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

Description

This handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning.  No other framework can provide comparable accuracy and precision to imaging and vision. Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists. 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