Statistical and computational methods in brain image analysis

The massive amount of nonstandard high-dimensional brain imaging data being generated is often difficult to analyze using current techniques. This challenge in brain image analysis requires new computational approaches and solutions. But none of the research papers or books in the field describe the quantitative techniques with detailed illustrations of actual imaging data and computer codes. Using MATLAB® and case study data sets, Statistical and Computational Methods in Brain Image Analysis is the first book to explicitly explain how to perform statistical analysis on brain imaging data.

The book focuses on methodological issues in analyzing structural brain imaging modalities such as MRI and DTI. Real imaging applications and examples elucidate the concepts and methods. In addition, most of the brain imaging data sets and MATLAB codes are available on the author's website.

By supplying the data and codes, this book enables researchers to start their statistical analyses immediately. Also suitable for graduate students, it provides an understanding of the various statistical and computational methodologies used in the field as well as important and technically challenging topics.

Author Notes

Moo K. Chung, Ph.D. is an associate professor in the Department of Biostatistics and Medical Informatics at the University of Wisconsin-Madison. He is also affiliated with the Waisman Laboratory for Brain Imaging and Behavior. He has won the Vilas Associate Award for his applied topological research (persistent homology) to medical imaging and the Editor's Award for best paper published in Journal of Speech, Language, and Hearing Research. Dr. Chung received a Ph.D. in statistics from McGill University. His main research area is computational neuroanatomy, concentrating on the methodological development required for quantifying and contrasting anatomical shape variations in both normal and clinical populations at the macroscopic level using various mathematical, statistical, and computational techniques.

