Riemannian Geometric Statistics in Medical Image Analysis

Cover......Page 1
RIEMANNIAN
GEOMETRIC
STATISTICS IN
MEDICAL IMAGE
ANALYSIS
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Copyright
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Contents......Page 7
Contributors......Page 14
Contents......Page 17
Part 1: Foundations of geometric statistics
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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