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 The Coarsest-Level Minimization (k=L+1)
Machine/Deep Learning Techniques for Image Segmentation
Machine Learning with Region-Based Active Contour Models in Medical Image Segmentation
Proposed Framework
Regularization for Classifier Probability Score
ResBCU-Net: Deep Learning Approach for Segmentation of Skin Images
Proposed Work
Encoding
Decoding
Conclusion
References
13 On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models
Contents
Introduction
Basic Notation
Augmented Lagrangian Method for Total Variation-Related Image Restoration Models
Augmented Lagrangian Method for TV-L2 Restoration
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. y
Convergence Analysis
Augmented Lagrangian Method for TV-L2 Restoration with Box Constraint
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. (y,z)
Augmented Lagrangian Method for TV Restoration with Non-quadratic Fidelity
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. (y,z)
Extension to Multichannel Image Restoration
The Multichannel TV Restoration Model
Augmented Lagrangian Method for Multichannel TV Restoration
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. y
Extension to High-Order Models
Augmented Lagrangian Method for Second-Order Total Variation Model
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. ŷ
Augmented Lagrangian Method for Total Generalized Variation Model
The Solution to Sub-problem w.r.t. (x,w)
The Solution to Sub-problem w.r.t. y,z
Augmented Lagrangian Method for Euler Elastic-Based Model
The Solution to Sub-problem w.r.t. x
The Solution to Sub-problem w.r.t. y
The Solution to Sub-problem w.r.t. m
The Solution to Sub-problem w.r.t. n
Augmented Lagrangian Method for Mean Curvature-Based Model
Numerical Experiments
Conclusions
References
14 Sparse Regularized CT Reconstruction: An Optimization Perspective
Contents
Introduction
Tomographic Imaging
Mathematics of Sparse Tomography
Lambert Beer's Law
The Radon Transform and Its Discretization
The Filtered Back Projection Algorithm
Model-Based Approaches for Sparse-View CT
From Lambert-Beer's Law to a Linear System
Implementation of the Forward Operator M
The Optimization Framework
Iterative Algorithms for Optimization
Regularization: Little or Too Much?
Toward the Convergence of the Iterative Method
New Frontiers of CT Reconstruction with Deep Learning
Case Study: Reconstruction of Digital Breast Tomosynthesis Images
DBT 3D Imaging
Model and Analysis
TV-Based Framework
Measure and Graphics of Merits for 3D Tomography
Reconstructions of the Accreditation Phantom
Reconstructions of a Human Dataset
Distance-Driven Approach for 3D CT Imaging
Code Parallelization
Conclusion
References
15 Recent Approaches for Image Colorization
Contents
Context and Modeling
Challenge
Mathematical Modeling of Colorization
Range of Chrominance
Description of the Range
Orthogonal Projection onto the Convex Range
Color Diffusion
State-of-the-Art of Color Diffusion
Coupled Total Variation for Image Colorization
Constrained TV-L2 Debiasing Algorithm
The CLEAR Method 55:deledalle2016clear
Direct Extension of CLEAR to Constrained Problems
Direct Debiasing Process
Adaptive Debiasing Model for Constrained Problems
Computation of the Oblique Projection
Exemplar-Based Colorization
Morphing-Based Approach
Segmentation-Based Techniques
Patch-Based Methods
A Variational Model for Image Colorization with Channel Coupling
Colorization from Dataset
Coupled Approaches
Coupling Manual Approach with Exemplar-Based Colorization
Coupling CNN with a Variational Approach
Coupling the CNN with a Variational Method
Numerical Results
Conclusion and Future Works
References
16 Numerical Solution for Sparse PDE Constrained Optimization
Contents
Introduction
