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

2015 | OriginalPaper | Buchkapitel

3. Apparent Contours of Embedded Surfaces

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 adapt the notions introduced in Chap. 2 to the special case of the apparent contour of a smooth, possibly nonconnected, compact surface \(\Sigma \) without boundary embedded in \(\mathbb{R}^{3}\). Embeddedness allows to enrich an apparent contour with a labelling, which, in particular, permits to define the visible contour.

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Vorheriges Kapitel Stable Maps and Morse Descriptions of an Apparent Contour
Nächstes Kapitel Solving the Completion Problem
Fußnoten
1
Recall that closed here means compact without boundary.
 
2
We advise the reader that in some of the figures, it will be convenient to imagine the x 1 x 2 plane as horizontal, with the z direction being vertical, and e 3 pointing downwards.
 
3
We shall show a completion theorem starting only from the visible part of the apparent contour.
 
4
We shall show a reconstruction theorem of a three-dimensional shape starting from the knowledge of the whole apparent contour, and of a consistent labelling on it.
 
5
Following, e.g., [9], a smooth embedding of M into \(\mathbb{R}^{3}\) is a smooth injective map having differential of rank 2 (maximal) at all points of M.
 
6
See [13] for stability theorems of composite mappings.
 
7
This could be achieved also in terms of small changes in the viewing direction, as in [16, 17].
 
8
The critical curve divides, with the terminology of [16], the “anterior” surfaces from the “posterior” ones.
 
9
Compare with Remark 2.​2.​6 which deals with a more general case.
 
10
For simplicity, in the present example, the set E is not disjoint from the retinal plane. Clearly, this assumption is irrelevant.
 
11
Note that e 3 is the tangent vector to the critical curve at the origin, thus coinciding with the kernel of \(d\pi _{\vert \Sigma }\).
 
12
The function \(d_{\Sigma }\) is not defined at a crossing, where it could be defined as a multifunction taking two nonnegative integer values: we shall not need such an extension.
 
13
For results concerning the number of cusps of apparent contours in general contexts, see [15, p. 409], [14, p. 84].
 
14
As already anticipated in the Introduction, the existence of a labelling satisfying all compatibility conditions makes possible the construction of an abstract smooth surface M and a smooth embedding \(\mathrm{e}: M \rightarrow \mathbb{R}^{3}\) so that \(\Sigma = \mathrm{e}(M)\); see Theorem 5.​3.​1 for a precise statement.
 
15
Recall that a closed surface embedded in \(\mathbb{R}^{3}\) encloses an interior, hence an outward normal is well defined; see, for instance, [10, p. 89].
 
16
Notice carefully that the signs + and − in Fig. 8.​1 refer to the embedding sign of a cusp Definition 8.​1.​2), and not to the notion of positivity and negativity of Definition 2.​2.​12.
 
17
Hence h 1 preserves the orientation of the arcs.
 
18
This list of properties originates the definition of visible contour graph given in Chap. 4: indeed, the visible contour is a visible contour graph in the sense of Definition 4.​1.​8.
 
Literatur
1.
Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vassiliev, V.A.: Dynamical Systems VIII. In: Arnold, V.I. (ed.) Singularity Theory. II. Applications. Springer, Berlin (1993)
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Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, Volume I. Birkhäuser, Boston (1985)CrossRefMATH
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Bellettini, G., Beorchia, V., Paolini, M.: Topological and variational properties of a model for the reconstruction of three-dimensional transparent images with self-occlusions. J. Math. Imaging Vision 32, 265–291 (2008)CrossRefMathSciNet
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Bruce, J.W., Giblin, P.J.: Curves and Singularities. A Geometrical Introduction to Singularity Theory, 2nd edn. Cambridge University Press, Cambridge (1992)
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Carter, J.S., Kamada, S., Saito, M.: Surfaces in 4-Space, Encyclopaedia of Mathematical Sciences, vol. 142. Springer, New York (2004)
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Carter, J.S., Saito, M.: Knotted Surfaces and Their Diagrams, Mathematical Surveys Monographs, vol. 55. American Mathematical Society, Providence, RI (1998)
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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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Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, NJ/Prentice-Hall (1974)MATH
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Huffman, D.A.: Impossible objects as nonsense sentences. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 6. American Elsevier, New York (1971)
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Koenderink, J.J.: Solid Shape. MIT, Cambridge, MA/London (1990)
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Whitney, H.: On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)MATHMathSciNet
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Williams, L.R.: Perceptual completion of occluded surfaces. Ph.D. dissertation, Department of Computer Science, University of Massachusetts, Amherst, MA (1994)
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Williams, L.R.: Topological reconstruction of a smooth manifold-solid from its occluding contour. Int. J. Comput. Vision 23, 93–108 (1997)CrossRef
Metadaten
Titel
Apparent Contours of Embedded Surfaces
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_3