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- δ) : Gaussian Markov random fields - . - . - . - . - 54 - δ) : Laplace Markov random fields - . - . - . - . - . 65

This document is the contents page for a book on inverse problems. It lists the six chapters of the book and their section headings. Chapter 1 introduces characteristics of inverse problems and provides an illustrative example. Chapter 2 describes regularization by spectral filtering methods and parameter selection. Chapter 3 presents two-dimensional test cases of image deblurring and computed tomography. Chapter 4 discusses Bayes' law, Markov random field priors, and maximum a posteriori estimation. Chapter 5 covers Markov chain Monte Carlo methods for linear inverse problems. Chapter 6 extends these methods to nonlinear inverse problems.

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Nguyễn Quang Huy
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47 views2 pages

- δ) : Gaussian Markov random fields - . - . - . - . - 54 - δ) : Laplace Markov random fields - . - . - . - . - . 65

This document is the contents page for a book on inverse problems. It lists the six chapters of the book and their section headings. Chapter 1 introduces characteristics of inverse problems and provides an illustrative example. Chapter 2 describes regularization by spectral filtering methods and parameter selection. Chapter 3 presents two-dimensional test cases of image deblurring and computed tomography. Chapter 4 discusses Bayes' law, Markov random field priors, and maximum a posteriori estimation. Chapter 5 covers Markov chain Monte Carlo methods for linear inverse problems. Chapter 6 extends these methods to nonlinear inverse problems.

Uploaded by

Nguyễn Quang Huy
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Contents

Preface vii

1 Characteristics of Inverse Problems 1


1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The least squares estimator . . . . . . . . . . . . . . . . . . . . . 6
1.3 The statistical properties of xLS and ill-posedness . . . . . . . . . 8
1.4 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . 11
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Regularization by Spectral Filtering 15


2.1 Spectral filtering methods . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Regularization parameter selection methods . . . . . . . . . . . . 19
2.3 Periodic and data-driven boundary conditions . . . . . . . . . . . 25
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Two-Dimensional Test Cases 33


3.1 Two-dimensional image deblurring . . . . . . . . . . . . . . . . . 33
3.2 Computed tomography . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 The preconditioned conjugate gradient iteration . . . . . . . . . . 45
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Bayes’ Law, Markov Random Field Priors, and MAP Estimation 53


4.1 Bayes’ law and regularization . . . . . . . . . . . . . . . . . . . . 53
4.2 Choosing p(x|δ): Gaussian Markov random fields . . . . . . . . . 54
4.3 Choosing p(x|δ): Laplace Markov random fields . . . . . . . . . . 65
4.4 The infinite-dimensional limit . . . . . . . . . . . . . . . . . . . . 68
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Markov Chain Monte Carlo Methods for Linear Inverse Problems 77


5.1 Sampling from high-dimensional Gaussian random vectors . . . . 77
5.2 Hierarchical modeling of λ and δ and sampling from p(x, λ, δ|b) . 81
5.3 Alternative MCMC methods for sampling from p(x, λ, δ|b) . . . . 89
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Markov Chain Monte Carlo Methods for Nonlinear Inverse Problems 105
6.1 A general setup for nonlinear inverse problems . . . . . . . . . . . 105
6.2 Levenburg–Marquardt nonlinear least squares optimization . . . . 106

v
vi Contents

6.3 Randomize-then-optimize as a proposal for Metropolis–Hastings . 109


6.4 Nonlinear test cases . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Hierarchical modeling of λ and δ and sampling from p(x, λ, δ|b) . 118
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Bibliography 125

Index 131

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