A stopping criterion for iterative regularization methods

Landi, Germana; Piccolomini, Elena Loli; Tomba, Ivan

doi:10.1016/j.apnum.2016.03.006

Cited by 19 publications

(6 citation statements)

References 47 publications

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“…But, for other Krylov processes [11], such as LSQR, identical stop requirements can be formulated. In order to create stopping criteria, which, given an a priori probability, stop the conjugate gradient process, we will use the stochastic qualities of the linear regression pattern if stochastic variables can be considered with lesser probability of iteration running algorithm termination [42,43].…”

Section: Discussionmentioning

confidence: 99%

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Jaffri

Almogren

Ahmad

et al. 2021

Mathematical Problems in Engineering

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Electrical capacitance tomography (ECT) has been used to measure flow by applying gas-solid flow in coal gasification, pharmaceutical, and other industries. ECT is also used for creating images of physically confined objects. The data collected by the acquisition system to produce images undergo blurring because of ambient conditions and the electronic circuitry used. This research includes the principle of ECT techniques for deblurring images that were created during measurement. The data recorded by the said acquisition system ascends a large number of linear equations. This system of equations is sparse and ill-conditioned and hence is ill-posed in nature. A variety of reconstruction algorithms with many pros and cons are available to deal with ill-posed problems. Large-scale systems of linear equations resulting during image deblurring problems are solved using iterative regularization algorithms. The conjugate gradient algorithm for least-squares problems (CGLS), least-squares QR factorization (LSQR), and the modified residual norm steepest descent (MRNSD) algorithm are the famous variations of iterative algorithms. These algorithms exhibit a semiconvergence behavior; that is, the computed solution quality first improves and then reduces as the error norm decreases and later increases with each iteration. In this work, soft thresholding has been used for image deblurring problems to tackle the semiconvergence issues. Numerical test problems were executed to indicate the efficacy of the suggested algorithms with criteria for optimal stopping iterations. Results show marginal improvement compared to the traditional iterative algorithms (CGLS, LSQR, and MRNSD) for resolving semiconvergence behavior and image restoration.

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Section: Discussionmentioning

confidence: 99%

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Jaffri

Almogren

Ahmad

et al. 2021

Mathematical Problems in Engineering

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“…The normal equations of (2) can be solved by a fast iterative method, such as the Conjugate Gradient Least-Squares (CGLS) algorithm, stopped after few iterations, far before convergence [7]. Several criteria have been proposed to suitably stop the iterations before noise enters the solution [8,9].…”

Section: Regularization By Iterationmentioning

confidence: 99%

A comparison of regularization models for few-view CT image reconstruction

Piccolomini

2022

Ann Univ Ferrara

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In this paper I analyse some regularization models for the reconstruction of X-rays Computed Tomography images from few-view projections. It is well known that the widely used low-cost Filtered Back Projection method is not suitable in case of low-dose data, since it produces images with noise and artifacts. Iterative reconstruction methods based on the model discretization are preferred in this case. However, since the problem has infinite possible solutions and is ill-posed, regularization is necessary to obtain a good solution. Different iterative regularization methods have been proposed in literature, but an organized comparison among them is not available. We compare some regularization approaches in the case of few-view tomography by means of simulated projections from both a phantom and a real image.

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“…A truncation parameter k satisfying the DPC can be selected by visual inspection of the so-called Picard plot (i.e., a plot of the quantities

and

versus i ). Alternatively, an index k satisfying the DPC can also be selected by using automatic techniques such as those described in [ 21 , 22 ].…”

Section: The Proposed Hybrid Algorithmmentioning

confidence: 99%

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

2021

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This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

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A stopping criterion for iterative regularization methods

Cited by 19 publications

References 47 publications

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

A comparison of regularization models for few-view CT image reconstruction

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

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