| Title:
|
Closed surfaces with different shapes that are indistinguishable by the SRNF (English) |
| Author:
|
Klassen, Eric |
| Author:
|
Michor, Peter W. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
107-114 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in $\mathbb{R}^3$, and equipping the set of these immersions with a “distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of $\mathbb{R}^3$. Thus, it induces a distance function on the shape space of immersions, i.e., the space of immersions modulo reparametrizations and rigid motions of $\mathbb{R}^3$. In this paper, we give examples of the degeneracy of this distance function, i.e., examples of immersed surfaces (some closed and some open) that have the same SRNF, but are not the same up to reparametrization and rigid motions. We also prove that the SRNF does distinguish the shape of a standard sphere from the shape of any other immersed surface, and does distinguish between the shapes of any two embedded strictly convex surfaces. (English) |
| Keyword:
|
shape space |
| Keyword:
|
square root normal field |
| MSC:
|
53A05 |
| MSC:
|
58D15 |
| idZBL:
|
Zbl 07217116 |
| idMR:
|
MR4115086 |
| DOI:
|
10.5817/AM2020-2-107 |
| . |
| Date available:
|
2020-05-21T08:50:14Z |
| Last updated:
|
2020-08-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148136 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Hirsch, M.W.: Differential Topology.Springer-Verlag, 1996. MR 1336822 |
| Reference:
|
[4] Jermyn, I., Kurtek, S., Laga, H., Srivastava, A: Elastic shape analysis of three-dimensional objects.Synthesis Lectures on Computer Vision 7 (2017), 1–185. 10.2200/S00785ED1V01Y201707COV012 |
| Reference:
|
[5] Jermyn, I.H., Kurtek, S., Klassen, E., Srivastava, A.: Elastic shape matching of parameterized surfaces using square root normal fields.Computer Vision – ECCV 2012 (2012), 804–817. |
| Reference:
|
[6] Laga, H., Qian, X., Jermyn, I., Srivastava, A.: Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces.IEEE Transactions on Pattern Analysis and Machine Intelligence 39 (2016), 2451–2464. 10.1109/TPAMI.2016.2647596 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
