Object-Image Correspondence for Algebraic Curves under Projections
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camer...
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| Дата: | 2013 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149227 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Object-Image Correspondence for Algebraic Curves under Projections / J.M. Burdis, I.A. Koga, H. Hong // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149227 |
|---|---|
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dspace |
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irk-123456789-1492272019-02-20T01:23:08Z Object-Image Correspondence for Algebraic Curves under Projections Burdis, J.M. Kogan, I.A. Hong, H. We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane. 2013 Article Object-Image Correspondence for Algebraic Curves under Projections / J.M. Burdis, I.A. Koga, H. Hong // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H50; 14Q05; 14L24; 53A55; 68T45 DOI: http://dx.doi.org/10.3842/SIGMA.2013.023 http://dspace.nbuv.gov.ua/handle/123456789/149227 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane. |
| format |
Article |
| author |
Burdis, J.M. Kogan, I.A. Hong, H. |
| spellingShingle |
Burdis, J.M. Kogan, I.A. Hong, H. Object-Image Correspondence for Algebraic Curves under Projections Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Burdis, J.M. Kogan, I.A. Hong, H. |
| author_sort |
Burdis, J.M. |
| title |
Object-Image Correspondence for Algebraic Curves under Projections |
| title_short |
Object-Image Correspondence for Algebraic Curves under Projections |
| title_full |
Object-Image Correspondence for Algebraic Curves under Projections |
| title_fullStr |
Object-Image Correspondence for Algebraic Curves under Projections |
| title_full_unstemmed |
Object-Image Correspondence for Algebraic Curves under Projections |
| title_sort |
object-image correspondence for algebraic curves under projections |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149227 |
| citation_txt |
Object-Image Correspondence for Algebraic Curves under Projections / J.M. Burdis, I.A. Koga, H. Hong // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 33 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:32:17Z |
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2023-05-20T17:32:17Z |
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