Handbook of mathematical methods in imaging

Cover
Titule Page
Copyright
Preface
	Second Edition
About the Editor
Contents
Contributors
Part I Inverse Problems – Methods
	Linear Inverse Problems
		Contents
		1 Introduction
		2 Background
		3 Mathematical Modeling and Analysis
			A Platonic Inverse Problem
			Cormack\'s Inverse Problem
			Forward and Reverse Diffusion
			Deblurring as an Inverse Problem
			Extrapolation of Band-Limited Signals
			PET
			Some Mathematics for Inverse Problems
				Weak Convergence
				Linear Operators
				Compact Operators and the SVD
				The Moore–Penrose Inverse
				Alternating Projection Theorem
		4 Numerical Methods
			Tikhonov Regularization
			Iterative Regularization
			Discretization
		5 Conclusion
		Cross-References
		References
	Large-Scale Inverse Problems in Imaging
		1 Introduction
		2 Background
			Model Problems
			Imaging Applications
				Image Deblurring and Deconvolution
				Multi-Frame Blind Deconvolution
				Tomosynthesis
		3 Mathematical Modeling and Analysis
			Linear Problems
				SVD Analysis
				Regularization by SVD Filtering
				Variational Regularization and Constraints
				Iterative Regularization
				Hybrid Iterative-Direct Regularization
				Choosing Regularization Parameters
			Separable Inverse Problems
				Fully Coupled Problem
				Decoupled Problem
				Variable Projection Method
			Nonlinear Inverse Problems
		4 Numerical Methods and Case Examples
			Linear Example: Deconvolution
			Separable Example: Multi-Frame Blind Deconvolution
			Nonlinear Example: Tomosynthesis
		5 Conclusion
		References
	Regularization Methods for Ill-Posed Problems
		1 Introduction
		2 Theory of Direct Regularization Methods
			Tikhonov Regularization in Hilbert Spaces with Quadratic Misfit and Penalty Terms
			Variational Regularization in Banach Spaces with Convex Penalty Term
			Some Specific Results for Hilbert Space Situations
			Further Convergence Rates Under Variational Inequalities
		3 Examples
		4 Conclusion
		References
	Distance Measures and Applications to Multimodal Variational Imaging
		1 Introduction
		2 Distance Measures
			Deterministic Pixel Measure
			Morphological Measures
			Statistical Distance Measures
			Statistical Distance Measures (Density Based)
				Density Estimation
				Csiszár Divergences (f-Divergences)
			f-Information
			Distance Measures Including Statistical Prior Information
		3 Mathematical Models for Variational Imaging
		4 Registration
		5 Recommended Reading
		6 Conclusion
		References
	Energy Minimization Methods
		1 Introduction
			Background
			The Main Features of the Minimizers as a Function of the Energy
			Organization of the Chapter
		2 Preliminaries
			Notation
			Reminders and Definitions
		3 Regularity Results
			Some General Results
			Stability of the Minimizers of Energies with Possibly Nonconvex Priors
				Local Minimizers
				Global Minimizers of Energies with for Possibly Nonconvex Priors
			Nonasymptotic Bounds on Minimizers
		4 Nonconvex Regularization
			Motivation
			Assumptions on Potential Functions ϕ
			How It Works on R
			Either Smoothing or Edge Enhancement
		5 Nonsmooth Regularization
			Main Theoretical Result
			Examples and Discussion
			Applications
		6 Nonsmooth Data Fidelity
			General Results
			Applications
		7 Nonsmooth Data Fidelity and Regularization
			The L1-TV Case
				Denoising of Binary Images and Convex Relaxation
				Multiplicative Noise Removal
			1 Data Fidelity with Regularization Concave on R+
				Motivation
				Main Theoretical Results
				Applications
		8 Conclusion
		References
	Compressive Sensing
		1 Introduction
		2 Background
			Early Developments in Applications
			Sparse Approximation
			Information-Based Complexity and Gelfand Widths
			Compressive Sensing
			Developments in Computer Science
		3 Mathematical Modelling and Analysis
			Preliminaries and Notation
			Sparsity and Compression
			Compressive Sensing
			The Null Space Property
			The Restricted Isometry Property
			Coherence
			RIP for Gaussian and Bernoulli Random Matrices
			Random Partial Fourier Matrices
			Compressive Sensing and Gelfand Widths
			Extensions of Compressive Sensing
				Affine Low-Rank Minimization
				Nonlinear Measurements
			Applications
		4 Numerical Methods
			A Primal-Dual Algorithm
			Iteratively Re-weighted Least Squares
				Weighted 2-Minimization
				An Iteratively Re-weighted Least Squares Algorithm (IRLS)
				Convergence Properties
				Rate of Convergence
			Numerical Experiments
			Extensions to Affine Low-Rank Minimization
		5 Open Questions
			Deterministic Compressed Sensing Matrices
			Removing Log-Factors in the Fourier-RIP Estimate
			Compressive Sensing with Nonlinear Measurements
		6 Conclusion
		References
	Duality and Convex Programming
		1 Introduction
			Linear Inverse Problems with Convex Constraints
			Imaging with Missing Data
			Image Denoising and Deconvolution
			Inverse Scattering
			Fredholm Integral Equations
		2 Background
			Lipschitzian Properties
			Subdifferentials
		3 Duality and Convex Analysis
			Fenchel Conjugation
			Fenchel Duality
			Applications
			Optimality and Lagrange Multipliers
			Variational Principles
			Fixed Point Theory and Monotone Operators
		4 Case Studies
			Linear Inverse Problems with Convex Constraints
			Imaging with Missing Data
			Inverse Scattering
			Fredholm Integral Equations
		5 Open Questions
		6 Conclusion
		References
	EM Algorithms
		1 Maximum Likelihood Estimation
		2 The Kullback–Leibler Divergence
		3 The EM Algorithm
			The Maximum Likelihood Problem
			The Bare-Bones EM Algorithm
			The Bare-Bones EM Algorithm Fleshed Out
			The EM Algorithm Increases the Likelihood
		4 The EM Algorithm in Simple Cases
			Mixtures of Known Densities
			A Deconvolution Problem
			The Deconvolution Problem with Binning
			Finite Mixtures of Unknown Distributions
			Empirical Bayes Estimation
		5 Emission Tomography
			Flavors of Emission Tomography
			The Emission Tomography Experiment
			The Shepp–Vardi EM Algorithm for PET
