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ITK Deformable Registration B-Splines Free-Form

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Deformable Registration

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Deformable Transforms BSplines

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Deformable Transformation y Fixed Image Transform x y Moving Image x

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Deformable Transformation y Fixed Image Transform x y Moving Image x

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Deformable Transformation y x

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y x

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Image Resampling Fixed Image Moving Image Transform Interpolator Resample Image Filter Deformed Image BSpline 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 -2120

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BSplines Area under the product curve -2120

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BSplines -2120

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BSplines Order One -2120

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

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BSplines Convolution -2120

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BSplines Convolution -2120

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BSplines Order Two -2120

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

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BSplines Convolution -2120

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BSplines Convolution -2120

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BSplines Order Three -2120

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

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BSplines Interpolating with BSplines

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BSplines Interpolation Zero OrderNearest Neighbor

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BSplines Interpolation Zero OrderNearest Neighbor

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BSplines Interpolation Zero OrderNearest Neighbor

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BSplines Interpolation Zero OrderNearest Neighbor

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BSplines Interpolation First OrderLinear Interpolation

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BSplines Interpolation First OrderLinear Interpolation

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BSplines Interpolation First OrderLinear Interpolation

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BSplines Interpolation First OrderLinear Interpolator

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BSplines Interpolation Second OrderQuadratic Interpolation

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BSplines Interpolation Second OrderQuadratic Interpolation

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BSplines Interpolation Second OrderQuadratic Interpolation

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BSplines Interpolation Second OrderQuadratic Interpolator

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BSplines Interpolation Third OrderCubic Interpolation

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BSplines Interpolation Third OrderCubic Interpolation

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BSplines Interpolation Third OrderCubic Interpolation

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BSplines Interpolation Third OrderCubic Interpolator

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BSplines Cubic BSplines in Two 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 = 4 N

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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 x BSpline Grid Resampling Grid

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BSplines Grid & Image Grid

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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 Interpolating Vectors

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y x BSpline Grid Resampling Grid

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Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

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Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

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Interpolating Vectors y x BSpline Grid Resampling Grid Y Compo- nents

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Interpolating Vectors y x BSpline Grid Resampling Grid X Compo- nents

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

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

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

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Feeding the BSpline Transform x BSpline Transform Input Point Output Point 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….… 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 N y N z N Input File Parameters Array

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BSplines Interpolation Image Resampling BSpline Warping

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

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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 LenaDeformed Lena

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Deformable Registration Image Registration BSpline Transform

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Image Registration Framework Fixed Image Moving Image Metric Transform Interpolator Optimizer 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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Enjoy ITK !

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