Abstract
Following [1], in this chapter we show how to solve the completion problem, namely we characterize those oriented plane graphs that are visible part of an apparent contour (Theorem 4.3.1). The proof is generalized to the case where the background is not reduced to the external region. In our presentation we need some elementary concepts of the theory of oriented graphs, and the Morse description of a graph, as outlined in Sect. 2.5.3 In Chap. 9 we describe a code that automates the construction of the proof of Theorem 4.3.1.
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Notes
- 1.
With kind permission from SIAM Journal on Imaging Sciences: in this chapter we illustrate the results and report some of the figures from the quoted paper [1].
- 2.
In particular, the local and global structures of an apparent contour and the labelling.
- 3.
As in Chap. 2, an arc of H is relatively open.
- 4.
The presence of closed arcs makes our notion of graph different from the classical one.
- 5.
The image of [0, 1] is an oriented arc (possibly oriented loop) and the image of \(\mathbb{S}^{1}\) is an oriented closed arc.
- 6.
Compare Sect. 2.2.
- 7.
- 8.
- 9.
It is not difficult to see that the conditions defining a visible contour graph K are necessary for K to be the visible part of an apparent contour; compare, for instance, Sect. 3.6.
- 10.
- 11.
Recall also that \(\mathfrak{m}_{0}([0, 1]) \cap K = \mathfrak{m}_{1}([0, 1]) \cap K = \varnothing \).
- 12.
- 13.
We need to avoid that the Morse line passing through p has locally both arcs of the completed contour on the same side, an inconvenient fact since it makes p to behave like a local maximum/minimum which is undesirable, since p (being a node of K) already forces the Morse line through p to be critical.
- 14.
For example, if the dangling arc is oriented downwards (respectively upwards), the local labelling at the corresponding crossing is the same as in the penultimate (respectively last) picture of Fig. 3.11.
- 15.
If the dangling arc is oriented downwards, then the new formed cusp points on the left.
- 16.
Notice that in the proof of Theorem 4.3.1 the connectedness of ext(K) is not used.
- 17.
With a left–right reflection. Notice that reflecting the values of f still results in an orientation of the two arcs from right to left.
References
Bellettini, G., Beorchia, V., Paolini, M.: Completion of visible contours. SIAM J. Imag. Sci. 2, 777–799 (2009)
Karpenko, O.A., Hughes, J.F.: SmoothSketch: 3D free-form shapes from complex sketches, The 33rd. International Conference and Exhibition on Computer Graphics and Imaging Techniques, Boston, Massachusetts, 2006, SIGGRAPH 2006, pp. 589–598. ACM, New York (2006)
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Solving the Completion Problem. 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_4
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DOI: https://doi.org/10.1007/978-3-662-45191-5_4
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