1. ITK Deformable Registration
Finite Elements
Methods

2. Open Source Disclaimer
Many of the slides in this talk were taken from the ITK course presented at MICCAI 2003 by
Dr. James Gee (U. Penn) Brian Avants (U. Penn) Tessa Sundaram (U. Penn) Dr. Lydia Ng (AllenInstitue)
Of course, any errors… are mine.

3. Deformable Registration
Finite Element Methods for Deformable Registration

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

5. Deformable Transformationy y
Transform x x
Fixed Image
Moving Image

6. Deformable Transformationy x

7. Deformable Transformationy x

8. FEM Grid y
FEM
Grid
Resampling
Grid x

9. FEM Grid y
FEM
Grid x

10. FEM Grid y
FEM
Grid
Computed
Deformation x

11. FEM Grid y
FEM
Grid
Displacements
Forces x

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

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

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

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

16. FEM Energy Formulation
Find registration transformation that maximizes

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

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

19. Deformable Registration
Linear Elements

20. Deformable Registrationα1 φ1
Element
Shape Function

21. Deformable Registration
Element α2 φ2
Shape Function

22. Deformable Registrationα3 φ3
Element
Shape Function

23. Deformable Registrationu α3 φ3 α1 φ1
Element α2 φ2
Shape Functions

24. Deformable Registration
Higher Order Elements

25. Deformable Registrationα1 φ1
Element
Shape Function

26. Deformable Registrationα4 φ4
Element
Shape Function

27. Deformable Registration
Element α2 φ2
Shape Function

28. Deformable Registrationφ5 α5
Element
Shape Function

29. Deformable Registrationα3 φ3
Element
Shape Function

30. Deformable Registrationφ6 α6
Element
Shape Function

31. Deformable Registrationα4 φ4 u α3 φ3 α1 φ1 α6 φ6
Element α5 φ5 α2 φ2
Shape Functions

32. Deformable Registration
Substitute uh into E, then minimizing with respect to αi:

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

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

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

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

37. 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

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

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

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

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

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

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

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

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

46. 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

47. ITK FEM Library
Applications
Mechanical modeling
Image registration

48. FEM Basics Mesh Loads Boundary conditions Nodes Elements
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

49. 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

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

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

52. Solvers
Use linear system wrappers to link FEM classes to an external numeric library
Any numeric library can be used to solve the systems of linear equations in FEM problems
VNL and ITPACK currently supported

53. Setting Up a FEM Problem
Four-step process
Select element classes
Discretize problem domain
Specify boundary conditions
Specify/Apply external loads
Two options
Directly  create proper objects in code
Indirectly  read object definitions from a file

54. Deformable Registration
FEM-Base Registration Parameters

55. Parameter File : Part 1 %
% Parameters for the single- or multi-resolution techniques
1 % Number of levels in the multi-resolution pyramid (1 = single-res)
1 % Highest level to use in the pyramid
1 1 % Scaling at lowest level for each image dimension
8 % Number of pixels per element
1.e5 % Elasticity (E)
1.e4 % Density (RhoC)
1. % Image energy scaling
4 % NumberOfIntegrationPoints
1 % WidthOfMetricRegion
25 % MaximumIterations %
% Parameters for the registration
% Similarity metric (0=mean sq, 1=ncc, 2=pattern int, 3=MI)
% Alpha
% DescentDirection
% DoLineSearch (0=never, 1=always, 2=if needed)
1.e1 % TimeStep
1.e-15 % Energy Reduction Factor

56. Parameter File : Part 2 % ----------------------------------
% Information about the image inputs
% ImageDimension
% Nx (image x dimension)
% Ny (image y dimension)
% Nz (image z dimension - not used if 2D)
brain_slice1.mhd % ReferenceFileName
brain_slice1warp.mhd % TargetFileName %
% The actions below depend on the values of the flags preceding them.
% For example, to write out the displacement fields, you have to set
% the value of WriteDisplacementField to 1.
% UseLandmarks?
% LandmarkFileName
brain_result % ResultsFileName (prefix only)
% WriteDisplacementField?
brain_disp % DisplacementsFileName (prefix only)
% ReadMeshFile?
brain_mesh.fem % MeshFileName
END

