maximization

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PARALLEL APPLICATION OF block-iterative METHODS IN MEDICAL IMAGING AND RADIATION-THERAPY

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MATHEMATICAL PROGRAMMING 1988年第2期42卷 307-325页

作者： CENSOR, Y HOSP UNIV PENN DEPT RADIOLMED IMAGE PROC GRPPHILADELPHIAPA 19104

Some row-action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems. Recently developed block-iterative versions of such special-purpose methods enable parallel computation when the underlying problem is appropriately decomposed. This opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning, all in their fully discretized modelling approach. Since there is more than one way of deriving block-iterative versions of any row-action method, the choice has to be made with reference to the underlying real-world problem.

关键词： Full discretization computerized tomography image reconstruction radiotherapy treatment planning block-iterative algorithms parallel computations Cimmino's algorithm entropy maximization classification of algorithms

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

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INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH 2009年第4期16卷 505-524页

作者： Davidi, R. Herman, G. T. Censor, Y. CUNY Grad Ctr Dept Comp Sci New York NY 10016 USA Univ Haifa Dept Math IL-31905 Haifa Israel

A block-iterative projection algorithm for solving the consistent convex feasibility problem in a finite-dimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in each iterative step of the algorithm) is proposed. This resilience can be used to steer the iterative process towards a feasible point that is superior in the sense of some functional on the points in the Euclidean space having a small value. The potential usefulness of this is illustrated in image reconstruction from projections, using both total variation and negative entropy as the functional.

关键词： block-iterative algorithms projection methods image reconstruction convex feasibility superiorization perturbation resilience product space

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