Riemannian Geometric Statistics in Medical Image Analysis (The Elsevier and Miccai Society)

Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods. Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. The foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology Content includes: The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs Applications of statistics on manifolds and shape spaces in medical image computing Diffeomorphic deformations and their applications As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science. A complete reference covering both the foundations and state-of-the-art methods Edited and authored by leading researchers in the field Contains theory, examples, applications, and algorithms Gives an overview of current research challenges and future applications Cover 1 RIEMANNIAN GEOMETRIC STATISTICS IN MEDICAL IMAGE ANALYSIS 5 Copyright 6 Contents 7 Contributors 14 Introduction 17 Introduction 17 Contents 17 Part 1: Foundations of geometric statistics 21 1 Introduction to differential and Riemannian geometry 23 1.1 Introduction 23 1.2 Manifolds 24 1.2.1 Embedded submanifolds 24 1.2.2 Charts and local euclideaness 26 1.2.3 Abstract manifolds and atlases 28 1.2.4 Tangent vectors and tangent space 29 1.2.5 Differentials and pushforward 32 1.3 Riemannian manifolds 32 1.3.1 Riemannian metric 34 1.3.2 Curve length and Riemannian distance 35 1.3.3 Geodesics 35 1.3.4 Levi-Civita connection 36 1.3.5 Completeness 37 1.3.6 Exponential and logarithm maps 37 1.3.7 Cut locus 39 1.4 Elements of analysis in Riemannian manifolds 40 1.4.1 Gradient and musical isomorphisms 40 1.4.2 Hessian and Taylor expansion 41 1.4.3 Riemannian measure or volume form 42 1.4.4 Curvature 42 1.5 Lie groups and homogeneous manifolds 44 1.5.1 One-parameter subgroups 47 1.5.2 Actions 47 1.5.3 Homogeneous spaces 48 1.5.4 Invariant metrics and geodesics 49 1.6 Elements of computing on Riemannian manifolds 49 1.7 Examples 51 1.7.1 The sphere 51 1.7.2 2D Kendall shape space 51 1.7.3 Rotations 52 1.8 Additional references 55 Introductory texts on differential and Riemannian geometry 55 Advanced differential and Riemannian geometry 55 Lie groups 56 References 56 2 Statistics on manifolds 59 2.1 Introduction 59 2.2 The Fréchet mean 60 2.2.1 Existence and uniqueness of the Fréchet mean 62 2.2.2 Estimation of the Fréchet mean 64 2.3 Covariance and principal geodesic analysis 64 2.3.1 Principal component analysis 65 2.3.2 Principal geodesic analysis 67 2.3.3 Estimation: tangent approximation and exact PGA 68 2.3.4 Further extensions of PCA to manifolds 69 2.4 Regression models 70 2.4.1 Regression in Euclidean space 71 2.4.1.1 Multilinear regression 71 2.4.1.2 Univariate kernel regression 73 2.4.2 Regression on Riemannian manifolds 73 2.4.2.1 Geodesic regression 73 Least squares estimation 74 R2 statistics and hypothesis testing 76 2.4.2.2 Kernel regression on manifolds 77 2.4.3 Example of regression on Kendall shape space 78 2.5 Probabilistic models 80 2.5.1 Normal densities on manifolds 80 2.5.1.1 Maximum-likelihood estimation of μ 81 2.5.1.2 Estimation of the dispersion parameter, τ 82 2.5.1.3 Sampling from a Riemannian normal distribution 83 2.5.1.4 Sphere example 85 2.5.2 Probabilistic principal geodesic analysis 86 2.5.2.1 Probability model 87 2.5.2.2 Inference 87 2.5.2.3 E-step: HMC 87 M-step: gradient ascent 89 Gradient for τ 89 Gradient for μ 89 Gradient for Λ 89 Gradient for W 89 2.5.2.4 PPGA of simulated sphere data 90 References 91 3 Manifold-valued image processing with SPD matrices 95 3.1 Introduction 95 Chapter organization 97 3.2 Exponential, logarithm, and square root of SPD matrices 99 3.2.1 Differential of the matrix exponential 100 3.2.2 Differential of the matrix logarithm 101 3.3 Affine-invariant metrics 101 3.3.1 Affine-invariant distances 102 3.3.2 An invariant Riemannian metric 103 3.3.2.1 Tangent vectors 103 3.3.2.2 Riemannian metric 103 3.3.2.3 A symmetric space structures 103 3.3.2.4 Geodesics 104 3.3.2.5 Riemannian exponential and log maps 106 3.3.3 The one-parameter family of affine-invariant metrics 106 3.3.3.1 GL(n)-invariant metrics 107 3.3.3.2 Different metrics for a unique affine connection 108 3.3.3.3 Orthonormal coordinate systems 109 A field of orthonormal bases for β=0 110 A field of orthonormal bases for β=0 110 3.3.4 Curvature of affine-invariant metrics 112 3.3.4.1 Sectional curvature 112 3.3.4.2 