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## Über dieses Buch

Motivated by a variational model concerning the depth of the objects in a picture and the problem of hidden and illusory contours, this book investigates one of the central problems of computer vision: the topological and algorithmic reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour.

The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to diffeomorphic deformations of three dimensional scenes.

A large part of the book is devoted to the algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants. This book is addressed to theoretical and applied scientists working in the field of mathematical models of image segmentation.

## Inhaltsverzeichnis

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

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.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 2. Stable Maps and Morse Descriptions of an Apparent Contour

In this chapter we recall the notion of stable map between two manifolds.1 It is convenient to introduce the terminology in arbitrary dimension, and in a rather abstract setting.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 3. Apparent Contours of Embedded Surfaces

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.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 4. Solving the Completion Problem

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.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 5. Topological Reconstruction of a Three-Dimensional Scene

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.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 6. Completeness of Reidemeister-Type Moves on Labelled Apparent Contours

In this chapter we illustrate the results and report the figures from the paper [3]. More specifically, we shall prove that there exists a finite set of simple, or elementary, moves (also called rules) on labelled apparent contours, such that the following property holds: the images $$\Sigma _{1}$$ and $$\Sigma _{2}$$ of two stable embeddings of a closed smooth (not necessarily connected) surface M in $$\mathbb{R}^{3}$$ are isotopic if and only if their apparent contours can be connected using finitely many isotopies of $$\mathbb{R}^{2}$$, and a finite sequence of elementary moves or of their inverses (sometimes called “reverses”).
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 7. Invariants of an Apparent Contour

The aim of this chapter is to illustrate some interesting invariants of apparent contours and labelled apparent contours. These invariants can be numbers, groups, polynomials; invariance here means that the they are insensitive to certain transformations, that will be specified case by case.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 8. Elimination of Cusps

In this chapter we show that the apparent contour of a stable embedded closed smooth (not necessarily connected) surface can be modified, using some of the moves illustrated in Chap. 6, to obtain an apparent contour without cusps.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 9. The Program “Visible”

In this chapter we describe an actual implementation of the constructive proof of the completion result Theorem& 4.​3.​1). The corresponding software code, a free software program, called visible, is written in C language and is part of the appcontour project described in Chap. 10 [1]. It is hosted on sourceforge.net, its home page is http://​appcontour.​sourceforge.​net/​, from where the source code can be downloaded, compiled and installed following the standard procedure for Unix projects.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 10. The Program “Appcontour”: User’s Guide

In this chapter we describe a software code developed by the authors and capable to manipulate the topological structure of apparent contours [15].
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Chapter 11. Variational Analysis of the Model on Labelled Graphs

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.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli

### Backmatter

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