Content

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   Contents
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   1 Introduction
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  2 Mathematical Preliminaries
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   2.1 Mathematical Statements and Notations
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  2.2 Analysis of Different Data-Terms
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   2.2.1 The L2-Data-Term
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   2.2.2 The Weighted L2-Data-Term
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   2.2.3 Kullback-Leibler Data-Term
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   I Graph Cut-Pursuit
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  3 Introduction to Graph Processing
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  3.1 Finite Weighted Graphs
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   3.1.1 Basic Graph Terminology
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   3.1.2 Vertex and Edge Functions
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   3.1.3 First-Order Partial Difference Operators on Graphs
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   3.1.4 Total Variation Regularization on Graphs
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   3.2 Graph p-q-Laplace Operator
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   3.3 Optimization Problems on Graphs
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   3.3.1 ROF on Graphs
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   3.3.2 Weighted ROF on Graphs
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   3.4 Graph Cuts for Energy Minimization
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  Appendix
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   3.A Reformulating p-q-Laplace Operator
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   3.B Derivative of the p-q-TV Regularizer
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   3.C Proximity Operator of the p-q-TV Regularizer
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   4 Cut-Pursuit
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   4.1 Introduction to Partitioning and Reduced Problems
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  4.2 Finer Partitioning via Graph Cuts
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   4.2.1 Refining the Partition
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   4.2.2 Solving the Partition Problem
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   4.2.3 Direction for Isotropic Cut-Pursuit
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   4.3 Cut-Pursuit Algorithm
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  4.4 Cut-Pursuit Algorithm for ROF Problems
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   4.4.1 Reduced ROF Problem
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   4.4.2 Partition Problem of the ROF Problem
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   4.4.3 Directions for the ROF Partition Problem
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  Appendix
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   4.A General Regularizer for ROF
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  4.B Reduced Terms
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   4.B.1 Reduced (Weighted) L2-Data-Term
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   4.B.2 Reduced p-q-TV Regularizer
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   4.C Derivation of Partition Problem
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   5 Cut-Pursuit for Minimal Partition Problems
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   5.1 Minimal Partition Cut-Pursuit Algorithm
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   5.1.1 Solving the Partition Problems
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   5.1.2 Solving the Reduced Minimal Partition Problem
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   5.2 Cut-Pursuit for L0-ROF
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   5.2.1 Solving the Partition Problems
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   5.2.2 Solving the Reduced Minimal Partition Problem
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  Appendix
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   5.A Proof: Merging Values
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   6 Numerics: Cut-Pursuit for TV Problems
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   6.1 Iterative Behavior of the Cut-Pursuit Algorithm
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   6.2 Cut-Pursuit Convergence and Runtime
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   6.3 Isotropic Cut-Pursuit and Choosing Directions
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   6.4 Debiasing
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  7 Numerics: Minimal Partition Problem
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   7.1 Comparing Different Constant Step Sizes
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   7.2 Different Partition Optimization Strategies
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   7.3 Comparison to Other Algorithms
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   7.4 Discretization via Minimal Partitions
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   7.4.1 Related Superpixel Methods
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   7.4.2 Minimal Partition as a Superpixel Method
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   7.4.3 Compare Superpixel Methods
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   7.4.4 Solving Problems on Discretization
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   7.4.5 Denoising Minimal Partitions
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  8 Numerics: Point Cloud Sparsification via Cut-Pursuit
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   8.1 Anisotropic L1-TV Regularization
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   8.2 Comparison of Anisotropic L1-TV and L0-TV Regularization
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   8.3 Visual Comparison of Different TV Regularizations
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   II Dynamic PET Image Reconstruction
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  9 Introduction to PET Reconstruction
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   9.1 Positron Emission Tomography
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   9.2 Expectation Maximization Reconstruction
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   10 Total Variation Regularization on Reconstruction
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   10.1 First-Order Primal-Dual for PET-TV
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   10.2 Forward-Backward EM-TV
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   11 Dynamic PET Reconstruction
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   11.1 Listmode PET Data and EM Reconstruction
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   11.2 Fully4D Reconstruction
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   11.2.1 Update for the Spatial Weights
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   11.2.2 Update for the Basis Functions
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   11.2.3 Fully4D Algorithm and Implementation Details
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   11.3 Regularized Fully4D
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   12 Numerics: Dynamic PET Reconstruction
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   12.1 Dynamic Data without Motion
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   12.2 Dynamic Data with Motion
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   III Cut-Pursuit Based Reconstruction
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  13 Cut-Pursuit on PET Reconstruction
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   13.1 Graph Setting for Reconstructions
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   13.2 Cut-Pursuit for Regularized PET Reconstruction
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   13.3 Efficient Reduced Operator Computation
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   13.4 Solving the Reduced Problem
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   13.4.1 Primal-Dual for Reduced Problem
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   13.4.2 FB-EM-TV for Reduced Problem
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  14 Numerics
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   14.1 Implementation Details
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   14.2 Solving the Reduced Problem
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   14.3 Full Operator Evaluations
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   14.4 Numerical Comparison
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   14.5 Comparison of Cut-Pursuit with Direct Algorithms
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  15 Summary and Conclusion
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   15.1 Efficient Cut-Pursuit Algorithm
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   15.2 Dynamic PET and Regularized Fully4D
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   15.3 Cut-Pursuit Based Reconstruction
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   List of Figures
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   Bibliography
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   Leere Seite