The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis
We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of t...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149235 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis / M. Boutin, S. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862597649352359936 |
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| author | Boutin, M. Huang, S. |
| author_facet | Boutin, M. Huang, S. |
| citation_txt | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis / M. Boutin, S. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The motivating application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries.
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| first_indexed | 2025-11-27T17:06:57Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149235 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T17:06:57Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Boutin, M. Huang, S. 2019-02-19T19:04:43Z 2019-02-19T19:04:43Z 2013 The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis / M. Boutin, S. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E05; 57S25; 68T10 DOI: http://dx.doi.org/10.3842/SIGMA.2013.031 https://nasplib.isofts.kiev.ua/handle/123456789/149235 We define the Pascal triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The motivating application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 This research was supported in parts by NSF grant CCF-0728929 en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis Article published earlier |
| spellingShingle | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis Boutin, M. Huang, S. |
| title | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis |
| title_full | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis |
| title_fullStr | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis |
| title_full_unstemmed | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis |
| title_short | The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis |
| title_sort | pascal triangle of a discrete image: definition, properties and application to shape analysis |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149235 |
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