Anatomical Standardization of the Human Brain in Euclidean 3-Space and on the Cortical 2-Manifold

315 Pages·2004·12.09 MB·English

Anatomical Standardization of the Human Brain in
Euclidean 3-Space and on the Cortical 2-Manifold
Steven M. Robbins
School of Computer Science
McGill University, Montreal
October 2003
A thesis submitted to McGill University in partial fulfilment of the requirements of
the degree of Doctor of Philosophy.
(cid:13)c Steven M. Robbins 2003
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Abstract
Anatomical standardization (also called spatial normalization) is a key process in
cross-sectional studies of brain structure and function using MRI, fMRI, PET and
other imaging techniques. This process has two components: (i) specification of a
3D template brain, which defines a common coordinate space for analysis of any
subsequent datasets; and (ii) a method to align the template with an individual 3D
brain image, thereby associating each point of the standard template to a point on
the individual. The association should be able to consistently match a particular
template location to an anatomically corresponding location on each individual of a
population.
Standardization methods in widespread use employ a 3D affine spatial transform-
ation to map from the individual to the template, which matches only overall size
and gross shape of the input brain. A wide range of more flexible image deformation
algorithms have been developed in order to better match fine detail. All such al-
gorithms involve design choices that are subject to debate, and most have numerical
parameters whose value must be specified by the user. In order to provide guidance
for such choices, the first part of this thesis develops two measures of alignment con-
sistency that are used to evaluate performance of a standardization method. The
performance of different choices for algorithm design, numerical parameters, and
template selection strategy for 3D normalization are compared.
Since the processing of brain function occurs on a thin, highly convoluted sheet
of cortex along the surface of the brain, there has been much recent interest in
studying the structure and function along the brain cortex only, modelled as a 2D
manifold. The second part of this thesis proposes an algorithm for highly-flexible
deformation in 2D of a template cortex to an individual. The alignment consistency
measuresdevelopedfor3Darereformulatedforthe2Dmanifoldandusedtoevaluate
the algorithm design and numerical parameters. Finally, the question of whether it
is better to standardize the 3D images or the 2D cortical manifold is addressed,
identifying the problem classes which are best suited to each type of normalization.
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Pr´ecis
La standardisation anatomique (´egalement appell´ee normalisation spatiale) est un
processus-cl´edansles´etudestrans-sectionnellesdestructureetdefonctionsc´er´ebrales
en imagerie par r´esonance magn´etique (IRM), IRM fonctionnelle, (IRMf), tomo-
graphie par´emission de positron (TEP) ainsi que d’autres techniques d’imagerie. Ce
processus est compos´e de deux parties: (i) la sp´ecification d’un cadre de r´eference
tri-dimensionnel(3D),quid´efinitunespacedecoordonn´eescommunespourl’analyse
subs´equentes des donn´ees; et (ii) une m´ethode pour aligner ce cadre avec l’image 3D
d’un cerveau individuel. Cette derni`ere permet d’associer `a chaque point du cadre
de r´eference commun un point dans un cerveau individuel. Cette association devrait
ˆetre capable de mettre en correspondance de fa¸con constante un endroit particulier
du cadre `a un endroit anatomique correspondant et ce, pour chaque cerveau dans
une population donn´ee.
Les m´ethodes de standardisation les plus employ´ees utilisent une transformation
spatiale affine en 3D pour mettre en correspondance le cerveau d’un individu avec
le cadre de r´eference commun. Cette m´ethode ne fait qu’ajuster la taille totale et la
forme globale du cerveau. Un large ´eventail d’algorithmes de d´eformation d’images,
plus flexibles, ont ´et´e d´evelopp´es pour faciliter l’ajustement de d´etails locaux. Tous
ces algorithmes impliquent des choix m´ethodologiques qui sont sujet `a d´ebat, et la
plupart ont des param`etres num´eriques dont les valeurs doivent ˆetre sp´ecifi´ees par
l’usager. Pour pouvoir guider ces choix, deux m´ethodes de mesure de la constance de
l’alignement, utilis´ees pour´evaluer la performance de la m´ethode de standardisation,
ont ´et´e d´evelopp´ees et sont pr´esent´ees dans la premi`ere partie de cette th`ese. La
performance de diff´erents choix m´ethodologiques pour les algorithmes, de diff´erents
param`etres num´eriques et de strat´egie de s´election du cadre de r´ef´erence commun
pour la normalisation 3D est ensuite compar´ee.