Ricci curvature 114 3.4 Basic statistical operations on SPD matrices 116 3.4.1 Computing the mean and the covariance matrix 116 3.4.2 Tangent PCA and PGA of SPD matrices 119 3.4.3 Gaussian distributions on SPD matrices 120 3.5 Manifold-valued image processing 123 3.5.1 Interpolation 123 3.5.2 Gaussian and Kernel-based filtering 124 3.5.3 Harmonic regularization 126 3.5.4 Anisotropic diffusion 128 3.5.5 Inpainting and extrapolation of sparse SPD fields 129 3.6 Other metrics on SPD matrices 130 3.6.1 Log-Euclidean metrics 130 3.6.2 Cholesky metrics 133 3.6.3 Square root and Procrustes metrics 133 3.6.4 Extrinsic "distances" 134 3.6.5 Power-Euclidean metrics 135 3.6.6 Which metric for which problem? 136 3.7 Applications in diffusion tensor imaging (DTI) 137 3.8 Learning brain variability from Sulcal lines 142 Acknowledgments 148 References 148 4 Riemannian geometry on shapes and diffeomorphisms 155 4.1 Introduction 155 4.2 Shapes and actions 156 4.3 The diffeomorphism group in shape analysis 158 4.3.1 Fréchet-Lie groups 159 4.3.2 Geodesic flows 160 4.4 Riemannian metrics on shape spaces 163 4.4.1 Shapes and descending metrics 164 4.4.2 Constructing diffeomorphisms 166 4.4.3 Reproducing kernel Hilbert spaces 168 4.5 Shape spaces 170 4.5.1 Landmarks 173 4.5.2 Images 174 4.6 Statistics in LDDMM 176 4.6.1 Random orbit model 177 4.6.2 Template estimation 178 4.6.3 Estimation in a general setting 179 4.6.4 Stochastics 180 4.7 Outer and inner shape metrics 181 4.7.1 Sobolev geodesics 183 4.8 Further reading 184 References 185 5 Beyond Riemannian geometry 189 5.1 Introduction 189 5.2 Affine connection spaces 192 5.2.1 Affine connection as an infinitesimal parallel transport 193 5.2.2 Geodesics 194 5.2.3 Levi-Civita connection of a Riemannian metric 195 5.2.4 Statistics on affine connection spaces 196 Mean values with exponential barycenters 197 Covariance matrix and higher-order moments 200 Mahalanobis distance 201 Open problems for generalizing other statistical tool 201 5.3 Canonical connections on Lie groups 202 5.3.1 The lie group setting 202 Adjoint group 203 Matrix lie group exponential and logarithm 204 One-parameter subgroups and group exponential 204 Baker-Campbel-Hausdorff (BCH) formula 205 5.3.2 Cartan-Schouten (CCS) connections 206 Torsion and curvature of Cartan-Schouten connections 207 5.3.3 Group geodesics, parallel transport 209 Parallel transport along geodesics 210 5.4 Left, right, and biinvariant Riemannian metrics on a Lie group 211 5.4.1 Levi-Civita connections of left-invariant metrics 211 5.4.2 Canonical connection of bi-invariant metrics 212 Compactness, commutativity, and existence of biinvariant metrics 213 Biinvariant pseudo-Riemannian metrics (quadratic Lie groups) 214 5.4.3 Example with rigid-body transformations 214 5.5 Statistics on Lie groups as symmetric spaces 215 5.5.1 Biinvariant means with exponential barycenters of the CCS connection 215 5.5.2 Existence and uniqueness results in specific matrix groups 218 5.5.2.1 The Heisenberg group 218 5.5.2.2 Scaling and translations ST(d) 219 5.5.2.3 Scaled upper unitriangular matrix group 220 5.5.2.4 General rigid-body transformations 221 Example with 2D rigid transformations 223 5.6 The stationary velocity fields (SVF) framework for diffeomorphisms 224 5.6.1 Parameterizing diffeomorphisms with the Flow of SVFs 225 5.6.2 SVF-based setting: properties and algorithm 226 5.6.2.1 Exponential of an SVF 227 5.6.2.2 Inversion of spatial transformations 227 5.6.2.3 Computing differential quantities 228 Integration of the Jacobian determinant in the region of interest 228 Flux of the deformation field across the boundary of the region 229 5.6.2.4 Composing transformations parameterized by SVF 230 5.6.3 SVF-based diffeomorphic registration with the log-demons 231 Image similarity in the log-demons 231 Regularization of SVF parameters 232 5.6.4 Optimizing the log-demons algorithm 232 5.7 Parallel transport of SVF deformations 233 5.7.1 Continuous and discrete parallel transport methods 234 5.7.2 Discrete ladders for the registration of image sequences 235 5.7.2.1 Schild's ladder 236 5.7.2.2 Pole ladder 237 5.7.2.3 Theoretical accuracy: pole ladder is a third-order scheme 237 5.7.2.4 Effective ladders on SVF-deformations 239 5.7.3 Longitudinal analysis of brain deformations in Alzheimer's disease 240 Data analysis and results [47] 241 5.8 Historical notes and additional references 