ITK Deformable Registration BSplines FreeForm Deformable Registration Deformable

ITK Deformable Registration B-Splines Free-Form

Deformable Registration

Deformable Transforms BSplines

Deformable Transformation y y Transform x Fixed Image x Moving Image

Deformable Transformation y x

Image Resampling Fixed Image Moving Image Interpolator Resample Image Filter BSpline Transform Deformed Image

Deformable Transforms BSplines in Geometrical Terms

BSplines Order Zero -2 -1 0 1 2

BSplines Order Zero Piece-Wise Y= 1 -2 -1 0 1 2

BSplines Convolution -2 -1 0 1 2

BSplines Area under the product curve -2 -1 0 1 2

BSplines -2 -1 0 1 2

BSplines Order One -2 -1 0 1 2

BSplines Order One Piece-Wise Y=(X+1) -2 -1 Y=(1 -X) 0 1 2

BSplines Convolution -2 -1 0 1 2

BSplines Order Two -2 -1 0 1 2

BSplines Order Two Piece-Wise Y = ( 1 – 2 X 2 ) Y = ( X + 3/2 )2 / 2 Y = ( X – 3/2 )2 / 2 -2 -1 0 1 2

BSplines Convolution -2 -1 0 1 2

BSplines Order Three -2 -1 0 1 2

BSplines Order Three Piece-Wise Y = ( - 3 X 3 - 6 X 2 + 4 )/6 Y= (2+X)3 -2 Y = ( 3 X 3 - 6 X 2 + 4 )/6 Y = (2 -X)3 / 6 /6 -1 0 1 2

BSplines Interpolating with BSplines

BSplines Interpolation Zero Order -2 Nearest Neighbor -1 0 1 2

BSplines Interpolation First Order -2 Linear Interpolation -1 0 1 2

BSplines Interpolation First Order -2 Linear Interpolator -1 0 1 2

BSplines Interpolation Second Order -2 -1 Quadratic Interpolation 0 1 2

BSplines Interpolation Second Order -2 -1 Quadratic Interpolator 0 1 2

BSplines Interpolation Third Order -2 -1 Cubic Interpolation 0 1 2

BSplines Interpolation Third Order -2 Cubic Interpolator -1 0 1 2

BSplines Cubic BSplines in Two Dimensions

BSplines in 2 D Tensor Product

BSplines in 2 D Tensor Product Separable

BSplines in 2 D Support Nodes in Region of Influence #Nodes = 4 N

BSplines in 2 D Interpolation BSpline Grid Resampling Grid

Sampling Grid y x

BSpline Grid y BSpline Grid Resampling Grid x

BSplines Grid & Image Grid

BSplines Grid & Image Grid BSpline Grid

BSplines Grid & Image Grid BSpline Grid Valid Region

BSplines Grid & Image Grid BSpline Grid

BSplines Grid & Image Grid BSpline Grid Valid Region

BSplines Interpolation Interpolating Vectors

Interpolating Vectors y BSpline Grid Resampling Grid x

Interpolating Vectors y BSpline Grid X Components Resampling Grid x

Interpolating Vectors y BSpline Grid Y Components Resampling Grid x

Interpolating Vectors y BSpline Grid X Components Resampling Grid x

Feeding the BSpline Transform Y X Grid Z Grid …. . N Grid Dimension N Input Point BSpline Transform Output Point x

Feeding the BSpline Transform Array of Parameters Input Point BSpline Transform Output Point x

Feeding the BSpline Transform x 1 x 2 x 3 x 4 x 5. . . y 1 y 2 y 3 y 4 y 5 …. …. z 1 z 2 z 3 z 4…. … Array of Parameters Input Point BSpline Transform Output Point x

Feeding the BSpline Transform x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 …. x. Ny. Nz. N Input File Parameters Array x 1 x 2 x 3 x 4 x 5. . . Input Point y 1 y 2 y 3 y 4 y 5 …. …. z 1 z 2 z 3 z 4…. … BSpline Transform Output Point x