Puisque le processus de l’activit´e c´er´ebrale se d´eroule sur une mince couche de
cortex tr`es pliss´ee le long de la surface du cerveau, un grand int´erˆet existe dans
l’´etude de la structure et de l’activit´e le long du cortex c´er´ebral, mod´elis´e comme
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une vari´et´e 2D. La deuxi`eme partie de cette th`ese propose un algorithme de d´eforma-
tion 2D ultra-flexible d’un cadre de r´ef´erence cortical commun `a un cortex individuel.
Les mesures de constance de l’alignement d´evelopp´ees en 3D sont reformul´ees pour la
vari´et´e 2D et utilis´ees pour ´evaluer la conception de l’algorithme ainsi que ses para-
m`etres num´eriques. Finalement, la question `a savoir si il est mieux de standardiser
des images 3D ou une vari´et´e corticale 2D est pr´esent´ee, en identifiant les classes de
probl`emes pour lesquelles chaque type de normalisation serait le mieux adapt´e.
Acknowledgements
I’d like to thank, first of all, my co-supervisors Sue Whitesides and Alan Evans. This
thesis would not exist without their guidance and support over the years.
This thesis builds on the wealth of software created at the MNI Brain Imaging
Centre, specifically the MINC tools developed by Peter Neelin, the surface extraction
program, ASP, of David MacDonald, and the 3D registration code, ANIMAL, of
Louis Collins. I’m grateful to them all, for sharing both their code and their insights.
Thanks also are due to Jean-Fran¸cois Malouin, Sylvain Milot, and Dale Einarson
for keeping the computing lab going over the years, even under the strain of the
thousands of experiments that this thesis represents.
I owe a great debt to Jurgen Germann, who generously shared his painstaking
labelling of the frontal lobe sulci. Additional sulcal data kindly supplied by Arnaud
Cachia and Jean-Fran¸cois Mangin was a great help.
Above all, it is the people, both in computer science and at the BIC who make
these departments great places in which to work. I’ve had informative and enjoy-
able discussions with many in the lab, notably Jason Lerch, Alex Zijdenbos, Chris
Cocosco, Serge Dumoulin, Andrew Janke, Vivek Singh, Richard Webster, Yasser
Ad-Dab’bagh, and Oliver Lyttelton. I’m grateful to them all.
Finally, I’d like to thank my family for their understanding and for trying hard
not to ask “is it done yet?” too often during the final stages. Most appreciated
of all during my long sojourn has been the love, support, and patience of my wife,
Sutanuka Bhattacharya.
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Contents
1 Introduction 1
1.1 Brain Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Magnetic Resonance Images . . . . . . . . . . . . . . . . . . . 2
1.1.2 Spatial Normalization . . . . . . . . . . . . . . . . . . . . . . 4
Talairach Coordinates . . . . . . . . . . . . . . . . . . . . . . 5
Transformation Function . . . . . . . . . . . . . . . . . . . . . 6
1.2 3D Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Cortical Surface Registration . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 About the Thesis Organization . . . . . . . . . . . . . . . . . . . . . 9
1.5 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I 3D Spatial Normalization 11
2 Spatial Transformation 13
2.1 Image Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Auxiliary Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Low-Dimensional Warping . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 High-Dimensional Warping . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Global Transformations . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Local Transformations . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Searching for an Optimal Transformation 25
3.1 Coordinate Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Data Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Labelled Features . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Unlabelled Features . . . . . . . . . . . . . . . . . . . . . . . . 32
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Geometric Feature . . . . . . . . . . . . . . . . . . . . . . . . 32
Image Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Regional Similarity Measure . . . . . . . . . . . . . . . . . . . 37
3.2.3 Multiple Features . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Model Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Regularization Rationale . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 Low-Order Differential Regularizers . . . . . . . . . . . . . . . 41
3.4 Iterative Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Landmark Matching . . . . . . . . . . . . . . . . . . . . . . . 43
Discretized Euler-Lagrange Equations . . . . . . . . . . . . . . 44
Discretized Objective Function . . . . . . . . . . . . . . . . . 45
3.4.2 Two-Step Registration . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3 Four-Step Registration . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Coarse-to-Fine Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Brain Image Registration Literature . . . . . . . . . . . . . . . . . . . 52
3.6.1 Labelled Features . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.2 Unlabelled Features . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.3 Multiple Features . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Algorithm for 3D Experiments: ANIMAL . . . . . . . . . . . . . . . 56
3.7.1 Outer Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.7.2 Inner Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8.1 Data Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8.2 Model Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8.3 Registration Algorithm . . . . . . . . . . . . . . . . . . . . . . 62
4 Quantifying Spatial Variability 63
4.1 Segmentation Variability . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 Application: Evaluating Algorithm Design . . . . . . . . . . . 65
Outer Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Data Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Miscellaneous Parameters . . . . . . . . . . . . . . . . . . . . 73
Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75