Finite Element Approximation and Error Estimates
An Inexact Heterogeneous ADMM Algorithm
An Inexact Heterogeneous ADMM Algorithm
Convergence Results of ihADMM
An Inexact Majorized Accelerated Block Coordinate Descent Method for (Dh)
An Inexact Block Symmetric Gauss-Seidel Iteration
Inexact Majorized Accelerate Block Coordinate Descent (imABCD) Method
A sGS-imABCD Algorithm for (Dh)
Numerical Results
Algorithmic Details
Examples
Conclusion
References
17 Game Theory and Its Applications in Imaging and Vision
Contents
Introduction to Game Theory and Paradigm
Applications of Game Theory in Image Restoration and Segmentation
Applications of Game Theory in Image Registration
Introduction to Image Registration
Application of Game Theory to a Simple Registration Model
Coupled Measures: Nongame Approach
Coupled Measures: Game Approach
Examples
Application of Game Theory to Registering Images Requiring Bias Correction
Non-game Approach
Game Model
The Potential Game
The Non-potential Game
Iterative Algorithm
Examples
Example 1: MRI Images
Example 2: Application to Perfusion CT Registration
Game Models in Deep Learning
Generative Adversarial Networks (GANs)
Generative vs Discriminative Algorithms
Theory and Numerics
GANs for Image Generation: A Two-Player Game
Examples
GANs for Image Segmentation: A Two-Player Game
Generator and Discriminator
Model Loss
Training
Example
Conclusion
References
18 First-Order Primal–Dual Methods for Nonsmooth Non-convex Optimization
Contents
Introduction
Sample Problems
Outline
Bregman Divergences
Primal–Dual Proximal Splitting
Optimality Conditions and Proximal Points
Algorithm Formulation
Block Adaptation
Convergence Theory
A Fundamental Estimate
Ellipticity of the Bregman Divergences
Ellipticity for Block-Adapted Methods
Nonsmooth Second-Order Conditions
Second-Order Growth Conditions for Block-Adapted Methods
Convergence of Iterates
Convergence of Gaps in the Convex-Concave Setting
Inertial Terms
A Generalization of the Fundamental Theorem
Inertia (Almost) as Usually Understood
Improvements to the Basic Method Without Dual Affinity
Further Directions
Acceleration
Stochastic Methods
Alternative Bregman Divergences
Alternative Approaches
Functions on Manifolds and Hadamard Spaces
Glossary
References
Part II Model- and Data-Driven VariationalImaging Approaches
19 Learned Iterative Reconstruction
Contents
Introduction
Deep Learning
Architectures
Gradient-Based Architectures
Variational Networks
Proximal-Based Architectures
Primal-Dual Networks
Other Schemes
Training Procedure
Engineering Aspects
Architectures for Learned Operator
Initialization
Parameter Sharing
Further Memory
Preconditioning
Learned Step Length
Scalable Training
Putting It All Together
Conclusions
References
20 An Analysis of Generative Methods for Multiple Image Inpainting
Contents
Introduction
A Walk Through the Image Inpainting Literature
How to Achieve Multiple and Diverse Inpainting Results?
Generative Adversarial Networks
Variational Autoencoders and Conditional Variational Autoencoders
Autoregressive Models
Image Transformers
High-Fidelity Pluralistic Image Completion with Transformers119:wan2021high
From Single-Image Evaluation Metrics to Diversity Evaluation
Experimental Results
Experimental Settings
Datasets
Quantitative Performance
Proximity to Ground Truth
Perceptual Quality
Inpainting Diversity
Qualitative Performance
Conclusions
Appendix
Additional Quantitative Results
Additional Qualitative Results
References
21 Analysis of Different Losses for Deep Learning Image Colorization
Contents
Introduction
Losses in the Colorization Literature
Error-Based Losses
Generative Adversarial Network-Based Losses
Distribution-Based Losses
Proposed Colorization Framework
Detailed Architecture
Quantitative Evaluation Metrics Used in Colorization Methods
