Hidden Markov model

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Table of contents

Introduction 
I Image Processing and Digital Geometry 
1 Variational methods in Image Processing 
1.1 Inverse problems in imaging
1.2 Bayesian rationale and total variation
1.2.1 Tikhonov regularization
1.2.2 Euler-Lagrange equation
1.2.3 Total variation regularization
1.3 Standard techniques
1.3.1 Curve evolution
1.3.2 Level set
1.3.3 Minimum path
1.3.4 Convex relaxation
2 Discrete methods in Image Processing 
2.1 Markov Random Fields
2.1.1 Clique factorization and Gibbs energy
2.1.2 Hidden Markov model
2.1.3 Grid graph and Tikhonov denoising revisited
2.1.4 Potts and Ising models
2.2 Pseudo-boolean functions
2.2.1 PBF optimization
2.2.2 Submodularity
2.3 Graph cut models
2.3.1 Binary segmentation
2.3.2 Geodesics computation
3 Curvature as a regularizer 
3.1 Curvature and the curve-shortening flow
3.1.1 Definitions
3.1.2 Curve-shortening flow
3.2 Diffusion and level curves motion
3.2.1 Curvature in denoising and image segmentation
3.2.2 Curvature and the connectivity principle applied to inpainting
3.3 Elastica curve
3.3.1 Imaging models using the elastica
3.4 Discrete methods and squared curvature
3.4.1 Discrete elastica
3.4.2 Linear programming model for image segmentation using the discrete elastica
3.4.3 Unconstrained formulations
4 Digital Geometry 
4.1 Ground concepts
4.1.1 Digital grid, digitization and digital line
4.1.2 Exact sampling versus digitization
4.2 Geometric measurements in digital objects
4.2.1 Multigrid convergence and perimeter estimation
4.2.2 Tangent and multigrid convergence of local quantities
4.2.3 Multigrid convergent estimators of curvature
4.3 Conclusion
II Contributions 
5 A combinatorial model for digital elastica shape optimization 
5.1 Digital elastica
5.2 Local combinatorial scheme
5.3 Experimental results
5.3.1 Free elastica
5.3.2 Constrained elastica
5.3.3 Running time
5.4 Global optimization
5.4.1 Simplified digital elastica
5.4.2 Optimization model for simplified digital elastica
5.4.3 Topological constraints
5.4.4 Linear relaxation of P1
5.4.5 Unconstrained version of P1
5.5 Conclusion
6 A 2-step evolution model driven by digital elastica minimization 
6.1 FlipFlow model
6.1.1 Definitions
6.1.2 Model and algorithm
6.1.3 Algorithm discussion
6.2 Optimization method
6.3 Evaluation across m-rings
6.4 Data term and image segmentation
6.5 Conclusion
7 A single step evolution model driven by digital elastica minimization
7.1 BalanceFlow model
7.1.1 Definitions
7.1.2 Algorithm
7.2 Relation with FlipFlow
7.3 Conclusion
8 Digital elastica minimization via graph cuts 
8.1 GraphFlow model
8.1.1 Candidate graphs and solution candidates set
8.1.2 GraphFlow algorithm
8.2 Conclusion
9 Experimental analysis 
9.1 Free elastica
9.1.1 Exp-General
9.1.2 Exp-Radius
9.2 Constrained elastica
9.2.1 Discussion
9.3 Image segmentation
9.3.1 Influence of parameters
9.3.2 Comparison
9.4 Conclusion
10 Conclusion and perspectives 
Appendices
A Curvature and distant disks
B Pixel incidence matrix