Registration

• Align images through spatial transformations and allow comparisons

• Algorithm: Sampling/Interpolation —> Transformation Model —> Loss Function —> Optimization

• Moving image (transformed) and traget image (attain)

• Spatial transformation to match moving and target image

• Forward transformation: x’=T(x)

• Backward transformation: x=T-1(x’) —> much easier to use with interpolation

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• Image resampling —> interpolation to compute new intenities

• Examples: spatial transformations, resizing, etc.

• Problems: intensities defined centers —> calculate value inbetween

• Assign intensity value of closest pixel with known intensity

• Very fast, no new intensities

• Introduce artifacts in gray level images, not continuous or diffbar

• Use closest four pixel intenities

• Fast, less artifacts than linear

• Introduce mew intensity values, continuous but not diffbar, smooth/blur images —> lower resolution

• Use clostest 4 pixels and derivatives of this points

• Convolutional form: using catmull-rom Kernel K

• Better in retaining gray values, continuous and diffbar

• Highest computitional effort, produces extra values

• General form: x’=T(x) = Ax + t (matrix A, vector t)

• Only linear in hoogeneous coordinates, same Jacobian

• Small number of parameters compared to non-linear transformations

• Number of parameters is independent on the number of pixels

• Inverse transformation if A is invertable

  

• In homogeneous coordinates (x,y, w); w describes scaling

• Rotation of xy plane around origin + transation t

• 3 parameters; in 3D: 3 rotations and a trranslatoon

• Can push object out of FOX, change visible anatomy

• performs bilinear/trilinear interpolation

• Important: define centers (center of img vs. origin)

• Die Reihenfolge der Transfomationen spielt eine rolle —> nicht commutativ

• Rotation + Traslation + Scaling (zoom in and out)

• x'=SRx x'=RSx (order is not important)

• Extending with anisotropic scaling: ifferent factors in different dimensions:

—> Shapes are not preserved —> no similarity transformation

—> Order is important!

• Rotation + Translation + anisotropic scaling + shearing

• Shapes are no longer preserved

• Num of parameters 2D: 1 rotation, 2 scaling, 1 shaer, 2 translation —> 6

• Full affine transformation: reflection through negative scaling

• Can not be lonear, direct modeling of inverse transformation to from output to input domain

• Naive: Independent displacement vector for each pixel

• Number of parameters 2D: 2x numbr of pixels

• - large number of parameters, noisy and unrealistic results

• Optimization by parametrization:

  

• Define displacement vectors for sparse set of control oiubts and interpolate between with smooth model

• Radial basis function:

• x_n are the control points; phi the basis function, d non linear coefficient

• Using Thin plate spines (TPS) or B-splines

• Sometimes defined together with affine transformations

• Control points: uniformly, randomly spaced or landmarks

• Num of parameters: nUm of control points x2

• TPS: nonlinear, deformable with few parameters, smooth, global, used for landmark based registration

• - increased complex if more control points

• Free Form Deformations (FFD)

• Control points placed in a uniform grid with spacing d

• Dislacement vectors d are given on the control points only

• u(x) is interpolated for the image grid points

• Smooth transfomrations even with large control points

• Pixel wise modeling - siplacement of each grid point

• u(x’): displacement vector for each pixel

• u(x’) must stisfy model equation

• Very large number of parameters, restricted by model

• Elastic body models (linear

• Diffusion models

• Curvature models

• Transformation at each point: ordinary DE (ODE)

solved by:

• If v(x,t) smooth —> transformation is diffeomorphism

• Diffeomorphism: bijektiv, stetig diffbar, umkehrbar

• Variation: without dependence on time

• Different subjects: alignment & transformation changes

• Different modalities: need of different loss functions —> e.g. intensity based

• Difficulty also depends on the anatomy

• DIfficulty due to deformation of anatomy during motion

• Difficulty due to pathologies in inter-subject registration

• Landmark: prominent locations, distinct from their surroundings, easily identifieable, orientation, manual

• More specific landmarks —> more accurate alignment

• Landmark based loss function:

• Measures the distance between the landmarks in image 1 and 2

• T is a linear transformation with matrix A and tranlation t

• Minimize to find the best transformation parameters A and t

• Displacement vectors defined through same landmarks

• Loss is minimized by d —> use displacement vectors and whole imahe

• Large degrees of freedom —> better alignment (also for landmarks)

• Further opportunities: more robust loss functions, automatic determination of landmarks (ML)

• Intensity based loss function:

• d Distance metric between I and J after transformation

• More difficult than landmark based registration

• d can be the Sum of Squared distances (SSD)

• 0 when I and J are the same

• Derivative with only one parameter:

• Only changes parallel to the image gradient can change/reduce loss function

• Alternative: intensities extracted from larger patches

• Not usable for inter modality registration —> intesities may not be comparable —> use statistical loss functions NCC

• 2D version of Pearson correlation coefficient

• Assumes pixel intensities as random variables —> 1 if I and J are linearly correlared

• Kullback-Leibler divergence measures distance between 2 distributions

• 0 if p(I,J) =p(I)p(J) —> independent intensities —> maximize

• Powerful, capzture complicated relationships, no ripples for large angles in rotation example

• Difficult to compute, NCC easier

• Diffiult in general —> Regularization for nonlinear transformations:

• r prefers some sort of transformations over other ones

• l2: prefers smoothly vaying displacement fields

• L1: prefers piecewise constant u(x)

• Optmization via continuous methods:

• alpha: step size param, t iterations

• Gradient free techniques are also possible

• Sequential optimization: Linear registration followed by deformable registration

• Multi-scale optimization: avoid to stuck in local minima (niedrigere resolution —> immer höhere)