Experimental Analysis
Quantitative Evaluation
Qualitative Evaluation
Generalization to Archive Images
Conclusion
References
22 Influence of Color Spaces for Deep Learning Image Colorization
Contents
Introduction
Related Work
On Color Spaces
Review of Colorization Methods
Scribble-Based Image Colorization
Exemplar-Based Image Colorization
Deep Learning Methods for Image Colorization
Datasets Used in Literature
Proposed Colorization Framework
Detailed Architecture
Training and Testing Images
Learning Strategy for Different Color Spaces
Analysis of the Influence of Color Spaces
Quantitative Evaluation
Qualitative Evaluation
Generalization to Archive Images
Conclusion
References
23 Variational Model-Based Deep Neural Networks for Image Reconstruction
Contents
Introduction
Learned Algorithm for Specified Optimization Problem
Structured Image Reconstruction Networks
Proximal Point Network
ISTA-Net
ADMM-Net
Variational Network
Primal-Dual Network
Learnable Descent Algorithm
Concluding Remarks
Appendix: Back-Propagation in ISTA-Net
References
24 Bilevel Optimization Methods in Imaging
Contents
Introduction
Variational Inverse Problems Setting
Image Reconstruction as an Inverse Problem
Regularizers
Restoration Models
Optimality and Duality
Solution Methods
Bilevel Optimization in Imaging
Total Variation Gaussian Denoising
Failure of Standard Constraint Qualification Conditions
Alternative Optimality Conditions
Solution Algorithms
Infinite-Dimensional Case
Existence and Other Properties
Stationarity Conditions
Dualization
Nonlocal Problems
Neural Network Optimization
Deep Neural Networks as a Further Regularizer
Deep Unrolling Within Optimization
Numerical Experiments
Conclusions
References
25 Multi-parameter Approaches in Image Processing
Contents
Introduction
PDE-Based Approaches
Dictionary-Based Approaches
Parameter Selection
Multiparameter Discrepancy Principle
Balancing Principle and Balanced Discrepancy Principle
L-Hypersurface
Generalized Lasso Path
Parameter Learning
Numerical Solution
Numerical Examples
Conclusion
References
26 Generative Adversarial Networks for Robust Cryo-EM Image Denoising
Contents
Introduction
Robust Denoising in Deep Learning
Challenges of Cryo-EM Image Denoising
Outline
Background: Data Representation and Mapping
Autoencoder
GAN
JS-GAN
WGAN and WGANgp
Robust Denoising Method
Huber Contamination Noise Model
Robust Denoising Method
Robust Recovery via β-GAN
Robust Recovery Theory
Stabilized Robust Denoising by Joint Autoencoder and β-GAN
Stability of Combining Autoencoder into GAN
Application: Robust Denoising of Cryo-EM Images
Datasets
RNAP: Simulation Dataset
EMPIAR-10028: Real Dataset
Evaluation Method
Network Architecture and Hyperparameter
Results for RNAP
Denoising Without Contamination
Robustness Under Contamination
Results for EMPIAR-10028
Conclusion
Appendix
Influence of Parameter(α, β) Brings in β-GAN
Clustering to Solve the Conformational Heterogeneity
Convolution Network
Test RNAP Dataset with PGGAN Strategy
Influence of the Regularization Parameter: λ
References
27 Variational Models and Their Combinations with Deep Learning in Medical Image Segmentation: A Survey
Contents
Introduction
Conventional Algorithms Based on Variational Methods
The Data Term
The Boundary Information
The Regional Information
The Regularization Term
Generic Regularization
Targeted Regularization Terms Arising from Object Properties
Variational Models Meet Deep Learning in Medical Image Segmentation
Variational Models Guided Deep Learning
Variational Model-Inspired Network Modules
Variational Model-Inspired Loss Functions
Deep Learning-Driven Variational Models
Learning Hyperparameters in Two-Stage Framework
Learning Hyperparameters in End-to-End Framework
Conclusion
References