Preface	p. xv
1 Introduction to Brain and Medical Images	p. 1
1.1 Image Volume Data	p. 2
1.1.1 Amygdala Volume Data	p. 4
1.2 Surface Mesh Data	p. 6
1.2.1 Topology of Surface Data	p. 6
1.2.2 Amygdala Surface Data	p. 9
1.3 Landmark Data	p. 11
1.3.1 Affine Transforms	p. 11
1.3.2 Least Squares Estimation	p. 13
1.4 Vector Data	p. 15
1.5 Tensor and Curve Data	p. 17
1.6 Brain Image Analysis Tools	p. 19
1.6.1 Surf Stat	p. 20
1.6.2 Public Image Database	p. 20
2 Bernoulli Models for Binary Images	p. 21
2.1 Sum of Bernoulli Distributions	p. 21
2.2 Inference on Proportion of Activation	p. 23
2.2.1 One Sample Test	p. 23
2.2.2 Two Sample Test	p. 26
2.3 HAT LAB Implementation	p. 27
3 General Linear Models	p. 29
3.1 General Linear Models	p. 29
3.1.1 R-square	p. 31
3.1.2 CLM for Whole Brain Images	p. 32
3.2 Voxel-Based Morphometry	p. 32
3.2.1 Mixture Models	p. 34
3.2.2 EM-Algorithm	p. 36
3.2.3 Two-Components Gaussian Mixture	p. 37
3.3 Case Study: VBM in Corpus Callosum	p. 40
3.3.1 White Matter Density Maps	p. 42
3.3.2 Manipulating Density Maps	p. 42
3.3.3 Numerical Implementation	p. 46
3.4 Testing Interactions	p. 49
4 Gaussian Kernel Smoothing	p. 51
4.1 Kernel Smoothing	p. 51
4.2 Gaussian Kernel Smoothing	p. 52
4.2.1 Fullwidth at Half Maximum	p. 53
4.3 Numerical Implementation	p. 54
4.3.1 Smoothing Scalar Functions	p. 54
4.3.2 Smoothing Image Slices	p. 55
4.4 Case Study: Smoothing of DWI Stroke Lesions	p. 57
4.5 Effective FWHM	p. 59
4.6 Checking Gaussianness	p. 60
4.6.1 Quantile-Quantile Plots	p. 60
4.6.2 Quantiles	p. 60
4.6.3 Empirical Distribution	p. 61
4.6.4 Normal Probability Plots	p. 61
4.6.5 MATLAB Implementation	p. 62
4.7 Effect of Ganssianness on Kernel Smoothing	p. 63
5 Random Fields Theory	p. 67
5.1 Random Fields	p. 67
5.1.1 Gaussian Fields	p. 68
5.1.2 Derivative of Gaussian Fields	p. 69
5.1.3 Integration of Gaussian Fields	p. 70
5.1.4 t, F and x 2 Fields	p. 70
5.2 Simulating Gaussian Fields	p. 71
5.3 Statistical Inference on Fields	p. 73
5.3.1 Bonferroni Correction	p. 75
5.3.2 Rice Formula	p. 76
5.3.3 Poisson Clumping Heuristic	p. 78
5.4 Expected Euler Characteristics	p. 78
5.4.1 Intrinsic Volumes	p. 79
5.4.2 Euler Characteristic Density	p. 80
5.4.3 Numerical Implementation of Euler Characteristics	p. 82
6 Anisotropic Kernel Smoothing	p. 85
6.1 Anisotropic Gaussian Kernel Smoothing	p. 85
6.1.1 Truncated Gaussian Kernel	p. 87
6.2 Probabilistic Connectivity in DTI	p. 88
6.3 Riemannian Metric Tensors	p. 89
6.4 Chapman-Kolmogorov Equation	p. 91
6.5 Cholesky Factorization of DTI	p. 95
6.6 Experimental Results	p. 97
6.7 Discussion	p. 98
7 Multivariate General Linear Models	p. 101
7.1 Multivariate Normal Distributions	p. 101
7.1.1 Checking Bivariate Normality of Data	p. 103
7.1.2 Covariance Matrix Factorization	p. 104
7.2 Deformation-Based Morphometry (DBM)	p. 105
7.3 Hotelling's T 2 Statistic	p. 108
7.4 Multivariate General Linear Models	p. 111
7.4.1 SurfStat	p. 113
7.5 Case Study: Surface Deformation Analysis	p. 114
7.5.1 Univariate Tests in SurfStat	p. 116
7.5.2 Multivariate Tests in SurfStat	p. 119
8 Cortical Surface Analysis	p. 121
8.1 Introduction	p. 121
8.2 Modeling Surface Deformation	p. 123
8.3 Surface Parameterization	p. 126
8.3.1 Quadratic Parameterization	p. 126
8.3.2 Numerical Implementation	p. 129
8.4 Surface-Based Morphological Measures	p. 130
8.4.1 Local Surface Area Change	p. 131
8.4.2 Local Gray Matter Volume Change	p. 132
8.4.3 Cortical Thickness Change	p. 134
8.4.4 Curvature Change	p. 135
8.5 Surface-Based Diffusion Smoothing	p. 135
8.6 Statistical Inference on the Cortical Surface	p. 139
8.7 Results	p. 142
8.7.1 Gray Matter Volume Change	p. 143
8.7.2 Surface Area Change	p. 143
8.7.3 Cortical Thickness Change	p. 144
8.7.4 Curvature Change	p. 147
8.8 Discussion	p. 148
9 Heat Kernel Smoothing on Surfaces	p. 149
9.1 Introduction	p. 149
9.2 Heat Kernel Smoothing	p. 150
9.3 Numerical Implementation	p. 156
9.4 Random Field Theory on Cortical Manifold	p. 159
9.5 Case Study: Cortical Thickness Analysis	p. 160
9.6 Discussion	p. 162
10 Cosine Series Representation of 3D Curves	p. 163
10.1 Introduction	p. 163
10.2 Parameterization of 3D Curves	p. 166
10.2.1 Eigenfunctions of ID Laplacian	p. 167
10.2.2 Cosine Representation	p. 167
10.2.3 Parameter Estimation	p. 168
10.2.4 Optimal Representation	p. 169
10.3 Numerical Implementation	p. 170
10.4 Modeling a Family of Curves	p. 172
10.4.1 Registering 3D Curves	p. 172
10.4.2 Inference on a Collection of Curves	p. 173
10.5 Case Study: White Matter Fiber Tracts	p. 175
10.5.1 Image Acquisition	p. 175
10.5.2 Image Processing	p. 176
10.5.3 Cosine Series Representation	p. 177
10.5.4 Two Sample T-test	p. 177
10.5.5 Hotelling's T-square Test	p. 178
10.5.6 Simulating Curves	p. 178
10.6 Discussion	p. 180
