Implicit regularization of the incomplete oblique projections method

The aim of this paper is to improve the performance of the incomplete oblique projections method (IOP), previously introduced by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method employs incomplete oblique projections onto the set of solu...

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Autores principales:	Scolnik, H.D., Echebest, N.E., Guardarucci, M.T.
Formato:	JOUR
Materias:	Computerized tomographies Image reconstruction Incomplete projections Least squares problems Minimum norm solution Regularizing
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_09696016_v16_n4_p525_Scolnik
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_09696016_v16_n4_p525_Scolnik
record_format	dspace
spelling	todo:paper_09696016_v16_n4_p525_Scolnik2023-10-03T15:55:22Z Implicit regularization of the incomplete oblique projections method Scolnik, H.D. Echebest, N.E. Guardarucci, M.T. Computerized tomographies Image reconstruction Incomplete projections Least squares problems Minimum norm solution Regularizing The aim of this paper is to improve the performance of the incomplete oblique projections method (IOP), previously introduced by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method employs incomplete oblique projections onto the set of solutions of the augmented system Ax−r=b, and converges to a weighted least squares solution of the system Ax=b. Many tomographic image reconstruction problems are such that the limitation of the range of rays makes the model underdetermined, the discretized linear system is rank-deficient, the nullspace is non-trivial, and the minimal norm least squares solution may be far away from the true image. In a previous paper, we have added a quadratic term reflecting neighboring pixel information to the standard least squares model for improving the quality of the reconstructed images. In this paper we replace the quadratic function by a more general regularizing function avoiding the modification of the original system. The key idea is to perform a joint optimization of the norm of the residual and of the regularizing function in each iteration. The theoretical properties of this new algorithm are analyzed, and numerical experiments are presented comparing its performance with other well-known methods. They show that the new approach improves the quality of the reconstructed images. © 2009 International Federation of Operational Research Societies. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_09696016_v16_n4_p525_Scolnik
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Computerized tomographies Image reconstruction Incomplete projections Least squares problems Minimum norm solution Regularizing
spellingShingle	Computerized tomographies Image reconstruction Incomplete projections Least squares problems Minimum norm solution Regularizing Scolnik, H.D. Echebest, N.E. Guardarucci, M.T. Implicit regularization of the incomplete oblique projections method
topic_facet	Computerized tomographies Image reconstruction Incomplete projections Least squares problems Minimum norm solution Regularizing
description	The aim of this paper is to improve the performance of the incomplete oblique projections method (IOP), previously introduced by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method employs incomplete oblique projections onto the set of solutions of the augmented system Ax−r=b, and converges to a weighted least squares solution of the system Ax=b. Many tomographic image reconstruction problems are such that the limitation of the range of rays makes the model underdetermined, the discretized linear system is rank-deficient, the nullspace is non-trivial, and the minimal norm least squares solution may be far away from the true image. In a previous paper, we have added a quadratic term reflecting neighboring pixel information to the standard least squares model for improving the quality of the reconstructed images. In this paper we replace the quadratic function by a more general regularizing function avoiding the modification of the original system. The key idea is to perform a joint optimization of the norm of the residual and of the regularizing function in each iteration. The theoretical properties of this new algorithm are analyzed, and numerical experiments are presented comparing its performance with other well-known methods. They show that the new approach improves the quality of the reconstructed images. © 2009 International Federation of Operational Research Societies.
format	JOUR
author	Scolnik, H.D. Echebest, N.E. Guardarucci, M.T.
author_facet	Scolnik, H.D. Echebest, N.E. Guardarucci, M.T.
author_sort	Scolnik, H.D.
title	Implicit regularization of the incomplete oblique projections method
title_short	Implicit regularization of the incomplete oblique projections method
title_full	Implicit regularization of the incomplete oblique projections method
title_fullStr	Implicit regularization of the incomplete oblique projections method
title_full_unstemmed	Implicit regularization of the incomplete oblique projections method
title_sort	implicit regularization of the incomplete oblique projections method
url	http://hdl.handle.net/20.500.12110/paper_09696016_v16_n4_p525_Scolnik
work_keys_str_mv	AT scolnikhd implicitregularizationoftheincompleteobliqueprojectionsmethod AT echebestne implicitregularizationoftheincompleteobliqueprojectionsmethod AT guardaruccimt implicitregularizationoftheincompleteobliqueprojectionsmethod
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Implicit regularization of the incomplete oblique projections method

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