28 Bidirectional Texture Function Modeling
Contents
Introduction
Visual Texture
Bidirectional Texture Function
BTF Measurement
Compound Markov Model
Principal Markov Model
Principal Single Model Markov Random Field
Non-parametric Markov Random Field
Non-parametric Markov Random Field with Iterative Synthesis
Iterative Principal Field Synthesis
Non-parametric Markov Random Field with Fast Iterative Synthesis
Iterative Principal Field Synthesis
Potts Markov Random Field
Potts-Voronoi Markov Random Field
Bernoulli Distribution Mixture Model
Gaussian Mixture Model
Local Markov and Mixture Models
3D Causal Simultaneous Autoregressive Model
3D Moving Average Model
Spatial 3D Gaussian Mixture Model
Parameter Estimation
Applications
Texture Synthesis and Enlargement
Texture Compression
Texture Editing
Illumination Invariants
(Un)supervised Image Recognition
Multispectral/Multi-channel Image Restoration
Conclusion
References
29 Regularization of Inverse Problems by Neural Networks
Contents
Introduction
Ill-Posedness
Data-Driven Reconstruction
Outline
Preliminaries
Right Inverses
Regularization Methods
Deep Learning
Regularizing Networks
Null-Space Networks
Convergence Analysis
Extensions
The NETT Approach
Learned Regularization Functionals
Convergence Analysis
Related Methods
Conclusion and Outlook
References
30 Shearlets: From Theory to Deep Learning
Contents
Introduction
The Applied Harmonic Analysis Viewpoint
Frame Theory Comes into Play
Wavelets
From Wavelets to Shearlets
From Inverse Problems to Deep Learning
Outline
Continuous Shearlet Systems
Classical Continuous Shearlet Systems
Cone-Adapted Continuous Shearlet Systems
Resolution of the Wavefront Set
Discrete Shearlet Systems
Cone-Adapted Discrete Shearlet Systems
Frame Properties
Band-Limited Shearlets
Compactly Supported Shearlets
Sparse Approximation
Extensions of Shearlets
Higher Dimensions
α-Molecules
Universal Shearlets
Digital Shearlet Systems
Digital 2D Shearlet Transform
Extensions of the Digital 2D Shearlet Transform and ShearLab3D
Applications of Shearlets
Sparse Regularization Using Shearlets
Image Separation
Image Inpainting
Shearlets Meet Deep Learning
Limited-Angle Computed Tomography
Wavefront Set Detection
Conclusion
References
31 Learned Regularizers for Inverse Problems
Contents
Introduction
Shallow Learned Regularizers
Bilevel Learning
Dictionary Learning
Deep Regularizers
Regularization Properties of Learned Regularizers
Adversarial Regularization
Total Deep Variation
Summary and Outlook
Conclusion
Outlook
References
32 Filter Design for Image Decomposition and Applications to Forensics
Contents
Introduction
Applications and Challenges for Automated Image Decomposition
Diffusion Methods
Fourier and Wavelet Methods
Variational Problems
Non-linear Spectral Decompositions
Texture Information
Machine Learning
Adaptive Balancing
Adapting the Data-Fidelity-Norm
Connection to the G-Norm
Other Choices of M
Connections with Machine Learning
Solving via the ADMM/AL-Algorithm
Interpretation via a Feasibility Problem
A General Learning Problem
Filter Design Using Factor Families
Conclusion
References
33 Deep Learning Methods for Limited Data Problems in X-Ray Tomography
Contents
Introduction
Background
Tomographic Image Reconstruction
Analytic Reconstruction Methods
Iterative Reconstruction Methods
Deep Learning
Case Examples in X-Ray CT
Limited Angle Computed Tomography
Reduction of Metal Artefacts
Low-Dose Computed Tomography
Further Methods
Conclusion
References
34 MRI Bias Field Estimation and Tissue Segmentation Using Multiplicative Intrinsic Component Optimization and Its Extensions
Contents
Introduction
Multiplicative Intrinsic Component Optimization
Decomposition of MR Images into Multiplicative Intrinsic Components
