The main part of the thesis concerns the development of a spatio-temporal point process model for functional magnetic resonance imaging (fMRI) data. FMRI is a non-invasive technique for studying the active human brain. With fMRI, time series of images showing the changing blood flow in the brain associated with neural activation are acquired. FMRI data is a realisation of a complex spatio-temporal process with many sources of variation, both biological and technical. In order to model the activation of interest, it is therefore usually necessary to use highly controlled set of stimuli where the stimuli is repeated several times with resting periods in between. The aim of the analysis is then to find those areas of the brain showing increased or decreased activation during the epochs of stimuli.

With the success of experiments of this type, there is a growing interest within the neuroscience community to extend the experimental paradigm to more complex and more natural stimuli. Examples of the questions asked here are what happens in the brain during rest, meditation, or the viewing of a motion picture? Data of this type is to date usually analysed using simple correlation analysis or data driven methods such as independent component analysis. Such an analysis will though not reveal the more complicated interaction structure of the activation. This may be investigated using the spatio-temporal point process modelling approach introduced in this thesis. Here, the activation is modelled as a marked spatio-temporal point process where for each point, the location in space defines the centre of the given activation, the location in time defines the starting time of the activation, and the mark describes the duration and spatial extension. Modelling framework of this type allows for simultaneous uncertainty about both the time points and locations of activation and permits great flexibility in both the experimental design and the type of inference questions asked.

Further work presented in the thesis includes a Bayesian procedure for removing noise from images that can be viewed as noisy realisations of random sets in the plane. This procedure is based on recent advances in configuration theory and assumptions on the mean normal measure of the set are used to obtain prior probabilities of observing the different boundary configurations.