Introduction

Geometric registration and motion correction are important stages in the analysis of functional brain imaging studies. Consequently, it is important that these stages perform robustly and accurately. Furthermore, for large imaging studies it is desirable that they be fully automated. There has been a considerable amount of research into registration and motion correction of brain images, and many different methods have been proposed [14]. Most methods in common usage are based on the mathematical framework of optimising an intensity-based cost function. However, although much work has concentrated on how the choice of cost function affects registration performance, there has been far less examination of the effect of the optimisation method. Moreover, when optimisation methods are discussed, global methods are often ignored and local methods compared purely on the basis of speed [4]. One of the most common and serious problems for registration methods is the presence of local minima in the cost function. These cause local optimisation methods to ``get stuck'' and hence to fail to find the desired the global minimum. Most registration methods attempt to solve this problem by incorporating a local optimisation strategy within a multi-resolution framework. Such a multi-resolution framework, which typically involves starting with low resolution images (containing only gross features) and working progressively through to higher resolutions, aims to avoid the local minima ``traps''. As we show later, this simple multi-resolution approach is not always sufficient for avoiding local minima, and that by using more sophisticated optimisation methods, the chances of becoming ``trapped'' in these local minima can be substantially reduced. Two types of local minima commonly occur for the cost functions used in image registration: large scale basins and small scale dips. The first type, the large scale basin, is responsible for large mis-registrations since the local minimum is often far from the global minimum. The second type, small scale dips, can cause the optimisation to get stuck at any stage and so are responsible for large mis-registrations at low resolutions and for small mis-registrations at high resolutions. We propose two methods for dealing with the local minima problem. These are: cost function apodization, which reduces or eliminates small scale dips; and a hybrid global-local optimisation technique which utilises prior knowledge about brain registration to create an optimisation technique that combines the speed of local optimisation with the robustness of global optimisation. The following sections of this paper are: background theory, methods (including both cost function apodization and the hybrid optimisation method), results and discussion. The results section contains a number of experiments on real, whole brain images which demonstrate the effectiveness of the registration in two different settings: (1) structural image registration (inter-modal/inter-subject) of an anatomical image to a standard template; and (2) functional image motion correction (intra-modal/intra-subject) which registers each image in a time-series to a particular example image from that time-series. The first case is examined using a robustness study (as accuracy is hard to define for inter-subject registration, and robustness is a more important issue in this context), while the second case is examined using an accuracy study (as, in this context, it is accuracy that is more important). In each case real brain image data is used. Comparisons with some commonly used methods are also included (in both cases) which demonstrate the superior robustness and accuracy which can be obtained using this approach.