Mathematical Description of Multiplicative Intrinsic Components
Energy Formulation for MICO
Optimization of Energy Function and Algorithm
Optimization of c
Optimization of w and Bias Field Estimation b̂
Optimization of u
Numerical Stability Using Matrix Analysis
Execution of MICO
Some Extensions
Introduction of Spatial Regularization in MICO
The Proposed TV-Based MICO Model and Its Solver
Formulation of Proposed Model
ADMM and Its Numerical Analysis
p-Subproblem
v-Subproblem
u-Subproblem
Solutions for Subproblems c, w, and Bias Field Estimation b
Spatiotemporal Regularization for 4D Segmentation
Modified MICO Formulation with Weighting Coefficients for Different Tissues
Results and Discussions
Conclusion
References
35 Data-Informed Regularization for Inverse and Imaging Problems
Contents
Introduction
A Data-Informed Regularization (DI) Approach
Data-Informed Regularization Derivation
A Statistical Data-Informed (DI) Inverse Framework
Properties of the DI Regularization Approach
Deterministic Properties
Statistical Properties
Applications to Imaging Problems
Image Deblurring
Image Denoising
X-Ray Tomography
Conclusions
References
36 Randomized Kaczmarz Method for Single Particle X-Ray Image Phase Retrieval
Contents
Introduction
The Phase Retrieval Problem
Challenges of X-Ray Data Processing
Phase Retrieval with Noisy or Incomplete Measurements
Outline
Background: Phase Retrieval and Stochastic Optimization
Phase Retrieval
Stochastic Optimization and the Kaczmarz Method
Variance-Reduced Randomized Kaczmarz (VR-RK) Method
Application: Robust Phase Retrieval of the Single-Particle X-Ray Images
Synthetic Single-Particle Data Recovery Experiment
Recovery Efficiency Under Constraints
Results of the PR772 Dataset
Conclusion
Appendix
References
37 A Survey on Deep Learning-Based Diffeomorphic Mapping
Contents
Introduction
Background and Motivation
Diffeomorphic Mapping
Large Deformation Diffeomorphic Metric Mapping
Stationary Vector Field
Problem Statement and Framework Overview
Deep Learning-Based Methods
Unsupervised Methods
Supervised Methods
Related Deep Network Introduction
Convolutional Neural Networks
Fully Convolutional Network
U-Net
Autoencoders
Recurrent Neural Networks and Long Short-Term Memory Networks
Unsupervised Methods
Loss Function
Similarity Metrics
Regularization for Diffeomorphic Mapping
CNN-Based Methods
VAE-Based Methods
More Related Works
Supervised Methods
Loss Function
Similarity Metrics
Regularization for Diffeomorphic Mapping
CNN-Based Methods
More Related Works
Discussion and Future Direction
Achievements and Applications
Challenges
Future Directions
Conclusions
References
Part III Shape Spaces and Geometric Flows
38 Stochastic Shape Analysis
Contents
Introduction
Key Concepts from Shape Analysis
Large Deformation Diffeomorphic Metric Mapping
Metric and Variational Formulations
Hamiltonian Systems and Noise
Hamiltonian Systems and Landmark Dynamics
Noise from a Statistical Physics Perspective
Non-dissipative Stochastic Shape Models
Riemannian Brownian Motion
Lagrangian Noise
Eulerian Noise
Stochastic Euler-Poincaré Reduction and Its Infinite Dimensional Extension
Reduction by Symmetry
Stochastic EPDiff
Other Stochastic Shape Models
Stochastic Models in Shape Statistics
Random Orbits with Time-Continuous Noise
Noise Inference from Evolution of Moments
Likelihood-Based Inference and Bridge Sampling
Likelihood Maximisation and Automatic Differentiation
Applications and Extensions
Conclusion and Outlook
References
39 Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation
Contents
Introduction
Matching of Geometric Curves Based on Reparametrization-Invariant Riemannian Metrics
General Framework
The SRV Framework
Curves in Rd
Curves in Lie Groups
Curves in Homogenous Spaces
Curves in Riemannian Manifolds
Implementation
