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E-raamat: Shape Reconstruction from Apparent Contours: Theory and Algorithms

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  • Formaat: PDF+DRM
  • Sari: Computational Imaging and Vision 44
  • Ilmumisaeg: 25-Feb-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662451915
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Shape Reconstruction from Apparent Contours: Theory and AlgorithmsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Computational Imaging and Vision 44
  • Ilmumisaeg: 25-Feb-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662451915
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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.
1 A Variational Model on Labelled Graphs with Cusps and Crossings
1(24)
1.1 The Reconstruction Problem
1(3)
1.2 The Mumford--Shah Model
4(2)
1.3 The Nitzberg--Mumford Model
6(5)
1.4 Other Curvature-Depending Functionals
11(2)
1.5 The Variational Model on Labelled Graphs
13(12)
References
21(4)
2 Stable Maps and Morse Descriptions of an Apparent Contour
25(28)
2.1 Stability of Maps
25(6)
2.2 Stable Maps from a Two-Manifold to the Plane
31(5)
2.3 Ambient Isotopies
36(5)
2.4 Ambient Isotopic and Diffeomorphically Equivalent Apparent Contours
41(1)
2.5 Morse Descriptions of an Apparent Contour
42(11)
2.5.1 Genericity of Morse Lines in Case of No Cusps
44(1)
2.5.2 Morse Lines in Case of Cusps: Markers
45(3)
2.5.3 The Morse Description
48(2)
2.5.4 Recovering the Shape from a Morse Description
50(1)
References
51(2)
3 Apparent Contours of Embedded Surfaces
53(20)
3.1 Three-Dimensional Scenes
53(2)
3.1.1 Splitting of R3
54(1)
3.2 Apparent Contours of Embedded Surfaces
55(4)
3.3 The Function fΣ
59(5)
3.4 Labelling an Apparent Contour: The Function dΣ
64(5)
3.5 Ambient Isotopic and Diffeomorphically Equivalent Labelled Apparent Contours
69(1)
3.6 Visible Contours
70(3)
References
72(1)
4 Solving the Completion Problem
73(28)
4.1 Some Concepts from Graph Theory
73(4)
4.1.1 Contour Graphs and Visible Contour Graphs
75(2)
4.2 Complete Contour Graphs and Labelling
77(3)
4.3 Statement of the Completion Theorem
80(2)
4.4 Morse Descriptions of a Visible Contour Graph
82(2)
4.4.1 Localization
84(1)
4.5 Proof of the Completion Theorem
84(12)
4.5.1 Analysis at the Global Maximum and at Local Maxima
86(1)
4.5.2 Analysis at Terminal Points
87(1)
4.5.3 Analysis at T-Junctions
88(4)
4.5.4 Analysis at Local Minima and at the Global Minimum
92(4)
4.6 Examples
96(5)
References
100(1)
5 Topological Reconstruction of a Three-Dimensional Scene
101(30)
5.1 Statement of the Reconstruction Theorem
101(2)
5.1.1 Depth-Equivalent Scenes
102(1)
5.2 Proof of Existence
103(15)
5.2.1 Glueing
104(4)
5.2.2 Smooth Local Embedding of T in R3
108(4)
5.2.3 Smooth Global Embedding of M in R3
112(5)
5.2.4 Definition of the 3D-Shape
117(1)
5.3 Proof of Uniqueness
118(13)
5.A Appendix
125(3)
References
128(3)
6 Completeness of Reidemeister-Type Moves on Labelled Apparent Contours
131(26)
6.1 Moves on a Labelled Apparent Contour
133(3)
6.1.1 List of All Simple Rules
134(2)
6.2 Stratifications and Stratified Morse Functions
136(4)
6.2.1 Stratifications Induced by a Stable Map
137(3)
6.3 Informal Statement
140(2)
6.4 Rigorous Statement
142(5)
6.5 Proof of the Completeness Theorem
147(6)
6.6 Completeness of Moves
153(4)
References
155(2)
7 Invariants of an Apparent Contour
157(38)
7.1 Definition of B(appcon(φ))
158(2)
7.2 Definition of BL(appcon(φ))
160(2)
7.3 Coincidence Between B(appcon(φ)) and BL(appcon(φ))
162(9)
7.3.1 Proof of Coincidence Up to a Constant
163(4)
7.3.2 Proof of Coincidence
167(4)
7.4 Euler--Poincare Characteristic of δE
171(5)
7.5 Cell Complexes and Fundamental Groups
176(5)
7.5.1 Cell Complexes
178(1)
7.5.2 Fundamental Groups
179(2)
7.6 Alexander Polynomials and Invariants of Fundamental Groups
181(2)
7.7 Free Differential Calculus
183(6)
7.8 Links with Two Components: Deficiency One
189(2)
7.9 Surfaces with Genus 2: Deficiency Two
191(4)
References
192(3)
8 Elimination of Cusps
195(14)
8.1 Embedding Sign of a Cusp
196(2)
8.2 Connectable Cusps in an Open Set
198(3)
8.3 Statement of the Elimination Theorem
