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

2015 | OriginalPaper | Buchkapitel

11. Variational Analysis of the Model on Labelled Graphs

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, essentially following [2],1 we discuss some coerciveness and semicontinuity properties of the functional \(\mathcal{F}\) introduced in Sect. 1.5 and motivating our study of apparent contours and three-dimensional shapes.

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Vorheriges Kapitel The Program “Appcontour”: User’s Guide
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Fußnoten
1
With kind permission from Springer Science+Business Media, in this chapter we report some of the results and figures from the quoted paper [2].
 
2
We recall that the model described in Sect. 1.​5 requires p ∈ (1, 2). The results of the present chapter hold under the less stringent assumption (11.1).
 
3
See, e.g., [4, Theorem 8.8] and [5, Corollary 8.31].
 
4
Accordingly, a subinterval of \([0,\mathcal{L}]\) corresponds to a connected subset of \(\mathbb{R}/(\mathcal{L}\mathbb{Z})\); we shall adopt this convention, for instance, when integrating functions on such subintervals.
 
5
See Definition 4.​1.​4.
 
6
We could equivalently require that the opposite arcs join in a \(\mathcal{C}^{1}\) way: indeed, if \(0 < \mathcal{L}_{1} < \mathcal{L}_{2}\), and \(\gamma \in W^{2,p}((0,\mathcal{L}_{1}), \mathbb{R}^{2}) \cap W^{2,p}((\mathcal{L}_{1},\mathcal{L}_{2}), \mathbb{R}^{2}) \cap \mathcal{C}^{1}((0,\mathcal{L}_{2}), \mathbb{R}^{2})\), an integration by parts shows that \(\gamma \in W^{2,p}((0,\mathcal{L}_{2}), \mathbb{R}^{2})\).
 
7
Referring to the final discussion in Sect. 1.​5, suppose that the infimization of \(\mathcal{F}\) has a solution; then condition (G5) in Definition 4.​2.​2 is not necessarily satisfied. As a consequence, if we adapt the proofs of Theorems 5.​1.​1 and 5.​1.​4 to this case, the reconstructed three-dimensional scene E is not necessarily of class \(\mathcal{C}^{\infty }\) anymore.
 
8
A sequence (K n ) of compact subsets of the plane converges to K in the sense of Kuratowski, if the following two conditions hold:
  • any x ∈ K is the limit of a sequence (x n ) with x n  ∈ K n for any \(n \in \mathbb{N}\),
  • if x n  ∈ K n for any \(n \in \mathbb{N}\), then any limit point of (x n ) belongs to K.
See [8, Chapter 2, par. 20, Section VI] and [6, Definition 4.10] for more information.
 
9
See also [1, Theorem 6.1].
 
10
The bound (11.15) can be proven with an inequality similar to that used in (11.37) below, which implies that a complete turn without cusps around a point has a fixed cost in terms of the action. As a consequence, an unbounded number (as n → +∞) of complete turns is forbidden, in view of (11.8). If cusps are present, it is sufficient to recall that their number must be uniformly bounded with respect to n, again due to assumption (11.8).
 
11
See [3, Lemma 3.3 (iii)] for similar arguments.
 
12
See, for instance, [7, Theorem 1, Section 1.9].
 
13
For instance, in the Hausdorff distance.
 
14
Sometimes called also (sequential) relaxation of \(\mathcal{F}\).
 
15
See, for instance, [1, Theorem 3.23].
 
16
Equality in (11.25) in general does not hold, as in the case of Fig. 11.2 with the choices \(\mathrm{d}_{1} =\mathrm{ d}_{2} =\mathrm{ d}_{3} =\mathrm{ d}_{4} =\mathrm{ d}_{5} =\mathrm{ d}_{6} = 0\).
 
17
See Definition 11.A.3, below.
 
18
It is also of class \(\mathcal{C}^{1}(\overline{\mathrm{I}})\), where I is any connected component of the complement of cusp parameters.
 
19
For notational simplicity, we shall keep the symbol γ to denote an arc-length reparametrization.
 
20
For example, if the common tangent line at the two cusps is {x 2 = 0}, then one can use a transformation of the form \((x_{1},x_{2}) \rightarrow (\varepsilon x_{1},\varepsilon ^{2}x_{2})\), and then let \(\varepsilon \rightarrow 0\).
 
21
The structure of \(\mathrm{par}_{\mathrm{cusp}}(\gamma )\) can be described (see (11.44), below).
 
22
See [4, Theorem 8.8 and Proposition 3.5].
 
23
That is, at t there is a collision of \(\#\mathcal{J}_{t}\) cusp parameters of γ n as n → +∞.
 
Literatur
1.
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)MATH
2.
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
3.
Bellettini, G., Dal Maso, G., Paolini, M.: Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20, 247–299 (1993)
4.
Brezis, H.:Functional Analysis and Partial Differential Equations. Springer, New York (2010)
5.
Bressan, A.: Lecture Notes on Functional Analysis. With Applications to Linear Partial Differential Equations. Graduate Studies in Mathematics, vol. 143. American Mathematical Society, Providence (2013)
6.
7.
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)MATH
8.
Kuratowski, K.: Topology. Academic Press, New York (1968)
Metadaten
Titel
Variational Analysis of the Model on Labelled Graphs
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_11