1. Non-Rigid Registration

2. 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

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

4. Feature Space  Geometric landmarks: Points Edges Contours Surfaces, etc.  Intensities: Raw pixel values 2335 2456

5. Transformation

6.  Rigid transformation: 3DOF (2D) 6 DOF (3D)  Affine transformation: 12 DOF

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

8. Transformation  For each pixel (voxel), one 2d(3d) vector to describe local deformation.  Parameters of non-rigid >> that of rigid

9. 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

10. 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 = -C similarity +C deformation

11. Search Strategy  Powell’s direction set method  Downhill simplex method  Dynamic programming  Relaxation matching Combined with  Multi-resolution techniques

12. Registration Scheme

13. Non-rigid Registration  Feature-based –Control Points: TPS –Curve/Edge/Contour –Surface  Intensity-based –Elastic model –Viscous fluid model –Others

14. Thin-plate splines (TPS)  Come from Physics: TPS has the property of minimizing the bending energy.

15. TPS (cont.)  Splines based on radial basis functions  Surface interpolation of scattered data

16. Description of the Approach  Select the control points in the images.  Calculate the coefficients for the TPS.  Apply the TPS transformation on the whole image.

17. Synthetic Images T1 T2

18. TPS-Results(1)

19. TPS-Results(2)

20. Rigid and non-rigid registration  Rigid Registration as pre-processing (global alignment)  Non-rigid registration for local alignment

21. Next time  Affine-mapping technique