e
FAIR: Fl e xi b l Algorithms for
Image Registration
Jan Modersitzki
Computing And Software, McMaster University
1280 Main Street West, Hamilton On, L8S 4K1, Canada
[email protected]
August 13, 2008
Under Construction
c Jan Modersitzki
2
FAIR
[August 13, 2008, 13:12]
Contents
1
2
3
Introduction
1.1
Image Registration . . . . . .
1.2
Brief Outline . . . . . . . . . .
1.3
Literature and Links . . . . .
1.3.1
Image Registration
1.3.2
Optimization . . .
1.3.3
Matlab . . . . .
1.3.4
Other Software . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13
13
14
15
15
15
16
16
FAIRConcepts
2.1
FAIR Theory . . . . . . . . . . . . . . . . . . . .
2.1.1
Images and Transformations . . . .
2.1.2
Distances and Regularization . . . .
2.2
FAIR Numerics . . . . . . . . . . . . . . . . . .
2.2.1
Discretize then Optimize . . . . . .
2.2.2
A Family of Nested Approximations
2.2.3
Optimization Techniques . . . . . .
2.3
FAIR Matlab . . . . . . . . . . . . . . . . . . .
2.3.1
Comments on Comments . . . . . .
2.3.2
Notation and Conventions . . . . . .
2.3.3
Coordinate System . . . . . . . . . .
2.3.4
Arguments, Parameter, and Defaults
2.3.5
Overwriting Default Parameter . . .
2.3.6
Matlab’s “@” Constructor . . . . .
2.3.7
FAIR Administration . . . . . . . . .
2.3.8
Memory versus Clarity . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
17
18
20
20
20
21
21
21
21
21
22
22
23
23
23
24
Image Interpolation
3.1
Cells, Grids, and Numbering . . . . . . . . . . . . . . . . . . .
3.2
Next Neighbor Interpolation . . . . . . . . . . . . . . . . . . .
3.3
Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
Linear Interpolation for One Dimensional Data . .
3.3.2
Linear Interpolation for Higher Dimensional Data
3.4
Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
25
25
29
29
29
30
31
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
c Jan Modersitzki
3.5
3.6
3.7
3.8
3.9
4
5
6
4
FAIR
3.4.1
Spline Interpolation for One Dimensional Data . .
3.4.2
Spline Interpolation for Higher Dimensional Data
Derivatives of Interpolation Schemes . . . . . . . . . . . . . .
3.5.1
Derivative of the Interpolant . . . . . . . . . . . .
3.5.2
Derivative of Multivariate Interpolant . . . . . . .
3.5.3
Testing the Implementation of a Derivative . . . .
Multiscale Spline Interpolation . . . . . . . . . . . . . . . . . .
3.6.1
Multiscale Interpolation in One Dimension . . . .
3.6.2
Truncating High Frequencies . . . . . . . . . . . .
3.6.3
Multiscale Interpolation in Higher Dimensions . .
Multiresolution of Data . . . . . . . . . . . . . . . . . . . . . .
Summarizing the Interpolation Toolbox . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
32
34
35
35
36
36
37
38
40
41
42
43
45
Transforming Images by Parameterized Transformations
4.1
Translations . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Affine Linear Transformations . . . . . . . . . . . . . . .
4.3
Rigid Transformations . . . . . . . . . . . . . . . . . . .
4.4
Rotations About the Domain Center . . . . . . . . . . .
4.5
Spline Based Transformations . . . . . . . . . . . . . . .
4.6
More Bizarre Transformations . . . . . . . . . . . . . . .
4.7
Derivatives of Parameterized Transformations . . . . . .
4.8
Summarizing the Parameterized Transformations . . . .
4.9
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
49
49
50
51
52
52
54
54
56
57
Landmark Based Registration
5.1
Affine Linear Landmark Based Registration
5.2
Quadratic Landmark Based Registration . .
5.3
Thin-Plate-Spline Registration . . . . . . . .
5.4
Summarizing Landmark Based Registration
5.5
Exercises . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
59
60
61
63
64
66
Parametric Image Registration
6.1
Discretizing an Integral . . . . . . . . . . . . . . . . . . . .
6.2
Sum of Squared Differences . . . . . . . . . . . . . . . . . .
6.2.1
The Continuous SSD . . . . . . . . . . . . . . .
6.2.2
The Discretized SSD . . . . . . . . . . . . . . .
6.2.3
SSD and Parametric Transformations . . . . .
6.3
Numerical Optimization of Parametric Image Registrations
6.3.1
The Objective Function . . . . . . . . . . . . .
