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2015 | OriginalPaper | Buchkapitel

1. A Variational Model on Labelled Graphs with Cusps and Crossings

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 we review some of the variational models appearing in the mathematical literature of image segmentation. We will mainly focus attention on those models related to the problem of reconstructing a notion of order between the various objects in a three-dimensional scene.

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Fußnoten
1
With kind permission from Springer Science+Business Media, in this chapter and in Chap. 11 we illustrate the results and report some of the figures from the quoted paper [14].
 
2
In the context of what we shall call “visible contours”, there are (arcs) fractures with terminal points ending inside a region; these arcs are not part of the boundary of a segmentation.
 
3
The problem of edge detection is extensively studied in computer vision; see, for instance, [24, 45, 56, 67].
 
4
It is not the aim of this chapter to give a complete overview on this argument. We refer the reader, for instance, to [3, 27, 53, 60, 66, 72, 79] for some of the topics that are not treated here, and for a more complete list of references.
 
5
It is worthwhile to observe that a contour may be, in general, endowed with several different labellings.
 
6
Basically, two scenes E and F are depth-equivalent if they consist of the same number of connected components, and each connected component F i of F is obtained from the corresponding connected component E i of E through a strictly monotone map in the view direction, continuously depending on the position in \(\Omega \). In particular, if it happens that E i is in front of the connected component E j , then the same depth ordering is preserved for the corresponding connected components F i and F j of F. See Definition 5.​1.​2 for the details.
 
7
That is, \(\mathcal{H}^{1}(K)\) is the length of K when K is sufficiently smooth, see [43].
 
8
Other norms different from the L 2-norm in (1.3) have been considered in the literature; also, suitable functions of u different from the identity can be taken into account: we refer the reader, for instance, to [44, 52] and the references therein.
 
9
This is a subspace of the space \(BV (\Omega )\) of functions with bounded variation in \(\Omega \), and it is called the space of special functions of bounded variation in \(\Omega \). An example of a function in \(SBV (\Omega )\) is given by the characteristic function of a finite perimeter set in \(\Omega \).
 
10
We recall also the variational model where the total variation \(\alpha \int _{\Omega }\vert Du\vert\) is considered, in place of the terms \(\alpha \int _{\Omega \setminus J_{u}}\vert \nabla u\vert ^{2}\ dx +\beta \mathcal{H}^{1}(J_{u})\); see [70]. The advantage of this model (originally introduced in the context of image denoising) is that the functional involved is convex; the disadvantage is its lackness of differentiability at zero and its linear growth at infinity. The discontinuities recovered by the total variation method appear less sharp with respect to the ones recovered by the Mumford–Shah functional (and seems not to be suited for the reconstruction of T-junctions). The total variation model has found (in one variant or another) many applications as a tool to compute the minimum of geometric functionals, in surface reconstruction, in the development of more sophisticated anisotropic total variation models, as a test example to develop efficient numerical schemes for nonlinear and non-differentiable functionals, or as inspiration for edge preserving regularizers. See also the book [3] for related questions and references.
 
11
Similarly, corners in case of nonsmooth shapes are smoothed out: however, in this book we will never be concerned with nonsmooth (polyhedral, for instance) 3D scenes. Apart from the discussion related to the functional in (1.5), all contours that we shall consider will be without corners.
 
12
We are not concerned here with functionals depending on the Hessian of u, see [18] for more information.
 
13
In this reference the Nitzberg–Mumford model was developed further, together with a related computer algorithm which, however, does not implement a direct minimization of the functional NM in (1.4).
 
14
If δ is too large, a minimizing configuration could, in principle, destroy a T-junction, transforming it into a smoothed corner.
 
15
Thus giving a first guess on the completion of the occluded contours.
 
16
Substituting ϕ 2 in place of ϕ NM does not modify this positive feature of the model.
 
17
However, as in the Mumford–Shah functional, in a minimizing segmentation of g corners are smoothed out due to the presence of the term measuring the length of the contours. Indeed, it is still convenient to smooth a corner appearing in the jump set of g and then reduce the length term, at the expense of slightly increasing the other terms in the functional. This implies that corners are not sharply reconstructed, which is a phenomenon that would obviously happen also if one replaces ϕ NM with ϕ 2 in (1.4).
 
