Optimal shift invariant spaces and their Parseval frame generators

Given a set of functions F = {f1, ..., fm} ⊂ L2 (Rd), we study the problem of finding the shift-invariant space V with n generators {φ1, ..., φn} that is "closest" to the functions of F in the sense thatV = under(arg min, V′ ∈ Vn) underover(∑, i = 1, m) wi {norm of matrix} fi - PV′ fi {nor...

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Detalles Bibliográficos
Autores principales: Cabrelli, Carlos Alberto, Molter, Ursula Maria
Publicado: 2007
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v23_n2_p273_Aldroubi
http://hdl.handle.net/20.500.12110/paper_10635203_v23_n2_p273_Aldroubi
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_10635203_v23_n2_p273_Aldroubi
record_format dspace
spelling paper:paper_10635203_v23_n2_p273_Aldroubi2023-06-08T16:03:31Z Optimal shift invariant spaces and their Parseval frame generators Cabrelli, Carlos Alberto Molter, Ursula Maria Given a set of functions F = {f1, ..., fm} ⊂ L2 (Rd), we study the problem of finding the shift-invariant space V with n generators {φ1, ..., φn} that is "closest" to the functions of F in the sense thatV = under(arg min, V′ ∈ Vn) underover(∑, i = 1, m) wi {norm of matrix} fi - PV′ fi {norm of matrix}2, where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-rays), from the observation of a set of m signals or images f1, ..., fm, which may be theoretical samples, or experimental data. © 2007 Elsevier Inc. All rights reserved. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2007 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v23_n2_p273_Aldroubi http://hdl.handle.net/20.500.12110/paper_10635203_v23_n2_p273_Aldroubi
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description Given a set of functions F = {f1, ..., fm} ⊂ L2 (Rd), we study the problem of finding the shift-invariant space V with n generators {φ1, ..., φn} that is "closest" to the functions of F in the sense thatV = under(arg min, V′ ∈ Vn) underover(∑, i = 1, m) wi {norm of matrix} fi - PV′ fi {norm of matrix}2, where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-rays), from the observation of a set of m signals or images f1, ..., fm, which may be theoretical samples, or experimental data. © 2007 Elsevier Inc. All rights reserved.
author Cabrelli, Carlos Alberto
Molter, Ursula Maria
spellingShingle Cabrelli, Carlos Alberto
Molter, Ursula Maria
Optimal shift invariant spaces and their Parseval frame generators
author_facet Cabrelli, Carlos Alberto
Molter, Ursula Maria
author_sort Cabrelli, Carlos Alberto
title Optimal shift invariant spaces and their Parseval frame generators
title_short Optimal shift invariant spaces and their Parseval frame generators
title_full Optimal shift invariant spaces and their Parseval frame generators
title_fullStr Optimal shift invariant spaces and their Parseval frame generators
title_full_unstemmed Optimal shift invariant spaces and their Parseval frame generators
title_sort optimal shift invariant spaces and their parseval frame generators
publishDate 2007
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v23_n2_p273_Aldroubi
http://hdl.handle.net/20.500.12110/paper_10635203_v23_n2_p273_Aldroubi
work_keys_str_mv AT cabrellicarlosalberto optimalshiftinvariantspacesandtheirparsevalframegenerators
AT molterursulamaria optimalshiftinvariantspacesandtheirparsevalframegenerators
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