The Geodesic Boundary Value Problem on Parametrized Curves
Normalization by Isometries
Minimization over the Reparametrization Group
Dynamic Programming Approach
Discretizing the Diffeomorphism Group and Using Gradient-Based Methods
Iterative ``Horizontalization'' Method
Relaxation of the Exact Matching Problem
Open-Source Implementations
Conclusion
References
40 An Overview of SaT Segmentation Methodology and Its Applications in Image Processing
Contents
Introduction
SaT Methodology
SaT-Based Methods and Applications
T-ROF Method
Two-Stage Method for Poisson or Gamma Noise
SLaT Method for Color Images
Two-Stage Method for Hyperspectral Images
Tight-Frame-Based Method for Images with Vascular Structures
Wavelet-Based Segmentation Method for Spherical Images
Three-Stage Method for Images with Intensity Inhomogeneity
Conclusions
References
41 Recent Development of Medical Shape Analysis via Computational Quasi-conformal Geometry
Contents
Introduction
The Quasi-conformal Teichmüller Theory
Conformal Mappings
Quasi-conformal Mappings
Teichmüller Mappings
Medical Image Segmentation and Registration by Quasi-conformal Theory
Image Segmentation
Image Registration and Fusion
Other Imaging Applications
Surface Analysis for Medical Applications
3D Surface Registration
High-Dimensional Shape Deformation
Disease Diagnosis and Classification by Quasi-conformal Geometry
Classification of the Alzheimer's Disease
Other Classification Model
Conclusion
References
42 A Survey of Topology and Geometry-Constrained Segmentation Methods in Weakly Supervised Settings
Contents
Introduction
Geometrical Constraints
Characterization of Geometrical Constraints
Model 1: A Simple Variational Model
Model 2: A Moving Band Model
Model 3: A Dual Level Model
Model 4: The Use of Moment Constraint for Segmentation
Model 5: Convex Segmentation Models
Model 6: Convex Models Based on Geodesic Distances
Other Possible Models
Topological Prior Knowledge
Topology Prescription
Digital Topology
Purely Continuous Methods
Regularization Enforcement on the Evolving Front
Regularization by Geometric Flows
Higher-Order Schemes for Level Set-Based Segmentation Models
Joint Segmentation and Registration Models
Motivations
Overview of Existing Methods
A Mixed Segmentation/Registration Model Based on a Nonlocal Characterization of Weighted Total Variation
Other Related Models
Optimal Flow Frameworks
Shape Priors
Deep Learning Models
Multi-modal Problems
Conclusion
References
43 Recent Developments of Surface Parameterization Methods Using Quasi-conformal Geometry
Contents
Introduction
Previous Works on Surface Parameterization
Mesh Parameterization
Point Cloud Parameterization
Mathematical Background
Conformal Maps
Quasi-conformal Maps
Linear Beltrami Solver (LBS)
Beltrami Holomorphic Flow (BHF)
Teichmüller Maps
Mesh Parameterization Using Quasi-conformal Geometry
Genus-0 Closed Triangle Meshes
Conformal Parameterization
Quasi-conformal Parameterization
Simply Connected Open Triangle Meshes
Conformal Parameterization
Quasi-conformal Parameterization
Multiply Connected Open Triangle Meshes
Conformal Parameterization
Quasi-conformal Parameterization
Point Cloud Parameterization Using Conformal and Quasi-conformal Geometry
Genus-0 Point Clouds
Point Clouds with Disk Topology
Applications
Conclusion
References
44 Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection
Contents
Introduction
Scores on Lie Groups G=Rd T and the Motivation for Left-Invariant Processing and a Left-Invariant Connection on T(G)
Motivation: Choosing a Cartan Connection for Geometric (PDE-Based) Image Processing via Scores
Structure and Contributions of the Article
A Parameterized Class of Cartan Connections and Their Duals
Expressing the Lie-Cartan Connection (and Its Dual) in Left-Invariant Coordinates
