Contents -- Preface -- 1. Statistical Preliminary -- 1.1 General Linear Models -- 1.2 Random Fields -- 1.2.1 Covariance Functions -- 1.2.2 Gaussian Random Fields -- 1.2.3 Differentiation and Integration of Fields -- 1.2.4 Statistical Inference on Fields -- 1.3 Multiple Comparisons -- 1.3.1 Bonferroni Correction -- 1.3.2 Random Fields Theory -- 1.3.3 Poisson Clumping Heuristic -- 1.3.4 Euler Characteristic Method -- 1.3.5 Intrinsic Volume -- 1.3.6 Euler Characteristic Density -- 1.4 Statistical Power Analysis -- 1.4.1 Statistical Power at a Voxel -- 1.4.2 Statistical Power under Multiple Comparisons -- 2. Deformation-Based Morphometry -- 2.1 Image Registration -- 2.2 Deformation-Based Morphometry -- 2.3 Displacement Vector Fields -- 2.3.1 Dynamic Model on Displacement -- 2.3.2 Local Inference via Hotelling's T2-Field -- 2.3.3 Detecting Local Brain Growth -- 2.4 Global Inference via Integral Statistic -- 2.4.1 Karhunen-Lo eve Expansion -- 2.4.2 Mercer's Theorem -- 2.4.3 Integral Statistic on Displacement -- 3. Tensor-Based Morphometry -- 3.1 Jacobian Determinant -- 3.2 Distributional Assumptions -- 3.3 Local Volume Changes -- 3.4 Longitudinal Modeling -- 3.4.1 Normal Brain Development in Children -- 3.5 Global Inference via Divergence Theorem -- 3.6 Second Order Tensor Fields -- 3.6.1 Membrane Spline Energy -- 3.6.2 Vorticity Tensor Fields -- 3.6.3 Generalized Variance Field -- 4. Voxel-Based Morphometry -- 4.1 Image Segmentation -- 4.1.1 Mumford-Shah Model -- 4.1.2 Level Sets -- 4.1.3 Active Contours -- 4.1.4 Deformable Surface Models -- 4.1.5 Thin-Plate Spline Thresholding -- 4.2 Mixture Models -- 4.2.1 Bayesian Segmentation -- 4.2.2 Mixture Models -- 4.2.3 Expectation Maximization Algorithm -- 4.2.4 Two Components Gaussian Mixtures -- 4.3 Voxel-Based Morphometry -- 4.3.1 ROI Volume Estimation in VBM -- 4.3.2 Limitations of Witelson Partition.

4.3.3 General Linear Models on Tissue Densities -- 4.3.4 2D VBM Applied to Corpus Callosum -- 5. Geometry of Cortical Manifolds -- 5.1 Surface Parameterization -- 5.1.1 B-Spline Parameterization -- 5.1.2 B-Spline Curves -- 5.1.3 Quadratic Parameterization -- 5.1.4 Fourier Descriptors -- 5.2 Surface Normals and Curvatures -- 5.2.1 Surface Normals -- 5.2.2 Gaussian and Mean Curvatures -- 5.2.3 Curvatures of Polynomial Surfaces -- 5.3 Laplace-Beltrami Operator -- 5.3.1 Eigenfunctions of Laplace-Beltrami Operator -- 5.3.2 Multiplicity of Eigenfunctions -- 5.3.3 Laplace-Beltrami Shape Descriptors -- 5.3.4 Second Eigenfunctions -- 5.3.5 Dirichlet Energy -- 5.3.6 Fiedler's Vector -- 5.4 Finite Element Methods -- 5.4.1 Pieacewise Linear Functions -- 5.4.2 Mass and Stiffness Matrices -- 6. Smoothing on Cortical Manifolds -- 6.1 Gaussian Kernel Smoothing -- 6.1.1 Isotropic Gaussian Kernel -- 6.1.2 Anisotropic Gaussian Kernel -- 6.2 Diffusion Smoothing -- 6.2.1 Diffusion in Euclidean Space -- 6.2.2 Diffusion in 1D -- 6.2.3 Diffusion on Triangular Mesh -- 6.2.4 Finite Difference Scheme -- 6.3 Heat Kernel Smoothing -- 6.3.1 Heat Kernel -- 6.3.2 Heat Kernel Smoothing -- 6.3.3 Iterated Kernel Smoothing -- 6.3.4 Smoothing via Laplace-Beltrami Eigenfunctions -- 6.4 Smoothness of Random Fields -- 6.4.1 Resels of Field -- 6.4.2 Effective Bandwidth -- 6.4.3 Unbiased Estimator of eFWHM -- 6.5 Gaussianness of Random Fields -- 6.5.1 Quantiles -- 6.5.2 Empirical Distribution -- 6.5.3 Quantile Quantile Plots -- 6.5.4 Checking Gaussianness in Cortical Thickness -- 7. Surface-Based Morphometry -- 7.1 Surface Flattening -- 7.2 Cortical Thickness -- 7.2.1 Cortical Thickness via Laplace Equation -- 7.2.2 Cortical Thickness vs. Gray Matter Density -- 7.2.3 Distance Map -- 7.3 Partial Correlation Mapping -- 7.3.1 Partial Correlations -- 7.3.2 Statistical Inference on Correlations.

