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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1702.02780 |
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| _version_ | 1866916890628063232 |
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| author | Benn, James Marsland, Stephen McLachlan, Robert I Modin, Klas Verdier, Olivier |
| author_facet | Benn, James Marsland, Stephen McLachlan, Robert I Modin, Klas Verdier, Olivier |
| contents | The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1702_02780 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Currents and finite elements as tools for shape space Benn, James Marsland, Stephen McLachlan, Robert I Modin, Klas Verdier, Olivier Numerical Analysis 32U40, 62M40, 65D18, 74S05 The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples. |
| title | Currents and finite elements as tools for shape space |
| topic | Numerical Analysis 32U40, 62M40, 65D18, 74S05 |
| url | https://arxiv.org/abs/1702.02780 |