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2015 | OriginalPaper | Buchkapitel

5. Topological Reconstruction of a Three-Dimensional Scene

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

Following closely [1],^, in this chapter we characterize those planar graphs contained in \(\Omega \) that are apparent contours of a stable smooth 3D scene \(E \subset Q = \Omega \times (-1,1)\). As we shall see, the conditions imposed on a graph for being a complete labelled contour graph are sufficient for our purposes.

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Vorheriges Kapitel Solving the Completion Problem

Nächstes Kapitel Completeness of Reidemeister-Type Moves on Labelled Apparent Contours

With kind permission from Springer Science+Business Media, in this chapter and in Chap. 11 we illustrate some results and report some of the figures from the quoted paper [1].

See also the panelling construction in [11] and [2‐6, 12].

See again [11, 12], and also [8].

Remember that, by definition, the regions are open.

We recall that, if \(X_{1},\ldots,X_{m}\) are sets, the disjoint union ∐_i = 1 ^m X _i is defined as \(\cup _{i=1}^{m}\{(x,i): x \in X_{i}\} = \cup _{i=1}^{m}(X_{i} \times \{ i\})\), and the disjoint union topology on ∐_i = 1 ^m X _i is defined as follows: A ⊂ ∐_i = 1 ^m X _i is open if A ∩ (X _i ×{ i}) is open for any \(i = 1,\ldots,m\).

Remember from Definitions 4.2.1 and 4.2.2 that f(x) is independent of the choice of x ∈ R _i.

Recall that a stratum of R _i is a pair (R _i, r) with \(r \in \{ 1,\ldots,f(R_{i})\}\); see Definition 3.3.1.

For the glueing concerning the penultimate picture of Fig. 3.11, it is sufficient to repeat items 3.1–3.5, with \(i_{+-}\) replaced by \(i_{-+}\).

The case of the second picture of Fig. 3.11 can be treated in a similar manner.

That is, if \(q: D \rightarrow \mathcal{T}\) is the quotient map, then \(U \subseteq \mathcal{T}\) is open if and only if q ⁻¹(U) is open in D.

Observe that z _r ∈ (−1, 1), so that all points that we consider belong to Q. Moreover \(z_{r_{1}} < z_{r_{2}}\) if \(r_{1},r_{2} \in \{ 1,\ldots,f(0, 0)\}\) and r ₁ < r ₂.

If f _min > 0, it is enough to consider the strata that are transverse in correspondence of U, and that are either in front of a parametrized stratum, or behind it, in dependence of the index r and of the value of d.

We suppose, as usual, that | z | < 1.

The reason being that the function \(\rho \in (0, +\infty ) \rightarrow \sqrt{\rho }\) is of class \(\mathcal{C}^{\infty }\).

For simplicity, here f takes odd positive integer values: in order our discussion to be included in the standard framework where f takes values in \(2\mathbb{N}\), it is enough to add a transversal layer at the proper depth.

For example, let us check that \(h \in \mathcal{C}^{1}(I)\). For z ≠ 0 we have \(h^{{\prime}}(z) = \frac{z\theta ^{{\prime}}(z)-2\theta (z)} {z^{3}}\), so that, applying twice de l’Hôpital’s theorem, we have \(\lim _{z\rightarrow 0}h^{{\prime}}(z) =\lim _{z\rightarrow 0}\frac{\theta ^{{\prime\prime\prime}}(z)} {6} = \frac{\theta ^{{\prime\prime\prime}}(0)} {6}\), and therefore h is differentiable at the origin. In a similar manner, one proves that all derivatives of h are continuous in I.

For z > 0 we have \(\theta (z) = \frac{z^{2}} {2} \theta ^{{\prime\prime}}(\tau )\) and θ ^′(z) = z θ ^{′ ′}(ν), for two suitable points τ, ν ∈ (0, z). Hence \(\delta ^{{\prime}}(z) = \frac{\theta ^{{\prime}}(z)} {2\sqrt{\theta (z)}} = \frac{\theta ^{{\prime\prime}}(\nu )} {\sqrt{2\theta ^{{\prime\prime} } (\tau )}}\), and therefore \(\delta ^{{\prime}}(0) =\lim _{z\rightarrow 0^{+}}\delta ^{{\prime}}(z) = \sqrt{\frac{\theta ^{{\prime\prime} } (0)} {2}} > 0\).

Recall that \(\zeta _{i}^{+} =\zeta _{ i}^{-}\) at (x ₁, g _a(x ₁)).