243 References 243 Part 2: Statistics on manifolds and shape spaces 251 6 Object shape representation via skeletal models (s-reps) and statistical analysis 253 6.1 Introduction to skeletal models 253 6.2 Computing an s-rep from an image or object boundary 256 6.3 Skeletal interpolation 259 6.4 Skeletal fitting 262 6.5 Correspondence 265 6.6 Skeletal statistics 265 6.6.1 Probability distribution estimation 266 6.6.1.1 Tangent plane statistical analysis methods 266 6.6.1.2 Principal nested spheres 268 6.6.1.3 Composite principal nested spheres 270 6.6.1.4 Polysphere PCA 271 6.6.2 Classification 271 6.7 How to compare representations and statistical methods 273 6.7.1 Classification accuracy 273 6.7.2 Hypothesis testing power 273 6.7.3 Specificity, generalization, and compactness 274 6.7.4 Compression into few modes of variation 274 6.8 Results of classification, hypothesis testing, and probability distribution estimation 276 6.8.1 Classification of mental illness via hippocampus and caudate s-reps 276 6.8.2 Hypothesis testing via s-reps 280 6.8.3 Shape change statistics 280 6.8.4 Correspondence evaluation via entropy minimization 282 6.8.5 Segmentations by posterior optimization 283 6.9 The code and its performance 285 6.10 Weaknesses of the skeletal approach 286 Acknowledgments 287 References 287 7 Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications 293 7.1 Introduction 293 7.2 Riemannian geometry of the hypersphere 294 7.3 Weak consistency of iFME on the sphere 296 7.4 Experimental results 303 7.5 Application to the classification of movement disorders 304 7.6 Riemannian geometry of the special orthogonal group 307 7.7 Weak consistency of iFME on so(n) 308 7.8 Experimental results 311 7.9 Conclusions 313 Acknowledgments 314 References 314 8 Statistics on stratified spaces 319 8.1 Introduction to stratified geometry 319 8.1.1 Examples 320 8.1.2 Metric spaces 323 8.1.3 Curvature in metric spaces 324 8.2 Least squares models 327 8.2.1 Least squares statistics and stickiness 327 8.2.2 The principal component and the mean 329 8.3 BHV tree space 330 8.3.1 Geometry in BHV tree space 330 8.3.2 Statistical methodology in BHV tree space 337 8.3.2.1 Fréchet mean and variance 338 8.3.2.2 Principal component analysis 339 8.3.2.3 Other approaches 341 8.4 The space of unlabeled trees 342 8.4.1 What is an unlabeled tree? 342 8.4.2 Geodesics between unlabeled trees 345 8.4.2.1 Mappings, geodesics, and compatible edges 345 8.4.2.2 Link between QED geodesics and BHV geodesics 346 8.4.3 Uniqueness of QED geodesics 348 8.5 Beyond trees 353 8.5.1 Variable topology data 354 8.5.2 More general quotient spaces 356 8.5.3 Open problems 357 Acknowledgments 358 References 358 9 Bias on estimation in quotient space and correction methods 363 9.1 Introduction 363 9.2 Shapes and quotient spaces 364 9.2.1 Group actions 364 9.2.2 Orbit, isotropy group, quotient space 366 9.2.3 Proper and effective group actions 368 9.2.4 Principal and singular orbits 369 9.2.5 Metric structure 371 9.3 Template estimation 374 9.3.1 Generative model 374 Observations in a Hilbert space 375 Observations in a finite-dimensional Riemannian manifold 375 9.3.2 An iterative estimation procedure 376 9.3.3 Convergence to the Fréchet mean in the quotient space 377 9.3.4 Other convergence(s) 378 9.3.5 Bias of the procedure 378 9.4 Asymptotic bias of template estimation 379 9.4.1 Intuition on examples 380 9.4.2 Bias on quotient of finite-dimensional Riemannian manifold 381 9.4.3 Bias on quotient of (in)finite-dimensional Hilbert space 383 Asymptotic bias for a very large noise 384 9.5 Applications to statistics on organ shapes 385 9.5.1 Shapes defined by landmarks 385 Different types of Procrustean analyses 385 Mean form and mean shape for 2D/3D landmarks 386 9.5.2 Brain images 387 Brain template 387 Spatial bias on the brain template image 387 9.6 Bias correction methods 388 9.6.1 Riemannian bootstrap 389 9.6.2 Brain images: topologically constrained template 391 Constrain the topology to control the bias 391 Applying topological denoising on the brain template's 392 9.7 Conclusion 392 Acknowledgment 393 References 393 10 Probabilistic approaches to geometric statistics 397 10.1 Introduction 397 10.2 Parametric probability distributions on manifolds 400 10.2.1 Probabilistic PCA 400 10.2.2 Riemannian normal distribution and probabilistic PGA 402 10.2.3 Transition distributions and