ITK Deformable Registration B-Splines Free-Form. Deformable Registration.

ITK Deformable Registration

B-SplinesFree-Form

Deformable Registration


Deformable Transforms

BSplines

Deformable Transformation

y

Fixed Image

Transform

x

y

Moving Image

x

Deformable Transformationy

x

Image Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage

BSplineTransform

Deformable Transforms

BSplines in

Geometrical Terms

BSplines

-2 -1 1 20

Order Zero

BSplines

-2 -1 1 20

Order Zero Piece-Wise

Y = 1

BSplinesConvolution

-2 -1 1 20

BSplinesArea under the

product curve

-2 -1 1 20

BSplines

-2 -1 1 20

BSplinesOrder One

-2 -1 1 20

BSplinesOrder One

-2 -1 1 20

Piece-Wise

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

BSplinesConvolution

-2 -1 1 20

BSplinesOrder Two

-2 -1 1 20

BSplinesOrder Two

-2 -1 1 20

Piece-Wise

Y = ( X – 3/2 )2 / 2

Y = ( 1 – 2 X2 )

Y = ( X + 3/2 )2 / 2

BSplinesConvolution

-2 -1 1 20

BSplinesOrder Three

-2 -1 1 20

BSplinesOrder 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

BSplines

Interpolating

with

BSplines

BSplines Interpolation

-2 -1 1 20

Zero Order Nearest Neighbor


-2 -1 1 20

First Order Linear Interpolation


-2 -1 1 20

First Order Linear Interpolator


-2 -1 1 20

Second Order Quadratic Interpolation


-2 -1 1 20

Second Order Quadratic Interpolator


-2 -1 1 20

Third Order Cubic Interpolation


-2 -1 1 20

Third Order Cubic Interpolator

BSplines

Cubic BSplines

in

Two Dimensions

BSplines in 2DTensor Product

BSplines in 2DTensor Product Separable

BSplines in 2DSupport Nodes in Region

of Influence

#Nodes = 4N

BSplines in 2DInterpolation

BSplineGrid

ResamplingGrid

Sampling Gridy

x

BSpline Gridy

x

BSplineGrid

ResamplingGrid

BSplines Grid & Image Grid


BSplineGrid


BSplineGrid

ValidRegion


BSplineGrid


BSplineGrid

ValidRegion


Interpolating

Vectors

Interpolating Vectorsy

x

BSplineGrid

ResamplingGrid


x

BSplineGrid

ResamplingGrid

XCompo-nents


x

BSplineGrid

ResamplingGrid

YCompo-nents


x

BSplineGrid

ResamplingGrid

XCompo-nents

X

Grid

Feeding the BSpline Transform

x

BSpline

Transform

Dimension N

N

Grid

Y

Grid…..

Z

Grid

Input Point

Output Point

Array of Parameters


x

BSpline

Transform

Input Point

Output Point

Array of Parameters


x

BSpline

Transform

Input Point

Output Point

x1 x2 x3 x4 x5... y1 y2 y3 y4 y5 ….…. z1 z2 z3 z4….…


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 Resampling

BSpline Warping

Image Resampling

FixedImage

MovingImage

Transform

Interpolator

ResampleImageFilter

DeformedImage


Original Lena


Deformed with BSpline Transform


Original LenaDeformed Lena


Image Registration

BSpline Transform

Image Registration Framework

FixedImage

MovingImage

Metric

Transform

Interpolator Optimizer

Parameters Array

BSplines Deformable Registration

Deformed with BSpline Transform


Registered with BSpline Transform


Original Lena


Difference After Registration

Difference Before Registration

Enjoy ITK !

ITK Deformable Registration B-Splines Free-Form. Deformable Registration.

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