Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration

Autores
Mazzieri, Gisela Luciana; Spies, Ruben Daniel; Temperini, Karina Guadalupe
Año de publicación
2010
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
The Tikhonov-Phillips method is widely used for regularizing ill-posed problems due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. The purpose of this article is twofold. First we study the problem of determining general sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and finnd conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.
Fil: Mazzieri, Gisela Luciana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Temperini, Karina Guadalupe. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Materia
Inverse problem
Ill-Posed
Regularization
Tikhonov-Phillips
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/75188
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/75188
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image RestorationMazzieri, Gisela LucianaSpies, Ruben DanielTemperini, Karina GuadalupeInverse problemIll-PosedRegularizationTikhonov-Phillipshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1The Tikhonov-Phillips method is widely used for regularizing ill-posed problems due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. The purpose of this article is twofold. First we study the problem of determining general sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and finnd conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.Fil: Mazzieri, Gisela Luciana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Temperini, Karina Guadalupe. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaAsociación Argentina de Mecánica Computacional2010-11info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/75188Mazzieri, Gisela Luciana; Spies, Ruben Daniel; Temperini, Karina Guadalupe; Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration; Asociación Argentina de Mecánica Computacional; Mecánica Computacional; 29; 11-2010; 6275-62832591-3522CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.amcaonline.org.ar/ojs/index.php/mc/article/viewFile/3448/3365info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-10T13:03:52Zoai:ri.conicet.gov.ar:11336/75188instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-10 13:03:52.632CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
title Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
spellingShingle Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
Mazzieri, Gisela Luciana
Inverse problem
Ill-Posed
Regularization
Tikhonov-Phillips
title_short Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
title_full Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
title_fullStr Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
title_full_unstemmed Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
title_sort Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration
dc.creator.none.fl_str_mv Mazzieri, Gisela Luciana
Spies, Ruben Daniel
Temperini, Karina Guadalupe
author Mazzieri, Gisela Luciana
author_facet Mazzieri, Gisela Luciana
Spies, Ruben Daniel
Temperini, Karina Guadalupe
author_role author
author2 Spies, Ruben Daniel
Temperini, Karina Guadalupe
author2_role author
author
dc.subject.none.fl_str_mv Inverse problem
Ill-Posed
Regularization
Tikhonov-Phillips
topic Inverse problem
Ill-Posed
Regularization
Tikhonov-Phillips
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv The Tikhonov-Phillips method is widely used for regularizing ill-posed problems due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. The purpose of this article is twofold. First we study the problem of determining general sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and finnd conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.
Fil: Mazzieri, Gisela Luciana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
Fil: Temperini, Karina Guadalupe. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
description The Tikhonov-Phillips method is widely used for regularizing ill-posed problems due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. The purpose of this article is twofold. First we study the problem of determining general sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and finnd conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.
publishDate 2010
dc.date.none.fl_str_mv 2010-11
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/75188
Mazzieri, Gisela Luciana; Spies, Ruben Daniel; Temperini, Karina Guadalupe; Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration; Asociación Argentina de Mecánica Computacional; Mecánica Computacional; 29; 11-2010; 6275-6283
2591-3522
CONICET Digital
CONICET
url http://hdl.handle.net/11336/75188
identifier_str_mv Mazzieri, Gisela Luciana; Spies, Ruben Daniel; Temperini, Karina Guadalupe; Regularization of Inverse Ill-Posed Problems with General Penalizing Terms: Applications to Image Restoration; Asociación Argentina de Mecánica Computacional; Mecánica Computacional; 29; 11-2010; 6275-6283
2591-3522
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.amcaonline.org.ar/ojs/index.php/mc/article/viewFile/3448/3365
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Asociación Argentina de Mecánica Computacional
publisher.none.fl_str_mv Asociación Argentina de Mecánica Computacional
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1842980112998334464
score 12.993085

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