CSci 6971: Image Registration Lecture 26: BSpline Transforms April 20, 2004

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CSci 6971 Image Registration Lecture 26
BSpline TransformsApril 20, 2004
Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware
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Deformable Registration
Deformable Registration
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Deformable Transforms
BSplines
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Deformable Transformation
y
y
Transform
x
x
Fixed Image
Moving Image
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Deformable Transformation
y
y
Transform
x
x
Fixed Image
Moving Image
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Deformable Transformation
y
x
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Deformable Transformation
y
x
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Image Resampling
Interpolator
FixedImage
Resample Image Filter
MovingImage
BSpline Transform
DeformedImage
Transform
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Deformable Transforms
BSplines in Geometrical Terms
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BSplines
Order Zero
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BSplines
Order Zero
Piece-Wise
Y 1
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BSplines
Convolution
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BSplines
Area under the product curve
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BSplines
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BSplines
Order One
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BSplines
Order One
Piece-Wise
Y ( 1 - X )
Y ( X 1 )
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BSplines
Convolution
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BSplines
Convolution
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BSplines
Order Two
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BSplines
Order Two
Piece-Wise
Y ( 1 2 X2 )
Y ( X 3/2 )2 / 2
Y ( X 3/2 )2 / 2
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BSplines
Convolution
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BSplines
Convolution
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BSplines
Order Three
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BSplines
Order Three
Piece-Wise
Y ( - 3X3 - 6X2 4 )/6
Y ( 3X3 - 6X2 4 )/6
Y (2-X)3 / 6
Y (2X)3 / 6
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BSplines
Interpolatingwith BSplines
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BSplines Interpolation
Zero Order
Nearest Neighbor
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BSplines Interpolation
Zero Order
Nearest Neighbor
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BSplines Interpolation
Zero Order
Nearest Neighbor
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BSplines Interpolation
Zero Order
Nearest Neighbor
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BSplines Interpolation
First Order
Linear Interpolation
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BSplines Interpolation
First Order
Linear Interpolation
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BSplines Interpolation
First Order
Linear Interpolation
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BSplines Interpolation
First Order
Linear Interpolator
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BSplines Interpolation
Second Order
Quadratic Interpolation
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BSplines Interpolation
Second Order
Quadratic Interpolation
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BSplines Interpolation
Second Order
Quadratic Interpolation
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BSplines Interpolation
Second Order
Quadratic Interpolator
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BSplines Interpolation
Third Order
Cubic Interpolation
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BSplines Interpolation
Third Order
Cubic Interpolation
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BSplines Interpolation
Third Order
Cubic Interpolation
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BSplines Interpolation
Third Order
Cubic Interpolator
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BSplines
Cubic BSplinesinTwo Dimensions
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BSplines in 2D
Tensor Product
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BSplines in 2D
Tensor Product
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BSplines in 2D
Tensor Product
Separable
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BSplines in 2D
Support
Nodes in Region of Influence
Nodes 4N
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BSplines in 2D
Interpolation
BSpline Grid
Resampling Grid
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BSplines in 2D
Interpolation
BSpline Grid
Resampling Grid
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Sampling Grid
y
x
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BSpline Grid
y
BSpline Grid
Resampling Grid
x
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BSplines Grid Image Grid
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BSplines Grid Image Grid
BSpline Grid
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BSplines Grid Image Grid
BSpline Grid
Valid Region
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BSplines Grid Image Grid
BSpline Grid
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BSplines Grid Image Grid
BSpline Grid
Valid Region
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BSplines Interpolation
InterpolatingVectors
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Interpolating Vectors
y
BSpline Grid
Resampling Grid
x
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Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
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Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
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Interpolating Vectors
y
BSpline Grid
Y Compo- nents
Resampling Grid
x
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Interpolating Vectors
y
BSpline Grid
X Compo- nents
Resampling Grid
x
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Feeding the BSpline Transform
Z Grid
Y Grid
N Grid
X Grid
..
Dimension N
BSpline Transform
Input Point
Output Point
x
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Feeding the BSpline Transform
Array of Parameters
BSpline Transform
Input Point
Output Point
x
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Feeding 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
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Feeding 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
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BSplines Interpolation
Image ResamplingBSpline Warping
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Image Resampling
Interpolator
FixedImage
Resample Image Filter
MovingImage
DeformedImage
Transform
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BSplines Interpolation
Original Lena
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BSplines Interpolation
Deformed with BSpline Transform
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BSplines Interpolation
Original Lena
Deformed Lena
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Deformable Registration
Image RegistrationBSpline Transform
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Image Registration Framework
FixedImage
Metric
MovingImage
Interpolator
Optimizer
Transform
Parameters Array
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BSplines Deformable Registration
Deformed with BSpline Transform
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BSplines Deformable Registration
Registered with BSpline Transform
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BSplines Deformable Registration
Original Lena
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BSplines Deformable Registration
Difference After Registration
Difference Before Registration
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End
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