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. Introduction -- Chapter 1.a Variational Model On Labelled Graphs With Cusps And Crossings -- Chapter 2.stable Maps And Morse Descriptions Of An Apparent Contour -- Chapter 3.apparent Contours Of Embedded Surfaces -- Chapter 4.solving The Completion Problem -- Chapter 5.topological Reconstruction Of A Three-dimensional Scene -- Chapter 6.completeness Of Reidemeister-type Moves On Labelled Apparent Contours -- Chapter 7.invariants Of An Apparent Contour -- Chapter 8.elimination Of Cusps -- Chapter 9.the Program “visible” -- Chapter 10.the Program “appcontour”: User’s Guide -- Chapter 11.variational Analysis Of The Model On Labelled Graphs -- Bibliography -- Nomenclature -- Index. By Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Franco Pasquarelli.

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

Valentina Beorchia; Franco Pasquarelli; Maurizio Paolini; Giovanni Bellettini

Springer Spektrum. in Springer-Verlag GmbH

Steinkopff. in Springer-Verlag GmbH

Bellettini Giovanni; Springer

Computational imaging and vision, Vol. 44, Heidelberg New York NY Dordrecht London Berlin, 2015

Computational imaging and vision, 44, 1st ed. 2015, Berlin, Heidelberg, 2015

Computational imaging and vision, volume 44, Heidelberg, 2015

Springer Nature, Heidelberg, 2015

Germany, Germany

Feb 26, 2015

eBook

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Source title: Shape Reconstruction from Apparent Contours: Theory and Algorithms (Computational Imaging and Vision)

