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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todo:paper_10635203_v23_n2_p273_Aldroubi2023-10-03T16:01:10Z Optimal shift invariant spaces and their Parseval frame generators Aldroubi, A. Cabrelli, C. Hardin, D. Molter, U. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_10635203_v23_n2_p273_Aldroubi |
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Universidad de Buenos Aires |
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I-28 |
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R-134 |
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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. |
| format |
JOUR |
| author |
Aldroubi, A. Cabrelli, C. Hardin, D. Molter, U. |
| spellingShingle |
Aldroubi, A. Cabrelli, C. Hardin, D. Molter, U. Optimal shift invariant spaces and their Parseval frame generators |
| author_facet |
Aldroubi, A. Cabrelli, C. Hardin, D. Molter, U. |
| author_sort |
Aldroubi, A. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_10635203_v23_n2_p273_Aldroubi |
| work_keys_str_mv |
AT aldroubia optimalshiftinvariantspacesandtheirparsevalframegenerators AT cabrellic optimalshiftinvariantspacesandtheirparsevalframegenerators AT hardind optimalshiftinvariantspacesandtheirparsevalframegenerators AT molteru optimalshiftinvariantspacesandtheirparsevalframegenerators |
| _version_ |
1807316565834596352 |