stochastic differential equations 402 10.3 The Brownian motion 404 10.3.1 Brownian motion on Riemannian manifolds 405 10.3.2 Lie groups 406 10.4 Fiber bundle geometry 407 10.4.1 The frame bundle 409 10.4.2 Horizontality 410 10.4.3 Development and stochastic development 412 10.5 Anisotropic normal distributions 413 10.5.1 Infinitesimal covariance 414 10.5.2 Isotropic noise 415 10.5.3 Euclideanization 416 10.6 Statistics with bundles 418 10.6.1 Normal distributions and maximum likelihood 418 10.6.2 Infinitesimal PPCA 419 10.6.3 Regression 419 10.7 Parameter estimation 420 10.7.1 Anisotropic least squares 420 10.7.2 Method of moments 421 10.7.3 Bridge sampling 422 10.7.4 Bridge sampling on manifolds 425 10.8 Advanced concepts 427 10.8.1 Curvature 427 10.8.2 Sub-Riemannian geometry 428 10.8.3 Most probable paths 429 10.8.4 Bundles without rotation 431 10.8.5 Flows with special structure 431 10.9 Conclusion 433 10.10 Further reading 434 References 434 11 On shape analysis of functional data 437 11.1 Introduction 437 11.2 Registration problem and elastic approach 439 11.2.1 The L2 norm and associated problems 439 Lack of invariance under L2 norm 442 11.2.2 SRVFs and curve registration 444 11.3 Shape space and geodesic paths 446 Shape spaces of closed curves 448 11.4 Statistical summaries and principal modes of shape variability 450 Multiple alignment algorithm 451 11.5 Summary and conclusion 454 Appendix: Mathematical background 454 References 456 Part 3: Deformations, diffeomorphisms and their applications 459 12 Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles 461 12.1 Introduction 461 12.2 General setting and notations 463 12.3 Currents 463 12.3.1 Curves and surfaces as currents 464 12.3.2 Kernel metrics on currents 464 12.3.3 The discrete model 467 12.3.4 Examples of registration using currents metrics 468 12.4 Varifolds 470 12.4.1 Representation by varifolds 471 12.4.2 Kernel metrics 471 12.4.3 Discrete model 474 12.4.4 Examples and applications 475 12.5 Normal cycles 477 12.5.1 Differential forms and currents 478 12.5.2 Unit normal bundle and normal cycle 478 12.5.3 Normal cycles for discrete curves or surfaces 479 12.5.4 Kernel metrics on normal cycles 481 12.5.5 Discrete inner product 481 12.5.6 Examples and applications 483 12.6 Computational aspects 485 12.6.1 Fast kernel computations 485 12.6.1.1 Exact computations 486 Linear algebra library 486 Graphics processing unit (GPU) 487 12.6.1.2 Approximate methods 488 Grid method and nonuniform fast Fourier transform (NFFT) 488 Fast multipole methods (FMM) 489 12.6.2 Compact approximations 489 12.6.3 Available implementations 491 12.7 Conclusion 492 Acknowledgments 493 References 493 13 A discretize-optimize approach for LDDMM registration 499 13.1 Introduction 499 13.2 Background and related work 503 13.3 Continuous mathematical models 508 13.3.1 Relaxation with transport of images (IBR) 510 13.3.2 Relaxation with transport of maps (MBR) 510 13.3.3 Shooting with maps using EPDiff (MBS) 511 13.3.4 Distance measures 512 13.4 Discretization of the energies 513 13.4.1 Discretization on grids 513 13.4.2 Discretization of the regularizer 514 13.4.3 Discretization of the distance measures 516 13.5 Discretization and solution of PDEs 516 13.5.1 Runge-Kutta methods 517 13.5.2 Runge-Kutta methods for the adjoint system 519 13.5.3 Application to the IBR model 521 13.5.4 Application to the MBR model 523 13.5.5 Application to the MBS model using EPDiff 524 13.6 Discretization in multiple dimensions 526 13.6.1 Discretization of the regularizer 526 13.6.2 Integrands and associated gradients for the Runge-Kutta methods 528 13.6.2.1 IBR model 528 13.6.2.2 MBR model 529 13.6.2.3 MBS model 531 13.7 Multilevel registration and numerical optimization 532 13.8 Experiments and results 535 13.8.1 2D registration of hand radiographs 535 13.8.2 3D registration of lung CT data 537 13.9 Discussion and conclusion 543 References 544 14 Spatially adaptive metrics for diffeomorphic image matching in LDDMM 553 14.1 Introduction to LDDMM 553 14.1.1 Problem definition 553 14.1.2 Properties 554 14.1.3 Implementation 557 14.2 Sum of kernels and semidirect product of groups 559 14.2.1 Introduction 559 14.2.2 Multiscale kernels 560 14.2.3 Distinguishing the deformations at different scales 563 14.3 Sliding motion constraints 565 14.3.1 Introduction 565 14.3.2 Methodology 566 14.3.3 Results and discussion 