			Prehistory of the Shepp–Vardi EM Algorithm
		6 Electron Microscopy
			Imaging Macromolecular Assemblies
			The Maximum Likelihood Problem
			The EM Algorithm, up to a Point
			The Ill-Posed Weighted Least-Squares Problem
		7 Regularization in Emission Tomography6pt
			The Need for Regularization
			-1pc Smoothed EM Algorithms
			Good\'s Roughness Penalization
			Gibbs Smoothing
		8 Convergence of EM Algorithms
			The Two Monotonicity Properties
			Monotonicity of the Shepp–Vardi EM Algorithm
			Monotonicity for Mixtures
			Monotonicity of the Smoothed EM Algorithm
			Monotonicity for Exact Gibbs Smoothing
		9 EM-Like Algorithms
			Minimum Cross-Entropy Problems
			Nonnegative Least Squares
			Multiplicative Iterative Algorithms
		10 Accelerating the EM Algorithm
			The Ordered Subset EM Algorithm
			The ART and Cimmino–Landweber Methods
			The MART and SMART Methods
			Row-Action and Block-Iterative EM Algorithms
		References
	EM Algorithms from a Non-stochastic Perspective
		1 Introduction
		2 A Non-stochastic Formulation of EM
			The Non-stochastic EM Algorithm
				The Continuous Case
				The Discrete Case
		3 The Stochastic EM Algorithm
			The E-Step and M-Step
			Difficulties with the Conventional Formulation
			An Incorrect Proof
			Acceptable Data
		4 The Discrete Case
		5 Missing Data
		6 The Continuous Case
			Acceptable Preferred Data
			Selecting Preferred Data
			Preferred Data as Missing Data
		7 The Continuous Case with Y=h(X)
			An Example
			Censored Exponential Data
			A More General Approach
		8 A Multinomial Example
		9 The Example of Finite Mixtures
		10 The EM and the Kullback-Leibler Distance
			Using Acceptable Data
		11 The Approach of Csiszár and Tusnády
			The Framework of Csiszár and Tusnády
			Alternating Minimization for the EM Algorithm
		12 Sums of Independent Poisson Random Variables
			Poisson Sums
			The Multinomial Distribution
		13 Poisson Sums in Emission Tomography
			The SPECT Reconstruction Problem
				The Preferred Data
				The Incomplete Data
			Using the KL Distance
		14 Nonnegative Solutions for Linear Equations
			The General Case
			Regularization
			Acceleration
			Using Prior Bounds on λ
				The ABMART Algorithm
				The ABEMML Algorithm
		15 Finite Mixture Problems
			Mixtures
			The Likelihood Function
			A Motivating Illustration
			The Acceptable Data
			The Mix-EM Algorithm
			Convergence of the Mix-EM Algorithm
		16 More on Convergence
		17 Open Questions
		18 Conclusion
		References
	Iterative Solution Methods
		1 Introduction
		2 Preliminaries
			Conditions on F
			Source Conditions
			Stopping Rules
		3 Gradient Methods
			Nonlinear Landweber Iteration
			Landweber Iteration in Hilbert Scales
			Steepest Descent and Minimal Error Method
			Further Literature on Gradient Methods
				Iteratively Regularized Landweber Iteration
				A Derivative Free Approach
				Generalization to Banach Spaces
		4 Newton Type Methods
			Levenberg–Marquardt and Inexact Newton Methods
			Further Literature on Inexact Newton Methods
			Iteratively Regularized Gauss–Newton Method
			Further Literature on Gauss–Newton Type Methods
				Generalizations of the IRGNM
				Generalized Source Conditions
				Other A-posteriori Stopping Rules
				Stochastic Noise Models
				Generalization to Banach Space
				Efficient Implementation
		5 Nonstandard Iterative Methods
			Kaczmarz and Splitting Methods
			EM Algorithms
			Bregman Iterations
		6 Conclusion
		References
	Level Set Methods for Structural Inversion and Image Reconstruction
		1 Introduction
			Level Set Methods for Inverse Problems and Image Reconstruction
			Images and Inverse Problems
			The Forward and the Inverse Problem
		2 Examples and Case Studies
			Example 1: Microwave Breast Screening
			Example 2: History Matching in Petroleum Engineering
			Example 3: Crack Detection
		3 Level Set Representation of Images with Interfaces
			The Basic Level Set Formulation for Binary Media
			Level Set Formulations for Multivalued and Structured Media
				Different Levels of a Single Smooth Level Set Function
				Piecewise Constant Level Set Function
				Vector Level Set
				Color Level Set
				Binary Color Level Set
			Level Set Formulations for Specific Applications
				A Modification of Color Level Set for Tumor Detection
				A Modification of Color Level Set for Reservoir Characterization
				A Modification of the Classical Level Set Technique for Describing Cracks or Thin Shapes
		4 Cost Functionals and Shape Evolution
			General Considerations
			Cost Functionals
			Transformations and Velocity Flows
			Eulerian Derivatives of Shape Functionals
			The Material Derivative Method
			Some Useful Shape Functionals
			The Level Set Framework for Shape Evolution
		5 Shape Evolution Driven by Geometric Constraints
			Penalizing Total Length of Boundaries
			Penalizing Volume or Area of Shape
		6 Shape Evolution Driven by Data Misfit
			Shape Deformation by Calculus of Variations
				Least Squares Cost Functionals and Gradient Directions
				Change of b Due to Shape Deformations
				Variation of Cost Due to Velocity Field v(x)
				Example: Shape Variation for TM-Waves
				Example: Evolution of Thin Shapes (Cracks)
			Shape Sensitivity Analysis and the Speed Method
				Example: Shape Sensitivity Analysis for TM-Waves
				Shape Derivatives by a Min–Max Principle
			Formal Shape Evolution Using the Heaviside Function
				Example: Breast Screening–Smoothly Varying Internal Profiles
				Example: Reservoir Characterization–Parameterized Internal Profiles
		7 Regularization Techniques for Shape Evolution Driven by Data Misfit
			Regularization by Smoothed Level Set Updates
			Regularization by Explicitly Penalizing Rough Level Set Functions
			Regularization by Smooth Velocity Fields
			Simple Shapes and Parameterized Velocities
		8 Miscellaneous On-Shape Evolution
			Shape Evolution and Shape Optimization
			Some Remarks on Numerical Shape Evolution with Level Sets
			Speed of Convergence and Local Minima
			Topological Derivatives
		9 Case Studies