Contents
Introduction
References
1 A Variational Model on Labelled Graphs with Cuspsand Crossings
1.1 The Reconstruction Problem
1.2 The Mumford–Shah Model
1.3 The Nitzberg–Mumford Model
1.4 Other Curvature-Depending Functionals
1.5 The Variational Model on Labelled Graphs
References
2 Stable Maps and Morse Descriptions of an Apparent Contour
2.1 Stability of Maps
2.2 Stable Maps from a Two-Manifold to the Plane
2.3 Ambient Isotopies
2.4 Ambient Isotopic and Diffeomorphically Equivalent Apparent Contours
2.5 Morse Descriptions of an Apparent Contour
2.5.1 Genericity of Morse Lines in Case of No Cusps
2.5.2 Morse Lines in Case of Cusps: Markers
2.5.3 The Morse Description
2.5.4 Recovering the Shape from a Morse Description
References
3 Apparent Contours of Embedded Surfaces
3.1 Three-Dimensional Scenes
3.1.1 Splitting of R3
3.2 Apparent Contours of Embedded Surfaces
3.3 The Function f
3.4 Labelling an Apparent Contour: The Function d-Sigma
3.5 Ambient Isotopic and Diffeomorphically Equivalent Labelled Apparent Contours
3.6 Visible Contours
References
4 Solving the Completion Problem
4.1 Some Concepts from Graph Theory
4.1.1 Contour Graphs and Visible Contour Graphs
4.2 Complete Contour Graphs and Labelling
4.3 Statement of the Completion Theorem
4.4 Morse Descriptions of a Visible Contour Graph
4.4.1 Localization
4.5 Proof of the Completion Theorem
4.5.1 Analysis at the Global Maximum and at Local Maxima
4.5.2 Analysis at Terminal Points
4.5.3 Analysis at T-Junctions
4.5.4 Analysis at Local Minima and at the GlobalMinimum
4.6 Examples
References
5 Topological Reconstruction of a Three-Dimensional Scene
5.1 Statement of the Reconstruction Theorem
5.1.1 Depth-Equivalent Scenes
5.2 Proof of Existence
5.2.1 Glueing
5.2.2 Smooth Local Embedding of T in R3
5.2.3 Smooth Global Embedding of M in R3
5.2.3.1 Partition of Unity
5.2.4 Definition of the 3D-Shape
5.3 Proof of Uniqueness
5.A Appendix
References
6 Completeness of Reidemeister-Type Moves on Labelled Apparent Contours
6.1 Moves on a Labelled Apparent Contour
6.1.1 List of All Simple Rules
6.2 Stratifications and Stratified Morse Functions
6.2.1 Stratifications Induced by a Stable Map
6.3 Informal Statement
6.4 Rigorous Statement
6.5 Proof of the Completeness Theorem
6.6 Completeness of Moves
References
7 Invariants of an Apparent Contour
7.1 Definition of B(appcon(phi))
7.2 Definition of BL(appcon(φ))
7.3 Coincidence Between B(appcon(phi)) and BL(appcon(φ))
7.3.1 Proof of Coincidence Up to a Constant
7.3.2 Proof of Coincidence
7.4 Euler–Poincaré Characteristic of dE
7.5 Cell Complexes and Fundamental Groups
7.5.1 Cell Complexes
7.5.2 Fundamental Groups
7.6 Alexander Polynomials and Invariantsof Fundamental Groups
7.7 Free Differential Calculus
7.8 Links with Two Components: Deficiency One
7.9 Surfaces with Genus 2: Deficiency Two
References
8 Elimination of Cusps
8.1 Embedding Sign of a Cusp
8.2 Connectable Cusps in an Open Set
8.3 Statement of the Elimination Theorem
8.4 Proof of the Elimination Theorem
8.5 Application to Closed Embedded Surfaces
References
9 The Program ``Visible''
9.1 An Example
9.2 Encoding a Morse Description of the Visible Contour
9.2.1 Encoding the Morse Events
9.2.2 Implicit Orientation
9.2.3 The ``e'' Region Marking
9.3 Using the Program
9.4 Encoding a Morse Description of the Constructed Apparent Contour
9.5 Some Examples
Reference
10 The Program ``Appcontour'': User's Guide
10.1 An Overview of the Software
10.2 Region Description
10.2.1 Extended Arcs
10.2.1.1 Streaming the Description of an Arc
10.2.2 Describing a Region
10.2.2.1 Streaming the Description of a Region
10.2.3 Completeness of the Region Description
10.3 Encoding an Apparent Contour with Labelling
10.3.1 Region Description as a Stream of Characters
10.3.2 Morse Description
10.3.3 Knot Description
10.4 The Rules (Reidemeister-Type Moves)
10.4.1 Simple Rules
10.4.1.1 The K Rules (kasanie)
10.4.1.2 The T Rules
10.4.1.3 The L Rule (Lip)
10.4.1.4 The B Rule (Beak-to-Beak)
10.4.1.5 The S Rule (Swallow's Tail)
10.4.1.6 The C Rule (Cusp-Fold)
10.4.2 A Nonlocal Effect of the B Rule
10.4.3 Composite Rules
10.4.3.1 The CR0 Composite Rule (B-1S)
10.4.3.2 The CR1 Composite Rule
10.4.3.3 The CR2 Composite Rule
10.4.3.4 The CR3 Composite Rules
10.4.3.5 The CR4 Composite Rules
10.4.3.6 The A1 Composite Rule
10.4.3.7 The A2 Composite Rule
10.4.3.8 The TI Composite Rule
10.4.4 Inverse Rules
10.5 Surgeries on Apparent Contours
10.5.1 Vertical Surgery
10.5.2 Horizontal Surgery
10.6 Canonical Description and Comparison
10.6.1 On the Isomorphism Problem for Graphs
10.6.2 The ``Regions'' Graph: R-Graph
10.6.3 The Depth-First Search of an R-Graph
10.6.4 The Canonization Procedure
10.6.4.1 Canonical Description of an Extended Arc
10.6.4.2 Sorting Non-Equivalent Arcs
10.6.4.3 Using DFS and Lexicographic Comparison
10.6.4.4 Sorting Regions and Arcs
10.6.4.5 Postprocessing
10.6.5 Comparison of Apparent Contours
10.7 Fundamental Groups and Cell Complexes
10.7.1 Computing the Euler–Poincaré Characteristic and the Number of Connected Components
10.7.2 Fundamental Groups
10.7.3 Invariants of Finitely Presented Groups and the Alexander Polynomial
10.7.4 Alexander Polynomials and Alexander Ideals in Two Indeterminates
10.7.4.1 Canonization of a Laurent Polynomial
10.7.4.2 Canonization of a Laurent Ideal
10.8 The Mendes Graph
10.9 Invariants
10.9.1 Euler–Poincaré Characteristic
10.9.2 Bennequin Invariant
10.9.3 Examples of Invariants Computation
10.9.3.1 Projective Plane
10.9.3.2 Milnor Curve and Millett Immersion
10.9.3.3 Torus
10.9.3.4 A knotted Genus-2 Surface
10.10 contour Reference Guide
10.10.1 Informational Commands
10.10.2 Operating Commands
10.10.3 Conversion and Standardization Commands
10.10.4 Cell Complex and Fundamental Group Commands
10.10.5 Options Specific to Fundamental GroupComputations
10.10.6 Common Options
10.10.7 Direct Input of a Finitely Presented Group or an Alexander Ideal
10.11 showcontour Reference Guide
10.11.1 Producing a Proper Morse Description
10.11.2 From the Morse description to a polygonal drawing
10.11.3 Discrete Optimization of the Polygonal Drawing
10.11.4 Dynamic Smoothing of the Polygonal
10.12 Using contour in Scripts
10.12.1 contour_interact.sh
10.12.2 contour_describe.sh
10.12.3 contour_transform.sh
10.13 Example: knotted Surface of Genus 2
10.14 Example: Knots in a Solid Torus
10.15 Example: Klein Bottle and the ``House with Two Rooms''
10.16 Example: Mixed Internal/External Knot
10.17 Using appcontour on Apparent Contours WithoutLabelling
10.17.1 Haefliger Sphere
10.17.2 Boy Surface
10.17.3 Milnor Curve
10.17.4 Millett curve
10.17.5 Klein bottle
10.A Appendix: Practical Canonization of Laurent Polynomials
10.A.1 One-Dimensional Support
10.A.2 Two-Dimensional Support
References
11 Variational Analysis of the Model on Labelled Graphs
11.1 The Action Functional
11.1.1 Graphs with Cusps and Curvature in Lp
11.1.2 The Functional
11.1.3 A Notion of Convergence
11.2 Lower Semicontinuity
11.3 On the Lower Semicontinuous Envelope of the Action
11.3.1 Limits of Labellings
11.3.2 Sufficient Conditions: An Example
11.A Appendix A: Systems of Curves
11.A.1 Curves of Class pwrcp
11.A.2 Systems of Curves
11.A.3 Parametrizations of Complete Contour Graphs
11.B Appendix B: Convergence and Compactnessof Systems of Curves
11.B.1 Convergence
References
Nomenclature
Index

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.
Erscheinungsdatum: 23.03.2015

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℗l 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℗l algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written℗lfor 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

2021-12-05