567 14.4 Left-invariant metrics 568 14.4.1 Introduction 568 14.4.2 Methodology 569 14.4.3 Results and discussion 570 14.5 Open directions 571 14.5.1 Learning the metric 571 14.5.2 Other models 573 References 574 15 Low-dimensional shape analysis in the space of diffeomorphisms 577 15.1 Introduction 577 15.2 Background 579 15.2.1 Flows of diffeomorphisms and geodesics 579 15.2.2 Fourier representation of velocity fields 580 15.2.3 Geodesic shooting in finite-dimensional spaces 582 15.3 PPGA of diffeomorphisms 583 15.4 Inference 584 15.4.1 Reduced adjoint Jacobi fields in bandlimited velocity spaces 584 MAP 586 MCEM 587 15.5 Evaluation 588 15.6 Results 589 15.7 Discussion and conclusion 591 Acknowledgments 594 References 594 16 Diffeomorphic density registration 599 16.1 Introduction 599 16.2 Diffeomorphisms and densities 600 16.2.1 α-actions 602 16.3 Diffeomorphic density registration 603 16.4 Density registration in the LDDMM-framework 604 16.5 Optimal information transport 607 16.5.1 Application: random sampling from nonuniform distribution 610 16.6 A gradient flow approach 611 16.6.1 Thoracic density registration 617 Acknowledgments 622 References 623 Index 627 Back Cover 637 Cover......Page 1 RIEMANNIAN GEOMETRIC STATISTICS IN MEDICAL IMAGE ANALYSIS ......Page 5 Copyright ......Page 6 Contents......Page 7 Contributors......Page 14 Contents......Page 17 Part 1: Foundations of geometric statistics ......Page 21 1.1 Introduction......Page 23 1.2.1 Embedded submanifolds......Page 24 1.2.2 Charts and local euclideaness......Page 26 1.2.3 Abstract manifolds and atlases......Page 28 1.2.4 Tangent vectors and tangent space......Page 29 1.3 Riemannian manifolds......Page 32 1.3.1 Riemannian metric......Page 34 1.3.3 Geodesics......Page 35 1.3.4 Levi-Civita connection......Page 36 1.3.6 Exponential and logarithm maps......Page 37 1.3.7 Cut locus......Page 39 1.4.1 Gradient and musical isomorphisms......Page 40 1.4.2 Hessian and Taylor expansion......Page 41 1.4.4 Curvature......Page 42 1.5 Lie groups and homogeneous manifolds......Page 44 1.5.2 Actions......Page 47 1.5.3 Homogeneous spaces......Page 48 1.6 Elements of computing on Riemannian manifolds......Page 49 1.7.2 2D Kendall shape space......Page 51 1.7.3 Rotations......Page 52 Advanced differential and Riemannian geometry......Page 55 References......Page 56 2.1 Introduction......Page 59 2.2 The Fréchet mean......Page 60 2.2.1 Existence and uniqueness of the Fréchet mean......Page 62 2.3 Covariance and principal geodesic analysis......Page 64 2.3.1 Principal component analysis......Page 65 2.3.2 Principal geodesic analysis......Page 67 2.3.3 Estimation: tangent approximation and exact PGA......Page 68 2.3.4 Further extensions of PCA to manifolds......Page 69 2.4 Regression models......Page 70 2.4.1.1 Multilinear regression......Page 71 2.4.2.1 Geodesic regression......Page 73 Least squares estimation......Page 74 R2 statistics and hypothesis testing......Page 76 2.4.2.2 Kernel regression on manifolds......Page 77 2.4.3 Example of regression on Kendall shape space......Page 78 2.5.1 Normal densities on manifolds......Page 80 2.5.1.1 Maximum-likelihood estimation of μ......Page 81 2.5.1.2 Estimation of the dispersion parameter, τ......Page 82 2.5.1.3 Sampling from a Riemannian normal distribution......Page 83 2.5.1.4 Sphere example......Page 85 2.5.2 Probabilistic principal geodesic analysis......Page 86 2.5.2.3 E-step: HMC......Page 87 Gradient for W......Page 89 2.5.2.4 PPGA of simulated sphere data......Page 90 References......Page 91 3.1 Introduction......Page 95 Chapter organization......Page 97 3.2 Exponential, logarithm, and square root of SPD matrices......Page 99 3.2.1 Differential of the matrix exponential......Page 100 3.3 Affine-invariant metrics......Page 101 3.3.1 Affine-invariant distances......Page 102 3.3.2.3 A symmetric space structures......Page 103 3.3.2.4 Geodesics......Page 104 3.3.3 The one-parameter family of affine-invariant metrics......Page 106 3.3.3.1 GL(n)-invariant metrics......Page 107 3.3.3.2 Different metrics for a unique affine connection......Page 108 3.3.3.3 Orthonormal coordinate systems......Page 109 A field of orthonormal bases for β=0......Page 110 3.3.4.1 Sectional curvature......Page 112 3.3.4.2 Ricci curvature......Page 114 3.4.1 Computing the mean and the covariance matrix......Page 116 3.4.2 Tangent PCA and PGA of SPD matrices......Page 