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

Image Registration Lecture 26 2

Deformable RegistrationDeformable Registration

Deformable Registration

Image Registration Lecture 26 3

Deformable TransformsDeformable Transforms

BSplines

Image Registration Lecture 26 4

Deformable TransformationDeformable Transformation

y

Fixed Image

Transform

x

y

Moving Image

x

Image Registration Lecture 26 5

Deformable TransformationDeformable Transformation

y

Fixed Image

Transform

x

y

Moving Image

x

Image Registration Lecture 26 6

Deformable TransformationDeformable Transformation

y

x

Image Registration Lecture 26 7

Deformable TransformationDeformable Transformation

y

x

Image Registration Lecture 26 8

Image ResamplingImage Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage

BSplineTransform

Image Registration Lecture 26 9

Deformable TransformsDeformable Transforms

BSplines in

Geometrical Terms

Image Registration Lecture 26 10

BSplinesBSplines

-2 -1 1 20

Order Zero

Image Registration Lecture 26 11

BSplinesBSplines

-2 -1 1 20

Order Zero Piece-Wise

Y = 1

Image Registration Lecture 26 12

BSplinesBSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 13

BSplinesBSplines

Area under the

product curve

-2 -1 1 20

Image Registration Lecture 26 14

BSplinesBSplines

-2 -1 1 20

Image Registration Lecture 26 15

BSplinesBSplines

Order One

-2 -1 1 20

Image Registration Lecture 26 16

BSplinesBSplines

Order One

-2 -1 1 20

Piece-Wise

Y = ( 1 - X )Y = ( X + 1 )

Image Registration Lecture 26 17

BSplinesBSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 18

BSplinesBSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 19

BSplinesBSplines

Order Two

-2 -1 1 20

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

Image Registration Lecture 26 21

BSplinesBSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 22

BSplinesBSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 23

BSplinesBSplines

Order Three

-2 -1 1 20

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

Image Registration Lecture 26 25

BSplinesBSplines

Interpolating

with

BSplines

Image Registration Lecture 26 26

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 27

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 28

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 29

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 30

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 31

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 32

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 33

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

First Order Linear Interpolator

Image Registration Lecture 26 34

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 35

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 36

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 37

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolator

Image Registration Lecture 26 38

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 39

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 40

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 41

BSplines InterpolationBSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolator

Image Registration Lecture 26 42

BSplinesBSplines

Cubic BSplines

in

Two Dimensions

Image Registration Lecture 26 43

BSplines in 2DBSplines in 2D

Tensor Product

Image Registration Lecture 26 44

BSplines in 2DBSplines in 2D

Tensor Product

Image Registration Lecture 26 45

BSplines in 2DBSplines in 2D

Tensor Product Separable

Image Registration Lecture 26 46

BSplines in 2DBSplines in 2D

Support Nodes in Region of Influence

#Nodes = 4N

Image Registration Lecture 26 47

BSplines in 2DBSplines in 2D

Interpolation

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 48

BSplines in 2DBSplines in 2D

Interpolation

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 49

Sampling GridSampling Grid

y

x

Image Registration Lecture 26 50

BSpline GridBSpline Grid

y

x

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 51

BSplines Grid & Image GridBSplines Grid & Image Grid

Image Registration Lecture 26 52

BSplines Grid & Image GridBSplines Grid & Image Grid

BSplineGrid

Image Registration Lecture 26 53

BSplines Grid & Image GridBSplines Grid & Image Grid

BSplineGrid

ValidRegion

Image Registration Lecture 26 54

BSplines Grid & Image GridBSplines Grid & Image Grid

BSplineGrid

Image Registration Lecture 26 55

BSplines Grid & Image GridBSplines Grid & Image Grid

BSplineGrid

ValidRegion

Image Registration Lecture 26 56

BSplines InterpolationBSplines Interpolation

Interpolating

Vectors

Image Registration Lecture 26 57

Interpolating VectorsInterpolating Vectors

y

x

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 58

Interpolating VectorsInterpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 59

Interpolating VectorsInterpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 60

Interpolating VectorsInterpolating Vectors

y

x

BSplineGrid

ResamplingGrid

YCompo-

nents

Image Registration Lecture 26 61

Interpolating VectorsInterpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 62

X

Grid

Feeding the BSpline TransformFeeding the BSpline Transform

x

BSpline

Transform

Dimension N

N

Grid

Y

Grid…..

Z

Grid

Input Point

Output Point

Image Registration Lecture 26 63

Array of Parameters

Feeding the BSpline TransformFeeding the BSpline Transform

x

BSpline

Transform

Input Point

Output Point

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….…

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

Image Registration Lecture 26 66

BSplines InterpolationBSplines Interpolation

Image Resampling

BSpline Warping

Image Registration Lecture 26 67

Image ResamplingImage Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage

Image Registration Lecture 26 68

BSplines InterpolationBSplines Interpolation

Original Lena

Image Registration Lecture 26 69

BSplines InterpolationBSplines Interpolation

Deformed with BSpline Transform

Image Registration Lecture 26 70

BSplines InterpolationBSplines Interpolation

Original LenaDeformed Lena

Image Registration Lecture 26 71

Deformable RegistrationDeformable Registration

Image Registration

BSpline Transform

Image Registration Lecture 26 72

Image Registration FrameworkImage Registration Framework

FixedImage

MovingImage

Metric

Transform

Interpolator Optimizer

Parameters Array

Image Registration Lecture 26 73

BSplines Deformable RegistrationBSplines Deformable Registration

Deformed with BSpline Transform

Image Registration Lecture 26 74

BSplines Deformable RegistrationBSplines Deformable Registration

Registered with BSpline Transform

Image Registration Lecture 26 75

BSplines Deformable RegistrationBSplines Deformable Registration

Original Lena

Image Registration Lecture 26 76

BSplines Deformable RegistrationBSplines Deformable Registration

Difference After Registration

Difference Before Registration

Image Registration Lecture 26 77

EndEnd

Enjoy ITK !