In Section 1.8 we showed there were advantages to performing Procrustes registration using only the midline landmarks when the template for the other landmarks was incorrect. In Chapter 2 we consider the application of Procrustes methods to a subset of the available landmarks in more detail. We formulate results regarding the distribution of the landmarks in the sub- set and in the complement when matching to both the correct and incorrect templates. Further, we extend Procrustes methods to enable registration with respect to any symmetric positive definite weighting matrix, Σ.
The choice of Σ will depend on the application and aim of the analysis but in a general setting Σ would represent the variance-covariance matrix of the data. Typically, estimation of shape variability is carried out af- ter isotropic Procrustes registration. However, Chapter 3 seeks to jointly estimate the registration parameters, the mean shape and the shape vari- ability, with the aim of using the estimated shape variability to influence the registration. Both maximum likelihood and Bayesian methods are de- veloped. Their ability to succeed relies on a priori information in the form of constraints in the maximum likelihood case, and prior distributions for the Bayesian MCMC algorithm. The methods are compared to isotropic Procrustes methods through a simulation study.
Section 1.8 also introduced the concept of asymmetry commonly found in the human brain. In Chapter 4, we apply a measure of symmetry to a data set consisting of MRI scans. The volunteers in the study were either healthy controls or patients with schizophrenia, and we use asymmetry to establish differences in the shape of the cortical surface between the two patient groups. Comparisons are only possible if the images have been cor- rectly registered to a template, so we develop brain registration algorithms, again using both maximum likelihood and Bayesian frameworks. Both al- gorithms make use of symmetry around the midline given that asymmetric features on the cortical surface have already been seen to lead to inaccurate
registrations. The technique for analysis we develop is compared to the existing technique of voxel-based morphometry.
In Chapter 5 we consider the more complicated problem of analysing a 4-dimensional data set created by a series of brain images acquired over a few minutes. This presents the additional challenge of registering the data temporally, as well as spatially. Following registration, however, we can analyse brain function by examining the resultant time series of oxy- gen levels at each voxel. We develop a statistical model for analysing the changes in response to a repeated stimuli, with errors correlated in time as well as space, and use the EM algorithm to maximise the likelihood of the model. The aims are to establish which voxels are activated by the stimulus and to quantify the changes in response through time. Improvements to current data pre-processing methods are also suggested, and the analysis is compared to the results of existing techniques that use statistical linear models and independent components analysis. The creation and analysis of functional imaging data is still developing. The 7T MRI scanner at the University of Nottingham, which came online in September 2005, is one of the two most powerful scanners in Europe. The increased power raises the signal to noise ratio, allowing single trial variability to be examined for the first time.
Lastly, in Chapter 6 we draw some conclusions and discuss potential areas for future research.
Calculations, graphics and results in this thesis have been conducted and produced using software packages R (R Development Core Team, 2005), including the shapes library, and Matlab (MathWorks, Natick, MA, USA). In Chapters 4 and 5, existing software packages, SPM2 (e.g. Friston et al., 1995a) and FSL (Smith et al., 2004), have also been used for data pre- processing and making comparisons between methods.
Weighted Procrustes analysis
2.1
Introduction
The standard approach to Procrustes analysis, outlined in Chapter 1, as- sumes that all landmarks are included in the analysis and an equal weighting is given to each. We previously demonstrated in Section 1.8 the need for techniques that do not make these assumptions. In this chapter we develop a weighted Procrustes registration method for shapes with k landmarks in m dimensions. We start by examining a special case, where we divide the landmarks into two sets and use only one subset to estimate the registration parameters. We formulate the distribution of the sum of squared Euclidean distances between a registered shape and a template for both subsets.
This application of isotropic Procrustes is then generalised to allow for a weighted Procrustes technique based on a km × km matrix, Σ, where the only restriction is that Σ is symmetric positive definite. This method is formulated for matching one shape to a template (ordinary Procrustes) and for registering multiple shapes to each other (generalised Procrustes). An earlier summary of these methods is presented in Brignell et al. (2005). A weighted registration is applicable in brain registration where the cortical surface is known to be more variable than other areas. Richmond et al.
(2004) consider the registration of molecules where the landmarks (atoms) are defined by charge as well as geometrical location. One method, amongst others, of applying a weighted Procrustes registration could incorporate the compatibility of atom charges in the weighting matrix. The results given in this chapter lay the groundwork for Chapter 3, where we estimate the shape covariance matrix and recursively use this estimate in place of Σ to find optimum registration parameters.