Bayesian Statistical Models of Shape and Appearance for Subcortical Brain Segmentation

251 Pages·2008·14.57 MB·English

Bayesian Statistical Models of Shape and
Appearance for Subcortical Brain Segmentation
Brian Patenaude
Trinity Term, 2007
Oxford Centre for Functional Magnetic Resonance Imaging of the Brain
Department of Clinical Neurology
University of Oxford
To my loving wife and son, Amanda and Sebastian, the two lights of my life.
Bayesian Statistical Models of Shape and
Appearance for Subcortical Brain Segmentation
Brian Patenaude
Worcester College
University of Oxford
Abstract
Our motivation is to develop an automated technique for segmentation of sub-cortical
human brain structures from MR images. To this purpose, models of shape-and-
appearance are constructed and fit to new image data. The statistical models are
trained from 317 manually labelled T1-weighted MR images. Shape is modelled us-
ing a surface-based point distribution model (PDM) such that the shape space is
constrained to the linear combination of the mean shape and eigenvectors of the
vertex coordinates. In addition, to model intensity at the structural boundary, inten-
sities are sampled along the surface normal from the underlying image. We propose a
novel Bayesian appearance model whereby the relationship between shape and inten-
sity are modelled via the conditional distribution of intensity given shape. Our fully
probabilistic approach eliminates the need for arbitrary weightings between shape
and intensity as well as for tuning parameters that specify the relative contribution
between the use of shape constraints and intensity information. Leave-one-out cross-
validation is used to validate the model and fitting for 17 structures.
The PDM for shape requires surface parameterizations of the volumetric, manual la-
belssuchthatverticesretainaone-to-onecorrespondenceacrossthetrainingsubjects.
Surface parameterizations with correspondence are generated through the use of de-
formable models under constraints that embed the correspondence criterion within
the deformation process. A novel force that favours equal-area triangles throughout
the mesh is introduced. The force adds stability to the mesh such that minimal
smoothing or within-surface motion is required.
The use of the PDM for segmentation across a series of subjects results in a set
surfaces that retain point correspondence. The correspondence facilitates landmark-
based shape analysis. Amongst other metrics, vertex-wise multivariate statistics and
discriminant analysis are used to investigate local and global size and shape differ-
ences between groups. The model is fit, and shape analysis is applied to two clinical
datasets.
Thesis submitted for the degree Doctorate of Philosophy
at the University of Oxford
Trinity Term, 2007
Acknowledgements
We would like to thank David Kennedy and the Center for Morphometric Analysis
(CMA) for providing the manually labelled T1-weighted images, without which this
work would not have been possible. I would also like to thank Chirstian Haselgrove
for the pratical aspects of procuring the data. Also, thanks to all the individual re-
searchers that contributed their data through the CMA.
We are very grateful to Nick Fox and Richard Boyes, Dementia Research Centre,
Institute of Neurology, UCL, UK for the MIRIAD data.
Also, to Dr. Clare Mackay, Prof. Tim Crow and MRC program “Brain Development
and Structure in Psychosis” for providing the schizophrenia data that was used in
chapter 4.
I would also like to acknowledge the ESPRC, IBIM grant for funding. I would like
to thank the members of the IBIM collaboration, all of whom have influenced this
research in some facet.
I would like to give a special thanks to Mark and Steve for providing me the oppor-
tunity to pursue my D.Phil as well as for their guidance and input into this research.
Finally, I would like to thank Amanda and Sebastian for being troopers through the
final stages of thesis, and for helping me keep my sanity.
