csci 6971: image registration lecture 27: fem-based methods april 23, 2004

CSci 6971: Image Registration Lecture 27: FEM-Based Methods

April 23, 2004

CSci 6971: Image Registration Lecture 27: FEM-Based Methods

April 23, 2004

Prof. Chuck Stewart, RPIDr. Luis Ibanez, KitwareProf. Chuck Stewart, RPIDr. Luis Ibanez, Kitware

Image Registration Lecture 27 2

Open Source DisclaimerOpen 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 (Insightful Corp.)

Of course, any errors… are mine.


Deformable RegistrationDeformable Registration

Finite Element Methods

for

Deformable Registration


Deformable TransformationDeformable Transformation

y

Fixed Image

Transform

x

y

Moving Image

x



y

Fixed Image

Transform

x

y

Moving Image

x



y

x


FEM GridFEM Grid

y

x

FEMGrid

ResamplingGrid


FEM GridFEM Grid

y

x

FEMGrid


FEM GridFEM Grid

y

x

FEMGrid

ComputedDeformation


FEM GridFEM Grid

y

x

FEMGrid

Forces Displacements


FEM Iterative Linear SystemFEM Iterative Linear System


Forces

Displacements

Regularization

F

U

K

Vector N

Vector N

Matrix NxN




F

U

K

FUK● =



N = Number of Nodes

FUK

N x N N N● =



Iteratively Solving a Linear System

FUK● =

Image based forces

Linearizationof a Physical

ModelNode

Displacements


FEM Energy FormulationFEM Energy Formulation

( ) similarity( ( ), ( ))

smoothness ( )

deformation ( )

t t tE u I x I x u

u

u

Find registration transformation that maximizes


FEM Energy FormulationFEM Energy Formulation

Benefits

Intuitive; easier to express constraints Powerful numerical solutions available Optimality of solutions; easier to debug

Limitations

Difficult / overhead to implement



( ) ( )h i iu x s

To solve the deformation, consider only displacements of the form



Linear Elements



( ) ( )h i iu x s

φ1 Element

α1

Shape Function



( ) ( )h i iu x s

φ2

Element

α2 Shape Function



( ) ( )h i iu x s

φ3

Element

α3

Shape Function



( ) ( )h i iu x s

φ3

Elementφ1

φ2

u

α2

α3

Shape Functions

α1



Higher Order Elements



( ) ( )h i iu x s

φ1 Element

α1

Shape Function



( ) ( )h i iu x s

φ4

Element

α4

Shape Function



( ) ( )h i iu x s

φ2

Element

α2 Shape Function



( ) ( )h i iu x s

φ5

Elementα5

Shape Function



( ) ( )h i iu x s

φ3

Element

α3

Shape Function



( ) ( )h i iu x s

φ6

Element

α6

Shape Function



( ) ( )h i iu x s

φ3

Elementφ1

φ2

u

α2

α3

Shape Functions

α1 α6α5

α4

φ5

φ6

φ4



0, 1, ,i

Ei n

Substitute uh into E, then minimizing with respect to αi:

( ) similarity( ( ), ( ))

smoothness ( )

deformation ( )

t t tE u I x I x u

u

u


BSplines Grid & Image GridBSplines Grid & Image Grid

Calculation

are made in

an Element

by Element

basis



Elements are connected at Nodes at which the displacement is solved



Efficiency is gained by elemental computation



Domain subdivision (Mesh) can be tailored to the underlying geometry of the image.


FEM SolverFEM Solver

Image MetricDerivative

Physical Assumptions

SolveNewSolution

Start Iteration Loop

End Iteration Loop

Begin Loop by making physical assumptions and then taking the derivative of the similarity metric.

End loop when the solution stabilizes.



