Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision

Preface
Contents
About the Editors
Contributors
Part I Convex and Non-convex Large-Scale Optimization in Imaging
	1 Convex Non-convex Variational Models
		Contents
		Introduction
		Convex or Non-convex: Main Idea and Related Works
		Sparsity-Inducing Separable Regularizers
		CNC Models with Sparsity-Inducing Separable Regularizers
		Sparsity-Inducing Non-separable Regularizers
		CNC Models with Sparsity-Inducing Non-separable Regularizers
			Construction of Matrix B
		A Simple CNC Example
			Path of Solution Components
		Forward-Backward Minimization Algorithms
			FB Strategy for Separable CNC Models
			FB Strategy for Non-separable CNC Models
			Efficient Solution of the Backward Steps by ADMM
		Numerical Examples
			Examples Using CNC Separable Models
			Examples Using CNC Non-separable Models
		Conclusion
		References
	2 Subsampled First-Order Optimization Methods with Applications in Imaging
		Contents
		Introduction
		Convolutional Neural Networks
			Convolutional Layer
			Max Pooling Layer
		Stochastic Gradient and Variance Reduction Methods
		Gradient Methods with Adaptive Steplength Selection Based on Globalization Strategies
			Accuracy Requirements
			Stochastic Line Search
			Adaptive Regularization and Trust-Region
		Numerical Experiments
			The Neural Network in Action
			Training the Neural Network
			Implementation Details
			Results
		Conclusion
		References
	3 Bregman Methods for Large-Scale Optimization with Applications in Imaging
		Contents
		Introduction
		Bregman Proximal Methods
			A Unified Framework for Implicit and Explicit Gradient Methods
			Bregman Proximal Gradient Method
			Bregman Iteration
			Linearized Bregman Iteration as Gradient Descent
			Bregman Iterations as Iterative Regularization Methods
			Inverse Scale Space Flows
		Accelerated Bregman Methods
		Incremental and Stochastic Bregman Proximal Methods
			Stochastic Mirror Descent
			The Sparse Kaczmarz Method
			Deep Neural Networks
			Bregman Incremental Aggregated Gradient
		Bregman Coordinate Descent Methods
			The Bregman Itoh–Abe Method
			Equivalencies of Certain Bregman Coordinate Descent Methods
		Saddle-Point Methods
			Alternating Direction Method of Multipliers
			Primal-Dual Hybrid Gradient Method
		Applications
			Robust Principal Component Analysis
			Deep Learning
			Student-t Regularized Image Denoising
		Conclusions and Outlook
		References
	4 Fast Iterative Algorithms for Blind Phase Retrieval: A Survey
		Contents
		Introduction
		Mathematical Formula and Nonlinear Optimization Model for BPR
			Mathematical Formula
			Optimization Problems and Proximal Mapping
		Fast Iterative Algorithms
			Alternating Projection (AP) Algorithms
			ePIE-Type Algorithms
			Proximal Algorithms
			ADMM
			Convex Programming
			Second-Order Algorithm Using Hessian
			Subspace Method
		Discussions
			Experimental Issues
			Theoretical Analysis
			Further Discussions
		Conclusions
		References
	5 Modular ADMM-Based Strategies for Optimized Compression, Restoration, and Distributed Representations of Visual Data
		Contents
		Introduction
		Modular ADMM-Based Optimization: General Construction and Guidelines
			Unconstrained Lagrangian Optimizations via ADMM
			Employing Black-Box Modules
			Another Splitting Structure
		Image Restoration Based on Denoising Modules
		Modular Optimizations Based on Standard Compression Techniques
			Preliminaries: Lossy Compression via Operational Rate-Distortion Optimization
			Restoration by Compression
			Modular Strategies for Intricate Compression Problems
		Distributed Representations Using Black-Box Modules
			The General Framework
			Modular Optimizations for Holographic Compression of Images
		Conclusion
		Appendix: Operational Rate-Distortion Optimizations in Block-Based Architectures
		References
	6 Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors
		Contents
		Introduction
		First-Order Hamilton-Jacobi PDEs and Optimization Problems
			Single-Time HJ PDEs and Image Denoising Models
			Multi-time HJ PDEs and Image Decomposition Models
			Min-Plus Algebra for HJ PDEs and Certain Non-convex Regularizations
			Application to Certain Decomposition Problems
		Viscous Hamilton-Jacobi PDEs and Bayesian Estimation
			Viscous HJ PDEs and Posterior Mean Estimators for Log-Concave Models
			On Viscous HJ PDEs with Certain Non-log-Concave Priors
		Conclusion
		References
	7 Multi-modality Imaging with Structure-Promoting Regularizers
		Contents
		Introduction
			Application Examples
			Variational Regularization
			Contributions
			Related Work
				Joint Reconstruction
				Other Models for Similarity
		Mathematical Models for Structural Similarity
			Measuring Structural Similarity
		Structure-Promoting Regularizers
			Isotropic Models
			Anisotropic Models
		Algorithmic Solution
			Algorithm
			Prewhitening
		Numerical Comparison
			Software, Data, and Parameters
			Numerical Results
				Test Case x-ray
				Test Case Super-Resolution
			Discussion on Computational Cost
		Conclusions
			Open Problems
		References
	8 Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion
		Contents
		Introduction
			Contribution and Outline
		Experimental Setup
		Forward Models
			Incident Plane Wave
				The Born Approximation
				The Rytov Approximation
			Modeling the Total Field Using Line and Point Sources
				Point Source Far From Object
				Line Source
		Numerical Comparison of Forward Models
