Semiconvergence and Relaxation Parameters for Projected SIRT Algorithms

Abstract. We present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image processing. The Kaczmarz method is one of the most popular solvers for overdetermined systems of linear equations due to its speed and simplicity. Here we introduce and analyze an extension of the Kaczmarz method which iteratively projects the estimate onto a solution space given from two randomly selected rows. We show that this projection algorithm provides exponential convergence to the solution in expectation. The convergence rate significantly improves upon that of the standard randomized Kaczmarz method when the system has coherent rows. We also show that the method is robust to noise, and converges exponentially in expectation to the noise floor. Experimental results are provided which confirm that in the coherent case our method significantly outperforms the randomized Kaczmarz method.

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“…[5]). For example, in Figure 5 (d), the estimation error for the two-subspace method decreases to a point and then begins to increase.…”

Section: Discussionmentioning

confidence: 99%

“…This error is sharp in general [13]. Modified Kaczmarz algorithms can also be used to solve the least squares version of this problem, see for example [4,5,8,2] and the references therein.…”

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confidence: 99%

Two-Subspace Projection Method for Coherent Overdetermined Systems

Needell

Ward

2012

J Fourier Anal Appl

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“…During the first iterations the iterates approach the exact solutionx, and after this stage the iterates start to diverge fromx and instead converge to a solution that is dominated by noise, cf. [16], [18], [27].…”

Section: Thementioning

confidence: 99%

Multicore Performance of Block Algebraic Iterative Reconstruction Methods

Sørensen¹,

Hansen²

2014

SIAM J. Sci. Comput.

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Abstract. Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the Algebraic Reconstruction Techniques (ART) and the Simultaneous Iterative Reconstruction Techniques (SIRT), both of which rely on semi-convergence. Block versions of these methods, based on a partitioning of the linear system, are able to combine the fast semi-convergence of ART with the better multi-core properties of SIRT. These block methods separate into two classes: those that, in each iteration, access the blocks in a sequential manner, and those that compute a result for each block in parallel and then combine these results before the next iteration. The goal of this work is to demonstrate which block methods are best suited for implementation on modern multi-core computers. To compare the performance of the different block methods we use a fixed relaxation parameter in each method, namely, the one that leads to the fastest semi-convergence. Computational results show that for multi-core computers, the sequential approach is preferable.

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“…The semiconvergence behavior means that initially the iteration vector approaches a regularized solution, while continuing the iteration often leads to iteration vectors corrupted by noise [10]. Based on the analysis of the semiconvergence behavior, T. Elfving et al proposed two relaxation strategies [4]. Later, these two strategies were improved by them [11].…”

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confidence: 99%

“…Based on the analysis of the semiconvergence behavior, T. Elfving et al proposed two relaxation strategies [4]. Later, these two strategies were improved by them [11]. If the matrix 푇 is positive definite, as a special case of the Richardson iteration, based on minimizing the spectral radius of iterative matrix, the relaxation coefficient of the Landweber iteration (3) is given by…”

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confidence: 99%

Weighting Algorithm and Relaxation Strategies of the Landweber Method for Image Reconstruction

Han

Wang

2018

Mathematical Problems in Engineering

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The iterative approach is important for image reconstruction with ill-posed problem, especially for limited angle reconstruction. Most of iterative algorithms can be written in the general Landweber scheme. In this context, appropriate relaxation strategies and appropriately chosen weights are critical to yield reconstructed images of high quality. In this paper, based on reducing the condition number of matrix 푇 , we find one method of weighting matrices for the general Landweber method to improve the reconstructed results. For high resolution images, the approximate iterative matrix is derived. And the new weighting matrices and corresponding relaxation strategies are proposed for the general Landweber method with large dimensional number. Numerical simulations show that the proposed weighting methods are effective in improving the quality of reconstructed image for both complete projection data and limited angle projection data.

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Semiconvergence and Relaxation Parameters for Projected SIRT Algorithms

Cited by 63 publications

References 22 publications

Two-Subspace Projection Method for Coherent Overdetermined Systems

Two-Subspace Projection Method for Coherent Overdetermined Systems

Multicore Performance of Block Algebraic Iterative Reconstruction Methods

Weighting Algorithm and Relaxation Strategies of the Landweber Method for Image Reconstruction

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