6.3.2
A Gauß-Newton Scheme . . . . . . . . . . . . .
6.3.3
PIR Examples . . . . . . . . . . . . . . . . . .
6.4
Experiments on Fixed Levels . . . . . . . . . . . . . . . . .
6.4.1
Rotations about the Center of the Domain . .
6.4.2
Rigid Transformations . . . . . . . . . . . . . .
6.5
Regularized Parametric Image Registration . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
67
68
70
70
71
72
74
75
77
79
82
83
83
84
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
[August 13, 2008, 13:12]
c Jan Modersitzki
FAIR
6.6
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
89
90
91
91
92
Distance Measures
7.1
Sum of Squared Differences . . . . . . . . . . . . . . . . .
7.1.1
SSD and Forces . . . . . . . . . . . . . . . .
7.1.2
Discretized SSD . . . . . . . . . . . . . . . .
7.2
Cross-Correlation . . . . . . . . . . . . . . . . . . . . . .
7.2.1
Continuous Normalized Cross-Correlation . .
7.2.2
Discretized Normalized Cross-Correlation . .
7.3
Mutual Information . . . . . . . . . . . . . . . . . . . . .
7.3.1
Estimating the Joint Density, Principles . . .
7.3.2
Estimating the Joint Density of Two Images
7.3.3
Mutual Information . . . . . . . . . . . . . .
7.3.4
Discretizing Mutual Information . . . . . . .
7.4
Normalized Gradient Field . . . . . . . . . . . . . . . . .
7.4.1
Continuous Normalized Gradient Field . . .
7.4.2
Discretized Normalized Gradient Field . . . .
7.5
Derivatives of Distance Measures . . . . . . . . . . . . .
7.6
Summarizing the Distance Measures Toolbox . . . . . . .
7.7
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
95
95
96
97
97
97
99
100
102
105
106
106
107
107
109
110
111
116
Regularizer
8.1
Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
L2 -Norm Based Regularizer . . . . . . . . . . . . . . . . . .
8.3
Discretizing L2 -Norm Based Regularizer . . . . . . . . . . .
8.3.1
Discretizing First Order Derivatives . . . . . . .
8.3.2
The Discretized Diffusion and Elastic Operators
8.3.3
The Discretized Curvature Operator . . . . . . .
8.3.4
The Discretized L2 -Norm Based Regularizer . .
8.4
Matrix Free Operations . . . . . . . . . . . . . . . . . . . . .
8.4.1
Matrix Free Elastic Operator . . . . . . . . . . .
8.4.2
Matrix Free Curvature Operator . . . . . . . . .
8.5
Summarizing the Regularization . . . . . . . . . . . . . . . .
8.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
117
118
120
122
122
126
126
128
128
129
129
130
131
6.7
6.8
7
8
Multilevel Based Optimization . . . . . . . . . . . .
6.6.1
SSD and Rigid Transformations . . . .
6.6.2
SSD and Affine Linear Transformations
Summarizing Parametric Image Registrations . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
9
Numerical Methods for Non-Parametric Image Registration
133
9.1
The NPIR Objective Function . . . . . . . . . . . . . . . . . . . 133
9.1.1
Staggered Grids and Image Interpolation . . . . . . 134
9.1.2
Matrix Free Operations . . . . . . . . . . . . . . . . 135
10
Discussion and Examples
[August 13, 2008, 13:12]
137
5
c Jan Modersitzki
11
12
13
14
15
6
General Tools
11.1
Get Cell Centered Grids . . . . . . . . .
11.2
Visualize 2D Data . . . . . . . . . . . . .
11.3
Checking Derivatives . . . . . . . . . . .
11.4
Set-up Data from an Ultrasound Image .
11.5
Set-up the Hand Example . . . . . . . .
11.6
Set-up the HNSP Example . . . . . . . .
11.7
Set-up the PET/CT Example . . . . . .
11.8
Setting Persistent Options . . . . . . . .
11.9
Creating Multilevel Data Representation
FAIR
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
139
139
140
140
141
142
143
144
145
147
Image Interpolation Tools
12.1
Linear Interpolation in 1D using Matlab
12.2
Linear Interpolation in 2D using Matlab
12.3
Linear Interpolation in 3D using Matlab
12.4
Linear Interpolation in 1D, final . . . . . .
12.5
Linear Interpolation in 2D, final . . . . . .
12.6
Linear Interpolation in 3D, final . . . . . .
12.7
“Mother” Spline Function . . . . . . . . . .
12.8
Spline Interpolation in 1D, final . . . . . .
12.9
Spline Interpolation in 2D, final . . . . . .