119 3.4.3 Gaussian distributions on SPD matrices......Page 120 3.5.1 Interpolation......Page 123 3.5.2 Gaussian and Kernel-based filtering......Page 124 3.5.3 Harmonic regularization......Page 126 3.5.4 Anisotropic diffusion......Page 128 3.5.5 Inpainting and extrapolation of sparse SPD fields......Page 129 3.6.1 Log-Euclidean metrics......Page 130 3.6.3 Square root and Procrustes metrics......Page 133 3.6.4 Extrinsic "distances"......Page 134 3.6.5 Power-Euclidean metrics......Page 135 3.6.6 Which metric for which problem?......Page 136 3.7 Applications in diffusion tensor imaging (DTI)......Page 137 3.8 Learning brain variability from Sulcal lines......Page 142 References......Page 148 4.1 Introduction......Page 155 4.2 Shapes and actions......Page 156 4.3 The diffeomorphism group in shape analysis......Page 158 4.3.1 Fréchet-Lie groups......Page 159 4.3.2 Geodesic flows......Page 160 4.4 Riemannian metrics on shape spaces......Page 163 4.4.1 Shapes and descending metrics......Page 164 4.4.2 Constructing diffeomorphisms......Page 166 4.4.3 Reproducing kernel Hilbert spaces......Page 168 4.5 Shape spaces......Page 170 4.5.1 Landmarks......Page 173 4.5.2 Images......Page 174 4.6 Statistics in LDDMM......Page 176 4.6.1 Random orbit model......Page 177 4.6.2 Template estimation......Page 178 4.6.3 Estimation in a general setting......Page 179 4.6.4 Stochastics......Page 180 4.7 Outer and inner shape metrics......Page 181 4.7.1 Sobolev geodesics......Page 183 4.8 Further reading......Page 184 References......Page 185 5.1 Introduction......Page 189 5.2 Affine connection spaces......Page 192 5.2.1 Affine connection as an infinitesimal parallel transport......Page 193 5.2.2 Geodesics......Page 194 5.2.3 Levi-Civita connection of a Riemannian metric......Page 195 5.2.4 Statistics on affine connection spaces......Page 196 Mean values with exponential barycenters......Page 197 Covariance matrix and higher-order moments......Page 200 Open problems for generalizing other statistical tool......Page 201 5.3.1 The lie group setting......Page 202 Adjoint group......Page 203 One-parameter subgroups and group exponential......Page 204 Baker-Campbel-Hausdorff (BCH) formula......Page 205 5.3.2 Cartan-Schouten (CCS) connections......Page 206 Torsion and curvature of Cartan-Schouten connections......Page 207 5.3.3 Group geodesics, parallel transport......Page 209 Parallel transport along geodesics......Page 210 5.4.1 Levi-Civita connections of left-invariant metrics......Page 211 5.4.2 Canonical connection of bi-invariant metrics......Page 212 Compactness, commutativity, and existence of biinvariant metrics......Page 213 5.4.3 Example with rigid-body transformations......Page 214 5.5.1 Biinvariant means with exponential barycenters of the CCS connection......Page 215 5.5.2.1 The Heisenberg group......Page 218 5.5.2.2 Scaling and translations ST(d)......Page 219 5.5.2.3 Scaled upper unitriangular matrix group......Page 220 5.5.2.4 General rigid-body transformations......Page 221 Example with 2D rigid transformations......Page 223 5.6 The stationary velocity fields (SVF) framework for diffeomorphisms......Page 224 5.6.1 Parameterizing diffeomorphisms with the Flow of SVFs......Page 225 5.6.2 SVF-based setting: properties and algorithm......Page 226 5.6.2.2 Inversion of spatial transformations......Page 227 Integration of the Jacobian determinant in the region of interest......Page 228 Flux of the deformation field across the boundary of the region......Page 229 5.6.2.4 Composing transformations parameterized by SVF......Page 230 Image similarity in the log-demons......Page 231 5.6.4 Optimizing the log-demons algorithm......Page 232 5.7 Parallel transport of SVF deformations......Page 233 5.7.1 Continuous and discrete parallel transport methods......Page 234 5.7.2 Discrete ladders for the registration of image sequences......Page 235 5.7.2.1 Schild's ladder......Page 236 5.7.2.3 Theoretical accuracy: pole ladder is a third-order scheme......Page 237 5.7.2.4 Effective ladders on SVF-deformations......Page 239 5.7.3 Longitudinal analysis of brain deformations in Alzheimer's disease......Page 240 Data analysis and results [47]......Page 241 References......Page 243 Part 2: Statistics on manifolds and shape spaces ......Page 251 6.1 Introduction to skeletal models......Page 253 6.2 Computing an s-rep from an image or object boundary......Page 256 6.3 Skeletal