The Laplace-Beltrami on explicit surfaces

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Table of contents

1 Introduction 
2 State of the art 
2.1 Surface representations: Introducing the common frameworks
2.2 Bibliographical discussion
2.2.1 Explicit Surfaces
2.2.2 Level-Set methods
2.3 The Laplace Beltrami operator (¢S)
2.3.1 The Laplace-Beltrami on explicit surfaces
2.3.2 The Laplace-Beltrami on implicit surfaces
2.4 Summary and conclusions
3 Regularization on explicit surfaces 
3.1 Isotropic di®usion
3.1.1 An area-averaged Laplace-Beltrami estimation
3.1.2 Discrete Laplace-Beltrami via Discrete Exterior Calculus (DEC)
3.2 Anisotropic di®usion
3.2.1 Anisotropic di®usion on °at images
3.2.2 Area-averaged estimation
3.3 Extension to a more general case
3.4 Beltrami °ow
3.4.1 Introducing the Beltrami framework
3.4.2 Beltrami °ow over non-°at surfaces
3.4.3 Example: Gray image on the sphere
3.4.4 Estimation of the di®erential operator
3.4.5 Implementation details
3.4.6 Example
3.5 Summary and conclusions
4 Regularization on implicit surfaces 
4.1 Anisotropic di®usion
4.2 Beltrami °ow
4.2.1 Scalarfeld defened on a curve: Geometric derivation
4.2.2 The L1 and L2 limits
4.2.3 Implementing the regularization of scalarfelds on surfaces
4.2.4 Examples
4.3 Summary and conclusions
5 Comparison of methods and relationships 
5.1 From explicit to implicit via the Beltrami °ow
5.1.1 Implicit formulation
5.1.2 Intrinsic formulation
5.1.3 The implicit-explicit correspondence
5.2 Comparison of the two methods
5.2.1 From explicit to implicit
5.2.2 From implicit to explicit
5.2.3 Comments on the geometry of the mesh
5.2.4 Further comments and comparison for a particular application
5.3 Summary and conclusions
6 Vector image regularization 
6.1 Unconstrained vector regularization
6.1.1 Vector regularization: Explicit approach
6.1.2 Bi-dimensional vectorfeld on a surface: Implicit approach
6.2 Constrained regularization: Color images
6.2.1 Beltrami °ow, explicit-intrinsic approach
6.2.2 Isotropic and anisotropic regularization of color images on implicit surfaces
6.3 Summary and conclusions
7 Application to cortical images 
7.1 Occipital retinotopic areas extraction
7.1.1 Biological visual system
7.1.2 Extraction method
7.2 Regularization methods
7.2.1 Implicit Beltrami °ow scalar regularization
7.2.2 Explicit regularization
7.2.3 Explicit vector regularization
7.3 Summary and conclusions
8 Conclusions 
9 Author publications 
10 Appendix 
10.1 Stereographic direction di®usion
10.2 Computation of the implicit isotropic smoothing PDE
10.3 Implicit isotropic smoothing for color images
10.4 Numerical schemes for implicit regularization
10.5 Algorithm for explicit anisotropic smoothing