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Erschienen in:
A Variational Model on Labelled Graphs with Cusps and Crossings Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

6. Completeness of Reidemeister-Type Moves on Labelled Apparent Contours

verfasst von : Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

Erschienen in: Shape Reconstruction from Apparent Contours

Verlag: Springer Berlin Heidelberg

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Abstract

In this chapter we illustrate the results and report the figures from the paper [3]. More specifically, we shall prove that there exists a finite set of simple, or elementary, moves (also called rules) on labelled apparent contours, such that the following property holds: the images \(\Sigma _{1}\) and \(\Sigma _{2}\) of two stable embeddings of a closed smooth (not necessarily connected) surface M in \(\mathbb{R}^{3}\) are isotopic if and only if their apparent contours can be connected using finitely many isotopies of \(\mathbb{R}^{2}\), and a finite sequence of elementary moves or of their inverses (sometimes called “reverses”).

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Vorheriges Kapitel Topological Reconstruction of a Three-Dimensional Scene
Nächstes Kapitel Invariants of an Apparent Contour
Fußnoten
1
See Theorem 6.0.3 and Corollary 6.6.5 for a precise statement.
 
2
Namely, the fact that there are no other moves, besides those in the list of Sect. 6.1, necessary to connect two apparent contours of isotopic surfaces.
 
3
In [6, Definition 2] a different notion of equivalence between maps is introduced. Such a definition can be more suitable when the target space is the cartesian product of a two-dimensional manifold with \(\mathbb{R}\).
 
4
A realization in space of these moves involves two folds of the surface which can be “far one from the other”.
 
5
The move L can be realized in space by considering the surface in Fig. 1.​4, by gradually reducing the “hill”. The inverse of a move B can be realized by straightening the central part of a depression in a long “wave” with two parallels arcs corresponding to the crease and the valley of the wave.
 
6
Compare also with Definition 8.​1.​1.
 
7
See Chap. 5
 
8
See Theorem 2.​1.​14.
 
9
See, e.g., [9] and the references therein, or also [15, p. 597, 600, 601]. Usually, a stratified Morse function takes real values: for technical reasons, we consider here the slightly different case of a function taking values in \(\mathbb{S}^{1}\), but the definition is essentially the same.
 
10
Hence, the critical points of \(u_{\vert Y _{j}}^{}\) are nondegenerate.
 
11
By definition, points of Y 3 are considered as critical points of u.
 
12
The converse statement also holds true, as a consequence of the Isotopy Extension Theorem (see, for instance, [11, Theorem 1.3, p. 180], see also [19, pp. 157–201]). Namely, suppose that γ, e1 and e2 are as in (6.5). Then, γ induces an isotopy from e1(M) to \(\mathbb{R}^{3}\), which extends to an \(\mathbb{R}^{3}\)-ambient isotopy with compact support.
 
13
Notice that the map \(\alpha \in \mathcal{C}^{\infty }(\mathcal{S}, \mathbb{R}^{3}) \rightarrow F_{\alpha } \in \mathcal{C}^{\infty }(\mathcal{S},\mathcal{T} )\) is continuous.
 
14
Note also that, defining \(f_{\varphi _{t}}\) as in (2.​2), we have \(f_{\varphi _{t}}(x) = \#\{m \in M: F_{\gamma }(m,t) = (x,t)\}\) for any \((x,t) \in \mathcal{T}\).
 
15
For t ∈ [0, 1], we use the notation \(\tilde{\gamma }_{t}(\cdot ) =\tilde{\gamma } (\cdot,t)\) .
 
16
See, for instance, [8, p. 176, 177], and more generally [13, 14].
 
17
See [18, Proposition 5.4] for related problems.
 
18
To be more specific, [4, Fig. 9(row 1, column 1)] corresponds to move L; [4, Fig. 9(2,1)] corresponds to move B; [4, Fig. 9(3,1)] corresponds to move S; [4, Fig. 10(1,1)] corresponds to move K; [4, Fig. 10(1,2)] corresponds to move C; [4, Fig. 10(2,1)] corresponds to move T. Theorem [4, 3.2.3] is a simpler version of [4, Theorem 3.5.5], in which the height function is not considered, and hence is more close to Corollary 6.6.2. It follows by combining the classifications given by [10, 16, 21].
 
Literatur
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Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Boston (1985)CrossRef
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Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vassiliev, V.A. (eds.): Dynamical Systems VIII. Singularity Theory. II. Applications. Springer, Berlin (1993)
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Bellettini, G., Beorchia, V., Paolini, M.: Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22, 1–19 (2011)
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Carter, J.S., Rieger, J.H., Saito, M.: A combinatorial description of knotted surfaces and their isotopies. Adv. Math. 127, 1–51 (1997)CrossRefMATHMathSciNet
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Favaro, L.A., Mendes, C.M.: Global stability for diagrams of differentiable applications. Ann. Inst. Fourier (Grenoble) 36, 133–153 (1986)CrossRefMATHMathSciNet
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Fisher, G.M.: On the group of all homeomorphisms of a manifold. Trans. Am. Math. Soc. 97, 193–212 (1960)CrossRefMATH
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Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York/Heidelberg/Berlin (1974)
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Kosinski, A.A.: Differential Manifolds. Academic, New York (1993)MATH
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Morin, B.: Formes canoniques des singularities d’une application différentiable. C. R. Acad. Sci. Paris 260, 5662–5665 (1965)MATHMathSciNet
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Morin, B.: Formes canoniques des singularities d’une application différentiable. C. R. Acad. Sci. Paris 260, 6503–6506 (1965)MathSciNet
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Pignoni, R.: Density and stability of Morse functions on a stratified space. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6, 593–608 (1979)MATHMathSciNet
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Rieger, J.H.: On the complexity and computation of view graph of piecewise smooth algebraic surfaces. Phil. Trans. R. Soc. Lond. A 354, 1899–1940 (1996)CrossRefMATHMathSciNet
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Roseman, D.: Reidemeister-type moves for surfaces in four dimensional space. In: Jones, V.F.R., et al. (eds.) Knot Theory. Proceedings of the Mini-Semester, Warsaw, Poland 1995, vol. 42, pp. 347–380. Banach Center Publications, Warsaw (1998)
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Thom, R.: La classifications des immersions. Seminaire Bourbaki, 157-01–157-11 (1957)
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West, J.M.: The differential geometry of the crosscap. Ph.D. thesis, University of Liverpool, England (1995)
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Whitney, H.: Local properties of analytic varieties. In: Cairns, S.S. (ed.) Differential and Combinatorial Topology, pp. 205–244. Princeton University Press, Princeton (1965)
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Metadaten
Titel
Completeness of Reidemeister-Type Moves on Labelled Apparent Contours
verfasst von
Giovanni Bellettini
Valentina Beorchia
Maurizio Paolini
Franco Pasquarelli
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Buch
Shape Reconstruction from Apparent Contours

Print ISBN: 978-3-662-45190-8

Electronic ISBN: 978-3-662-45191-5

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-662-45191-5_6

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