12.10 Spline Interpolation in 3D, final . . . . . .
12.11 Computing the Spline Coefficients . . . . .
12.12 The Interpolator . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
149
149
149
150
150
151
151
152
152
153
154
155
156
Parameterization Tools
13.1
Translations in 2D . . . . . . . . . . . . . .
13.2
Affine Linear Transformations in 2D . . . .
13.3
Rigid Transformations in 2D . . . . . . . .
13.4
Rotations around the domain center in 2D
13.5
Spline Based Transformations in 2D . . . .
13.6
Affine Linear Transformations in 3D . . . .
13.7
Rigid Transformations in 3D . . . . . . . .
13.8
Generic Parameteric Transformation . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
157
157
157
158
158
159
160
160
161
Landmark Registration Tools
14.1
Plot Landmark Registration Results . . .
14.2
Affine Linear Landmark Registration . .
14.3
Quadratic Landmark Registration . . . .
14.4
Get Thin-Plate-Spline Coefficients . . . .
14.5
Evaluate Thin-Plate-Splines . . . . . . .
14.6
Thin-Plate-Spline Registration . . . . . .
14.7
Landmark Registration . . . . . . . . . .
14.8
Computing and Evaluating TPS Solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
163
163
163
164
164
164
165
165
165
Parametric Image Registration
.
.
.
.
.
.
.
.
167
[August 13, 2008, 13:12]
c Jan Modersitzki
FAIR
15.1
15.2
15.3
15.4
15.5
16
Objective Function . . . . . . . . . . . . . . . . . .
Regularization . . . . . . . . . . . . . . . . . . . . .
Armijo Line Search . . . . . . . . . . . . . . . . . .
Gauß-Newton Based Parametric Image Registration
MLPIR: Multilevel Parametric Image Registration .
Distance Measures
16.1
Sum of Squared Differences . .
16.2
Normalized Cross Correlation
16.3
Joint Density Estimator . . .
16.4
Mutual Information . . . . . .
16.5
Normalized Gradient Fields .
16.6
FAIR Distance . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
167
168
169
170
172
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
175
175
175
176
177
179
180
17
Regularization
181
17.1
The Elastic Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 183
17.2
The Curvature Matrix . . . . . . . . . . . . . . . . . . . . . . . 184
17.3
FAIR Regularizer . . . . . . . . . . . . . . . . . . . . . . . . . . 186
18
Non-Parametric Image Registration
18.1
Prolongation Operator . . . . . .
18.2
Staggered to Cell-center Operator
18.3
Objective Function . . . . . . . .
18.4
Staggered to Cell-Center Operator
[August 13, 2008, 13:12]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
187
187
189
191
192
7
c Jan Modersitzki
8
FAIR
[August 13, 2008, 13:12]
FAIR Listing
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
FAIR
1: Flexible Algorithms for Image Registration . . . . . . . . .
2: Interpolation Toolbox . . . . . . . . . . . . . . . . . . . . .
3: Parametric Transformations . . . . . . . . . . . . . . . . . .
4: Landmark Based Registration . . . . . . . . . . . . . . . . .
5: Thin-Plate-Spline Interpolation . . . . . . . . . . . . . . . .
6: Thin-Plate-Spline Approximation . . . . . . . . . . . . . . .
7: Sum of Squared Difference Distance Measure (SSD) . . . . .
8: Discretized Sum of Squared Difference (SSD) . . . . . . . .
9: Objective Function for Parametric Image Registration . . .
10: Parametric Image Registration (PIR) . . . . . . . . . . . .
11: The normalized cross-correlation distance measure (NCC)
12: Parzen-Window Estimator . . . . . . . . . . . . . . . . . .
13: The mutual information distance measure (MI) . . . . . .
14: The normalized gradient field distance measure (NGF) . .
15: Distance Measure Toolbox . . . . . . . . . . . . . . . . . .
16: L2 -Norm Based Continuous Regularizer . . . . . . . . . . .
17: Regularizer Toolbox . . . . . . . . . . . . . . . . . . . . . .
18: Objective Function for Non-Parametric Image Registration
9
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
45
56
60
63
64
70
72
77
79
99
105
106
109
111
120
130
134
c Jan Modersitzki
10
FAIR
[August 13, 2008, 13:12]
List of Figures
1.1
Modified slices from CT scans of a human knee. . . . . . . . . . .
14
2.1
2.2
Visualizations of an image and a transformation. . . . . . . . . . .
Transforming images. . . . . . . . . . . . . . . . . . . . . . . . . .
18
19
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.13
3.14
3.12
Discretization of a 1D domain Ω = (0, ω) ⊂ R. . . . . . .
