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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Notes
- 1.
- 2.
- 3.
- 4.
Remember that, by definition, the regions are open.
- 5.
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 \(i = 1,\ldots,m\).
- 6.
- 7.
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.
- 8.
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 \(i_{-+}\).
- 9.
The case of the second picture of Fig. 3.11 can be treated in a similar manner.
- 10.
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.
- 11.
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.
- 12.
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.
- 13.
We suppose, as usual, that | z | < 1.
- 14.
The reason being that the function \(\rho \in (0, +\infty ) \rightarrow \sqrt{\rho }\) is of class \(\mathcal{C}^{\infty }\).
- 15.
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.
- 16.
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.
- 17.
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 \(\delta ^{{\prime}}(0) =\lim _{z\rightarrow 0^{+}}\delta ^{{\prime}}(z) = \sqrt{\frac{\theta ^{{\prime\prime} } (0)} {2}} > 0\).
- 18.
Recall that \(\zeta _{i}^{+} =\zeta _{ i}^{-}\) at (x 1, g a (x 1)).
- 19.
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 \(x \in \Omega \).
- 20.
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.
- 21.
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 \(\Sigma \).
- 22.
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.
- 23.
Compare with Remark 3.3.2: function f, on \(G_{\Sigma }\), counts the actual number of intersections of the light ray with the surfaces.
- 24.
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.
- 25.
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.
- 26.
Recall that, by definition, the arcs are relatively open.
- 27.
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.
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Topological Reconstruction of a Three-Dimensional Scene. In: Shape Reconstruction from Apparent Contours. Computational Imaging and Vision, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45191-5_5
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