			Case Study: Microwave Breast Screening
			Case Study: History Matching in Petroleum Engineering
			Case Study: Reconstruction of Thin Shapes (Cracks)
		References
Part II Inverse Problems – Case Examples
	Expansion Methods
		1 Introduction
		2 Electrical Impedance Tomography for Anomaly Detection
			Physical Principles
			Mathematical Model
			Asymptotic Analysis of the Voltage Perturbations
			Numerical Methods for Anomaly Detection
				Detection of a Single Anomaly: A Projection-Type Algorithm
				Detection of Multiple Anomalies: A MUSIC-Type Algorithm
			Bibliography and Open Questions
		3 Ultrasound Imaging for Anomaly Detection
			Physical Principles
			Asymptotic Formulas in the Frequency Domain
			Asymptotic Formulas in the Time Domain
			Numerical Methods
				MUSIC-Type Imaging at a Single Frequency
				Backpropagation-Type Imaging at a Single Frequency
				Kirchhoff-Type Imaging Using a Broad Range of Frequencies
				Time-Reversal Imaging
			Bibliography and Open Questions
		4 Infrared Thermal Imaging
			Physical Principles
			Asymptotic Analysis of Temperature Perturbations
			Numerical Methods
				Detection of a Single Anomaly
				Detection of Multiple Anomalies: A MUSIC-Type Algorithm
			Bibliography and Open Questions
		5 Impediography
			Physical Principles
			Mathematical Model
			Substitution Algorithm
			Bibliography and Open Questions
		6 Magneto-Acoustic Imaging
			Magneto-Acousto-Electrical Tomography
				Physical Principles
				Mathematical Model
				Substitution Algorithm
			Magneto-Acoustic Imaging with Magnetic Induction
				Physical Principles
				Mathematical Model
				Reconstruction Algorithm
			Bibliography and Open Questions
		7 Magnetic Resonance Elastography
			Physical Principles
			Mathematical Model
			Asymptotic Analysis of Displacement Fields
			Numerical Methods
			Bibliography and Open Questions
		8 Photo-Acoustic Imaging of Small Absorbers
			Physical Principles
			Mathematical Model
			Reconstruction Algorithms
				Determination of Location
				Estimation of Absorbing Energy
				Reconstruction of the Absorption Coefficient
			Bibliography and Open Questions
		9 Conclusion
		References
	Sampling Methods
		1 Introduction
		2 The Factorization Method in Impedance Tomography
			Impedance Tomography in the Presence of Insulating Inclusions
			Conducting Obstacles
			Local Data
			Other Generalizations
				The Half-Space Problem
				The Crack Problem
		3 The Factorization Method in Inverse Scattering Theory
			Inverse Acoustic Scattering by a Sound-Soft Obstacle
			Inverse Electromagnetic Scattering by an Inhomogeneous Medium
			Historical Remarks and Open Questions
		4 Related Sampling Methods
			The Linear Sampling Method
			MUSIC
			The Singular Sources Method
			The Probe Method
		5 Conclusion
		6 Appendix
		References
	Inverse Scattering
		1 Introduction
		2 Direct Scattering Problems
			The Helmholtz Equation
			Obstacle Scattering
			Scattering by an Inhomogeneous Medium
			The Maxwell Equations
			Historical Remarks
		3 Uniqueness in Inverse Scattering
			Scattering by an Obstacle
			Scattering by an Inhomogeneous Medium
			Historical Remarks
		4 Iterative and Decomposition Methods in Inverse Scattering
			Newton Iterations in Inverse Obstacle Scattering
			Decomposition Methods
			Iterative Methods Based on Huygens\' Principle
			Newton Iterations for the Inverse Medium Problem
			Least-Squares Methods for the Inverse Medium Problem
			Born Approximation
			Historical Remarks
		5 Qualitative Methods in Inverse Scattering
			The Far-Field Operator and Its Properties
			The Linear Sampling Method
			The Factorization Method
			Lower Bounds for the Surface Impedance
			Transmission Eigenvalues
			Historical Remarks
		References
	Electrical Impedance Tomography
		1 Introduction
			Measurement Systems and Physical Derivation
			The Concentric Anomaly: A Simple Example
			Measurements with Electrodes
		2 Uniqueness and Stability of the Solution
			The Isotropic Case
				Calderón\'s Paper
				Uniqueness at the Boundary
				CGO Solutions for the Schrödinger Equation
				Dirichlet-to-Neumann Map and Cauchy Data for the Schrödinger Equation
				Global Uniqueness for n≥3
				Global Uniqueness in the Two-Dimensional Case
				Some Open Problems for the Uniqueness
				Stability of the Solution at the Boundary
				Global Stability for n≥3
				Global Stability for the Two-Dimensional Case
				Some Open Problems for the Stability
			The Anisotropic Case
				The Non-uniqueness
				Uniqueness up to Diffeomorphism
				Anisotropy Which Is Partially A Priori Known
			Some Remarks on the Dirichlet-to-Neumann Map
				EIT with Partial Data
				The Neumann-to-Dirichlet Map
		3 The Reconstruction Problem
			Locating Objects and Boundaries
			Forward Solution
			Regularized Linear Methods
			Regularized Iterative Nonlinear Methods
			Direct Nonlinear Solution
		4 Conclusion
		References
	Synthetic Aperture Radar Imaging
		1 Introduction
		2 Historical Background
		3 Mathematical Modeling
			Scattering of Electromagnetic Waves
			Basic Facts About the Wave Equation
			Basic Scattering Theory
				The Lippmann–Schwinger Integral Equation
				The Lippmann–Schwinger Equation in the Frequency Domain
				The Born Approximation
			The Incident Field
			Model for the Scattered Field
			The Matched Filter
			The Small-Scene Approximation
			The Range Profile
		4 Survey on Mathematical Analysis of Methods
			Inverse Synthetic Aperture Radar (ISAR)
				The Data-Collection Manifold
				ISAR in the Time Domain
			Synthetic Aperture Radar
				Spotlight SAR
				Stripmap SAR
			Resolution for ISAR and Spotlight SAR
				Down-Range Resolution in the Small-Angle Case
				Cross-Range Resolution in the Small-Angle Case
		5 Numerical Methods
			ISAR and Spotlight SAR Algorithms
			Range Alignment
		6 Open Problems
			Problems Related to Unmodeled Motion
			Problems Related to Unmodeled Scattering Physics
			New Applications of Radar Imaging
		7 Conclusion
		References
	Tomography
		1 Introduction
		2 Background
		3 Mathematical Modeling and Analysis
