CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI...
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Transcript of CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI...
![Page 1: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/1.jpg)
CSci 6971: Image Registration Lecture 26: BSpline Transforms
April 20, 2004
CSci 6971: Image Registration Lecture 26: BSpline Transforms
April 20, 2004
Prof. Chuck Stewart, RPIDr. Luis Ibanez, KitwareProf. Chuck Stewart, RPIDr. Luis Ibanez, Kitware
![Page 2: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/2.jpg)
Image Registration Lecture 26 2
Deformable RegistrationDeformable Registration
Deformable Registration
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Image Registration Lecture 26 3
Deformable TransformsDeformable Transforms
BSplines
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Image Registration Lecture 26 4
Deformable TransformationDeformable Transformation
y
Fixed Image
Transform
x
y
Moving Image
x
![Page 5: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/5.jpg)
Image Registration Lecture 26 5
Deformable TransformationDeformable Transformation
y
Fixed Image
Transform
x
y
Moving Image
x
![Page 6: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/6.jpg)
Image Registration Lecture 26 6
Deformable TransformationDeformable Transformation
y
x
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Image Registration Lecture 26 7
Deformable TransformationDeformable Transformation
y
x
![Page 8: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/8.jpg)
Image Registration Lecture 26 8
Image ResamplingImage Resampling
FixedImage
MovingImage
Transform
Interpolator
ResampleImageFilter
DeformedImage
BSplineTransform
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Image Registration Lecture 26 9
Deformable TransformsDeformable Transforms
BSplines in
Geometrical Terms
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Image Registration Lecture 26 10
BSplinesBSplines
-2 -1 1 20
Order Zero
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Image Registration Lecture 26 11
BSplinesBSplines
-2 -1 1 20
Order Zero Piece-Wise
Y = 1
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Image Registration Lecture 26 12
BSplinesBSplines
Convolution
-2 -1 1 20
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Image Registration Lecture 26 13
BSplinesBSplines
Area under the
product curve
-2 -1 1 20
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Image Registration Lecture 26 14
BSplinesBSplines
-2 -1 1 20
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Image Registration Lecture 26 15
BSplinesBSplines
Order One
-2 -1 1 20
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Image Registration Lecture 26 16
BSplinesBSplines
Order One
-2 -1 1 20
Piece-Wise
Y = ( 1 - X )Y = ( X + 1 )
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Image Registration Lecture 26 17
BSplinesBSplines
Convolution
-2 -1 1 20
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Image Registration Lecture 26 18
BSplinesBSplines
Convolution
-2 -1 1 20
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Image Registration Lecture 26 19
BSplinesBSplines
Order Two
-2 -1 1 20
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Image Registration Lecture 26 20
BSplinesBSplines
Order Two
-2 -1 1 20
Piece-Wise
Y = ( X – 3/2 )2 / 2
Y = ( 1 – 2 X2 )
Y = ( X + 3/2 )2 / 2
![Page 21: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/21.jpg)
Image Registration Lecture 26 21
BSplinesBSplines
Convolution
-2 -1 1 20
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Image Registration Lecture 26 22
BSplinesBSplines
Convolution
-2 -1 1 20
![Page 23: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/23.jpg)
Image Registration Lecture 26 23
BSplinesBSplines
Order Three
-2 -1 1 20
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Image Registration Lecture 26 24
BSplinesBSplines
Order Three
-2 -1 1 20
Piece-Wise
Y = ( 3X3 - 6X2 + 4 )/6
Y = (2-X)3 / 6Y = (2+X)3 / 6
Y = ( - 3X3 - 6X2 + 4 )/6
![Page 25: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/25.jpg)
Image Registration Lecture 26 25
BSplinesBSplines
Interpolating
with
BSplines
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Image Registration Lecture 26 26
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Zero Order Nearest Neighbor
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Image Registration Lecture 26 27
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Zero Order Nearest Neighbor
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Image Registration Lecture 26 28
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Zero Order Nearest Neighbor
![Page 29: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/29.jpg)
Image Registration Lecture 26 29
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Zero Order Nearest Neighbor
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Image Registration Lecture 26 30
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
First Order Linear Interpolation
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Image Registration Lecture 26 31
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
First Order Linear Interpolation
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Image Registration Lecture 26 32
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
First Order Linear Interpolation
![Page 33: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/33.jpg)
Image Registration Lecture 26 33
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
First Order Linear Interpolator
![Page 34: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/34.jpg)
Image Registration Lecture 26 34
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Second Order Quadratic Interpolation
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Image Registration Lecture 26 35
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Second Order Quadratic Interpolation
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Image Registration Lecture 26 36
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Second Order Quadratic Interpolation
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Image Registration Lecture 26 37
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Second Order Quadratic Interpolator
![Page 38: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/38.jpg)
Image Registration Lecture 26 38
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Third Order Cubic Interpolation
![Page 39: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/39.jpg)
Image Registration Lecture 26 39
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Third Order Cubic Interpolation
![Page 40: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/40.jpg)
Image Registration Lecture 26 40