7.3.3 Brain-Behavior Correlations -- 7.3.4 Facial Emotion Discrimination Tasks -- 7.4 Tensor-Based Surface Morphometry -- 7.4.1 Surface Deformation -- 7.4.2 Metric Tensor Computation on Surfaces -- 7.4.3 Statistical Inference on Surfaces -- 7.4.4 Quantifying Brain Growth -- 7.4.5 Tensor Computation via SPHARM -- 7.5 Multivariate General Linear Models -- 7.5.1 Roy's Maximum Root -- 7.5.2 SurfStat -- 7.6 Mixed Effect Models on Surface Shape Change -- 7.6.1 Longitudinal Imaging Data -- 7.6.2 Mixed Effect Models -- 7.6.3 Restricted Maximum Likelihood Estimation -- 7.6.4 Longitudinal Hippocampus Shape Model -- 7.6.5 Functional Mixed Effect Models -- 7.7 Sparse Surface Shape Recovery -- 7.7.1 Sparse Regression on Surface Data -- 7.7.2 Effect of Aging on Hippocampus Shape -- 8. Weighted Fourier Representation -- 8.1 Fourier Series in Hilbert Space -- 8.2 Weighted Fourier Representation -- 8.2.1 Cauchy Problem -- 8.2.2 Heat Kernel Smoothing -- 8.2.3 Kernel Regression -- 8.2.4 Iterative Residual Fitting Algorithm -- 8.2.5 Best Model Selection -- 8.3 Weighted Spherical Harmonic Representation -- 8.3.1 Spherical Harmonics -- 8.3.2 Spherical Harmonic Representation -- 8.3.3 Iterative Residual Fitting on Spherical Harmonics -- 8.4 Gibbs Phenomenon -- 8.4.1 Reduction of Gibbs Phenomenon -- 8.4.2 The Overshoot of Gibbs Phenomenon -- 8.5 SPHARM Correspondance -- 8.6 Cortical Asymmetry -- 8.6.1 Hemisphere Correspondence -- 8.6.2 Abnormal Cortical Asymmetry in Autism -- 8.6.3 FWHM of Heat Kernel -- 8.7 Logistic Discriminant Analysis on Cortical Surface -- 8.7.1 Logistic Model -- 8.7.2 Maximum Likelihood Estimation -- 8.7.3 Best Model Selection -- 8.7.4 Classification Accuracy -- 8.8 Tiling Surfaces with Orthonormal Basis -- 8.8.1 Orhonormal Basis on a Sphere -- 8.8.2 Orthonormal Basis on Manifolds -- 8.8.3 Numerical Implementation -- 8.8.4 Pullback Representation.

8.9 Basis Function Expansion on Multiple Shells -- 8.9.1 Eigenfunction Expansion in a Solid Ball -- 8.9.2 Iterative Residual Fitting -- 8.9.3 3D Resampling of 2D Surface Data -- 9. Structural Brain Connectivity -- 9.1 White Matter Fiber Tractography -- 9.1.1 Diffusion Tensors -- 9.1.2 Streamlines -- 9.1.3 Probabilistic Methods -- 9.2 Probabilistic Connectivity -- 9.3 Cosine Series Representation of Fiber Tracts -- 9.3.1 Cosine Basis in a Unit Interval -- 9.3.2 Cosine Series Representation of 3D Curves -- 9.3.3 Optimal Degree Selection -- 9.3.4 Distance Between Tracts -- 9.3.5 Tract Registration -- 9.3.6 Limitation of Cosine Series Representation -- 9.4 Parcellation-Free Brain Networks -- 9.4.1 Why Parcellation Free? -- 9.4.2 Epsilon Neighbor Networks -- 9.4.3 Connected Components -- 9.4.4 Epsilon Filtration -- 9.4.5 Electrical Circuit Model for Fiber Tracts -- 9.5 Structural Brain Connectivity without DTI -- 9.5.1 Correlating Jacobian Determinants -- 9.5.2 Seed-Based Connectivity -- 9.5.3 Parcellation-Based Connectivity -- 9.5.4 Validation -- 9.5.5 RV-Coefficient -- 9.6 Network Complexity Measures -- 9.6.1 Degree Distribution -- 9.6.2 Small-Worldness -- 9.6.3 Fractal Dimension -- 9.6.4 Clustering Coefficient -- 9.7 Sparse Brain Network Models -- 9.7.1 Correlation Thresholding -- 9.7.2 Sparse Partial Correlation -- 9.7.3 Sparse Network Recovery -- 9.8 Dynamic Network Modeling -- 10. Topological Data Analysis -- 10.1 Detecting Topological Defect in Images -- 10.2 Expected Euler Characteristic -- 10.3 Rips Complex -- 10.3.1 Topology -- 10.3.2 Simplex -- 10.3.3 Rips complex -- 10.4 Persistence Diagrams -- 10.4.1 Morse Functions -- 10.4.2 Persistence Diagrams -- 10.4.3 Persistence Diagram for Cortical Thickness -- 10.4.4 Inference on Persistent Diagrams -- 10.5 Min-Max Diagrams -- 10.5.1 Why Critical Values?.

10.5.2 Iterative Pairing and Deletion Algorithm -- 10.5.3 Statistical Inference on Mix-Max Diagrams -- 10.6 Graph Filtrations -- 10.6.1 Weighted Graphs -- 10.6.2 Single Linkage Matrix -- 10.6.3 Persistent Brain Networks -- Bibliography -- Index.