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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2013-04-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2013.031 |
| _version_ | 1811192738815672320 |
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| author | Mireille Boutin Shanshan Huang |
| author_facet | Mireille Boutin Shanshan Huang |
| author_sort | Mireille Boutin |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-04-11T23:57:27Z |
| format | Article |
| id | doaj.art-6350785d7ce5490592f54b76cbf76c99 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-04-11T23:57:27Z |
| publishDate | 2013-04-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-6350785d7ce5490592f54b76cbf76c992022-12-22T03:56:20ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592013-04-01903110.3842/SIGMA.2013.031The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape AnalysisMireille BoutinShanshan HuangWe 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.http://dx.doi.org/10.3842/SIGMA.2013.031momentssymmetry detectionmoving frameshape recognition |
| spellingShingle | Mireille Boutin Shanshan Huang The Pascal Triangle of a Discrete Image: Definition, Properties and Application to Shape Analysis Symmetry, Integrability and Geometry: Methods and Applications moments symmetry detection moving frame shape recognition |
| 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 |
| topic | moments symmetry detection moving frame shape recognition |
| url | http://dx.doi.org/10.3842/SIGMA.2013.031 |
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