Non-Rigid Registration

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Non-Rigid Registration
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Why Non-Rigid Registration

• In many applications a rigid transformation is
  sufficient. (Brain)
• Other applications
• Intra-subject tissue deformation
• Inter-subject anatomical variability across
  individuals
• Fast-Moving area Non-rigid

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Registration Framework

• In terms of L.Brown.(1992)
• Feature Space
• Transformation
• Similarity Measure
• Search Strategy (Optimization)
• Rigid vs. Non-rigid in the framework

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Feature Space

• Geometric landmarks
• Points
• Edges
• Contours
• Surfaces, etc.
• Intensities
• Raw pixel values

1. 35
2. 56

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Transformation
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Transformation

• Rigid transformation
• 3DOF (2D)
• 6 DOF (3D)
• Affine transformation
• 12 DOF

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Transformation

• Additional DOF.
• Second order polynomial-30 DOF
• Higher order
• third-60, fourth-105,fifth-168
• Model only global shape changes

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Transformation

• For each pixel (voxel), one 2d(3d) vector to
  describe local deformation.
• Parameters of non-rigid gtgt that of rigid

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Similarity Measure

• Point based
• ---The distance between features, such as
  points,curves,or surfaces of corresponding
  anatomical structure.
• --- Feature extraction.
• Voxel based
• ---Absolute Difference, Sum of squared
  differences, Cross correlation, or Mutual
  information

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Search Strategy

• Registration can be formulated as an optimization
  problem whose goal is to minimize an associated
  energy or cost function.
• General form of cost function
• C -CsimilarityCdeformation

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Search Strategy

• Powells direction set method
• Downhill simplex method
• Dynamic programming
• Relaxation matching
• Combined with
• Multi-resolution techniques

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Registration Scheme
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Non-rigid Registration

• Feature-based
• Control Points TPS
• Curve/Edge/Contour
• Surface
• Intensity-based
• Elastic model
• Viscous fluid model
• Others

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Thin-plate splines (TPS)

• Come from Physics TPS has the property of
  minimizing the bending energy.

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TPS (cont.)

• Splines based on radial basis functions
• Surface interpolation of scattered data

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Description of the Approach

1. Select the control points in the images.
2. Calculate the coefficients for the TPS.
3. Apply the TPS transformation on the whole image.

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Synthetic Images
T2
T1
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TPS-Results(1)
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TPS-Results(2)
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Rigid and non-rigid registration

• Rigid Registration as pre-processing (global
  alignment)
• Non-rigid registration for local alignment

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Next time

• Affine-mapping technique