Möbius Invariants of Shapes and Images
Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformatio...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147846 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Möbius Invariants of Shapes and Images / S. Marsland, R.I. McLachlan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862740935736033280 |
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| author | Marsland, S. McLachlan, R.I. |
| author_facet | Marsland, S. McLachlan, R.I. |
| citation_txt | Möbius Invariants of Shapes and Images / S. Marsland, R.I. McLachlan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the Möbius group PSL(2,C), which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known Möbius invariants, and then develop an algorithm by which shapes can be recognised that is Möbius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a Möbius-invariant signature of grey-scale images.
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| first_indexed | 2025-12-07T20:17:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147846 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:17:07Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Marsland, S. McLachlan, R.I. 2019-02-16T09:14:36Z 2019-02-16T09:14:36Z 2016 Möbius Invariants of Shapes and Images / S. Marsland, R.I. McLachlan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 68T45; 68U10 DOI:10.3842/SIGMA.2016.080 https://nasplib.isofts.kiev.ua/handle/123456789/147846 Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the Möbius group PSL(2,C), which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known Möbius invariants, and then develop an algorithm by which shapes can be recognised that is Möbius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a Möbius-invariant signature of grey-scale images. This research was supported by the Marsden Fund, and RM by a James Cook Research Fellowship,
 both administered by the Royal Society of New Zealand. SM would like to thank the
 Erwin Schr¨odinger International Institute for Mathematical Physics, Vienna, where some of this
 research was performed. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Möbius Invariants of Shapes and Images Article published earlier |
| spellingShingle | Möbius Invariants of Shapes and Images Marsland, S. McLachlan, R.I. |
| title | Möbius Invariants of Shapes and Images |
| title_full | Möbius Invariants of Shapes and Images |
| title_fullStr | Möbius Invariants of Shapes and Images |
| title_full_unstemmed | Möbius Invariants of Shapes and Images |
| title_short | Möbius Invariants of Shapes and Images |
| title_sort | möbius invariants of shapes and images |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147846 |
| work_keys_str_mv | AT marslands mobiusinvariantsofshapesandimages AT mclachlanri mobiusinvariantsofshapesandimages |