成像中的变分法

　　Imagmg is an interdisciplinary research area with profound
applications in many areas of science， engineering， technology， and
medicine. The most primitrve form of imaging is visual in，spection，
which has dominated the area before the technical and computer
revolution era. Today， computer imaging covers various aspects of
data tiltering， pattern recognition， feature extraction.， co’mputer
aided mspection， and medical diagnosis. The above mentioned areas
are treated in different scientific communities such as Imagin，g，
In，verse Problems， Computer Vision， Signal and Image Processing，…，
but all share the common thread of recovery of an object or one of
its properties.

Part I Fundamentals of Imagmg

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 Thermoacoustic Computerzed Tomography

1.6 Schlieren Tomography

2 Image and Noise Models 

2.1 Basic Concepts of Statistics

2.2 Digitzed （Discrete） Images

2.3 Noise Models

2.4 Priors for Images

2.5 Maximum A Posteriori Estimation

2.6 MAP Rstimation for Noisy Images

Part II Regularization

3 Variational R,egularzation Methods for the Solution of Inverse
Problems

3.1 Quadratic Tikhonov R;egularization in Hilbert Spaces

3.2 Variational R,egularization Methods in Banach Spaces

3.3 Regularization with Sparsity Constraints

3.4 Linear Inverse Problems with Convex Constraints

3.5 Schlieren Tomography.

3.6 Further 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 R,egularization

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 Serru-group Theory and Scale Spaces

6.1 Linear Semi-group Theory

6.2 Non-linear Semi-groups in Hilbert Spaces

6.3 Non-Iinear Semi-groups in Banach Spaces

6.4 Axiomatic Approach to Scale Spaces

6.5 Evolution by Non-convex Energy Functionals

6.6 Enhancing

7 Inverse Scale Spaces

7.1 Iterative Tikhonov Regularization

7.2 Iterative Regularization with Bregman Distances

7.3Recent 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 Differentiable 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

10 Convex Analysis and Calculus of Variations

References

Nomenclature

Index