Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
shape space; square root normal field
shape space; square root normal field
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.
References:
[1] Aledo, J.A., Alias, L.J., Romero, A.: A New Proof of Liebmann Classical Rigidity Theorem for Surfaces in Space Forms. Rocky Mountain J. Math. 35 (6) (2005), 1811–1824. DOI 10.1216/rmjm/1181069618 | MR 2210636
[2] Bauer, M., Charon, N., Harms, P.: Inexact Elastic Shape Matching in the Square Root Normal Field Framework. Geometric Science of Information (Nielsen, F., Barbaresco, F., eds.), 2019, pp. 13–21.
[3] Hirsch, M.W.: Differential Topology. Springer-Verlag, 1996. MR 1336822
[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. DOI 10.2200/S00785ED1V01Y201707COV012
[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.
[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. DOI 10.1109/TPAMI.2016.2647596
[7] Michor, P.W.: Topics in differential geometry. Graduate Studies in Mathematics, vol. 93, American Mathematical Society, Providence, RI, 2008. DOI 10.1090/gsm/093 | MR 2428390
[8] Michor, P.W.: Manifolds of mappings for continuum mechanics. Geometric Continuum Mechanics (Segev, R., Epstein, M., eds.), Birkhäuser, June 2020, arxiv:1909.00445, pp. 3–75. MR 2605800
[9] Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57 (4) (1903), 447–495. DOI 10.1007/BF01445180 | MR 1511220
Search
Partner of