interpolation......Page 259 6.4 Skeletal fitting......Page 262 6.6 Skeletal statistics......Page 265 6.6.1.1 Tangent plane statistical analysis methods......Page 266 6.6.1.2 Principal nested spheres......Page 268 6.6.1.3 Composite principal nested spheres......Page 270 6.6.2 Classification......Page 271 6.7.2 Hypothesis testing power......Page 273 6.7.4 Compression into few modes of variation......Page 274 6.8.1 Classification of mental illness via hippocampus and caudate s-reps......Page 276 6.8.3 Shape change statistics......Page 280 6.8.4 Correspondence evaluation via entropy minimization......Page 282 6.8.5 Segmentations by posterior optimization......Page 283 6.9 The code and its performance......Page 285 6.10 Weaknesses of the skeletal approach......Page 286 References......Page 287 7.1 Introduction......Page 293 7.2 Riemannian geometry of the hypersphere......Page 294 7.3 Weak consistency of iFME on the sphere......Page 296 7.4 Experimental results......Page 303 7.5 Application to the classification of movement disorders......Page 304 7.6 Riemannian geometry of the special orthogonal group......Page 307 7.7 Weak consistency of iFME on so(n)......Page 308 7.8 Experimental results......Page 311 7.9 Conclusions......Page 313 References......Page 314 8.1 Introduction to stratified geometry......Page 319 8.1.1 Examples......Page 320 8.1.2 Metric spaces......Page 323 8.1.3 Curvature in metric spaces......Page 324 8.2.1 Least squares statistics and stickiness......Page 327 8.2.2 The principal component and the mean......Page 329 8.3.1 Geometry in BHV tree space......Page 330 8.3.2 Statistical methodology in BHV tree space......Page 337 8.3.2.1 Fréchet mean and variance......Page 338 8.3.2.2 Principal component analysis......Page 339 8.3.2.3 Other approaches......Page 341 8.4.1 What is an unlabeled tree?......Page 342 8.4.2.1 Mappings, geodesics, and compatible edges......Page 345 8.4.2.2 Link between QED geodesics and BHV geodesics......Page 346 8.4.3 Uniqueness of QED geodesics......Page 348 8.5 Beyond trees......Page 353 8.5.1 Variable topology data......Page 354 8.5.2 More general quotient spaces......Page 356 8.5.3 Open problems......Page 357 References......Page 358 9.1 Introduction......Page 363 9.2.1 Group actions......Page 364 9.2.2 Orbit, isotropy group, quotient space......Page 366 9.2.3 Proper and effective group actions......Page 368 9.2.4 Principal and singular orbits......Page 369 9.2.5 Metric structure......Page 371 9.3.1 Generative model......Page 374 Observations in a finite-dimensional Riemannian manifold......Page 375 9.3.2 An iterative estimation procedure......Page 376 9.3.3 Convergence to the Fréchet mean in the quotient space......Page 377 9.3.5 Bias of the procedure......Page 378 9.4 Asymptotic bias of template estimation......Page 379 9.4.1 Intuition on examples......Page 380 9.4.2 Bias on quotient of finite-dimensional Riemannian manifold......Page 381 9.4.3 Bias on quotient of (in)finite-dimensional Hilbert space......Page 383 Asymptotic bias for a very large noise......Page 384 Different types of Procrustean analyses......Page 385 Mean form and mean shape for 2D/3D landmarks......Page 386 Spatial bias on the brain template image......Page 387 9.6 Bias correction methods......Page 388 9.6.1 Riemannian bootstrap......Page 389 Constrain the topology to control the bias......Page 391 9.7 Conclusion......Page 392 References......Page 393 10.1 Introduction......Page 397 10.2.1 Probabilistic PCA......Page 400 10.2.3 Transition distributions and stochastic differential equations......Page 402 10.3 The Brownian motion......Page 404 10.3.1 Brownian motion on Riemannian manifolds......Page 405 10.3.2 Lie groups......Page 406 10.4 Fiber bundle geometry......Page 407 10.4.1 The frame bundle......Page 409 10.4.2 Horizontality......Page 410 10.4.3 Development and stochastic development......Page 412 10.5 Anisotropic normal distributions......Page 413 10.5.1 Infinitesimal covariance......Page 414 10.5.2 Isotropic noise......Page 415 10.5.3 Euclideanization......Page 416 10.6.1 Normal distributions and maximum likelihood......Page 418 10.6.3 Regression......Page 419 10.7.1 Anisotropic least squares......Page 420 10.7.2 Method of moments......Page 421 10.7.3 Bridge sampling......Page 422 10.7.4 Bridge sampling on manifolds......Page 425 10.8.1 Curvature......Page 427 10.8.2 Sub-Riemannian geometry......Page 428 