1. ITK Deformable Registration Finite Elements Methods

2. Deformable Registration Finite Element Methodsfor Deformable Registration

3. Deformable Transformation y y Transform x x Fixed Image Moving Image

4. Deformable Transformation y y Transform x x Fixed Image Moving Image

5. Deformable Transformation y x

6. Deformable Transformation y x

7. FEM Grid y FEM Grid Resampling Grid x

8. FEM Grid y FEM Grid x

9. FEM Grid y FEM Grid Computed Deformation x

10. FEM Grid y FEM Grid Displacements Forces x

11. FEM Iterative Linear System Finite Element Methods F Vector N Forces U Vector N Displacements K Matrix NxN Regularization

12. FEM Iterative Linear System Finite Element Methods F K U F U = ● K

13. FEM Iterative Linear System N = Number of Nodes N x N N N = ● U F K

14. FEM Iterative Linear System Iteratively Solving a Linear System K U F = ● Linearization of a Physical Model Image based forces Node Displacements

15. FEM Energy Formulation Find registration transformation that maximizes

16. FEM Energy Formulation Benefits • Intuitive; easier to express constraints • Powerful numerical solutions available • Optimality of solutions; easier to debug Limitations • Difficult / overhead to implement

17. Deformable Registration To solve the deformation, consider only displacements of the form

18. Deformable Registration Linear Elements

19. Deformable Registration α1 Element φ1 Shape Function

20. Deformable Registration Element φ2 α2 Shape Function

21. Deformable Registration α3 φ3 Element Shape Function

22. Deformable Registration u α3 φ3 Element α1 φ1 φ2 α2 Shape Functions

23. Deformable Registration Higher Order Elements

24. Deformable Registration α1 Element φ1 Shape Function

25. Deformable Registration α4 φ4 Element Shape Function

26. Deformable Registration Element φ2 α2 Shape Function

27. Deformable Registration φ5 α5 Element Shape Function

28. Deformable Registration α3 φ3 Element Shape Function

29. Deformable Registration φ6 α6 Element Shape Function

30. Deformable Registration α4 φ4 u α3 φ3 Element φ6 α1 α6 φ1 φ5 α5 φ2 α2 Shape Functions

31. Deformable Registration Substitute uhinto E, then minimizing with respect to αi:

32. BSplines Grid & Image Grid Calculation are made in an Element by Element basis

33. BSplines Grid & Image Grid Elements are connected at Nodes at which the displacement is solved

34. BSplines Grid & Image Grid Efficiency is gained by elemental computation

35. BSplines Grid & Image Grid Domain subdivision (Mesh) can be tailored to the underlying geometry of the image.

36. FEM Solver Start Iteration Loop • Begin Loop by making physical assumptions and then taking the derivative of the similarity metric. • End loop when the solution stabilizes. Physical Assumptions New Solution Solve Image Metric Derivative End Iteration Loop

37. FEM Solver Start Iteration Loop K Physical Assumptions Solve New Solution U Image Metric Derivative F End Iteration Loop

38. K U F = ● FEM Solver Start Iteration Loop K U Unew F If ( Unew – Uold) < ε then Stop

39. KU=F in Code itk::FEMRegistrationFilter::IterativeSolve() itk::FEMSolver::AssembleK() FEMSolver :: AddSolution() FEMSolver:: Solve() FEMSolver::AssembleF() calls FEMImageMetricLoad::Fe()

40. FEM-Based Registration Options Element Type • Triangles • Quadrilaterals • Hexahedra • Tetrahedra

41. FEM-Based Registration Options Continuum / Physical Model • Linear elasticity • Membrane • Other specialized

42. FEM-Based Registration Options Mesh geometry • Uniform grid vs. adaptive • Anatomy-specific mesh

43. FEM-Based Registration Options Metric • Mean square • Normalized cross-correlation • Mutual information • Pattern intensity

44. ITK FEM Library Introduction to the ITK Finite Element Library

45. ITK FEM Library Library for solving general FEM problems • Object oriented • C++ classes are used to • specify the geometry and behavior of the elements • apply external forces and boundary conditions • solve problem and post-process the results

46. ITK FEM Library Applications • Mechanical modeling • Image registration

47. FEM Basics • Mesh • Nodes • Points in space where solutions are obtained • Elements • e.g., 2-D triangular elements • Loads • e.g., gravity (body) load • Boundary conditions • e.g., nodes fixed in space

48. ITK FEM Elements • Core of the library is the Element class • Code is in two functionally independent parts • Geometry and Physics • Arbitrarily combined to create new elements • Problem domain is specified by a mesh Geometry Physics

49. Loads Classes that apply external forces (loads) to elements • Various types • Easily extensible

50. Solvers • Provide functionality to obtain and process the solution • Different solution methods  different solver classes • Static problems • Time dependent - dynamic problems