		4 Numerical Methods and Case Examples
		5 Conclusion
		References
	Microlocal Analysis in Tomography
		1 Introduction
		2 Motivation
			X-Ray Tomography (CT) and Limited Data Problems
			Electron Microscope Tomography (ET) Over Arbitrary Curves
			Synthetic-Aperture Radar Imaging
				The Linearized Model in SAR Imaging
			General Observations
		3 Properties of Tomographic Transforms
			Function Spaces
			Basic Properties of the Radon Line Transform
			Continuity Results for the X-Ray Transform
			Filtered Backprojection (FBP) for the X-Ray Transform
			Limited Data Algorithms
			ROI Tomography
			Limited Angle CT
			Fan Beam and Cone Beam CT
			Algorithms in Conical Tilt ET
		4 Microlocal Analysis
			Singular Support and Wavefront Set
			Pseudodifferential Operators
			Fourier Integral Operators
		5 Applications to Tomography
			Microlocal Analysis in X-Ray CT
			Limited Data X-Ray CT
			Exterior X-Ray CT Data
			Limited Angle Data
			Region of Interest (ROI) Data
			Microlocal Analysis of Conical Tilt Electron Microscope Tomography (ET)
			SAR Imaging
				Monostatic SAR Imaging
				Common Offset Bistatic SAR Imaging
		6 Conclusion
		References
	Mathematical Methods in PET and SPECT Imaging
		1 Introduction
		2 Background
			The Importance of PET and SPECT
			The Mathematical Foundation of the IART
			A General Methodology for Constructing Transform Pairs
		3 The Inverse Radon Transform and the Inverse Attenuated Radon Transform
			The Construction of the Inverse Radon Transform
			The Construction of Inverse Attenuated Radon Transform
		4 SRT for PET
			Comparison between FBP and SRT for PET
				Simulated Data
				Real Data
		5 SRT for SPECT
		6 Conclusion
		7 Cross-References
		References
	Mathematics of Electron Tomography
		1 Introduction
		2 The Transmission Electron Microscope (TEM)
			Sample Preparation
		3 Basic Notation and Definitions
		4 The Forward Model
			Illumination
			Electron–Specimen Interaction
				Elastic Scattering
				Relativistic Corrections
				Inelastic Scattering
				Properties of the Scattering Potential
				Computationally Feasibility
				Geometrical Optics Approximation
				The Small Angle, Projection, and Weak Phase Object Approximations
				Other Approaches
			Optics
				The General Setting
				The Optical Set-Up
				Lens-Less Imaging
				Single Thin Lens with an Aperture
				Model Refinements
			Detection
				Intensity Generated by a Single Electron
				The Total Intensity and Its Detector Response
				Characteristics of the Noise
				The Measured Image Data
			Forward Operator for Combined Phase and Amplitude Contrast
				Standard Phase Contrast Model
				Phase Contrast Model with Lens-Less Imaging
				Phase Contrast Model with Ideal Detector Response and Optics
			Forward Operator for Amplitude Contrast Only
			Summary
		5 Data Acquisition Geometry
			Parallel Beam Geometries
			Examples Relevant for ET
		6 The Reconstruction Problem in ET
			Mathematical Formulation
			Notion of Solution
		7 Specific Difficulties in Addressing the Inverse Problem
			The Dose Problem
			Incomplete Data, Uniqueness, and Stability
				Standard Phase Contrast Model
				Amplitude Contrast Model
				General Inverse Scattering
			Nuisance Parameters
				Detector Parameters
				Illumination and Optics Parameters
				Specimen-Dependent Parameters
		8 Data Pre-processing
			Basic Pre-processing
			Alignment
			Deconvolving Detector Response
			Deconvolving Optics PSF
			Phase Retrieval
		9 Reconstruction Methods
			Analytic Methods
				Backprojection-Based Methods
				Electron Λ-Tomography (ELT)
				Generalized Ray Transform
			Approximative Inverse
			Iterative Methods with Early Stopping
				Comments and Discussion
			Variational Methods
				Entropy Regularization
				TV Type of Regularization
				Sparsity Promoting Regularization
			Other Reconstruction Schemes
		10 Validation
		11 Examples
			Balls
			Virions and Bacteriophages in Aqueous Buffer
		12 Conclusion
		References
	Optical Imaging
		1 Introduction
		2 Background
			Spectroscopic Measurements
			Imaging Systems
		3 Mathematical Modeling and Analysis
			Radiative Transfer Equation
			Diffusion Approximation
				Boundary Conditions for the DA
				Source Models for the DA
				Validity of the DA
				Numerical Solution Methods for the DA
			Hybrid Approaches Utilizing the DA
			Green\'s Functions and the Robin to Neumann Map
			The Forward Problem
			Schrödinger Form
			Perturbation Analysis
				Born Approximation
				Rytov Approximation
			Linearization
				Linear Approximations
				Sensitivity Functions
			Adjoint Field Method
				Time-Domain Case
			Light Propagation and Its Probabilistic Interpretation
		4 Numerical Methods and Case Examples
			Image Reconstruction in Optical Tomography
			Bayesian Framework for Inverse Optical Tomography Problem
				Bayesian Formulation for the Inverse Problem
				Inference
				Likelihood and Prior Models
				Nonstationary Problems
				Approximation Error Approach
			Experimental Results
				Experiment and Measurement Parameters
				Prior Model
				Selection of FEM Meshes and Discretization Accuracy
				Construction of Error Models
				Computation of the MAP Estimates
		5 Conclusion
		References
	Photoacoustic and Thermoacoustic Tomography: Image Formation Principles
		1 Introduction
		2 Imaging Physics and Contrast Mechanisms
			The Thermoacoustic Effect and Signal Generation
			Image Contrast in Laser-Based PAT
			Image Contrast in RF-Based PAT
			Functional PAT
		3 Principles of PAT Image Reconstruction
			PAT Imaging Models in Their Continuous Forms
			Universal Backprojection Algorithm
			The Fourier-Shell Identity
				Special Case: Planar Measurement Geometry
			Spatial Resolution from a Fourier Perspective
				Effects of Finite Transducer Bandwidth
				Effects of Nonpoint-Like Transducers
		4 Speed-of-Sound Heterogeneities and Acoustic Attenuation
			Frequency-Dependent Acoustic Attenuation
			Weak Variations in the Speed-of-Sound Distribution
		5 Data Redundancies and the Half-Time Reconstruction Problem