57. Configuring Parameters #1
this->DoMultiRes(true);
this->m_NumLevels = nlev;
this->m_MaxLevel = mlev;
for (jj=0; jj < ImageDimension; jj++) {
m_ImageScaling[jj] = dim; }
for (jj=0; jj < this->m_NumLevels; jj++) {
this->m_MeshPixelsPerElementAtEachResolution(jj) = p;
this->SetElasticity(e, jj);
this->SetRho(p, jj);
this->SetGamma(g, jj);
this->SetNumberOfIntegrationPoints(ip, jj);
this->SetWidthOfMetricRegion(w, jj);
this->SetMaximumIterations(mit, jj);

58. Configuring Parameters #2
this->SetDescentDirectionMinimize(); or
this->SetDescentDirectionMaximize();
this->DoLineSearch( n ); // n = 0, 1, 2
this->SetTimeStep( t );
this->SetEnergyReductionFactor( fbuf );

59. Configuring Parameters #3
this->m_ImageSize[0] = xdim;
this->m_ImageSize[1] = ydim;
if (dim == 3) this->m_ImageSize[2] = zdim;
this->SetReferenceFile( imgfile1 );
this->SetTargetFile( imgfile2 );
this->UseLandmarks( true );
this->SetLandmarkFile( lmfile );
this->SetResultsFile( resfile );
this->SetWriteDisplacements( true );
this->SetDisplacementsFile( dispfile );
this->m_ReadMeshFile = true;
this->m_MeshFileName = meshfile;

60. Deformable Registration
FEM-Based Registration: Writing the Code
../ Insight / Examples / Registration / DeformableRegistration1.cxx

61. Header Declarations #include "itkImageFileReader.h"
#include "itkImageFileWriter.h“
#include "itkFEM.h"
#include “itkFEMRegistrationFilter.h"

62. Type Definitions
typedef itk::Image< unsigned char, 2 > fileImageType;
typedef itk::Image< float, 2 > ImageType;
typedef itk::fem::Element2DC0LinearQuadrilateralMembrane ElementType;
typedef itk::fem::Element2DC0LinearTriangularMembrane ElementType2;
typedef itk::fem::FEMRegistrationFilter< ImageType, ImageType > RegistrationType;

63. Registering Objects
ElementType::LoadImplementationFunctionPointer fp1 =
& itk::fem::ImageMetricLoadImplementation< ImageLoadType >::ImplementImageMetricLoad;
DispatcherType::RegisterVisitor( (ImageLoadType*)0 , fp1 );
ElementType2::LoadImplementationFunctionPointer fp2 =
& itk::fem::ImageMetricLoadImplementation< ImageLoadType >::ImplementImageMetricLoad;
DispatcherType2::RegisterVisitor( (ImageLoadType*)0 , fp2 );

64. Input / Output
RegistrationType::Pointer registration = RegistrationType::New();
registration->SetConfigFileName( paramname );
registration->ReadConfigFile();

65. Material and Element Setup
// Create the material properties
itk::fem::MaterialLinearElasticity::Pointer m;
m = itk::fem::MaterialLinearElasticity::New();
m->GN = 0;
m->E = registration->GetElasticity();
m->A = 1.0; // Cross-sectional area
m->h = 1.0; // Thickness
m->I = 1.0; // Moment of inertia
m->nu = 0.; // Poisson's ratio
m->RhoC = 1.0; // Density
// Create the element type
ElementType::Pointer e1 = ElementType::New();
e1->m_mat=
dynamic_cast< itk::fem::MaterialLinearElasticity* >( m );
registration->SetElement( e1 );
registration->SetMaterial( m );

66. Running the Registration
registration->RunRegistration();
registration->WriteWarpedImage();
if ( registration->GetWriteDisplacements() ) {
registration->WriteDisplacementField( 0 ); // x
registration->WriteDisplacementField( 1 ); // y
registration->WriteDisplacementFieldMultiComponent(); }

67. FEM - Deformable Registration
Example #1

68. Fixed Image

69. Moving Image

70. Registered Image

71. Registered Image

72. FEM - Deformable Registration
Example #2

73. Fixed Image

74. Moving Image

75. Registered Image

76. Registered Image

77. FEM - Deformable Registration
Example #3

78. Fixed Image

79. Moving Image

80. Registered Image

81. Registered Image

82. FEM - Deformable Registration
Example #4 Elasticity value was doubled

83. Fixed Image

84. Moving Image

85. Registered Image

86. Registered Image

87. Enjoy ITK !