Inhalt

•  
  Introduction
•  
   Motivation and Contributions
•  
   Organization of the Work
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  Variational Methods
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   Motivation: Inverse Problems
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   Bayesian Modeling
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   Variational Calculus
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   Function Spaces and Total Variation
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   Differentiability and Optimality
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  Convex Analysis and Bregman Distances
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  Introduction
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   Subdifferential Calculus
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   Legendre-Fenchel Duality
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   Bregman distances
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  Algorithms and Error Estimation
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   Introduction
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   Primal Bregman Iteration
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   Dual Bregman Iteration
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   Bregman and Error Forgetting
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  Splitting Methods
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   Saddle Point Problems
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   Uzawa Methods
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   Augmented Lagrangian Methods
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   The Splitting Zoo
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   Forward-Backward Splitting and AMA
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   ADMM and DRS and Split Bregman
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   Bregmanized Operator Splitting and Inexact Uzawa
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  3D Imaging
•  
   Introduction
•  
  Modeling and EM Algorithm
•  
   Model for Data Acquisition
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   Statistical Problem Formulation of Image Reconstruction
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   Reconstruction Method: EM Algorithm
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  EM-TV Reconstruction Method
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   Algorithm FB-EM-TV
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   Forward-Backward-Splitting
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  Analysis
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   Assumptions, Definitions and Preliminary Results
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   Well-Posedness of the Minimization Problem
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   Positivity Preservation of FB-EM-TV
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   Convergence Results
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   Numerical Realization of weighted ROF
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   Dual Implementation
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  Contrast Enhancement via Bregman Iterations
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   Primal Bregman-EM-TV
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   Dual-Bregman-EM-TV
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  3D Reconstruction in Nanoscopy and PET
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  Applications
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   Optical Nanoscopy
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   Positron Emission Tomography (PET)
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  Results
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   2D Deconvolution in Optical Nanoscopy
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   2D PET Reconstruction in Nuclear Medicine
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   2D Primal and dual Bregman in Optical Nanoscopy
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   3D Deconvolution in Optical Nanoscopy
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  4D Imaging
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   Introduction
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   Optical Flow and Tracking
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   Optimal Transport and Mass Conservation
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   Full Joint 4D Model
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  Optical Flow and Tracking
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   Introduction
•  
   Model Derivation
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   Data Fidelities
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   Robustness
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  Regularization
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   Image-Driven Regularization
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   Flow-Driven Regularization
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  3D Optical Flow-TV
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   Model: Optical Flow-TV
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   Algorithm: Split Bregman Optical Flow-TV
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  Numerical Results
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   Results in Computed Tomography
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   Compressible versus Incompressible Flows
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  Optimal Transport
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   Continuum mechanics
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   Mass conservation
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   Monge´s Transport Problem
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   The Benamou-Brenier Formulation
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  4D Image Reconstruction in Nanoscopy and PET
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   4D Model - Reconstruction and Optimal Transport
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   4D Model - Space-Time L2 Regularization
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   4D Model - Space-Time TV regularization
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   Existence
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   Uniqueness
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   Numerical Realization: Newton-SQP for the L2 case
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   Optimality Conditions - KKT System
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   Newton-SQP Algorithm
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   Line-Search and Multigrid Preconditioning
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   Numerical Realization: Splitting Methods for the TV case
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   Inexact Uzawa & Bregmanized Operator Splitting
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   DCT and CUDA and Parallelization
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   Results - Denoising
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   Results - Deblurring
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   Results - Tomography
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   Summary and Open Questions
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   Bibliography