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Produktbild: Nonlinear Inverse Problems in Imaging

Nonlinear Inverse Problems in Imaging

133,99 €

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Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

26.12.2012

Verlag

John Wiley & Sons

Seitenzahl

374

Maße (L/B/H)

25,1/17,2/2,2 cm

Gewicht

706 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-0-470-66942-6

Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

26.12.2012

Verlag

John Wiley & Sons

Seitenzahl

374

Maße (L/B/H)

25,1/17,2/2,2 cm

Gewicht

706 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-0-470-66942-6

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: gpsr@libri.de

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  • Produktbild: Nonlinear Inverse Problems in Imaging
  • Preface xi
     
    List of Abbreviations xiii
     
    1 Introduction 1
     
    1.1 Forward Problem 1
     
    1.2 Inverse Problem 3
     
    1.3 Issues in Inverse Problem Solving 4
     
    1.4 Linear, Nonlinear and Linearized Problems 6
     
    References 7
     
    2 Signal and System as Vectors 9
     
    2.1 Vector Spaces 9
     
    2.1.1 Vector Space and Subspace 9
     
    2.1.2 Basis, Norm and Inner Product 11
     
    2.1.3 Hilbert Space 13
     
    2.2 Vector Calculus 16
     
    2.2.1 Gradient 16
     
    2.2.2 Divergence 17
     
    2.2.3 Curl 17
     
    2.2.4 Curve 18
     
    2.2.5 Curvature 19
     
    2.3 Taylor's Expansion 21
     
    2.4 Linear System of Equations 23
     
    2.4.1 Linear System and Transform 23
     
    2.4.2 Vector Space of Matrix 24
     
    2.4.3 Least-Squares Solution 27
     
    2.4.4 Singular Value Decomposition (SVD) 28
     
    2.4.5 Pseudo-inverse 29
     
    2.5 Fourier Transform 30
     
    2.5.1 Series Expansion 30
     
    2.5.2 Fourier Transform 32
     
    2.5.3 Discrete Fourier Transform (DFT) 37
     
    2.5.4 Fast Fourier Transform (FFT) 40
     
    2.5.5 Two-Dimensional Fourier Transform 41
     
    References 42
     
    3 Basics of Forward Problem 43
     
    3.1 Understanding a PDE using Images as Examples 44
     
    3.2 Heat Equation 46
     
    3.2.1 Formulation of Heat Equation 46
     
    3.2.2 One-Dimensional Heat Equation 48
     
    3.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion 50
     
    3.2.4 Boundary Conditions 51
     
    3.3 Wave Equation 52
     
    3.4 Laplace and Poisson Equations 56
     
    3.4.1 Boundary Value Problem 56
     
    3.4.2 Laplace Equation in a Circle 58
     
    3.4.3 Laplace Equation in Three-Dimensional Domain 60
     
    3.4.4 Representation Formula for Poisson Equation 66
     
    References 70
     
    Further Reading 70
     
    4 Analysis for Inverse Problem 71
     
    4.1 Examples of Inverse Problems in Medical Imaging 71
     
    4.1.1 Electrical Property Imaging 71
     
    4.1.2 Mechanical Property Imaging 74
     
    4.1.3 Image Restoration 75
     
    4.2 Basic Analysis 76
     
    4.2.1 Sobolev Space 78
     
    4.2.2 Some Important Estimates 81
     
    4.2.3 Helmholtz Decomposition 87
     
    4.3 Variational Problems 88
     
    4.3.1 Lax-Milgram Theorem 88
     
    4.3.2 Ritz Approach 92
     
    4.3.3 Euler-Lagrange Equations 96
     
    4.3.4 Regularity Theory and Asymptotic Analysis 100
     
    4.4 Tikhonov Regularization and Spectral Analysis 104
     
    4.4.1 Overview of Tikhonov Regularization 105
     
    4.4.2 Bounded Linear Operators in Banach Space 109
     
    4.4.3 Regularization in Hilbert Space or Banach Space 112
     
    4.5 Basics of Real Analysis 116
     
    4.5.1 Riemann Integrability 116
     
    4.5.2 Measure Space 117
     
    4.5.3 Lebesgue-Measurable Function 119
     
    4.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 123
     
    4.5.5 Differentiation Theory 125
     
    References 127
     
    Further Reading 127
     
    5 Numerical Methods 129
     
    5.1 Iterative Method for Nonlinear Problem 129
     
    5.2 Numerical Computation of One-Dimensional Heat Equation 130
     
    5.2.1 Explicit Scheme 132
     
    5.2.2 Implicit Scheme 135
     
    5.2.3 Crank-Nicolson Method 136
     
    5.3 Numerical Solution of Linear System of Equations 136
     
    5.3.1 Direct Method using LU Factorization 136
     
    5.3.2 Iterative Method using Matrix Splitting 138
     
    5.3.3 Iterative Method using Steepest Descent Minimization 140
     
    5.3.4 Conjugate Gradient (CG) Method 143
     
    5.4 Finite Difference Method (