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F I F T H P R O O F S “FAIR” 2009/10/8 page v

Contents

Listings xi

Examples xiii

List of Figures xvii

List of Tables xix

Preface xxi

1 Introduction 1

1.1 Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope and Aims of This Book . . . . . . . . . . . . . . . . . . . . 2

1.3 Brief Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Links to the Literature . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 (Medical) Image Registration . . . . . . . . . . . . . . . 4

1.4.2 Image Processing and Interpolation . . . . . . . . . . . . 5

1.4.3 Numerics and Linear Algebra . . . . . . . . . . . . . . . 5

1.4.4 Partial Diﬀerential Equations and Optimization . . . . . 6

1.5 Further Links and Software . . . . . . . . . . . . . . . . . . . . . 7

2 F

Concepts 9

2.1 F

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Images and Transformations . . . . . . . . . . . . . . . . 10

2.1.2 Distances and Regularization . . . . . . . . . . . . . . . 12

2.2 F

Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Discretize-then-Optimize . . . . . . . . . . . . . . . . . . 12

2.2.2 A Family of Nested Approximations . . . . . . . . . . . 13

2.2.3 Numerical Optimization . . . . . . . . . . . . . . . . . . 13

2.3 F

MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Comments on Comments . . . . . . . . . . . . . . . . . . 14

2.3.2 Notation and Conventions . . . . . . . . . . . . . . . . . 14

2.3.3 Coordinate System . . . . . . . . . . . . . . . . . . . . . 14

2.3.4 Arguments, Parameters, and Defaults . . . . . . . . . . . 14

2.3.5 Overwriting Default Parameters . . . . . . . . . . . . . . 15

Unauthorized reproduction of this article is prohibited.

F I F T H P R O O F S “FAIR” 2009/10/8 page vi

vi Contents

2.3.6 Using the MATLAB “@” Constructor . . . . . . . . . . . 15

2.3.7 F

Administration . . . . . . . . . . . . . . . . . . . . 15

2.3.8 Memory Versus Clarity . . . . . . . . . . . . . . . . . . . 16

2.4 F

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Image Interpolation 19

3.1 Cells, Grids, and Numbering . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Right-Handed Coordinate System . . . . . . . . . . . . . 22

3.1.2 Lexicographical Ordering . . . . . . . . . . . . . . . . . . 23

3.2 Next Neighbor Interpolation . . . . . . . . . . . . . . . . . . . . . 24

3.3 Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Linear Interpolation for 1D Data . . . . . . . . . . . . . 24

3.3.2 Linear Interpolation for Higher-Dimensional Data . . . . 25

3.3.3 Summarizing Linear Interpolation . . . . . . . . . . . . . 26

3.4 Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.1 Spline Interpolation for 1D Data . . . . . . . . . . . . . 27

3.4.2 Spline Interpolation for Higher-Dimensional Data . . . . 29

3.5 Derivatives of Interpolation Schemes . . . . . . . . . . . . . . . . 30

3.5.1 Derivatives of Interpolants . . . . . . . . . . . . . . . . . 31

3.5.2 Derivatives of Multivariate Interpolants . . . . . . . . . 31

3.5.3 Testing Implementations of Derivatives . . . . . . . . . . 32

3.6 Multiscale Spline Interpolation . . . . . . . . . . . . . . . . . . . 32

3.6.1 Multiscale Interpolation in One Dimension . . . . . . . . 33

3.6.2 Truncating High Frequencies . . . . . . . . . . . . . . . . 35

3.6.3 Multiscale Interpolation in Higher Dimensions . . . . . . 36

3.7 Multilevel Representation of Data . . . . . . . . . . . . . . . . . . 40

3.8 Summarizing the Interpolation Toolbox . . . . . . . . . . . . . . . 42

3.9 F

Tutorials on Interpolation . . . . . . . . . . . . . . . . . . . 43

3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Transforming Images by Parameterized Transformations 47

4.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Aﬃne Linear Transformations . . . . . . . . . . . . . . . . . . . . 49

