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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Notes
- 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.
- 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.
- 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.
- 17.
Hence h 1 preserves the orientation of the arcs.
- 18.
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Apparent Contours of Embedded Surfaces. 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_3
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