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CSci 6971: Image Registration Lecture 27: FEM-Based Methods April 23, 2004

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## CSci 6971: Image Registration Lecture 27: FEM-Based Methods April 23, 2004

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**CSci 6971: Image Registration Lecture 27: FEM-Based**MethodsApril 23, 2004 Prof. Chuck Stewart, RPI Dr. Luis Ibanez, Kitware**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 (Insightful Corp.) Of course, any errors… are mine. Lecture 27**Deformable Registration**Finite Element Methodsfor Deformable Registration Lecture 27**Deformable Transformation**y y Transform x x Fixed Image Moving Image Lecture 27**Deformable Transformation**y y Transform x x Fixed Image Moving Image Lecture 27**Deformable Transformation**y x Lecture 27**Deformable Transformation**y x Lecture 27**FEM Grid**y FEM Grid Resampling Grid x Lecture 27**FEM Grid**y FEM Grid x Lecture 27**FEM Grid**y FEM Grid Computed Deformation x Lecture 27**FEM Grid**y FEM Grid Displacements Forces x Lecture 27**FEM Iterative Linear System**Finite Element Methods F Vector N Forces U Vector N Displacements K Matrix NxN Regularization Lecture 27**FEM Iterative Linear System**Finite Element Methods F K U F U = ● K Lecture 27**FEM Iterative Linear System**N = Number of Nodes N x N N N = ● U F K Lecture 27**FEM Iterative Linear System**Iteratively Solving a Linear System K U F = ● Linearization of a Physical Model Image based forces Node Displacements Lecture 27**FEM Energy Formulation**Find registration transformation that maximizes Lecture 27**FEM Energy Formulation**Benefits • Intuitive; easier to express constraints • Powerful numerical solutions available • Optimality of solutions; easier to debug Limitations • Difficult / overhead to implement Lecture 27**Deformable Registration**To solve the deformation, consider only displacements of the form Lecture 27**Deformable Registration**Linear Elements Lecture 27**Deformable Registration**α1 Element φ1 Shape Function Lecture 27**Deformable Registration**Element φ2 α2 Shape Function Lecture 27**Deformable Registration**α3 φ3 Element Shape Function Lecture 27**Deformable Registration**u α3 φ3 Element α1 φ1 φ2 α2 Shape Functions Lecture 27**Deformable Registration**Higher Order Elements Lecture 27**Deformable Registration**α1 Element φ1 Shape Function Lecture 27**Deformable Registration**α4 φ4 Element Shape Function Lecture 27**Deformable Registration**Element φ2 α2 Shape Function Lecture 27**Deformable Registration**φ5 α5 Element Shape Function Lecture 27**Deformable Registration**α3 φ3 Element Shape Function Lecture 27**Deformable Registration**φ6 α6 Element Shape Function Lecture 27**Deformable Registration**α4 φ4 u α3 φ3 Element φ6 α1 α6 φ1 φ5 α5 φ2 α2 Shape Functions Lecture 27**Deformable Registration**Substitute uhinto E, then minimizing with respect to αi: Lecture 27**BSplines Grid & Image Grid**Calculation are made in an Element by Element basis Lecture 27**BSplines Grid & Image Grid**Elements are connected at Nodes at which the displacement is solved Lecture 27**BSplines Grid & Image Grid**Efficiency is gained by elemental computation Lecture 27**BSplines Grid & Image Grid**Domain subdivision (Mesh) can be tailored to the underlying geometry of the image. Lecture 27**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 Lecture 27**FEM Solver**Start Iteration Loop K Physical Assumptions Solve New Solution U Image Metric Derivative F End Iteration Loop Lecture 27**K**U F = ● FEM Solver Start Iteration Loop K U Unew F If ( Unew – Uold) < ε then Stop Lecture 27**KU=F in Code**itk::FEMRegistrationFilter::IterativeSolve() itk::FEMSolver::AssembleK() FEMSolver :: AddSolution() FEMSolver:: Solve() FEMSolver::AssembleF() calls FEMImageMetricLoad::Fe() Lecture 27**FEM-Based Registration Options**Element Type • Triangles • Quadrilaterals • Hexahedra • Tetrahedra Lecture 27**FEM-Based Registration Options**Continuum / Physical Model • Linear elasticity • Membrane • Other specialized Lecture 27**FEM-Based Registration Options**Mesh geometry • Uniform grid vs. adaptive • Anatomy-specific mesh Lecture 27**FEM-Based Registration Options**Metric • Mean square • Normalized cross-correlation • Mutual information • Pattern intensity Lecture 27**ITK FEM Library**Introduction to the ITK Finite Element Library Lecture 27**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 Lecture 27**ITK FEM Library**Applications • Mechanical modeling • Image registration Lecture 27**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 Lecture 27**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 Lecture 27**Loads**Classes that apply external forces (loads) to elements • Various types • Easily extensible Lecture 27