10.8.3 Most probable paths......Page 429 10.8.5 Flows with special structure......Page 431 10.9 Conclusion......Page 433 References......Page 434 11.1 Introduction......Page 437 11.2.1 The L2 norm and associated problems......Page 439 Lack of invariance under L2 norm......Page 442 11.2.2 SRVFs and curve registration......Page 444 11.3 Shape space and geodesic paths......Page 446 Shape spaces of closed curves......Page 448 11.4 Statistical summaries and principal modes of shape variability......Page 450 Multiple alignment algorithm......Page 451 Appendix: Mathematical background......Page 454 References......Page 456 Part 3: Deformations, diffeomorphisms and their applications ......Page 459 12.1 Introduction......Page 461 12.3 Currents......Page 463 12.3.2 Kernel metrics on currents......Page 464 12.3.3 The discrete model......Page 467 12.3.4 Examples of registration using currents metrics......Page 468 12.4 Varifolds......Page 470 12.4.2 Kernel metrics......Page 471 12.4.3 Discrete model......Page 474 12.4.4 Examples and applications......Page 475 12.5 Normal cycles......Page 477 12.5.2 Unit normal bundle and normal cycle......Page 478 12.5.3 Normal cycles for discrete curves or surfaces......Page 479 12.5.5 Discrete inner product......Page 481 12.5.6 Examples and applications......Page 483 12.6.1 Fast kernel computations......Page 485 Linear algebra library......Page 486 Graphics processing unit (GPU)......Page 487 Grid method and nonuniform fast Fourier transform (NFFT)......Page 488 12.6.2 Compact approximations......Page 489 12.6.3 Available implementations......Page 491 12.7 Conclusion......Page 492 References......Page 493 13.1 Introduction......Page 499 13.2 Background and related work......Page 503 13.3 Continuous mathematical models......Page 508 13.3.2 Relaxation with transport of maps (MBR)......Page 510 13.3.3 Shooting with maps using EPDiff (MBS)......Page 511 13.3.4 Distance measures......Page 512 13.4.1 Discretization on grids......Page 513 13.4.2 Discretization of the regularizer......Page 514 13.5 Discretization and solution of PDEs......Page 516 13.5.1 Runge-Kutta methods......Page 517 13.5.2 Runge-Kutta methods for the adjoint system......Page 519 13.5.3 Application to the IBR model......Page 521 13.5.4 Application to the MBR model......Page 523 13.5.5 Application to the MBS model using EPDiff......Page 524 13.6.1 Discretization of the regularizer......Page 526 13.6.2.1 IBR model......Page 528 13.6.2.2 MBR model......Page 529 13.6.2.3 MBS model......Page 531 13.7 Multilevel registration and numerical optimization......Page 532 13.8.1 2D registration of hand radiographs......Page 535 13.8.2 3D registration of lung CT data......Page 537 13.9 Discussion and conclusion......Page 543 References......Page 544 14.1.1 Problem definition......Page 553 14.1.2 Properties......Page 554 14.1.3 Implementation......Page 557 14.2.1 Introduction......Page 559 14.2.2 Multiscale kernels......Page 560 14.2.3 Distinguishing the deformations at different scales......Page 563 14.3.1 Introduction......Page 565 14.3.2 Methodology......Page 566 14.3.3 Results and discussion......Page 567 14.4.1 Introduction......Page 568 14.4.2 Methodology......Page 569 14.4.3 Results and discussion......Page 570 14.5.1 Learning the metric......Page 571 14.5.2 Other models......Page 573 References......Page 574 15.1 Introduction......Page 577 15.2.1 Flows of diffeomorphisms and geodesics......Page 579 15.2.2 Fourier representation of velocity fields......Page 580 15.2.3 Geodesic shooting in finite-dimensional spaces......Page 582 15.3 PPGA of diffeomorphisms......Page 583 15.4.1 Reduced adjoint Jacobi fields in bandlimited velocity spaces......Page 584 MAP......Page 586 MCEM......Page 587 15.5 Evaluation......Page 588 15.6 Results......Page 589 15.7 Discussion and conclusion......Page 591 References......Page 594 16.1 Introduction......Page 599 16.2 Diffeomorphisms and densities......Page 600 16.2.1 α-actions......Page 602 16.3 Diffeomorphic density registration......Page 603 16.4 Density registration in the LDDMM-framework......Page 604 16.5 Optimal information transport......Page 607 16.5.1 Application: random sampling from nonuniform distribution......Page 610 16.6 A gradient flow approach......Page 611 16.6.1 Thoracic density registration......Page 617 Acknowledgments......Page 622 References......Page 623 Index......Page 627 Back Cover......Page 637