4.3 Rigid Transformations . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Rotations About the Domain Center . . . . . . . . . . . . . . . . 50

4.5 Spline-Based Transformations . . . . . . . . . . . . . . . . . . . . 50

4.6 More Bizarre Transformations . . . . . . . . . . . . . . . . . . . . 51

4.7 Derivatives of Parameterized Transformations . . . . . . . . . . . 52

4.8 Summarizing the Parameterized Transformations . . . . . . . . . 54

4.9 F

Tutorials on Transformations . . . . . . . . . . . . . . . . . 55

4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Landmark-Based Registration 57

5.1 Aﬃne Linear Landmark-Based Registration . . . . . . . . . . . . 58

5.2 Quadratic Landmark-Based Registration . . . . . . . . . . . . . . 59

5.3 Thin-Plate-Spline Registration . . . . . . . . . . . . . . . . . . . . 61

Unauthorized reproduction of this article is prohibited.

F I F T H P R O O F S “FAIR” 2009/10/8 page vii

Contents vii

5.3.1 Thin-Plate-Spline Interpolation . . . . . . . . . . . . . . 61

5.3.2 Thin-Plate-Spline Approximation . . . . . . . . . . . . . 62

5.4 Summarizing Landmark-Based Registration . . . . . . . . . . . . 64

5.5 F

Tutorials on Landmark-Based Registration . . . . . . . . . 64

5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Parametric Image Registration 67

6.1 Numerical Integration—Discretizing Integrals . . . . . . . . . . . 68

6.2 Sum of Squared Diﬀerences . . . . . . . . . . . . . . . . . . . . . 71

6.2.1 Continuous SSD . . . . . . . . . . . . . . . . . . . . . . . 71

6.2.2 Discretized SSD . . . . . . . . . . . . . . . . . . . . . . . 72

6.2.3 SSD and Parametric Transformations . . . . . . . . . . . 72

6.3 Numerical Optimization of Parametric Image Registration . . . . 75

6.3.1 PIR Objective Function . . . . . . . . . . . . . . . . . . 75

6.3.2 Practical Issues in Coding the PIR Objective

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3.3 Gauss–Newton Scheme . . . . . . . . . . . . . . . . . . . 77

6.3.4 Brief Comments on a Visualization . . . . . . . . . . . . 79

6.3.5 PIR Examples . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4 PIR Experiments on Fixed Levels . . . . . . . . . . . . . . . . . . 83

6.5 Regularized Parametric Image Registration . . . . . . . . . . . . 87

6.6 Multilevel Parametric Image Registration . . . . . . . . . . . . . 89

6.7 Summarizing Parametric Image Registration Topics . . . . . . . . 92

6.8 F

Tutorials on Parametric Image Registration . . . . . . . . . 93

6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7 Distance Measures 95

7.1 Sum of Squared Diﬀerences . . . . . . . . . . . . . . . . . . . . . 95

7.1.1 SSD and Forces . . . . . . . . . . . . . . . . . . . . . . . 96

7.1.2 Discretized SSD . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2.1 Continuous Normalized Cross-Correlation . . . . . . . . 97

7.2.2 Discretized Normalized Cross-Correlation . . . . . . . . 99

7.3 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3.1 Estimating the Joint Density, Principles . . . . . . . . . 101

7.3.2 Estimating the Joint Density of Two Images . . . . . . . 105

7.3.3 Mutual Information . . . . . . . . . . . . . . . . . . . . . 105

7.3.4 Discretizing Mutual Information . . . . . . . . . . . . . . 106

7.4 Normalized Gradient Fields . . . . . . . . . . . . . . . . . . . . . 107

7.4.1 Continuous Normalized Gradient Fields . . . . . . . . . 107

7.4.2 Discretized Normalized Gradient Fields . . . . . . . . . . 108

7.5 Derivatives of Distance Measures . . . . . . . . . . . . . . . . . . 109

7.6 Summarizing the Distance Measures . . . . . . . . . . . . . . . . 110

7.7 F

Tutorials on Distance Measures . . . . . . . . . . . . . . . . 115

7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Unauthorized reproduction of this article is prohibited.