10.6.1 Similarly Shaped Tracts	p. 180
10.6.2 Gibbs Phenomenon	p. 180
11 Weighted Spherical Harmonic Representation	p. 185
11.1 Introduction	p. 185
11.2 Spherical Coordinates	p. 187
11.3 Spherical Harmonics	p. 188
11.3.1 Weighted Spherical Harmonic Representation	p. 190
11.3.2 Estimating Spherical Harmonic Coefficients	p. 194
11.3.3 Validation Against Heat Kernel Smoothing	p. 196
11.4 Weighted-SPHARM Package	p. 200
11.5 Surface Registration	p. 205
11.5.1 MATLAB Implementation	p. 207
11.6 Encoding Surface Asymmetry	p. 210
11.7 Case Study: Cortical Asymmetry Analysis	p. 214
11.7.1 Descriptions of Data Set	p. 214
11.7.2 Statistical Inference on Surface Asymmetry	p. 216
11.8 Discussion	p. 217
12 Multivariate Surface Shape Analysis	p. 210
12.1 Introduction	p. 219
12.2 Surface Parameterization	p. 222
12.2.1 Flattening of Simulated Cube	p. 224
12.3 Weighted Spherical Harmonic Representation	p. 226
12.3.1 Optimal Degree Selection	p. 226
12.4 Gibbs Phenomenon in SPHARM	p. 228
12.4.1 Overshoot in Gibbs Phenomenon	p. 232
12.4.2 Simulation Study	p. 233
12.5 Surface Normalization	p. 233
12.5.1 Validation	p. 236
12.6 Image and Data Acquisition	p. 237
12.7 Results	p. 239
12.7.1 Amygdala Volumetry	p. 239
12.7.2 Local Shape Difference	p. 239
12.7.3 Brain and Behavior Association	p. 241
12.8 Discussion	p. 242
12.8.1 Anatomical Findings	p. 242
12.9 Numerical Implementation	p. 243
13 Lap lace-Beltrami Eigenfunctions for Surface Data	p. 247
13.1 Introduction	p. 247
13.2 Heat Kernel Smoothing	p. 248
13.2.1 Heat Kernel Smoothing in 2D Images	p. 250
13.3 Generalized Eigenvalue Problem	p. 252
13.3.1 Finite Element Method	p. 252
13.3.2 Fourier Coefficients Estimation	p. 256
13.4 Numerical Implementation	p. 257
13.5 Experimental Results	p. 260
13.5.1 Image Acquisition and Preprocessing	p. 260
13.5.2 Validation of Heat Kernel Smoothing	p. 261
13.6 Case Study: Mandible Growth Modeling	p. 265
13.6.1 Diffeomorphic Surface Registration	p. 266
13.6.2 Random Field Theory	p. 267
13.6.3 Numerical Implementation	p. 268
13.7 Conclusion	p. 274
14 Persistent Homology	p. 275
14.1 Introduction	p. 275
14.2 Rips Filtration	p. 277
14.2.1 Topology	p. 277
14.2.2 Simplex	p. 279
14.2.3 Rips Complex	p. 279
14.2.4 Constructing Rips Filtration	p. 280
14.3 Heat Kernel Smoothing of Functional Signal	p. 282
14.4 Min-max Diagram	p. 283
14.4.1 Pairing Rule	p. 286
14.4.2 Algorithm	p. 287
14.5 Case Study: Cortical Thickness Analysis	p. 289
14.5.1 Numerical Implementation	p. 290
14.5.2 Statistical Inference	p. 292
14.6 Discussion	p. 293
15 Sparse Networks	p. 295
15.1 Introduction	p. 295
15.2 Massive Univariate Methods	p. 296
15.3 Why Are Sparse Models Needed?	p. 298
15.4 Persistent Structures for Sparse Correlations	p. 300
15.4.1 Numerical Implementation	p. 304
15.5 Persistent Structures for Sparse Likelihood	p. 306
15.6 Case Study: Application to Persistent Homology	p. 309
15.6.1 MRI Data and Univariate-TBM	p. 309
15.6.2 Multivariate-TBM via Barcodes	p. 311
15.6.3 Connection to DTI Study	p. 311
15.7 Sparse Partial Correlations	p. 312
15.7.1 Partial Correlation Network	p. 312
15.7.2 Sparse Network Recovery	p. 314
15.7.3 Sparse Network Modeling	p. 314
15.7.4 Application to Jacobian Determinant	p. 315
15.7.5 Limitations of Sparse Partial Correlations	p. 316
15.8 Summary	p. 317
16 Sparse Shape Models	p. 319
16.1 Introduction	p. 319
16.2 Amygdala and Hippocampus Shape Models	p. 320
16.3 Data Set	p. 321
16.4 Sparse Shape Representation	p. 322
16.5 Case Study: Subcortical Structure Modeling	p. 324
16.5.1 Traditional Volumetric Analysis	p. 325
16.5.2 Sparse Shape Analysis	p. 325
16.6 Statistical Power	p. 326
16.6.1 Type-II Error	p. 326
16.6.2 Statistical Power for t-Test	p. 327
16.7 Power Under Multiple Comparisons	p. 329
16.7.1 Type-I Error Under Multiple Comparisons	p. 330
16.7.2 Type-II Error Under Multiple Comparisons	p. 330
16.7.3 Statistical Power of Sparse Representation	p. 333
16.8 Conclusion	p. 333
17 Modeling Structural Brain Networks	p. 335
17.1 Introduction	p. 335
17.2 DTI Acquisition and Preprocessing	p. 336
17.3 ¿-Neighbor Construction	p. 337
17.4 Node Degrees	p. 340
17.5 Connected Components	p. 341
17.6 ¿-Filtration	p. 344
17.7 Numerical Implementation	p. 345
17.7.1 Fiber Bundle Visualization	p. 346
17.7.2 ¿-Neighbor Network Construction	p. 346
17.7.3 Network Computation	p. 349
17.8 Discussion	p. 349
18 Mixed Effects Models	p. 351
18.1 Introduction	p. 351
18.2 Mixed Effects Models	p. 352
18.2.1 Fixed Effects Model	p. 353
18.2.2 Random Effects Model	p. 354
18.2.3 Restricted Maximum Likelihood Estimation	p. 356
18.2.4 Case Study: Longitudinal Image Analysis	p. 357
18.2.5 Functional Mixed Effects Models	p. 360
Bibliography	p. 363
Index	p. 397

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