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Contents
1 Introduction 2
1.1 T -weighted MR Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 4
1
1.1.1 Bias Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Partial Volume Effects . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Neuroanatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Manual Segmentation Protocol . . . . . . . . . . . . . . . . . 12
1.2.2 The Brainstem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 The Thalamus . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.4 The Basal Ganglia . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.5 Limbic System . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iii
iv
1.2.6 Ventricular System . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3 Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Training Data, Pre-Processing and Surface Parameterization 40
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 The Point Distribution Model and Point Correspondence . . . . . . . 44
2.3 Mesh Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Marching Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.2 Deformable Models . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 Methods for Establishing Point Correspondence . . . . . . . . 53
2.4 Training Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Intensity Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Linear Subcortical Registration . . . . . . . . . . . . . . . . . . . . . 63
2.7 Surface Parameterization with Embedded Correspondence Using 3D
Deformable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.1 Structural Pre-Alignment . . . . . . . . . . . . . . . . . . . . 69
v
2.7.2 The Deformation Process . . . . . . . . . . . . . . . . . . . . . 71
2.7.3 Mesh Initialization . . . . . . . . . . . . . . . . . . . . . . . . 75
2.8 Evaluation of Parameterization Accuracy . . . . . . . . . . . . . . . . 79
2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3 Bayesian Models of Shape and Appearance 88
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.1.1 The Issue of Dimensionality . . . . . . . . . . . . . . . . . . . 93
3.1.2 Review of Segmentation Methods . . . . . . . . . . . . . . . . 95
3.2 Bayesian Shape and Appearance Model . . . . . . . . . . . . . . . . . 107
3.2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 108
3.2.2 Choice of priors . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2.3 Conditional distributions . . . . . . . . . . . . . . . . . . . . . 114
3.2.4 Parameterization of Bayesian Models from Finite Training Data 116
3.2.5 Bayesian Appearance Models . . . . . . . . . . . . . . . . . . 118
3.3 Model Fitting and Evaluation . . . . . . . . . . . . . . . . . . . . . . 120
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3.3.1 Posterior as a Cost Function . . . . . . . . . . . . . . . . . . . 120
3.3.2 Conditional Shape Priors . . . . . . . . . . . . . . . . . . . . . 123
3.3.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.3.4 Computational Simplifications . . . . . . . . . . . . . . . . . . 125
3.3.5 Validation and Accuracy . . . . . . . . . . . . . . . . . . . . . 127
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4 Size and Shape Analysis from Surfaces with Vertex Correspondence141
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2 The General Linear Model (GLM), Univariate and Multivariate Tests. 144
4.3 Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.3.1 Linear and Quadratic Discriminant Analysis . . . . . . . . . . 151
4.3.2 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3.3 Support Vector Machine . . . . . . . . . . . . . . . . . . . . . 154
4.3.4 Relevance Vector Machines . . . . . . . . . . . . . . . . . . . . 154
4.4 Assessment of Performance . . . . . . . . . . . . . . . . . . . . . . . . 156
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4.4.1 N-fold Cross-Validation . . . . . . . . . . . . . . . . . . . . . . 158
4.4.2 Boot Strapping . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.5 Literature Review of Discriminant Analysis in Neuroimaging. . . . . . 162
4.6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.6.1 Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.6.2 Structural Segmentation . . . . . . . . . . . . . . . . . . . . . 168
4.6.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 169
4.6.4 Choice of Discriminants and Assessment of Prediction Error . 169
4.6.5 Spatial Normalization and Metrics for Size and Shape . . . . 170
4.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.7.1 Controls versus AD patients . . . . . . . . . . . . . . . . . . . 177
4.7.2 Controls versus Schizophrenia . . . . . . . . . . . . . . . . . . 186
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5 Conclusions and Future Work 195
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
viii
5.1.1 Mesh parameterization using deformable models . . . . . . . . 195
5.1.2 Bayesian Shape and Appearance Models . . . . . . . . . . . . 198
5.1.3 Shape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A Registration of Statistical Shape Models 218
B Simplification of the Posterior 220
C Computational Simplifications 223
C.1 Conditional Mean as a Mode of Variation . . . . . . . . . . . . . . . . 224
C.2 Simplifying Conditional Covariance Operations . . . . . . . . . . . . 225
C.2.1 Simplifying (x −µ )TΣ−1(x −µ ) . . . . . . . . . . . . . . 226
2 2 22 2 2
C.2.2 Simplifying Σ . . . . . . . . . . . . . . . . . . . . . . . . . . 226
c11
C.2.3 Simplifying the calculation of (x −µ )TΣ−1(x −µ ) . . . 228
1 1|2 1|2 1 1|2
D Calculating Conditional Mode Parameters 230
E Results from Leave-One-Out Cross Validation 233