			Data Redundancies
			Mitigation of Image Artifacts Due to Acoustic Heterogeneities
		6 Discrete Imaging Models
			Continuous-to-Discrete Imaging Models
			Finite-Dimensional Object Representations
			Discrete-to-Discrete Imaging Models
				Numerical Example: Impact of Representation Error on Computed Pressure Data
			Iterative Image Reconstruction
				Numerical Example: Influence of Representation Error on Image Accuracy
		7 Conclusion
		References
	Mathematics of Photoacoustic and Thermoacoustic Tomography
		1 Introduction
		2 Mathematical Models of TAT
			Point Detectors and the Wave Equation Model
			Acoustically Homogeneous Media and Spherical Means
			Main Mathematical Problems Arising in TAT
			Variations on the Theme: Planar, Linear, and Circular Integrating Detectors
		3 Mathematical Analysis of the Problem
			Uniqueness of Reconstruction
				Acoustically Homogeneous Media
				Acoustically Inhomogeneous Media
			Stability
			Incomplete Data
				Uniqueness of Reconstruction
				``Visible\'\' (``Audible\'\') Singularities
				Stability of Reconstruction for Incomplete Data Problems
			Discussion of the Visibility Condition
				Visibility for Acoustically Homogeneous Media
				Visibility for Acoustically Inhomogeneous Media
			Range Conditions
				The Range of the Spherical Mean Operator M
				The Range of the Forward Operator W
			Reconstruction of the Speed of Sound
		4 Reconstruction Formulas, Numerical Methods, and Case Examples
			Full Data (Closed Acquisition Surfaces)
				Constant Speed of Sound
				Variable Speed of Sound
			Partial (Incomplete) Data
				Constant Speed of Sound
				Variable Speed of Sound
		5 Final Remarks and Open Problems
		References
	Mathematical Methods of Optical Coherence Tomography
		1 Introduction
		2 Basic Principles of OCT
		3 The Direct Scattering Problem
			Maxwell\'s Equations
			Initial Conditions
			The Measurements
		4 Solution of the Direct Problem
			Born and Far Field Approximation
			The Forward Operator
		5 The Inverse Scattering Problem
			The Isotropic Case
				Non-dispersive Medium in Full Field OCT
				Non-dispersive Medium with Focused Illumination
				Dispersive Medium
				Dispersive Layered Medium with Focused Illumination
			The Anisotropic Case
		6 Conclusion
		References
	Wave Phenomena
		1 Introduction
		2 Background
			Wave Imaging and Boundary Control Method
			Travel Times and Scattering Relation
			Curvelets and Wave Equations
		3 Mathematical Modeling and Analysis
			Boundary Control Method
				Inverse Problems on Riemannian Manifolds
				From Boundary Distance Functions to Riemannian Metric
				From Boundary Data to Inner Products of Waves
				From Inner Products of Waves to Boundary Distance Functions
				Alternative Reconstruction of Metric via Gaussian Beams
			Travel Times and Scattering Relation
				Geometrical Optics
				Scattering Relation
			Curvelets and Wave Equations
				Curvelet Decomposition
				Curvelets and Wave Equations
				Low-Regularity Wave Speeds and Volterra Iteration
		4 Conclusion
		References
	Sonic Imaging
		1 Introduction
		2 The Model Problem
		3 The Born Approximation
			An Explicit Formula for the Slab
			An Error Estimate for the Born Approximation
		4 The Nonlinear Problem in the Time Domain
			The Kaczmarz Method in the Time Domain
			Numerical Example (Transmission)
			Numerical Examples (Reflection)
		5 The Nonlinear Problem in the Frequency Domain
			Initial Value Techniques for the Helmholtz Equation
			The Kaczmarz Method in the Frequency Domain
		6 Initial Approximations
		7 Pecularities
			Missing Low Frequencies
			Caustics and Trapped Rays
			The Role of Reflectors
		8 Direct Methods
			Boundary Control
			Inverse Scattering
		9 Conclusion
		References
	Imaging in Random Media
		1 Introduction
		2 Main Text
		3 Basic Imaging
			The Forward Model
				Passive Array Data Model
				Active Array Data Model
			Least Squares Inversion
				Connection to Bayesian Inversion
				Imaging with Passive Arrays
				Imaging with Active Arrays
			The Normal Operator and the Time Reversal Process
				The Time Reversal Process
				The Normal Operator
			Imaging in Smooth and Known Media
				Passive Arrays
				Active Arrays
			Robustness to Additive Noise
		4 Challenges of Imaging in Complex Media
			Cluttered Media
			The Random Model
			Time Reversal Is Not Imaging
		5 The Random Travel Time Model of Wave Propagation
			Long Range Scaling and Gaussian Statistics
			Statistical Moments
				Loss of Coherence
				Statistical Decorrelation of the Waves
		6 Setup for Imaging
		7 Migration Imaging
			The Expectation
			The SNR
		8 CINT Imaging
			Analysis of the Cross-Correlations for a Point Source
				The Mean
				The SNR
			Resolution Analysis of the CINT Imaging Function
				The Mean Point Spread Function
				The SNR
			CINT Images for Passive Arrays as the Smoothed Wigner Transform
				Calculation of the Wigner Transform
			CINT Imaging with Active Arrays
				Numerical Simulations
		9 Appendix 1: Second Moments of the Random Travel Time
		10 Appendix 2: Second Moments of the Local Cross-Correlations
		11 Conclusion
		References
Part III Image Restoration and Analysis
	Statistical Methods in Imaging
		1 Introduction
		2 Background
			Images in the Statistical Setting
			Randomness, Distributions, and Lack of Information
			Imaging Problems
		3 Mathematical Modeling and Analysis
			Prior Information, Noise Models, and Beyond
			Accumulation of Information and Priors
			Likelihood: Forward Model and Statistical Properties of Noise
			Maximum Likelihood and Fisher Information
			Informative or Noninformative Priors?
			Adding Layers: Hierarchical Models
		4 Numerical Methods and Case Examples
			Estimators
				Prelude: Least Squares and Tikhonov Regularization
				Maximum Likelihood and Maximum A Posteriori
				Conditional Means
			Algorithms
				Iterative Linear Least Squares Solvers
				Nonlinear Maximization
				EM Algorithm
				Markov Chain Monte Carlo Sampling
			Statistical Approach: What Is the Gain?