We recall that if \(A_{1},\ldots,A_{n}\) is a finite covering of \(\Omega \), a partition of unity subordinated to the covering is given by a family of \(\mathcal{C}^{\infty }\) functions \(\lambda _{1},\ldots,\lambda _{n}: \Omega \rightarrow [0, 1]\) such that \(\sum _{i=1}^{n}\lambda _{i}(x) = 1\) for any \(x \in \Omega \).

Generic here means the following: the curve has only a finite number of intersection with the image of the embedding, and each intersection is transverse.

Since \(\Sigma \) is orientable, also M turns out to be orientable; in this book, we shall always choose the orientation on M consistently with the induced orientation on \(\Sigma \).

An example of a nontrivial covering can be constructed by taking the Klein bottle as M, constructed as the square [0, 1] × [0, 1] with identification of the two horizontal sides, the two vertical sides are also identified but with reversed orientation: (0, m ₂) is identified with (1, 1 − m ₂). The map \(\varphi\) can then be constructed as \((m_{1},m_{2}) \in M \rightarrow \rho (\cos \theta,\sin \theta )\) with θ = 2π m ₁ and \(\rho = 3 +\cos (2\pi m_{2})\). The apparent contour consists of two concentric circles of radii 2 and 4.

Compare with Remark 3.3.2: function f, on \(G_{\Sigma }\), counts the actual number of intersections of the light ray with the surfaces.

This can be done because the arcs of G are of class \(\mathcal{C}^{\infty }\), and in proximity of a cusp we are considering the region where f = f _min.

Namely, in the list of the z-coordinates of all intersections of ∂ E _h with π ⁻¹(y), we now insert also the depth \(\hat{z}_{-}^{h}(y)\) of the fictitious point.

Recall that, by definition, the arcs are relatively open.

If cusps are not present, the quotient space \(\mathcal{T}\) is a compact Hausdorff space, which is locally homeomorphic to an open 2-ball (see, e.g., [9, 12 Thms. 74.1, 77.5]). In case of a connected quotient surface \(\mathcal{T}\), we also recall (see, e.g., [10, Thm. 1]) that the genus of a single compact surface embedded in \(\mathbb{R}^{3}\) can be computed from the apparent contour.

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

Cipolla, R., Giblin, P.: Visual Motion of Curves and Surfaces. Cambridge University Press, Cambridge (1999)

Golubyatnikov, V.P.: On reconstruction of transparent surfaces from their apparent contours. J. Inverse Ill-Posed Probl. 6, 395–401 (1998)CrossRefMATHMathSciNet

Golubyatnikov, V.P.: Uniqueness questions in reconstruction multidimensional objects from tomography-type projection data. In: Inverse and Ill-Posed Problems Series. VSP, Utrecht (2000)CrossRef

Golubyatnikov, V.P., Karaca, I., Ozyilmaz, E., Tantay, B.: On determining the shapes of hypersurfaces from the shapes of their apparent contours and symplectic geometry measurements. Sib. Adv. Math. 10, 9–15 (2000)MATHMathSciNet

Golubyatnikov, V.P., Pekmen, U., Karaca, I., Ozyilmaz, E., Tantay, B.: On reconstruction of surfaces from their apparent contours and the stationary phase observations. In: Proceedings of International Conference on Shape Modeling and Applications (Shape Modeling International ’99), pp. 116–120 (1999)

Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, Prentice-Hall (1974)MATH

Karpenko, O.A., Hughes, J.F.: SmoothSketch: 3D free-form shapes from complex sketches. In: The 33rd International Conference and Exhibition on Computer Graphics and Imaging Techniques, Boston, Massachusetts, SIGGRAPH 2006, pp. 589–598. ACM, New York (2006)

Munkres, J.R.: Topology, vol. xvi, 2nd. edn. Upper Saddle River, Prentice Hall (2000)

10.

Pignoni, R.: On surfaces and their contour. Manuscripta Math. 72, 223–249 (1991)CrossRefMATHMathSciNet

11.

Williams, L.R.: Perceptual completion of occluded surfaces. Ph.D. dissertation, Department of Computer Science, University of Massachusetts, Amherst (1994)

12.

Williams, L.R.: Topological reconstruction of a smooth manifold-solid from its occluding contour. Int. J. Comput. Vision 23, 93–108 (1997)CrossRef

Titel: Topological Reconstruction of a Three-Dimensional Scene
verfasst von: Giovanni Bellettini
Valentina Beorchia
Maurizio Paolini
Franco Pasquarelli
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_5

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