Variational approach to eikonal function computation
The problem of calculating the eikonal function from the condition of focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is found that the cost function in the Monge-Kantorovich problem corresponds to the distance betwe...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Samara National Research University
2018-08-01
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| Series: | Компьютерная оптика |
| Subjects: | |
| Online Access: | http://computeroptics.smr.ru/KO/PDF/KO42-4/420405.pdf |
| _version_ | 1818274770509103104 |
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| author | Leonid Doskolovich Albert Mingazov Dmitry Bykov Evgeniy Andreev |
| author_facet | Leonid Doskolovich Albert Mingazov Dmitry Bykov Evgeniy Andreev |
| author_sort | Leonid Doskolovich |
| collection | DOAJ |
| description | The problem of calculating the eikonal function from the condition of focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is found that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the original region (in which the eikonal function is defined) and a point of the focal region. The formalism proposed in this work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. In this case, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed variational approaches are illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangular region. |
| first_indexed | 2024-12-12T22:19:08Z |
| format | Article |
| id | doaj.art-6ccb0799b9c5406484a65d72157bda12 |
| institution | Directory Open Access Journal |
| issn | 0134-2452 2412-6179 |
| language | English |
| last_indexed | 2024-12-12T22:19:08Z |
| publishDate | 2018-08-01 |
| publisher | Samara National Research University |
| record_format | Article |
| series | Компьютерная оптика |
| spelling | doaj.art-6ccb0799b9c5406484a65d72157bda122022-12-22T00:09:59ZengSamara National Research UniversityКомпьютерная оптика0134-24522412-61792018-08-0142455756710.18287/2412-6179-2018-42-4-557-567Variational approach to eikonal function computationLeonid Doskolovich0Albert Mingazov1Dmitry Bykov2Evgeniy Andreev3Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Samara, Russia; IPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001, Samara, RussiaIPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001, Samara, RussiaSamara National Research University, 34, Moskovskoye shosse, Samara, 443086, Samara, Russia; IPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001, Samara, RussiaSamara National Research University, 34, Moskovskoye shosse, Samara, 443086, Samara, Russia; IPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001, Samara, RussiaThe problem of calculating the eikonal function from the condition of focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is found that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the original region (in which the eikonal function is defined) and a point of the focal region. The formalism proposed in this work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. In this case, the calculation of the “ray mapping” corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed variational approaches are illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangular region.http://computeroptics.smr.ru/KO/PDF/KO42-4/420405.pdfgeometrical opticseikonal functionvariational problemMonge-Kantorovich mass transportation problem |
| spellingShingle | Leonid Doskolovich Albert Mingazov Dmitry Bykov Evgeniy Andreev Variational approach to eikonal function computation Компьютерная оптика geometrical optics eikonal function variational problem Monge-Kantorovich mass transportation problem |
| title | Variational approach to eikonal function computation |
| title_full | Variational approach to eikonal function computation |
| title_fullStr | Variational approach to eikonal function computation |
| title_full_unstemmed | Variational approach to eikonal function computation |
| title_short | Variational approach to eikonal function computation |
| title_sort | variational approach to eikonal function computation |
| topic | geometrical optics eikonal function variational problem Monge-Kantorovich mass transportation problem |
| url | http://computeroptics.smr.ru/KO/PDF/KO42-4/420405.pdf |
| work_keys_str_mv | AT leoniddoskolovich variationalapproachtoeikonalfunctioncomputation AT albertmingazov variationalapproachtoeikonalfunctioncomputation AT dmitrybykov variationalapproachtoeikonalfunctioncomputation AT evgeniyandreev variationalapproachtoeikonalfunctioncomputation |