				Beyond the Traditional Concept of Noise
				Sparsity and Hypermodels
		5 Conclusion
		References
	Supervised Learning by Support Vector Machines
		1 Introduction
		2 Historical Background
		3 Mathematical Modeling and Applications
			Linear Learning
				Linear Support Vector Classification
				Linear Support Vector Regression
				Linear Least Squares Classification and Regression
			Nonlinear Learning
				Kernel Trick
				Support Vector Classification
				Support Vector Regression
				Relations to Sparse Approximation in RKHSs, Interpolation by Radial Basis Functions, and Kriging
				Least Squares Classification and Regression
				Other Models
				Multi-class Classification and Multitask Learning
				Applications of SVMs
		4 Survey of Mathematical Analysis of Methods
			Reproducing Kernel Hilbert Spaces
			Quadratic Optimization
			Results from Generalization Theory
		5 Numerical Methods
		6 Conclusion
		References
	Total Variation in Imaging
		1 Introduction
		2 Notation and Preliminaries on BV Functions
			Definition and Basic Properties
			Sets of Finite Perimeter: The Co-area Formula
			The Structure of the Derivative of a BV Function
		3 The Regularity of Solutions of the TV Denoising Problem
			The Discontinuities of Solutions of the TV Denoising Problem
			Hölder Regularity Results
		4 Some Explicit Solutions
		5 Numerical Algorithms: Iterative Methods
			Notation
			Chambolle\'s Algorithm
			Primal-Dual Approaches
		6 Numerical Algorithms: Maximum-Flow Methods
			Discrete Perimeters and Discrete Total Variation
			Graph Representation of Energies for Binary MRF
		7 Other Problems: Anisotropic Total Variation Models
			Global Solutions of Geometric Problems
			A Convex Formulation of Continuous Multilabel Problems
		8 Other Problems: Image Restoration
			Some Restoration Experiments
			The Image Model
		9 Final Remarks: A Different Total Variation-Based Approach to Denoising
		10 Conclusion
		References
	Numerical Methods and Applications in Total Variation Image Restoration
		1 Introduction
		2 Background
		3 Mathematical Modeling and Analysis
			Variants of Total Variation
				Basic Definition
				Multichannel TV
				Matrix-Valued TV
				Discrete TV
				Nonlocal TV
			Further Applications
				Inpainting in Transformed Domains
				Superresolution
				Image Segmentation
				Diffusion Tensors Images
		4 Numerical Methods and Case Examples
			Dual and Primal-Dual Methods
				Chan-Golub-Mulet\'s Primal-Dual Method
				Chambolle\'s Dual Method
				Primal-Dual Hybrid Gradient Method
				Semi-smooth Newton\'s Method
				Primal-Dual Active-Set Method
			Bregman Iteration
				Original Bregman Iteration
				The Basis Pursuit Problem
				Split Bregman Iteration
				Augmented Lagrangian Method
			Graph Cut Methods
				Leveling the Objective
				Defining a Graph
			Quadratic Programming
			Second-Order Cone Programming
			Majorization-Minimization
			Splitting Methods
		5 Conclusion
		References
	Mumford and Shah Model and Its Applications to Image Segmentation and Image Restoration
		1 Introduction
		2 Background: The First Variation
			Minimizing in u with K Fixed
		3 Minimizing in K
		4 Mathematical Modeling and Analysis: The Weak Formulation of the Mumford and Shah Functional
		5 Numerical Methods: Approximations to the Mumford and Shah Functional
		6 Ambrosio and Tortorelli Phase-Field Elliptic Approximations
		7 Approximations of the Perimeter by Elliptic Functionals
		8 Ambrosio-Tortorelli Approximations
		9 Level Set Formulations of the Mumford and Shah Functional
		10 Piecewise-Constant Mumford and Shah Segmentation Using Level Sets
		11 Piecewise-Smooth Mumford and Shah Segmentation Using Level Sets
		12 Case Examples: Variational Image Restoration with Segmentation-BasedRegularization
		13 Non-blind Restoration
		14 Semi-blind Restoration
		15 Image Restoration with Impulsive Noise
		16 Color Image Restoration
		17 Space-Variant Restoration
		18 Level Set Formulations for Joint Restoration and Segmentation
		19 Image Restoration by Nonlocal Mumford-Shah Regularizers
		20 Conclusion
		21 Recommended Reading
		References
	Local Smoothing Neighborhood Filters
		1 Introduction
		2 Denoising
			Analysis of Neighborhood Filter as a Denoising Algorithm
			Neighborhood Filter Extension: The NL-Means Algorithm
			Extension to Movies
		3 Asymptotic
			PDE Models and Local Smoothing Filters
			Asymptotic Behavior of Neighborhood Filters (Dimension 1)
			The Two-Dimensional Case
			A Regression Correction of the Neighborhood Filter
			The Vector-Valued Case
				Interpretation
		4 Variational and Linear Diffusion
			Linear Diffusion: Seed Growing
			Linear Diffusion: Histogram Concentration
		5 Conclusion
		References
	Neighborhood Filters and the Recovery of 3D Information
		1 Introduction
		2 Bilateral Filters Processing Meshed 3D Surfaces
			Glossary and Notation
			Bilateral Filter Definitions
			Trilateral Filters
			Similarity Filters
			Summary of 3D Mesh Bilateral Filter Definitions
			Comparison of Bilateral Filter and Mean Curvature Motion Filter on Artificial Shapes
			Comparison of the Bilateral Filter and the Mean Curvature Motion Filter on Real Shapes
		3 Depth-Oriented Applications
			Bilateral Filter for Improving the Depth Map Provided by Stereo Matching Algorithms
			Bilateral Filter for Enhancing the Resolution of Low-Quality Range Images
			Bilateral Filter for the Global Integration of Local Depth Information
		4 Conclusion
		References
	Splines and Multiresolution Analysis
		1 Introduction
		2 Historical Notes
		3 Fourier Transform, Multiresolution, Splines, and Wavelets
			Mathematical Foundations
				Regularity and Decay Under the Fourier Transform
				Criteria for Riesz Sequences and Multiresolution Analyses
				Regularity of Multiresolution Analysis
				Order of Approximation
				Wavelets
			B-Splines
			Polyharmonic B-Splines
		4 Survey on Spline Families
			Schoenberg\'s B-Splines for Image Analysis: The Tensor Product Approach
			Fractional and Complex B-Splines
			Polyharmonic B-Splines and Variants
			Splines on Other Lattices
				Splines on the Quincunx Lattice
				Splines on the Hexagonal Lattice
		5 Numerical Implementation
		6 Open Questions
		7 Conclusion
		References
	Gabor Analysis for Imaging
