Fast algorithms for image processing and data assimilation

Didier Auroux

• Contents
• Introduction
• Image processing
  • 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
  
  
• Back and forth nudging
  • Introduction
  • ``Back and Forth Nudging'' (BFN) algorithm
    • Forward nudging
    • Backward nudging
    • BFN algorithm
    • Choice of the nudging matrices and interpretation
      • Variational interpretation of the nudging
      • Sequential interpretation
      • Pole assignment method and backward nudging matrix
  • Numerical experiments
    • Numerical choice of the nudging matrices
    • Experimental approach
    • Physical models
      • Lorenz equations
      • 1-D viscous Burgers' equation
      • Shallow water model
      • Layered quasi-geostrophic ocean model
    • Conclusions emerging from the numerical experiments
  • Theoretical convergence results
    • Linear case
    • Transport equations
      • Linear viscous transport
      • Viscous Burgers
      • Inviscid linear transport
      • Inviscid Burgers
      • Remarks on the theoretical results
  • 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
• Summary