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Habilitation
Contents
Introduction
Image processing by topological asymptotic analysis
Introduction
Topological asymptotic analysis
Presentation of the method
Main result
Inpainting
Crack localization problem
Dirichlet and Neumann formulations for the inpainting problem
Asymptotic expansion
Algorithm
Remarks
Restoration
Variational formulation
Topological gradient
Algorithm
Remarks
Extension to color images
Classification
Introduction to the classification problem
Restoration and classification coupling
Extension to unsupervised classification
Segmentation
From restoration to segmentation
Power series expansion
Algorithm
Complexity and speeding up
Discrete cosine transform
Preconditioned conjugate gradient
Coupling between the topological gradient and the minimal path technique
Minimal paths
Fast marching
Coupled algorithm
Conclusions and perspectives
Data assimilation: the Back and Forth Nudging (BFN) algorithm
Introduction
``Back and Forth Nudging'' (BFN) algorithm
Forward nudging
Backward nudging
BFN algorithm
Choice of the nudging matrices and interpretation
Numerical experiments
Numerical choice of the nudging matrices
Experimental approach
Physical models
Conclusions emerging from the numerical experiments
Theoretical convergence results
Linear case
Transport equations
Nudging and observers
Observers for a shallow water model
Invariant correction terms
Convergence study on a linearized simplified system
Numerical experiments
Conclusion
Image data assimilation
Introduction
Description of the algorithm
Constant brightness assumption
Cost function
Regularization
Muti-grid approach and optimization
Quality estimate
Numerical experiments
Simulated data
Experimental data
Conclusions
General conclusions and perspectives
Bibliography
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