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Handbook-of-Mathematical-Methods-in-Imaging.pdf
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
Otmar Scherzer
Editor
Handbook of
Mathematical
Methods in
Imaging
Second Edition
1 3 Reference
Handbook of Mathematical Methods
in Imaging
Otmar Scherzer
Editor
Handbook of
Mathematical Methods
in Imaging
Second Edition
With 472 Figures and 23 Tables
Editor
Otmar Scherzer
Computational Science Center
University of Vienna
Vienna, Austria
RICAM
Austrian Academy of Sciences
Linz, Austria
ISBN 978-1-4939-0789-2
ISBN 978-1-4939-0791-5 (print and electronic bundle)
DOI 10.1007/978-1-4939-0790-8
ISBN 978-1-4939-0790-8
(eBook)
Library of Congress Control Number: 2015936285
Mathematics Subject Classification: 65N22, 65R32, 62H35, 68U10
Springer New York Heidelberg Dordrecht London
© Springer Science+Business Media New York 2011, 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, express or implied, with respect to the material contained herein or for any
errors or omissions that may have been made.
Printed on acid-free paper
Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www.
springer.com)
Preface
Today, computer imaging covers various aspects of data filtering, pattern
recognition,
feature extraction, computer aided design, and computer aided
inspection and diagnosis.
Pillars of the field of computer imaging are advanced, stable, and reliable algo-
rithms. In addition, feasibility analysis is required to evaluate practical relevance
of the methods. To put these pillars on solid grounds, a significant amount of
mathematical tools are required. This handbook makes a humble attempt to provide
a survey of such mathematical tools.
We had the vision that this imaging handbook should contain individual chapters
which can serve as toolboxes, which, when aligned, form background material for
complete applied imaging problems. Therefore it should also give an impression on
the broad mathematical knowledge required to solve industrial and applied research
applications: The image formation process, very frequently, is assigned to the
inverse problems community, which is prominently represented in this handbook.
The subsequent step is image analysis. Nowadays, advanced Image Analysis,
and Image Processing in general, uses sophisticated methods from Geometry,
Differential Geometry, Convex Analysis, Numerical Analysis, to mention just a few.
In fact, by the rapid advance of Imaging, the mathematical areas have been pushed
forward heavily, and raised their impact in application sciences.
Second Edition
My special thanks go to all individual authors for their valuable contributions
and the referees for their help in improving the contributions and making detailed
comments. My sincere thanks go to the Springer’s editors and staff, Marc Strauss,
Annalea Manalili, Michael Hermann, and Saskia Ellis for their patience and their
constant support and encouragement over the last two years. My thanks also goes to
Vaishali Damle, who initialized the second edition of this handbook.
Finally, I would like to encourage the readers to submit suggestions regarding
this handbook’s content. For a project of this size, it is likely that essential topics
are missed. In a rapidly evolving area like Imaging it is likely that new areas will
v
vi
Preface
appear in a very short time and should be added to this handbook, as well as recent
development enforce modifications of existing contributions. We are committed to
issuing periodic updates and we look forward to the feedback from the community.
Otmar Scherzer
Computational Science Center
University of Vienna, Austria
and
RICAM
Austrian Academy of Sciences
Linz, Austria
About the Editor
Otmar Scherzer was born on June 10, 1964, in Vöcklabruck, Austria. He studied
technical mathematics at the University of Linz, Austria, and received his Diploma
in Mathematics in 1987. He received his Ph.D. in 1990 and his habilitation in
1995 from the same university. During 1995 and 1996, he visited Texas A&M
University and the University of Delaware in USA. From 1999 to 2001, he held
professorships at the Ludwig Maximilian University, Munich, and the University of
Bayreuth, Germany. Otmar then joined the University of Innsbruck, Austria, where
he served as full professor of Applied and Computer Oriented Mathematics, from
2001 to 2009. In 2009, he accepted a position at the University of Vienna, where
he currently heads the Computational Science Center which was created upon his
appointment. More information about the center can be found at: http://www.csc.
univie.ac.at/. He is also group leader of the Inverse Problems and Imaging groups
at the Radon Institute of Computational and Applied Mathematics in Linz, Austria.
More information on the institute can be found at: http://www.ricam.oeaw.ac.at/.
Otmar’s research interests include inverse problems, in particular photoacoustic
imaging, regularization, image processing, and PDEs. He is a prolific researcher,
with more than a 170 research articles published in several well-respected journals.
He is the coauthor of two monographs and had coedited 7 volumes, including this
handbook, and has served on the editorial board of many prominent journals.
vii
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