Discretization of a 2D domain Ω = (0, ω1 ) × (0, ω2 ) ⊂ R2 .
Next neighbor interpolation in 1D (dashed line). . . . . .
Linear interpolation in 1D. . . . . . . . . . . . . . . . . .
Linear interpolation in 2D. . . . . . . . . . . . . . . . . .
“Mother” spline b = b0 . . . . . . . . . . . . . . . . . . . .
Spline interpolation in 1D. . . . . . . . . . . . . . . . . .
Spline interpolation in 2D. . . . . . . . . . . . . . . . . .
A typical result of checkDerivative. . . . . . . . . . . .
Spline approximation in 1D. . . . . . . . . . . . . . . . .
Oscillations in spline interpolation. . . . . . . . . . . . . .
Multiresolution of an ultrasound image. . . . . . . . . . .
Interpolation on duty. . . . . . . . . . . . . . . . . . . . .
Spline approximations in 2D. . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
26
27
29
30
31
32
33
35
37
39
40
43
44
48
4.1
4.2
Translation, rigid, and affine linear transformations. . . . . . . . .
Spline based transformation and more. . . . . . . . . . . . . . . .
51
53
5.1
5.2
5.3
Reference and template with corresponding landmarks. . . . . . .
Linear and quadratic landmark based registrations. . . . . . . . .
Thin-Plate-Spline registration with various θ’s. . . . . . . . . . . .
60
62
65
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Midpoint quadrature rule in 1D. . . . . . .
Quadrature on duty. . . . . . . . . . . . . .
SSD versus rotations, coarse. . . . . . . . .
SSD versus rotations, fine. . . . . . . . . .
SSD versus translations. . . . . . . . . . . .
Plots from Parametric Image Registration.
PIR for SSD and rotations, m = (32, 16). .
PIR for SSD and rotations, m = (256, 128).
68
71
73
73
74
80
84
84
11
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
c Jan Modersitzki
12
FAIR
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
PIR iteration histories for SSD and rotations. . . . . . . . . .
PIR for SSD and rigid transformations, m = (32, 16). . . . .
PIR for SSD and rigid transformations, m = (256, 128). . . .
PIR iteration histories for SSD and rigid transformations. . .
PIR results for SSD and spline transformation. . . . . . . . .
Regularized PIR for SSD and spline transformations. . . . .
Multilevel representation of data and images. . . . . . . . . .
MLPIR iteration history for SSD and rigid transformations .
MLPIR results for SSD and rigid transformations . . . . . .
MLPIR iteration history for SSD and affine transformations .
MLPIR results for SSD and affine transformations . . . . . .
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
Force field of SSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Distance Measures for mon-modal images. . . . . . . . . . . . . . . 98
T1/T2 weighted MRI’s of a head. . . . . . . . . . . . . . . . . . . 99
Distance Measures for multi-modal images. . . . . . . . . . . . . . 100
Histograms of discretizations from n samples of a univariate function.103
Spline based Parzen-Window kernel k(·, σ). . . . . . . . . . . . . . 104
Parzen-Window density estimators. . . . . . . . . . . . . . . . . . 105
NGF for MRI’s of a head. . . . . . . . . . . . . . . . . . . . . . . . 108
Distance Measures for PET/CT images. . . . . . . . . . . . . . . . 113
Distance Measures for PET/CT images. . . . . . . . . . . . . . . . 114
8.1
8.2
8.3
8.4
Two images showing squares with texture. . . . .
Cell centered finite difference approximation of a
Staggered Grids in 2D. . . . . . . . . . . . . . .
Staggered Grids in 3D. . . . . . . . . . . . . . .
. . . . . .
derivative.
. . . . . .
. . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
85
85
86
86
87
88
89
90
90
91
92
119
122
123
124
[August 13, 2008, 13:12]
List of Tables
4.1
A more efficient implementation of the affine linear transformation
using a persistent variable Q. . . . . . . . . . . . . . . . . . . . . .
55
6.1
6.2
6.3
6.4
Implementations of the rotation and translation examples.
Paramatric Image Registration on a fixed level. . . . . . . .
Parametric Image Registration Results. . . . . . . . . . . .
Parametric Spline Registration. . . . . . . . . . . . . . . . .
75
80
81
87
7.1
Calling differentdistance measures. . . . . . . . . . . . . . . . . . . 112
8.1
8.2
8.3
8.4
Staggered grid based discrete derivatives. . . . . .
Staggered grid based discrete derivative operators.
Discretized elastic operator in matrix form. . . . .
Discretized curvature operator in matrix form. . .
13
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
125
125
127
127