		1 Introduction
		2 Tools from Functional Analysis
			The Pseudo-inverse Operator
			Bessel Sequences in Hilbert Spaces
			General Bases and Orthonormal Bases
			Frames and Their Properties
		3 Operators
			The Fourier Transform
			Translation and Modulation
			Convolution, Involution, and Reflection
			The Short-Time Fourier Transform
		4 Gabor Frames in L2(Rd)
		5 Discrete Gabor Systems
			Gabor Frames in 2(Z)
			Finite Discrete Periodic Signals
			Frames and Gabor Frames in CL
		6 Image Representation by Gabor Expansion
			2D Gabor Expansions
			Separable Atoms on Fully Separable Lattices
			Efficient Gabor Expansion by Sampled STFT
			Visualizing a Sampled STFT of an Image
			Non-separable Atoms on Fully Separable Lattices
		7 Historical Notes and Hint to the Literature
		References
	Shape Spaces
		1 Introduction
		2 Background
		3 Mathematical Modeling and Analysis
			Some Notation
			A Riemannian Manifold of Deformable Landmarks
				Interpolating Splines and RKHSs
				Riemannian Structure
				Geodesic Equation
				Metric Distortion and Curvature
				Invariance
			Hamiltonian Point of View
				General Principles
				Application to Geodesics in a Riemannian Manifold
				Momentum Map and Conserved Quantities
				Euler–Poincaré Equation
				A Note on Left Actions
				Application to the Group of Diffeomorphisms
				Reduction via a Submersion
				Reduction: Quotient Spaces
				Reduction: Transitive Group Action
			Spaces of Plane Curves
				Introduction and Notation
				Some Simple Distances
				Riemannian Metrics on Curves
				Projecting the Action of 2D Diffeomorphisms
			Extension to More General Shape Spaces
			Applications to Statistics on Shape Spaces
		4 Numerical Methods and Case Examples
			Landmark Matching via Shooting
			Landmark Matching via Path Optimization
			Computing Geodesics Between Curves
			Inexact Matching and Optimal Control Formulation
				Inexact Matching
				Optimal Control Formulation
				Gradient w.r.t. the Control
				Application to the Landmark Case
		5 Conclusion
		References
	Variational Methods in Shape Analysis
		1 Introduction
		2 Background
		3 Mathematical Modeling and Analysis
			Recalling the Finite-Dimensional Case
			Path-Based Viscous Dissipation Versus State-Based Elastic Deformation forNon-rigid Objects
				Path-Based, Viscous Riemannian Setup
				State-Based, Path-Independent Elastic Setup
				Conceptual Differences Between the Path- and State-Based Dissimilarity Measures
		4 Numerical Methods and Case Examples
			Elasticity-Based Shape Space
				Elastic Shape Averaging
				Elasticity-Based PCA
			Viscous Fluid-Based Shape Space
			A Collection of Computational Tools
				Shapes Described by Level Set Functions
				Shapes Described via Phase Fields
				Multi-scale Finite Element Approximation
		5 Conclusion
		References
	Manifold Intrinsic Similarity
		1 Introduction
			Problems
			Methods
			Chapter Outline
		2 Shapes as Metric Spaces
			Basic Notions
				Topological Spaces
				Metric Spaces
				Isometries
			Euclidean Geometry
			Riemannian Geometry
				Manifolds
				Differential Structures
				Geodesics
				Embedded Manifolds
				Rigidity
			Diffusion Geometry
				Diffusion Operators
			Diffusion Distances
		3 Shape Discretization
			Sampling
				Farthest Point Sampling
				Centroidal Voronoi Sampling
			Shape Representation
				Simplicial Complexes
				Parametric Surfaces
				Implicit Surfaces
		4 Metric Discretization
			Shortest Paths on Graphs
				Dijkstra\'s Algorithm
				Metrication Errors and Sampling Theorem
			Fast Marching
				Eikonal Equation
				Triangular Meshes
				Parametric Surfaces
				Parallel Marching
				Implicit Surfaces and Point Clouds
			Diffusion Distance
				Discretized Laplace–Beltrami Operator
				Computation of Eigenfunctions and Eigenvalues
				Discretization of Diffusion Distances
		5 Invariant Shape Similarity
			Rigid Similarity
				Hausdorff Distance
				Iterative Closest Point Algorithms
				Shape Distributions
				Wasserstein Distances
			Canonical Forms
				Multidimensional Scaling
				Eigenmaps
			Gromov–Hausdorff Distance
				Generalized Multidimensional Scaling
			Graph-Based Methods
				Probabilistic Gromov–Hausdorff Distance
			Gromov–Wasserstein Distances
				Numerical Computation
			Shape DNA
		6 Partial Similarity
			Significance
			Regularity
			Partial Similarity Criterion
			Computational Considerations
		7 Self-Similarity and Symmetry
			Rigid Symmetry
			Intrinsic Symmetry
			Spectral Symmetry
			Partial Symmetry
			Repeating Structure
		8 Feature-Based Methods
			Feature Descriptors
				Feature Detection
				Feature Description
				Heat Kernel Signatures
				Scale-Invariant Heat Kernel Signatures
			Bags of Features
			Combining Global and Local Information
		9 Conclusion
		References
	Image Segmentation with Shape Priors: Explicit Versus Implicit Representations
		1 Introduction
			Image Analysis and Prior Knowledge
			Explicit Versus Implicit Shape Representation
		2 Image Segmentation via Bayesian Inference
		3 Statistical Shape Priors for Parametric Shape Representations
			Linear Gaussian Shape Priors
			Nonlinear Statistical Shape Priors
		4 Statistical Priors for Level Set Representations
			Shape Distances for Level Sets
			Invariance by Intrinsic Alignment
				Translation Invariance by Intrinsic Alignment
				Translation and Scale Invariance via Alignment
			Kernel Density Estimation in the Level Set Domain
			Gradient Descent Evolution for the Kernel Density Estimator
			Nonlinear Shape Priors for Tracking a Walking Person
		5 Dynamical Shape Priors for Implicit Shapes
			Capturing the Temporal Evolution of Shape
			Level Set-Based Tracking via Bayesian Inference
			Linear Dynamical Models for Implicit Shapes
			Variational Segmentation with Dynamical Shape Priors
		6 Parametric Representations Revisited: Combinatorial Solutions for Segmentation with Shape Priors
		7 Conclusion
		References
	Optical Flow
		1 Introduction
			Motivation, Overview
			Organization
		2 Basic Aspects
			Invariance, Correspondence Problem
			Assignment Approach, Differential Motion Approach
				Definitions
				Common Aspects and Differences
				Differential Motion Estimation: Case Study (1D)
				Assignment or Differential Approach?