18
Again, here α is a nonnegative parameter. Note that letting α → + forces the function u to be piecewise constant. Notice also that, referring to the modification considered in (1.6), the term to be added reads as \(\sum _{i=1}^{m+1}\int _{R_{i}'}(u -\mathrm{ g})^{2}\ dx +\alpha \sum _{ i=1}^{m+1}\int _{R_{i}'}\vert \nabla u\vert ^{2}\ dx\).
 
19
See [16] and the references therein for the applications of the elastica functional to computer vision.
 
20
The functional A has been studied in [30] (see also [21] for further approximation properties), to which we refer for all details.
 
21
This means essentially that the jump set of u is contained in the jump set of χ.
 
22
See also [65, Chapter 6].
 
23
The reader can look through reference [23] for an introduction to apparent contours.
 
24
See Definition 3.​2.​1.
 
25
In computer vision, the description of the possible singularities of the visual mapping of a smooth manifold-solid onto the image plane under parallel projection can be found, for instance, in [54].
 
26
Notice that \(f_{\Sigma } \in BV (\Omega, 2\mathbb{N})\), the class of all functions of bounded variation in \(\Omega \) taking values in the even natural numbers (zero included).
 
27
This is the reason why, in some of the figures containing the values of \(f_{\Sigma }\), we do not display the orientation of the apparent contour.
 
28
In Fig. 2.​3 the function \(f_{\Sigma }\) is denoted by f since in the more general framework of Sect. 2.​2, there is not a surface \(\Sigma \) embedded in \(\mathbb{R}^{3}\).
 
29
The notion of ambient isotopically equivalence of two scenes is explained in Chap. 6.
 
30
See also Fig. 3.​14, which shows the labelling for Fig. 3 in the Introduction. Once we have given \(d_{\Sigma }\), we can define the visible contour of \(\Sigma \) as the closure of \(\{d_{\Sigma } = 0\}\). Its singularities are only terminal points (corresponding to cusps in the apparent contour) and T-junctions (corresponding to crossings).
 
31
Chapter 11 is devoted to the mathematical study of some aspects of the functional \(\mathcal{F}\), whose rigorous definition is given in Sect. 11.​1
 
32
As we shall see in Chap. 5 the exponent p = 2 is not allowed (close to the cusps). Notice that the canonical cusp of J f has the local form x 2 2 = x 1 3; hence locally around the origin on a branch of the cusp and for x 1 > 0, we have \(\kappa (x) = \frac{3} {4} \frac{1} {x_{1}^{1/2}(1+\frac{9} {4} x_{1})^{3/2}}\), which belongs to L p for p ∈ [1, 2) but not to L 2 in a neighbourhood of the origin. It may also be useful to observe that, as in the Nitzberg–Mumford model, the values of the parameters appearing in the expression of \(\mathcal{F}\) can be related to the size of the curvature of the contour.
 
33
Another configuration with finite action (both for NM and for \(\mathcal{F}\)) less favourable for a suitable range of parameters and also less natural, consists in splitting the set into two disjoint regions, by smoothing the two T-junctions and transforming them in two smoothed corners.
 
34
Namely, the continuation of hidden contours is computed solely from the endpoints and tangents to the visible contours at their terminal points, and not from a procedure taking also into account the global shape of the regions.
 
35
Concerning the “Gestalt school”, see, e.g., [50, 51, 59, 76].
 
36
For a circle \(S_{\varepsilon }\) of radius \(\varepsilon > 0\) and κ its curvature, we have \(\int _{S_{\varepsilon }}\vert \kappa \vert ^{p}\ d\mathcal{H}^{1} = 2\pi \varepsilon ^{1-p}\), which diverges as \(\varepsilon \rightarrow 0^{+}\).
 
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Metadaten
Titel
A Variational Model on Labelled Graphs with Cusps and Crossings
verfasst von
Giovanni Bellettini
Valentina Beorchia
Maurizio Paolini
Franco Pasquarelli
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-45191-5_1