(Partial) Lie-Cartan Connections for (Sub)-Riemannian Geometry
The Special Case of Interest ν=1 and Hamiltonian Flows for the Riemannian Geodesic Problem on G
The Homogeneous Space Md of Positions and Orientations
The Metric Models on Md: Shortest Curves and Spheres
Straight Curve Fits
Exponential Curve Fits of the Second Order Are Found by SVD of the Hessian
Inclusion of External Regularization
A Single Exponential Curve Fit Gives Rise to a Gauge Frame
Overview of Image Analysis Applications for G=SE(d)
Shortest Curve Application: Tracking of Blood Vessels
Shortest Curve Applications: Geodesic Vessel and Fibre Tracking in M3
Straight Curve Application: Biomarkers for Diabetes
Straight Curve Application: PDEs on M2 for Denoising
Straight Curve Application: PDEs on M3 for Denoising FODFs in DW-MRI
Straight Curve Application: PDEs on M3 for Denoising 3D X-Ray Data
Conclusion
Appendix A: Hamiltonian Flow of the Left-Invariant (Sub-)Riemannian Geodesic Problem on Lie Group G
Appendix B: Left-Invariant Vector Fields on SE(3) via Two Charts
Appendix C: Proofs of Results on Lie-Cartan Connections
Proof of Lemma 2
Proof of Lemma 3
References
45 PDE-Constrained Shape Optimization: Toward Product Shape Spaces and Stochastic Models
Contents
Introduction
Optimization Over Product Shape Manifolds
Optimization on Shape Spaces with Steklov–Poincaré Metric
Optimization of Multiple Shapes
Stochastic Multi-shape Optimization and the Stochastic Gradient Method
Numerical Investigations
Deterministic Model Problem
Stochastic Model Problem
Numerical Experiments
Deterministic Case: Behavior of Algorithm 2
Stochastic Case: Behavior of Algorithm 3
Robustness: Deterministic vs. Stochastic Model
Conclusion
References
46 Iterative Methods for Computing Eigenvectors of Nonlinear Operators
Contents
Introduction and Preliminaries
One-Homogeneous Functionals
Eigenvectors of Nonlinear Operators
Nossek-Gilboa (NG)
NG Flow Properties
NG Iteration Algorithm Properties
Aujol-Gilboa-Papadakis (AGP)
AGP Flow Properties
AGP Iteration Algorithm Properties
Feld-Aujol-Gilboa-Papadakis (FAGP)
Cohen-Gilboa (CG)
Bungert-Hait-Papadakis-Gilboa (BHPG)
Evaluation and Examples
Global and Local Measures
Numerical Examples
Conclusion, Discussion and Open Problems
References
47 Optimal Transport for Generative Models
Contents
Introduction
Related Works
Optimal Transport Map
Generative Models
Optimal Transport Theory
Monge's Problem
Kantorovich's Approach
Brenier's Approach
McCann's Displacement
Benamou-Brenier Dynamic Fluid
Otto's Calculus
Regularity of Optimal Transport Maps
Convex Target Domain
Non-convex Target Domain
Computational Algorithm
Semi-discrete Optimal Transport Map
Damping Newton's Method
Monte-Carlo Method
Manifold Distribution Principle
Manifold Learning
ReLu Deep Neural Network
AutoEncoder
Generative Adversarial Networks
Competition vs. Collaboration
Memorization vs. Learning
Mode Collapsing
AE-OT Model
Conclusion
References
48 Image Reconstruction in Dynamic Inverse Problems with Temporal Models
Contents
Introduction
Outline of Survey
Spatiotemporal Inverse Problems
Reconstruction Without Explicit Temporal Models
Reconstruction Using a Motion Model
Parametrized Motion Models
General Variational Formulation
Reconstruction Using a Deformable Template
Deformation Operators
General Variational Formulation
Time Discretized Data
Motion Models Based on Partial Differential Equations
Physical Motion Constraints
Joint Motion Estimation and Reconstruction
Implementation and Reconstruction
Deformable Templates Given by Diffeomorphisms
Flow of Diffeomorphisms and Intensities
Deformable Templates by Metamorphosis
Spatiotemporal Reconstruction with LDDMM
Data-Driven Approaches
Data-Driven Reconstruction Without Temporal Modelling