201(1)
8.4 Proof of the Elimination Theorem
202(4)
8.5 Application to Closed Embedded Surfaces
206(3)
References
207(2)
9 The Program "Visible"
209(16)
9.1 An Example
209(3)
9.2 Encoding a Morse Description of the Visible Contour
212(2)
9.2.1 Encoding the Morse Events
213(1)
9.2.2 Implicit Orientation
213(1)
9.2.3 The "e" Region Marking
214(1)
9.3 Using the Program
214(1)
9.4 Encoding a Morse Description of the Constructed Apparent Contour
215(1)
9.5 Some Examples
216(9)
Reference
223(2)
10 The Program "Appcontour": User's Guide
225(98)
10.1 An Overview of the Software
226(4)
10.2 Region Description
230(5)
10.2.1 Extended Arcs
231(1)
10.2.2 Describing a Region
232(2)
10.2.3 Completeness of the Region Description
234(1)
10.3 Encoding an Apparent Contour with Labelling
235(5)
10.3.1 Region Description as a Stream of Characters
235(1)
10.3.2 Morse Description
236(2)
10.3.3 Knot Description
238(2)
10.4 The Rules (Reidemeister-Type Moves)
240(8)
10.4.1 Simple Rules
240(4)
10.4.2 A Nonlocal Effect of the B Rule
244(1)
10.4.3 Composite Rules
244(3)
10.4.4 Inverse Rules
247(1)
10.5 Surgeries on Apparent Contours
248(1)
10.5.1 Vertical Surgery
248(1)
10.5.2 Horizontal Surgery
249(1)
10.6 Canonical Description and Comparison
249(9)
10.6.1 On the Isomorphism Problem for Graphs
250(1)
10.6.2 The "Regions" Graph: R-Graph
250(2)
10.6.3 The Depth-First Search of an R-Graph
252(1)
10.6.4 The Canonization Procedure
253(5)
10.6.5 Comparison of Apparent Contours
258(1)
10.7 Fundamental Groups and Cell Complexes
258(16)
10.7.1 Computing the Euler--Poincare Characteristic and the Number of Connected Components
259(1)
10.7.2 Fundamental Groups
260(2)
10.7.3 Invariants of Finitely Presented Groups and the Alexander Polynomial
262(5)
10.7.4 Alexander Polynomials and Alexander Ideals in Two Indeterminates
267(7)
10.8 The Mendes Graph
274(1)
10.9 Invariants
275(5)
10.9.1 Euler--Poincare Characteristic
275(1)
10.9.2 Bennequin Invariant
275(1)
10.9.3 Examples of Invariants Computation
276(4)
10.10 Contour Reference Guide
280(13)
10.10.1 Informational Commands
280(4)
10.10.2 Operating Commands
284(3)
10.10.3 Conversion and Standardization Commands
287(1)
10.10.4 Cell Complex and Fundamental Group Commands
288(1)
10.10.5 Options Specific to Fundamental Group Computations
289(1)
10.10.6 Common Options
290(1)
10.10.7 Direct Input of a Finitely Presented Group or an Alexander Ideal
291(2)
10.11 Showcontour Reference Guide
293(3)
10.11.1 Producing a Proper Morse Description
293(1)
10.11.2 From the Morse description to a polygonal drawing
294(1)
10.11.3 Discrete Optimization of the Polygonal Drawing
294(1)
10.11.4 Dynamic Smoothing of the Polygonal
295(1)
10.12 Using contour in Scripts
296(4)
10.12.1 contour_interact.sh
296(1)
10.12.2 contour_describe.sh
297(2)
10.12.3 contour_transform.sh
299(1)
10.13 Example: knotted Surface of Genus 2
300(1)
10.14 Example: Knots in a Solid Torus
301(3)
10.15 Example: Klein Bottle and the "House with Two Rooms"
304(5)
10.16 Example: Mixed Internal/External Knot
309(3)
10.17 Using appcontour on Apparent Contours Without Labelling
312(11)
10.17.1 Haefliger Sphere
312(1)
10.17.2 Boy Surface
313(1)
10.17.3 Milnor Curve
313(1)
10.17.4 Millett curve
314(1)
10.17.5 Klein bottle
315(2)
10.A Appendix: Practical Canonization of Laurent Polynomials
317(1)
10.A.1 One-Dimensional Support
318(1)
10.A.2 Two-Dimensional Support
318(3)
References
321(2)
11 Variational Analysis of the Model on Labelled Graphs
323(32)
11.1 The Action Functional
324(3)
11.1.1 Graphs with Cusps and Curvature in Lp
324(1)
11.1.2 The Functional
325(1)
11.1.3 A Notion of Convergence
326(1)
11.2 Lower Semicontinuity
327(5)
11.3 On the Lower Semicontinuous Envelope of the Action
332(23)
11.3.1 Limits of Labellings
335(2)
11.3.2 Sufficient Conditions: An Example
337(5)
11.A Appendix A: Systems of Curves
342(1)
11.A.1 Curves of Class pwrpc
342(2)
11.A.2 Systems of Curves
344(1)
11.A.3 Parametrizations of Complete Contour Graphs
345(1)
11.B Appendix B: Convergence and Compactness of Systems of Curves
346(1)
11.B.1 Convergence
347(6)
References
353(2)
Nomenclature 355(6)
Index 361
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