图像分析中的模型和逆问题

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

Ⅰ Spline Models

2 Nonparametric Spline Models

2.1 Definition

2.2 Optimiation

2.2.1 Bending Spline

2.2.2 Spline Under Tension

2.2.3 Robustness

2.3 Bayesian Interpretation

2.4 Choice of R,egularization 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 Projectionlndex

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

……

Ⅲ Modeling in Action

In the last decade of the past centurlr we witnessed an exceptional participation of mathematicians in the development of digital image processing as a science. These contributions have found a natural place at the low level of processing， with the advent of mathematical morphology， differential equations， Markov random fields， and wavelet theory. They are also reflected in the increasingly important role modeling has played in solving complex problems.
　　Although modeling is often a hidden stage of the solution， the benefits of correctly modeling an image processing problem are huge. Modeling is the very place where "sensitivity" steals a lead over "geometry"， to put it in Pascal's words， Correct modeling introduces information that cannot be expressed by data or deduced by equations， but refiects the subtle dependency or causality between the ingredients. Modeling has more to do with pots and pans than recipes and spices， to draw out the culinary metaphor. Bernard Chalmond's work is mainly dedicated to modeling issues. It does not fully cover this field， since it is mostly concerned with two types of model: Bayesian models issued from probability theory and energy-based models derived from physics and mechanics. Within the scope of these models， the book deeply explores the various consequences of the choice of a model; it compares their hypotheses， discusses their merits， explores their validity， and suggests possible fields of application.
　　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.