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Third Order Cubic Interpolation
![Page 41: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/41.jpg)
Image Registration Lecture 26 41
BSplines InterpolationBSplines Interpolation
-2 -1 1 20
Third Order Cubic Interpolator
![Page 42: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/42.jpg)
Image Registration Lecture 26 42
BSplinesBSplines
Cubic BSplines
in
Two Dimensions
![Page 43: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/43.jpg)
Image Registration Lecture 26 43
BSplines in 2DBSplines in 2D
Tensor Product
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Image Registration Lecture 26 44
BSplines in 2DBSplines in 2D
Tensor Product
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Image Registration Lecture 26 45
BSplines in 2DBSplines in 2D
Tensor Product Separable
![Page 46: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/46.jpg)
Image Registration Lecture 26 46
BSplines in 2DBSplines in 2D
Support Nodes in Region of Influence
#Nodes = 4N
![Page 47: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/47.jpg)
Image Registration Lecture 26 47
BSplines in 2DBSplines in 2D
Interpolation
BSplineGrid
ResamplingGrid
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Image Registration Lecture 26 48
BSplines in 2DBSplines in 2D
Interpolation
BSplineGrid
ResamplingGrid
![Page 49: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/49.jpg)
Image Registration Lecture 26 49
Sampling GridSampling Grid
y
x
![Page 50: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/50.jpg)
Image Registration Lecture 26 50
BSpline GridBSpline Grid
y
x
BSplineGrid
ResamplingGrid
![Page 51: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/51.jpg)
Image Registration Lecture 26 51
BSplines Grid & Image GridBSplines Grid & Image Grid
![Page 52: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/52.jpg)
Image Registration Lecture 26 52
BSplines Grid & Image GridBSplines Grid & Image Grid
BSplineGrid
![Page 53: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/53.jpg)
Image Registration Lecture 26 53
BSplines Grid & Image GridBSplines Grid & Image Grid
BSplineGrid
ValidRegion
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Image Registration Lecture 26 54
BSplines Grid & Image GridBSplines Grid & Image Grid
BSplineGrid
![Page 55: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/55.jpg)
Image Registration Lecture 26 55
BSplines Grid & Image GridBSplines Grid & Image Grid
BSplineGrid
ValidRegion
![Page 56: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/56.jpg)
Image Registration Lecture 26 56
BSplines InterpolationBSplines Interpolation
Interpolating
Vectors
![Page 57: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/57.jpg)
Image Registration Lecture 26 57
Interpolating VectorsInterpolating Vectors
y
x
BSplineGrid
ResamplingGrid
![Page 58: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/58.jpg)
Image Registration Lecture 26 58
Interpolating VectorsInterpolating Vectors
y
x
BSplineGrid
ResamplingGrid
XCompo-
nents
![Page 59: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/59.jpg)
Image Registration Lecture 26 59
Interpolating VectorsInterpolating Vectors
y
x
BSplineGrid
ResamplingGrid
XCompo-
nents
![Page 60: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/60.jpg)
Image Registration Lecture 26 60
Interpolating VectorsInterpolating Vectors
y
x
BSplineGrid
ResamplingGrid
YCompo-
nents
![Page 61: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/61.jpg)
Image Registration Lecture 26 61
Interpolating VectorsInterpolating Vectors
y
x
BSplineGrid
ResamplingGrid
XCompo-
nents
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X
Grid
Feeding the BSpline TransformFeeding the BSpline Transform
x
BSpline
Transform
Dimension N
N
Grid
Y
Grid…..
Z
Grid
Input Point
Output Point
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Array of Parameters
Feeding the BSpline TransformFeeding the BSpline Transform
x
BSpline
Transform
Input Point
Output Point
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Image Registration Lecture 26 64
Array of Parameters
Feeding the BSpline TransformFeeding the BSpline Transform
x
BSpline
Transform
Input Point
Output Point
x1 x2 x3 x4 x5... y1 y2 y3 y4 y5 ….…. z1 z2 z3 z4….…
![Page 65: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/65.jpg)
Image Registration Lecture 26 65
Feeding the BSpline TransformFeeding the BSpline Transform
x
BSpline
Transform
Input Point
Output Point
x1 x2 x3 x4 x5... y1 y2 y3 y4 y5 ….…. z1 z2 z3 z4….…
x1y1z1 x2y2z2 x3y3z3 x4y4z4 …. xNyNzN Input File
Parameters Array
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Image Registration Lecture 26 66
BSplines InterpolationBSplines Interpolation
Image Resampling
BSpline Warping
![Page 67: CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware Prof. Chuck Stewart, RPI.](https://reader035.fdocuments.us/reader035/viewer/2022062518/56649d405503460f94a19bcb/html5/thumbnails/67.jpg)
Image Registration Lecture 26 67
Image ResamplingImage Resampling
FixedImage
MovingImage
Transform
Interpolator
ResampleImageFilter
DeformedImage
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Image Registration Lecture 26 68
BSplines InterpolationBSplines Interpolation
Original Lena
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Image Registration Lecture 26 69
BSplines InterpolationBSplines Interpolation
Deformed with BSpline Transform
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Image Registration Lecture 26 70
BSplines InterpolationBSplines Interpolation
Original LenaDeformed Lena
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Image Registration Lecture 26 71
Deformable RegistrationDeformable Registration
Image Registration
BSpline Transform
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Image Registration Lecture 26 72
Image Registration FrameworkImage Registration Framework
FixedImage
MovingImage
Metric
Transform
Interpolator Optimizer
Parameters Array
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Image Registration Lecture 26 73
BSplines Deformable RegistrationBSplines Deformable Registration
Deformed with BSpline Transform
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Image Registration Lecture 26 74
BSplines Deformable RegistrationBSplines Deformable Registration
Registered with BSpline Transform
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Image Registration Lecture 26 75
BSplines Deformable RegistrationBSplines Deformable Registration
Original Lena
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BSplines Deformable RegistrationBSplines Deformable Registration
Difference After Registration
Difference Before Registration
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Image Registration Lecture 26 77
EndEnd
Enjoy ITK !