			Modeling the Scattered Field Assuming Incident Plane Waves
			Modeling the Total Field Using Line and Point Sources
		Fourier Diffraction Theorem
			Rotating the Object
			Varying Wave Number
			Rotating the Object with Multiple Wave Numbers
		Reconstruction Methods
			Reconstruction Using Full Waveform Inversion
			Reconstruction Based on the Born and Rytov Approximations
		Numerical Experiments
			Reconstruction of Circular Contrast with Various Amplitudes and Sizes
				Reconstruction Using FWI with Single-Frequency Datasets
				Reconstruction Using FWI with Multiple Frequency Datasets
				Reconstruction Using Born and Rytov Approximations
			Reconstruction of Embedded Shapes: Phantom 1
				Reconstruction Using FWI
				Reconstruction Using Born and Rytov Approximations
			Reconstruction of Embedded Shapes: Phantom 2
				Reconstruction Using FWI
				Reconstruction Using Born and Rytov Approximations
			Computational Costs
		Conclusion
		References
	9 Models for Multiplicative Noise Removal
		Contents
		Introduction
		Variational Methods with Different Data Fidelity Terms
			Statistical Property Based Models
			MAP-Based Models
			Root and Inverse Transformation-Based Models
		Variational Methods with Different Regularizers
			TV Regularization
			Sparse Regularization
			Nonconvex Regularization
		Multitasks
			Root Transformation
			Fractional Transformation
		Nonlocal Methods
			Indirect Method
			Direct Method
		DNN Method
			Indirect Method
			Direct Method
		Conclusion
		References
	10 Recent Approaches to Metal Artifact Reduction in X-Ray CT Imaging
		Contents
		Introduction
		Background: CT Image Formation and Metal Artifacts
		Methods
			Normalized Metal Artifact Reduction (NMAR)
				Inpainting of Metal Traces in the Normalized Sinogram
				NMAR Algorithm
			Surgery-Based Metal Artifact Reduction (SMAR)
				Preprocessing Step
				Iterative Reconstruction Step
			Convolutional Neural Network-Based MAR (CNN-MAR)
				Training of the Convolutional Neural Network
				CNN-MAR Method
			Industrial Application: 3D Cone Beam CT
				Data Preparation
				Registration via Shape Prior Chan-Vese Model
				Shape Prior SMAR Algorithm: Alignment and Registration
				Shape Prior SMAR Algorithm: CT Volume Reconstruction
		Simulations and Results
			Simulation Conditions
			NMAR vs. SMAR: Patient Image Simulations
			SMAR vs. CNN-MAR
				Data Acquisition
				Results
			NMAR vs. SMAR for 3D CBCT
				Phantoms and Hardware Specifications
				Test I: Performance Evaluation
				Test II: Practical Application – Air Bubble Detection Simulation
		Conclusion
		References
	11 Domain Decomposition for Non-smooth (in Particular TV) Minimization
		Contents
		Introduction
			Basic Idea of Domain Decomposition
				Non-overlapping Domain Decomposition
				Overlapping Domain Decomposition
			Difficulty for Non-smooth and Non-separable Optimization Problems
		Domain Decomposition for Smoothed Total Variation
			Direct Splitting Approach
			Decomposition Based on the Euler-Lagrange Equation
		Decomposition for Predual Total Variation
			Overlapping Domain Decomposition
			Non-overlapping Domain Decomposition
				Finite Difference Setting
				Approach via Finite Differences
				Finite Element Approach Based on FISTA
				A FETI Approach
		Decomposition for Primal Total Variation
			Basic Domain Decomposition Approach
				Convergence Properties
				Subspace Minimization
			Domain Decomposition Approach Based on the (Pre)Dual
				Derivation of the Methods
				Subspace Minimization
				Limit Case: Non-overlapping Decomposition
		Conclusion
		References
	12 Fast Numerical Methods for Image Segmentation Models
		Contents
		Introduction
		Mathematical Models for Image Segmentation
		Two-Phase Segmentation Models
			Snakes: Active Contour Model
			Geodesic Active Contour Model (GAC)
			Chan-Vese Model
				Level Set Representation of the Model
			Fast Numerical Methods:
				Semi-implicit Method
				Additive Operator Splitting (AOS) Method
				Multigrid Method
				The Full Approximation Scheme
				Smoother I: Local Smoother
				Smoother II: Global Smoother
				The Multigrid Algorithm
				Local Fourier Analysis of Smoothers
			Multigrid Solver for Solving a Class of Variational Problems with Application to Image Segmentation
				First Algorithm
			Sobolev Gradient Minimization of Curve Length in Chan-Vese Model
				Numerical Method
		Multiphase Image Segmentation
			Multigrid Method for Multiphase Segmentation Model
			Multigrid Method with Typical and Modified Smoother
			Local Fourier Analysis and a Modified Smoother
			Convex Multiphase Image Segmentation Model
				The Bregman Iterations
				Convex Multiphase Model
				Split Bregman Method for the Model
			A Three-Stage Approach for Multiphase Segmentation Degraded Color Images
				Stage 1: Restoration and Smoothing of Given Image
			Stage 2: Dimension Lifting with Secondary Color Space
				Stage 3: Segmentation
		Selective Segmentation Models
			Image Segmentation Under Geometrical Conditions
			Active Contour-Based Image Selective Model
			Dual-Level Set Selective Segmentation Model
			One-Level Selective Segmentation Model
			Reproducible Kernel Hilbert Space-Based Image Segmentation
				Global Segmentation Model
			An Optimization-Based Multilevel Algorithm for Selective Image Segmentation Models
				Multilevel Algorithm for Badshah-Chen (BC) Model
				The Finest-Level Local Minimization (k=1)
				The General-Level k Local Minimization (1