Multi-scale Riemann-Finsler geometry : applications to diffusion tensor imaging and high angular resolution diffusion imaging

In this brief summary, we reflect the goals and the achievements of this PhD-project by citing three sentences in the original project proposal. "This project investigates the exploitation of the scale degree of freedom in images from the vantage point of differential geometry and tensor calculus in scale space." Indeed, this thesis introduces the necessary tools for differential geometric tensor calculus and derives some theoretical as well as practical results. The scale is considered separately in two different settings. In this thesis, the weight is somewhat shifted from the problem of scale to problems in differential geometry and applications. The applications considered here are multi-directional medical images, namely diffusion weighted magnetic resonance images. The measurements are modeled using symmetric tensors, which is equivalent to a polynomial approximation. This thesis can be roughly separated in to two parts: one that studies second order tensor fields and one that uses higher order tensor fields. The second order tensor fields are studied with tools from Riemann geometry, and the higher order ones respectively with those from Finsler geometry. Each of these can be separated to a theoretical part, that contain the mathematical definitions and derivations and an applied part, where some geometric properties of synthetic, simulated or real data are computed and analyzed. "The goal is to couple geometry to image content based on a specific task." By attaching geometric meaning to the physical properties (of the imaged object) represented by data, we have derived some measures and algorithms to extract information from images. For example a novel method to do fiber tractography, i.e. extract neural connections from diffusion weighted images of brain, is introduced. This method has the special property, that it can propagate through voxels with complex fiber orientations. Techniques to measure relative diffusivity along a curve and to detect inhomogeneities in tensor field are some of the other examples. Since the real data is discrete, the interpolation of tensor fields is also considered."The objective is to foster specific applications in biomedical image analysis, and to extend these to multiple scales." In the Riemannian framework, the concept of scale is introduced in a Gaussian derivative scheme to second order tensor fields. In Finslerian context, a scale parameter was attached to higher order tensors by applying Laplace-Beltrami smoothing, solving the heat equation on the sphere