Title: CSci%206971:%20Image%20Registration%20Lecture%2026:%20BSpline%20Transforms%20April%2020,%202004
1CSci 6971 Image Registration Lecture 26
BSpline TransformsApril 20, 2004
Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware
2Deformable Registration
Deformable Registration
3Deformable Transforms
BSplines
4Deformable Transformation
y
y
Transform
x
x
Fixed Image
Moving Image
5Deformable Transformation
y
y
Transform
x
x
Fixed Image
Moving Image
6Deformable Transformation
y
x
7Deformable Transformation
y
x
8Image Resampling
Interpolator
FixedImage
Resample Image Filter
MovingImage
BSpline Transform
DeformedImage
Transform
9Deformable Transforms
BSplines in Geometrical Terms
10BSplines
Order Zero
11BSplines
Order Zero
Piece-Wise
Y 1
12BSplines
Convolution
13BSplines
Area under the product curve
14BSplines
15BSplines
Order One
16BSplines
Order One
Piece-Wise
Y ( 1 - X )
Y ( X 1 )
17BSplines
Convolution
18BSplines
Convolution
19BSplines
Order Two
20BSplines
Order Two
Piece-Wise
Y ( 1 2 X2 )
Y ( X 3/2 )2 / 2
Y ( X 3/2 )2 / 2
21BSplines
Convolution
22BSplines
Convolution
23BSplines
Order Three
24BSplines
Order Three
Piece-Wise
Y ( - 3X3 - 6X2 4 )/6
Y ( 3X3 - 6X2 4 )/6
Y (2-X)3 / 6
Y (2X)3 / 6
25BSplines
Interpolatingwith BSplines
26BSplines Interpolation
Zero Order
Nearest Neighbor
27BSplines Interpolation
Zero Order
Nearest Neighbor
28BSplines Interpolation
Zero Order
Nearest Neighbor
29BSplines Interpolation
Zero Order
Nearest Neighbor
30BSplines Interpolation
First Order
Linear Interpolation
31BSplines Interpolation
First Order
Linear Interpolation
32BSplines Interpolation
First Order
Linear Interpolation
33BSplines Interpolation
First Order
Linear Interpolator
34BSplines Interpolation
Second Order
Quadratic Interpolation
35BSplines Interpolation
Second Order
Quadratic Interpolation
36BSplines Interpolation
Second Order
Quadratic Interpolation
37BSplines Interpolation
Second Order
Quadratic Interpolator
38BSplines Interpolation
Third Order
Cubic Interpolation
39BSplines Interpolation
Third Order
Cubic Interpolation
40BSplines Interpolation
Third Order
Cubic Interpolation
41BSplines Interpolation
Third Order
Cubic Interpolator
42BSplines
Cubic BSplinesinTwo Dimensions
43BSplines in 2D
Tensor Product
44BSplines in 2D
Tensor Product
45BSplines in 2D
Tensor Product
Separable
46BSplines in 2D
Support
Nodes in Region of Influence
Nodes 4N
47BSplines in 2D
Interpolation
BSpline Grid
Resampling Grid
48BSplines in 2D
Interpolation
BSpline Grid
Resampling Grid
49Sampling Grid
y
x
50BSpline Grid
y
BSpline Grid
Resampling Grid
x
51BSplines Grid Image Grid
52BSplines Grid Image Grid
BSpline Grid
53BSplines Grid Image Grid
BSpline Grid
Valid Region
54BSplines Grid Image Grid
BSpline Grid
55BSplines Grid Image Grid
BSpline Grid
Valid Region
56BSplines Interpolation
InterpolatingVectors
57Interpolating Vectors
y
BSpline Grid
Resampling Grid
x
58Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
59Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
60Interpolating Vectors
y
BSpline Grid
Y Compo- nents
Resampling Grid
x
61Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
62Feeding the BSpline Transform
Z Grid
Y Grid
N Grid
X Grid
..
Dimension N
BSpline Transform
Input Point
Output Point
x
63Feeding the BSpline Transform
Array of Parameters
BSpline Transform
Input Point
Output Point
x
64Feeding the BSpline Transform
x1 x2 x3 x4 x5... y1 y2 y3 y4 y5 .. z1 z2
z3 z4.
Array of Parameters
BSpline Transform
Input Point
Output Point
x
65Feeding the BSpline Transform
x1y1z1 x2y2z2 x3y3z3 x4y4z4 . xNyNzN
Input File
Parameters Array
x1 x2 x3 x4 x5... y1 y2 y3 y4 y5 .. z1 z2
z3 z4.
BSpline Transform
Input Point
Output Point
x
66BSplines Interpolation
Image ResamplingBSpline Warping
67Image Resampling
Interpolator
FixedImage
Resample Image Filter
MovingImage
DeformedImage
Transform
68BSplines Interpolation
Original Lena
69BSplines Interpolation
Deformed with BSpline Transform
70BSplines Interpolation
Original Lena
Deformed Lena
71Deformable Registration
Image RegistrationBSpline Transform
72Image Registration Framework
FixedImage
Metric
MovingImage
Interpolator
Optimizer
Transform
Parameters Array
73BSplines Deformable Registration
Deformed with BSpline Transform
74BSplines Deformable Registration
Registered with BSpline Transform
75BSplines Deformable Registration
Original Lena
76BSplines Deformable Registration
Difference After Registration
Difference Before Registration
77End
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