				Basic Difficulties of Motion Estimation
			Two-View Geometry, Assignment and Motion Fields
				Two-View Geometry
				Assignment Fields
				Motion Fields
			Early Pioneering Work
			Benchmarks
		3 The Variational Approach to Optical Flow Estimation
			Differential Constraint Equations, Aperture Problem
			The Approach of Horn and Schunck
				Model
				Discretization
				Solving
				Examples
				Probabilistic Interpretation
			Data Terms
				Handling Violation of the Constancy Assumption
				Patch Features
				Multiscale
			Regularization
				Regularity Priors
				Distance Functions
				Adaptive, Anisotropic, and Nonlocal Regularization
			Further Extensions
				Spatiotemporal Approach
				Geometrical Prior Knowledge
				Physical Prior Knowledge
			Algorithms
				Smooth Convex Functionals
				Non-smooth Convex Functionals
				Non-convex Functionals
		4 The Assignment Approach to Optical Flow Estimation
			Local Approaches
			Assignment by Displacement Labeling
			Variational Image Registration
		5 Open Problems and Perspectives
			Unifying Aspects: Assignment by Optimal Transport
			Motion Segmentation, Compressive Sensing
			Probabilistic Modeling and Online Estimation
		6 Conclusion
		7 Basic Notation
		References
	Non-linear Image Registration
		1 Introduction
		2 The Mathematical Setting
			Variational Formulation of Image Registration
			Images and Transformation
			Length, Area, and Volume Under Transformation
			Distance Functionals
			Ill-Posedness and Regularization
			Elastic Regularization Functionals
			Hyperelastic Regularization Functionals
			Constraints
			Related Literature and Further Reading
		3 Existence Theory of Hyperelastic Registration
			Sketch of an Existence Proof
			Set of Admissible Transformations
			Existence Result for Unconstrained Image Registration
		4 Numerical Methods for Hyperelastic Image Registration
			Discretizing the Determinant of the Jacobian
				Finite Differences in 1D
				Finite Differences in 2D
				Finite Volume Discretization
			Galerkin Finite Element Discretization
			Multi-level Optimization Strategy
		5 Applications of Hyperelastic Image Registration
			Motion Correction of Cardiac PET
				Mass-Preservation
				Registration Results and Impact on Image Quality
				Summary and Further Literature
			Susceptibility Artefact Correction of Echo-Planar MRI
				Correction Results
		6 Conclusion
		References
	Starlet Transform in Astronomical Data Processing
		1 Introduction
			Source Detection
		2 Standard Approaches to Source Detection
			The Traditional Data Model
			PSF Estimation
			Background Estimation
			Convolution
			Detection
			Deblending/Merging
			Photometry and Classification
				Photometry
				Star–Galaxy Separation
				Galaxy Morphology Classification
		3 Mathematical Modeling
			Sparsity Data Model
			The Starlet Transform
			The Starlet Reconstruction
			Starlet Transform: Second Generation
			Sparse Modeling of Astronomical Images
				Selection of Significant Coefficients Through Noise Modeling
			Sparse Positive Decomposition
				Example 1: Sparse Positive Decomposition of NGC 2997
				Example 2: Sparse Positive Starlet Decomposition of a Simulated Image
		4 Source Detection Using a Sparsity Model
			Detection Through Wavelet Denoising
			The Multiscale Vision Model
				Introduction
				Multiscale Vision Model Definition
				From Wavelet Coefficients to Object Identification
				Multiresolution Support Segmentation
				Interscale Connectivity Graph
				Filtering
				Merging/Deblending
				Object Identification
			Source Reconstruction
				Partial Reconstruction as an Inverse Problem
			Examples
				Band Extraction
				Star Extraction in NGC 2997
				Galaxy Nucleus Extraction
		5 Deconvolution
			Statistical Approach to Deconvolution
			The Richardson–Lucy Algorithm
			Blind Deconvolution
			Deconvolution with a Sparsity Prior
				Constraints in the Object or Image Domains
				Example
			Detection and Deconvolution
			Object Reconstruction Using the PSF
			The Algorithm
			Space-Variant PSF
			Undersampled Point Spread Function
			Example: Application to Abell 1689 ISOCAM Data
		6 Conclusion
		References
	Differential Methods for Multi-dimensional Visual Data Analysis
		1 Introduction
		2 Modeling Data via Fiber Bundles
			Differential Geometry: Manifolds, Tangential Spaces, and Vector Spaces
				Tangential Vectors
				Covectors
				Tensors
				Exterior Product
				Visualizing Exterior Products
				Geometric Algebra
				Vector and Fiber Bundles
			Topology: Discretized Manifolds
			Ontological Scheme and Seven-Level Hierarchy
				Field Properties
				Topological Skeletons
				Non-topological Representations
		3 Differential Forms and Topology
			Differential Forms
				Chains
				Cochains
				Duality Between Chains and Cochains
			Homology and Cohomology
			Topology
		4 Geometric Algebra Computing
			Benefits of Geometric Algebra
				Unification of Mathematical Systems
				Uniform Handling of Different Geometric Primitives
				Simplified Geometric Operations
				More Efficient Implementations
			Conformal Geometric Algebra
			Computational Efficiency of Geometric Algebra Using Gaalop
		5 Feature-Based Vector Field Visualization
			Characteristic Curves of Vector Fields
			Derived Measures of Vector Fields
			Topology of Vector Fields
				Critical Points
				Separatrices
				Application
		6 Anisotropic Diffusion PDEs for Image Regularization and Visualization
			Regularization PDEs: A Review
				Local Multivalued Geometry and Diffusion Tensors
				Divergence-Based PDEs
				Trace-Based PDEs
				Curvature-Preserving PDEs
			Applications
				Color Image Denoising
				Color Image Inpainting
				Visualization of Vector and Tensor Fields
		7 Conclusion
		References
Index