1 Introduction Motivation and Aims Functional Imaging Computational Neuroanatomy... 12

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1 Contents 1 Introduction Motivation and Aims Functional Imaging Computational Neuroanatomy Overview of Chapters Rigid Body Registration Introduction Affine Transformations Parameterising a Rigid Body Transformation Working with Volumes of Differing or Anisotropic Voxel Sizes Left- and Right-handed Co-ordinate Systems Resampling Images Optimisation Within Modality Image Registration Methods Residual Artifacts from PET and fmri Between Modality Image Registration Methods Evaluation Image Warping using Basis Functions Introduction Methods A Maximum A Posteriori Solution Affine Registration Nonlinear Registration Linear Regularisation for Nonlinear Registration Templates and Intensity Transformations Evaluation Evaluation of the MAP Scheme for Affine Registration Comparing Spatial Normalisation both With and Without Nonlinear Deformations

2 CONTENTS Discussion High-Dimensional Image Warping Introduction Methods Bayesian Framework Likelihood Potentials Prior Potentials - 2D Prior Potentials - 3D The Optimisation Algorithm Inverting a Deformation Field Examples Two Dimensional Warping Using Simulated Data Registering Pairs of Images Registering to an Average Discussion Parameterising the Deformations The Matching Criterion The Priors The Optimisation Algorithm Image Segmentation Introduction Methods Estimating the Cluster Parameters Assigning Belonging Probabilities Estimating and Applying the Modulation Function Evaluation Stability With Respect to Misregistration with the Prior Probability Images Discussion Morphometry Introduction Multivariate Analysis of Covariance Canonical Correlation Analysis Voxel-Based Morphometry Methods Evaluations Deformation Based Morphometry Methods Results Tensor-Based Morphometry Theory...135

3 CONTENTS Data for Evaluations Morphometry on Jacobian Determinants Morphometry on Strain Tensors Discussion Discussion Original Contributions Modularity Hyper-parameter estimation References 157

4 List of Figures 1.1 Deformation- and tensor-based morphometry Left- and right-handed co-ordinate systems Image interpolation in two dimensions Sinc function in two dimensions The optimisation can be thought of as fitting a series of quadratics Example of registered PET and MRI Example of registered T1 and T2 weighted images Illustration of Bayes rule Different boundary conditions Discrete Cosine Transform basis functions Deformation fields consist of a linear combination of basis functions The fast algorithm Example template images Two dimensional histograms of template images Simulated images of T1, T2 and PD weighted images Average χ 2 for the images plotted against iteration number Number of iterations in which convergence to within 1% is reached Parameter estimates from reduced data versus those from complete data Means and standard deviations of spatially normalised images Illustration of the effect of regularisation Triangular mesh used for 2D registration Probability density functions Tetrahedral mesh used for 3D registration A comparison of the different cost functions The six triangles whos Jacobian matrices are influenced by the central point C code for computing the rate of change of the prior potential An illustration of how voxels are located within a tetrahedron Demonstration using simulated data Demonstration of the reversibility of the deformations Example of 2D registration of brain images Example of 3D registration of brain images

5 LIST OF FIGURES Deformation fields from 3D registration Symmetry of 3D deformation fields Mean and standard deviation images Affine registered images Basis function registered images High-dimensionally registered images Rendered brain surfaces showing equivalent locations Mappings obtained by combining warps A comparison of a symmetric with an asymmetric likelihood potential Rotation and translations of a region, keeping surrounding points stationary A shear has singular values not equal to one Prior probability images The segmentation model Aflow diagram for the tissue classification Algorithm for computing A kt A k and A kt b k in two dimensions Randomly generated modulation fields Example segmentation of real data Classification of the simulated BrainWeb image Recovery of modulation field Segmentation accuracy with respect to misregistration Effects of partial volume on the intensity histograms Example of automatically cleaned up segmented images Canonical correlation analysis using simulated data Histogram of correlation coefficients Histograms of t-scores from randomly generated tests Frequency of false positives Template and weighting images Means of spatially normalised images for each group Separation of subjects using canonical correlation analysis Caricatured shape differences Average shape differences Warping of same subject brain during Alzheimers disease progression Volume changes during progression of Alzheimers disease Polar decomposition Mean of 58 warped hippocampus images Randomly chosen warped images of hippocampi Jacobian determinant fields Histogram of correlation coefficients Histograms of t-scores from randomly generated tests More histograms of t-scores from randomly generated tests

6 LIST OF FIGURES Histogram of Q-Q plot correlation coefficients taken over tensor fields Histograms of Wilk's Λ statistics

7 List of Tables 2.1 Errors for PET-MRI registration Errors for PET-MRI registration from other methods » statistics computed from classified simulated images