Image MetricDerivative

Physical Assumptions

SolveNewSolution


End Iteration Loop F

U

K

UU

U

Old

New



F

KUUnew


If ( Unew – Uold) < ε then Stop

UU

U

Old

New

K FU● =


KU=F in CodeKU=F in Code

FEMSolver::AssembleF()

calls

FEMImageMetricLoad::Fe()

itk::FEMSolver::AssembleK()

FEMSolver::Solve()

FEMSolver ::

AddSolution()

itk::FEMRegistrationFilter::IterativeSolve()


FEM-Based Registration OptionsFEM-Based Registration Options

Element Type

Triangles Quadrilaterals Hexahedra Tetrahedra



Continuum / Physical Model

Linear elasticity Membrane Other specialized



Mesh geometry

Uniform grid vs. adaptive Anatomy-specific mesh



Metric

Mean square Normalized cross-correlation Mutual information Pattern intensity


ITK FEM LibraryITK FEM Library

Introduction to the ITK

Finite Element 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



Applications

Mechanical modeling Image registration


FEM BasicsFEM 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


ITK FEM ElementsITK 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


LoadsLoads

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


SolversSolvers

Provide functionality to obtain and process the solution

Different solution methods different solver classes Static problems Time dependent - dynamic problems


SolversSolvers

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


Setting Up a FEM ProblemSetting 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



FEM-Base Registration

Parameters


Parameter File : Part 1Parameter 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% -------------------------------0 1.0 % Similarity metric (0=mean sq, 1=ncc, 2=pattern int, 3=MI)1.0 % Alpha1 % DescentDirection2 % DoLineSearch (0=never, 1=always, 2=if needed)1.e1 % TimeStep1.e-15 % Energy Reduction Factor


Parameter File : Part 2Parameter File : Part 2

% ----------------------------------% Information about the image inputs% ----------------------------------2 % ImageDimension256 % Nx (image x dimension)256 % Ny (image y dimension)128 % Nz (image z dimension - not used if 2D)brain_slice1.mhd % ReferenceFileNamebrain_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.% -------------------------------------------------------------------0 % UseLandmarks?- % LandmarkFileNamebrain_result % ResultsFileName (prefix only)1 % WriteDisplacementField?brain_disp % DisplacementsFileName (prefix only)1 % ReadMeshFile?brain_mesh.fem % MeshFileName

END


Configuring Parameters #1Configuring 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);}



this->SetDescentDirectionMinimize();or

this->SetDescentDirectionMaximize();

this->DoLineSearch( n ); // n = 0, 1, 2 this->SetTimeStep( t );

this->SetEnergyReductionFactor( fbuf );



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;



FEM-Based Registration:

Writing the Code

../ Insight / Examples / Registration / DeformableRegistration1.cxx


Header DeclarationsHeader Declarations

#include "itkImageFileReader.h"

#include "itkImageFileWriter.h“

#include "itkFEM.h"

#include “itkFEMRegistrationFilter.h"


Type DefinitionsType 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;


Registering ObjectsRegistering 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 );


Input / OutputInput / Output

RegistrationType::Pointer registration = RegistrationType::New();

registration->SetConfigFileName( paramname );

registration->ReadConfigFile();


Material and Element SetupMaterial and Element Setup

// Create the material propertiesitk::fem::MaterialLinearElasticity::Pointer m;m = itk::fem::MaterialLinearElasticity::New();m->GN = 0;m->E = registration->GetElasticity();m->A = 1.0; // Cross-sectional aream->h = 1.0; // Thicknessm->I = 1.0; // Moment of inertiam->nu = 0.; // Poisson's ratiom->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 );


Running the RegistrationRunning the Registration

registration->RunRegistration();

registration->WriteWarpedImage();

if ( registration->GetWriteDisplacements() )

{

registration->WriteDisplacementField( 0 ); // x

registration->WriteDisplacementField( 1 ); // y

registration->WriteDisplacementFieldMultiComponent();

}


FEM - Deformable RegistrationFEM - Deformable Registration

Example #1


Fixed ImageFixed Image


Moving ImageMoving Image


Registered ImageRegistered Image



Example #2



Example #3



Example #4

Elasticity value was doubled


EndEnd

Enjoy ITK !

csci 6971: image registration lecture 27: fem-based methods april 23, 2004

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