Learning Deformation Operators
Learning Motion Models
Outlook and Conclusions
Note
References
49 Computational Conformal Geometric Methods for Vision
Contents
Fundamental Concepts
Riemann Surfaces
Conformal Maps
Uniformization
Quasi-conformal Maps
Holomorphic Quadratic Differential
Teichmüller Map
Teichmüller Space
Ricci Flow
Computational Methods
Concepts in Discrete Setting
A Discrete Conformal Geometry of Polyhedral Surfaces Derived from Vertex Scaling
A Discrete Conformal Geometry of Polyhedral Surfaces Derived from Circle Patterns
Harmonic Maps
Hodge Decomposition
Direct Applications
Shape Space
Surface Registration
Medical Imaging
Conclusion
References
50 From Optimal Transport to Discrepancy
Contents
Introduction
Preliminaries
Discrepancies
Optimal Transport and Wasserstein Distances
Regularized Optimal Transport
Sinkhorn Divergence
Numerical Approach and Examples
Conclusions
Basic Theorems
References
51 Compensated Convex-Based Transforms for Image Processing and Shape Interrogation
Contents
Introduction
Related Areas: Semiconvex Envelope
Related Areas: Proximity Hull
Related Areas: Mathematical Morphology
Related Areas: Quadratic Envelopes
Outline of the Chapter
Notation and Preliminaries
Compensated Convexity-Based Transforms
Smoothing Transform
Stable Ridge/Edge Transform
Basic Transforms
Extractable Corner Points
Interior Corners
Stable Multiscale Intersection Transform of Smooth Manifolds
Stable Multiscale Medial Axis Map
Approximation Transform
Numerical Algorithms
Convex-Based Algorithms
PDE-Based Algorithm
Biconjugate Algorithm
Moreau Envelope-Based Algorithms
Distance-Based Algorithm
Parabola Envelope-Based Algorithm
The Moreau Transform as Legendre–Fenchel Transform
Numerical Examples
Prototype Example: Upper Transform of a Singleton Set of R2
Intersection of Sampled Smooth Manifolds
Approximation Transform
Level Set Reconstruction
Salt and Pepper Noise Removal
Inpainting
Conclusions
References
52 The Potts Model with Different Piecewise Constant Representations and Fast Algorithms: A Survey
Contents
Introduction
Representation by Integer-Valued Labeling Function
Potts Model for Integer-Valued Functions
Graph Cuts for the Integer-Labeled Potts Model
Continuous Max-Flow Formulation for Integer-Valued Potts Model
Numerical Algorithms for the Integer-Valued Continuous Max-Flow Problems
Representation by Simplex-Constrained Vector Functions
Primal-Dual Formulation for Simplex-Constrained Potts Model
Dual Formulation for Simplex-Constrained Potts Model
Continuous Max-Flow Formulation for Simplex-Constrained Potts Model
Representation by Overlapping Functions
Potts Model with Overlapping Binary Functions Representation
Extension to More General Cases
Convex Relaxation via Convex Envelope for Overlapping Representation
Numerical Algorithms for Relaxed Potts Model via Overlapping Representation
A Continuous Max-Flow Approach for 4-Phase Overlapping Binary Representation
Extension to the High-Dimensional Graphical Models
Constructing a Graph for a Given Data Set
Graphical Potts Model with Simplex-Constrained Representation
Efficient Inference in CRF Model
Conclusion
References
53 Shape Spaces: From Geometry to Biological Plausibility
Contents
Introduction: Shape Spaces
Shape Spaces Under Diffeomorphic Action
Hybrid Models
Description
Elastic Metrics
Three-Dimensional Case
Elastic Metrics on Surfaces
Elastic Metrics on Curves
Growth Models
Introduction
Riemannian Viewpoint
Growth as an Internal Force
A Simple Example
Growth Due to External Action
Constraints, Deformation Modules, and Other Growth Models
Conclusion
Appendix A: Elastic Surface Metric as the Limit of the Laminar Model (section ``Elastic Metrics on Surfaces'')
Appendix B: Existence of Optimal Paths (section ``Riemannian Viewpoint'')
References
Index