CSci 6971: Image Registration Lecture 26: BSpline Transforms

April 20, 2004

Prof. Chuck Stewart, RPIDr. Luis Ibanez, Kitware

Image Registration Lecture 26 2

Deformable Registration

Deformable Registration

Image Registration Lecture 26 3

Deformable Transforms

BSplines

Image Registration Lecture 26 4

Deformable Transformation

y

Fixed Image

Transform

x

y

Moving Imagex

Image Registration Lecture 26 5

Deformable Transformation

y

Fixed Image

Transform

x

y

Moving Imagex

Image Registration Lecture 26 6

Deformable Transformation

y

x

Image Registration Lecture 26 7

Deformable Transformation

y

x

Image Registration Lecture 26 8

Image Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage

BSplineTransform

Image Registration Lecture 26 9

Deformable Transforms

BSplines in Geometrical Terms

Image Registration Lecture 26 10

BSplines

-2 -1 1 20

Order Zero

Image Registration Lecture 26 11

BSplines

-2 -1 1 20

Order Zero Piece-Wise

Y = 1

Image Registration Lecture 26 12

BSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 13

BSplines

Area under the product curve

-2 -1 1 20

Image Registration Lecture 26 14

BSplines

-2 -1 1 20

Image Registration Lecture 26 15

BSplines

Order One

-2 -1 1 20

Image Registration Lecture 26 16

BSplines

Order One

-2 -1 1 20

Piece-Wise

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

Image Registration Lecture 26 17

BSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 18

BSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 19

BSplines

Order Two

-2 -1 1 20

Image Registration Lecture 26 20

BSplines

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

BSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 22

BSplines

Convolution

-2 -1 1 20

Image Registration Lecture 26 23

BSplines

Order Three

-2 -1 1 20

Image Registration Lecture 26 24

BSplines

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

BSplines

Interpolatingwith

BSplines

Image Registration Lecture 26 26

BSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 27

BSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 28

BSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 29

BSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor

Image Registration Lecture 26 30

BSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 31

BSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 32

BSplines Interpolation

-2 -1 1 20

First Order Linear Interpolation

Image Registration Lecture 26 33

BSplines Interpolation

-2 -1 1 20

First Order Linear Interpolator

Image Registration Lecture 26 34

BSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 35

BSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 36

BSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolation

Image Registration Lecture 26 37

BSplines Interpolation

-2 -1 1 20

Second Order Quadratic Interpolator

Image Registration Lecture 26 38

BSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 39

BSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 40

BSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolation

Image Registration Lecture 26 41

BSplines Interpolation

-2 -1 1 20

Third Order Cubic Interpolator

Image Registration Lecture 26 42

BSplines

Cubic BSplinesin

Two Dimensions

Image Registration Lecture 26 43

BSplines in 2D

Tensor Product

Image Registration Lecture 26 44

BSplines in 2D

Tensor Product

Image Registration Lecture 26 45

BSplines in 2D

Tensor Product Separable

Image Registration Lecture 26 46

BSplines in 2D

Support Nodes in Region of Influence

#Nodes = 4N

Image Registration Lecture 26 47

BSplines in 2D

Interpolation

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 48

BSplines in 2D

Interpolation

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 49

Sampling Grid

y

x

Image Registration Lecture 26 50

BSpline Grid

y

x

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 51

BSplines Grid & Image Grid

Image Registration Lecture 26 52

BSplines Grid & Image Grid

BSplineGrid

Image Registration Lecture 26 53

BSplines Grid & Image Grid

BSplineGrid

ValidRegion

Image Registration Lecture 26 54

BSplines Grid & Image Grid

BSplineGrid

Image Registration Lecture 26 55

BSplines Grid & Image Grid

BSplineGrid

ValidRegion

Image Registration Lecture 26 56

BSplines Interpolation

InterpolatingVectors

Image Registration Lecture 26 57

Interpolating Vectors

y

x

BSplineGrid

ResamplingGrid

Image Registration Lecture 26 58

Interpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 59

Interpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 60

Interpolating Vectors

y

x

BSplineGrid

ResamplingGrid

YCompo-

nents

Image Registration Lecture 26 61

Interpolating Vectors

y

x

BSplineGrid

ResamplingGrid

XCompo-

nents

Image Registration Lecture 26 62

X Grid

Feeding the BSpline Transform

x

BSplineTransform

Dimension N

NGrid

YGrid

…..Z

Grid

Input Point

Output Point

Image Registration Lecture 26 63

Array of Parameters

Feeding the BSpline Transform

x

BSplineTransform

Input Point

Output Point

Image Registration Lecture 26 64

Array of Parameters

Feeding the BSpline Transform

x

BSplineTransform

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 Transform

x

BSplineTransform

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 Interpolation

Image Resampling

BSpline Warping

Image Registration Lecture 26 67

Image Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage

Image Registration Lecture 26 68

BSplines Interpolation

Original Lena

Image Registration Lecture 26 69

BSplines Interpolation

Deformed with BSpline Transform

Image Registration Lecture 26 70

BSplines Interpolation

Original LenaDeformed Lena

Image Registration Lecture 26 71

Deformable Registration

Image Registration

BSpline Transform

Image Registration Lecture 26 72

Image Registration Framework

FixedImage

MovingImage

Metric

Transform

Interpolator Optimizer

Parameters Array

Image Registration Lecture 26 73

BSplines Deformable Registration

Deformed with BSpline Transform

Image Registration Lecture 26 74

BSplines Deformable Registration

Registered with BSpline Transform

Image Registration Lecture 26 75

BSplines Deformable Registration

Original Lena

Image Registration Lecture 26 76

BSplines Deformable Registration

Difference After Registration

Difference Before Registration

Image Registration Lecture 26 77

End

Enjoy ITK !