《成像中的变分法》(作者斯科泽)致力于研究成像处理中的方法，本着数学的严谨，用逆问题的观点研究这个科目。更重要的，本书从确定性、几何和随机的观点研究变分法，架起了成像分析中规范理论和逆问题的桥梁，图形分析中的案例来解释变分法，如去噪、热声学、计算断层照相法中的应用，讨论了非凸变量微积分、形态学分析和水平集之间的联系。

Part I Fundamentals Of Imaging

1 Case Examples Of Imaging

1.1 Denoising

1.2 Chopping And Nodding

1.3 Image Inpainting

1.4 X-Ray-Based Computerized Tomography

1.5 Thermoaconstic Computerized Tomography

1.6 Schlieren Tomography

2 Image And Noise Models

2.1 Basic Concepts Of Statistics

2.2 Digitized (Discrete) Images

2.3 Noise Models

2.4 Priors For Images

2.5 Maximum A Posteriori Estimation

2.6 Map Estimation For Noisy Images

Part Ii Regularization

Variational Regularization Methods For The Solution Of Inverse Problems

3.1 Quadratic Tikhonov Regularization In Hilbert Spaces

3.2 Variational Regularization Methods In Banach Spaces

3.3 Regularization With Sparsity Constraints

3.4 Linear Inverse Problems With Convex Constraints

3.5 Schlieren Tomography

3.6 Fulrther Literature On Regularization Methods For Inverse Problems

4 Convex Regularization Methods For Denoising

4.1 The *-Number

4.2 Characterization Of Minimizers

4.3 One-Dimensional Results

4.4 Taut String Algorithm

4.5 Mumford-Shah Regularization

4.6 Recent Topics On Denoising With Variational Methods

5 Variational Calculus For Non-Convex Regularization

5.1 Direct Methods

5.2 Relaxation On Sobolev Spaces

5.3 Relaxation On Bv

5.4 Applications In Non-Convex Regularization

5.5 One-Dimensional Results

5.6 Examples

6 Semi-Group Theory And Scale Spaces

6.1 Linear Semi-Group Theory

6.2 Non-Linear Semi-Groups In Hilbert Spaces

6.3 Non-Linear Semi-Groups In Banach Spaces

6.4 Axiomatic Approach To Scale Spaces

6.5 Evolution By Non-Convex Energy Functionals

6.6 Enhancing

Inverse Scale Spaces

7.1 Iterative Tikhonov Regularization

7.2 Iterative Regularization With Bregm/M Distances

7.3 Recent Topics On Evolutionary Equations For Inverse Problems

Part Iii Mathematical Foundations

8 Functional Analysis

8.1 General Topology

8.2 Locally Convex Spaces

8.3 Bounded Linear Operators And Functionals

8.4 Linear Operators In Hilbert Spaces

8.5 Weak And Weak* Topologies

8.6 Spaces Of Differentiable Functions

9 Weakly Different/Able Functions

9.1 Measure And Integration Theory

9.2 Distributions And Distributional Derivatives

9.3 Geometrical Properties Of Functions And Domains

9.4 Sobolev Spaces

9.5 Convolution

9.6 Sobolev Spaces Of Fractional Order

9.7 Bochner Spaces

9.8 Functions Of Bounded Variation

10 Convex Analysis And Calculus Of Variations

10.1 Convex And Lower Semi-Continuous Functionals

10.2 Fenchel Duality And Subdifferentiability

10.3 Duality Mappings

10.4 Differentiability Of Functionals And Operators

10.5 Derivatives Of Integral Functionals On Lp(Ω)

References

Nomenclature

Index