This book fulfills a need in the field of computer science research and education. It is not intended for professional mathematicians, but it undoubtedly deals with applied mathematics. Most of the expectations of the topic are fulfilled: precision, exactness, completeness, and excellent references to the original historical works. However, for the sake of read-ability, many demonstrations are omitted. It is not a book on practical image processing, of which so many abound, although all that it teaches is directly concerned with image analysis and image restoration. It is the perfect resource for any advanced scientist concerned with a better un-derstanding of the theoretical models underlying the methods that have efficiently solved numerous issues in robot vision and picture processing.
Foreword by Henri Maitre
Acknowledgments
List of Figures
Notation and Symbols
1 Introduction
1.1 About Modeling
1.1.1 Bayesian Approach
1.1.2 Inverse Problem
1.1.3 Energy-Based Formulation
1.1.4 Models
1.2 Structure of the Book
Spline Models
2 Nonparametrie Spline Models
2.1 Definition
2.2 Optimization
2.2.1 Bending Spline
2.2.2 Spline Under Tension
2.2.3 Robustness
2.3 Bayesian Interpretation
2.4 Choice of Regularization Parameter
2.5 Approximation Using a Surface
2.5.1 L-Spline Surface
2.5.2 Quadratic Energy
2.5.3 Finite Element Optimization
3 Parametric Spline Models
3.1 Representation on a Basis of B-Splines
3.1.1 Approximation Spline
3.1.2 Construction of B-Splines
3.2 Extensions
3.2.1 Multidimensional Case
3.2.2 Heteroscedasticity
3.3 High-Dimensional Splines
3.3.1 Revealing Directions
3.3.2 Projection Pursuit Regression
4 Auto-Associative Models
4.1 Analysis of Multidimensional Data
4.1.1 A Classical Approach
4.1.2 Toward an Alternative Approach
4.2 Auto-Associative Composite Models
4.2.1 Model and Algorithm
4.2.2 Properties
4.3 Projection Pursuit and Spline Smoothing
4.3.1 Projection Index
4.3.2 Spline Smoothing
4.4 Illustration
Ⅱ Markov Models
5 Fundamental Aspects
5.1 Definitions
5.1.1 Finite Markov Fields
5.1.2 Gibbs Fields
5.2 Markov-Gibbs Equivalence
5.3 Examples
5.3.1 Bending Energy
5.3.2 Bernoulli Energy
5.3.3 Gaussian Energy
5.4 Consistency Problem
6 Bayesian Estimation
6.1 Principle
6.2 Cost Functions
6.2.1 Cost b-hnction Examples
6.2.2 Calculation Problems
7 Simulation and Optimization
7.1 Simulation
7.1.1 Homogeneous Markov Chain
7.1.2 Metropolis Dynamic
7.1.3 Simulated Gibbs Distribution
7.2 Stochastic Optimization
7.3 Probabilistic Aspects
7.4 Deterministic Optimization
7.4.1 ICM Algorithm
7.4.2 Relaxation Algorithms
8 Parameter Estimation
8.1 Complete Data
8.1.1 Maximum Likelihood
8.1.2 Maximum Pseudolikelihood
8.1.3 Logistic Estimation
8.2 Incomplete Data
8.2.1 Maximum Likelihood
8.2.2 Gibbsian EM Algorithm
8.2.3 Bayesian Calibration
Ⅲ Modeling in Action
9 Model-Building
9.1 Multiple Spline Approximation
9.1.1 Choice of Data and Image Characteristics
9.1.2 Definition of the Hidden Field
9.1.3 Building an Energy
9.2 Markov Modeling Methodology
9.2.1 Details for Implementation
10 Degradation in Imaging
10.1 Denoising
10.1.1 Models with Explicit Discontinuities
10.1.2 Models with Implicit Discontinuities
10.2 Deblurring
10.2.1 A Particularly Ill-Posed Problem
10.2.2 Model with Implicit Discontinuities
10.3 Scatter
10.3.1 Direct Problem
10.3.2 Inverse Problem
10.4 Sensitivity Functions and Image Fusion
10.4.1 A Restoration Problem
10.4.2 Transfer Function Estimation
10.4.3 Estimation of Stained Transfer Function
11 Detection of Filamentary Entities
11.1 Valley Detection Principle
11.1.1 Definitions
11.1.2 Bayes-Markov Formulation
11.2 Building the Prior Energy
11.2.1 Detection Term
11.2.2 Regularization Term
11.3 Optimization
11.4 Extension to the Case of an Image Pair
12 Reconstruction and Projections
12.1 Projection Model
12.1.1 Transmission Tomography
12.1.2 Emission Tomography
12.2 Regularized Reconstruction
12.2.1 Regularization with Explicit Discontinuities
12.2.2 Three-Dimensional Reconstruction
12.3 Reconstruction with a Single View
12.3.1 Generalized Cylinder
12.3.2 Training the Deformations
12.3.3 Reconstruction in the Presence of Occlusion
13 Matching
13.1 Template and Hidden Outline
13.1.1 Rigid Transformations
13.1.2 Spline Model of a Template
13.2 Elastic Deformations
13.2.1 Continuous Random Fields
13.2.2 Probabilistie Aspects
References
Author Index
Subject Index
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