A Riemannian manifold is a mathematical concept that generalizes curved surfaces to higher dimensions, giving a precise meaning to concepts like angle, length, area, volume and curvature. A glimpse of the consequences of a non-flat geometry is given on the sphere, where the shortest path between two points – a geodesic – is along a great circle. Different from Euclidean space, the angle sum of geodesic triangles on the sphere is always larger than 180 degrees. Signals and data found in applied research are sometimes naturally described by such curved spaces. This dissertation presents basic research and tools for the analysis, processing and visualization of such manifold-valued data, with a particular emphasis on future applications in medical imaging and visualization. Two-dimensional manifolds, i.e. surfaces, enter naturally into the geometric modelling of anatomical entities, such as the human brain cortex and the colon. In advanced algorithms for processing of images obtained from computed tomography (CT) and ultrasound imaging (US), images themselves and derived local structure tensor fields may be interpreted as two- or three-dimensional manifolds. In diffusion tensor magnetic resonance imaging (DT-MRI), the natural description of diffusion in the human body is a second-order tensor field, which can be related to the metric of a manifold. A final example is the analysis of shape variations of anatomical entities, e.g. the lateral ventricles in the brain, within a population by describing the set of all possible shapes as a manifold. Work presented in this dissertation include: Probabilistic interpretation of intrinsic and extrinsic means in manifolds. A Bayesian approach to filtering of vector data, removing noise from sampled manifolds and signals. Principles for the storage of tensor field data and learning a natural metric for empirical data. The main contribution is a novel class of algorithms called LogMaps, for the numerical estimation of logp (x) from empirical data sampled from a low-dimensional manifold or geometric model embedded in Euclidean space. The logp (x) function has been used extensively in the literature for processing data in manifolds, including applications in medical imaging such as shape analysis. However, previous approaches have been limited to manifolds where closed form expressions of logp (x) have been known. The introduction of the LogMap framework allows for a generalization of the previous methods. The application of LogMaps to texture mapping, tensor field visualization, medial locus estimation and exploratory data analysis is also presented. / The electronic version is corrected for grammatical and spelling errors.

Magnetic resonance imaging (MRI) is an imaging technique to visualize internal structures of the body. Diffusion MRI is an MRI modality that measures overall diffusion effect of molecules in vivo and non-invasively. Diffusion tensor imaging (DTI) is an extended technique of diffusion MRI. The major application of DTI is to measure the location, orientation and anisotropy of fiber tracts in white matter. It enables non-invasive investigation of major neural pathways of human brain, namely tractography. As spatial resolution of MRI is limited, it is possible that there are multiple fiber bundles within the same voxel. However, diffusion tensor model is only capable of resolving a single direction. The goal of this dissertation is to investigate complex anatomical structures using high angular resolution diffusion imaging (HARDI) data without any assumption on the parameters. The dissertation starts with a study of the noise distribution of truncated MRI data. The noise is often not an issue in diffusion tensor model. However, in HARDI studies, with many more gradient directions being scanned, the number of repetitions of each gradient direction is often small to restrict total acquisition time, making signal-to-noise ratio (SNR) lower. Fitting complex diffusion models to data with reduced SNR is a major interest of this study. We focus on fitting diffusion models to data using maximum likelihood estimation (MLE) method, in which the noise distribution is used to maximize the likelihood. In addition to the parameters being estimated, we use likelihood values for model selection when multiple models are fit to the same data. The advantage of carrying out model selection after fitting the models is that both the quality of data and the quality of fitting results are taken into account. When it comes to tractography, we extend streamline method by using covariance of the estimated parameters to generate probabilistic tracts according to the uncertainty of local tract orientations.