F I F T H P R O O F S “FAIR” 2009/10/8 page viii

viii Contents

8 Regularization 117

8.1 Ill-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.2 L

-Norm–Based Regularizers . . . . . . . . . . . . . . . . . . . . 120

8.2.1 Examples in One Dimension . . . . . . . . . . . . . . . . 120

8.2.2 Examples in Two Dimensions . . . . . . . . . . . . . . . 121

8.2.3 Extensions to Higher Dimensions . . . . . . . . . . . . . 122

8.2.4 Thin-Plate-Spline and Curvature Regularizers . . . . . . 123

8.3 Discretizing L

-Norm–Based Regularizers . . . . . . . . . . . . . 125

8.3.1 Discretizing First Order Derivatives . . . . . . . . . . . . 125

8.3.2 Discretized Diﬀusion and Elastic Operators . . . . . . . 128

8.3.3 Discretized Curvature Operator . . . . . . . . . . . . . . 129

8.3.4 Discretized L

-Norm–Based Regularizers . . . . . . . . . 130

8.4 Summarizing the Regularization . . . . . . . . . . . . . . . . . . . 130

8.5 Matrix-Free Operations . . . . . . . . . . . . . . . . . . . . . . . . 131

8.5.1 Matrix-Free Elastic Operator . . . . . . . . . . . . . . . 132

8.5.2 Matrix-Free Curvature Operator . . . . . . . . . . . . . 133

8.5.3 Matrix-Free Solver for the Linear Systems . . . . . . . . 134

8.6 F

Tutorials on Regularization . . . . . . . . . . . . . . . . . . 134

8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

9 Nonparametric Image Registration 137

9.1 Numerical Optimization of Nonparametric Image Registration . . 139

9.1.1 Grid to Grid Interpolation . . . . . . . . . . . . . . . . . 139

9.1.2 NPIR Objective Function . . . . . . . . . . . . . . . . . 140

9.1.3 Practical Issues in Coding the NPIR Objective

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.2 NPIR Experiments on Fixed Level . . . . . . . . . . . . . . . . . 142

9.3 Multiscale Image Registration . . . . . . . . . . . . . . . . . . . . 145

9.4 Multilevel Image Registration . . . . . . . . . . . . . . . . . . . . 145

9.4.1 Outline of MLIR . . . . . . . . . . . . . . . . . . . . . . 148

9.4.2 Prolongation Operator . . . . . . . . . . . . . . . . . . . 148

9.5 MLIR Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.6 Alternative Numerical Optimizers . . . . . . . . . . . . . . . . . . 153

9.6.1 `-BFGS . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.6.2 MLIR Using an `-BFGS Scheme . . . . . . . . . . . . . . 154

9.6.3 Trust-Region Methods . . . . . . . . . . . . . . . . . . . 158

9.7 Examples in Three Dimensions . . . . . . . . . . . . . . . . . . . 159

9.8 Summarizing the Nonparametric Image Registration . . . . . . . 163

9.9 F

Tutorials on Image Registration . . . . . . . . . . . . . . . . 163

9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10 Outlook 165

10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

10.1.1 Registration Modules . . . . . . . . . . . . . . . . . . . . 165

10.1.2 Multiscale and Multilevel Approaches . . . . . . . . . . . 166

10.1.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . 166

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内容反馈

YCFANG11

2014-06-22

是一本学习算法的好书。
xny2077

2015-06-12

不错，资源很难下到
qq_21059443

2015-03-27

医学图像处理中少有的Matlab软件
lrq841289802

2016-07-15

一本好书，非常不错。
joanrry

2015-12-16

在做特征点图像配准方向的研究，这是一本特别好的书，很详细的介绍了图像配准方法并附对应matlab完整代码，强力推荐。

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