Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved to be an indispensable tool for studying medical conditions as well as for general scientific research. For example, it allows to map fiber bundles, the major neuronal pathways through the brain. But all evaluation of scanned data depends on mathematical signal models that describe the raw signal output and map it to biologically more meaningful values. And here we find the most potential for improvement. In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This gives a more detailed quantification of fibers. These additional values can also be estimated by the Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles. This reduces the effects of fiber crossings and allows a more direct examination of fibers. Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of multi-fiber signal models to a continuous distribution of fiber directions. To this approach we introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally, in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the signal can be decomposed into fibers. This means, tractography and other methods that depend on identifying a discrete set of fiber directions greatly benefit from our constraint. Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry. In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces inside the brain. Although not completely obvious, there is a mathematical obstacle coming from differential topology, that prevents general tangential planes spanned by fiber directions at each point to be connected into consistent surfaces. Research into how well this constraint is fulfilled in our brain is hindered by the high precision and complexity needed by previous evaluation methods. This is why we present a drastically simpler method that negates the need for precisely finding fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use our new method to explore and improve streamsurface visualization.<br /

Streamsurface Smoke Effect for Visualizing Dragon Fly CFD Data in Modern OpenGL with an Emphasis on High Performance

Visualizing 3D, time dependent velocity vector fields is a difficult topic. Streamlines can be used to visualize 3D vector fields. A smoke effect where the streamline is faded out as time progresses can provide a better visualization of a time dependent flow. This work uses modern OpenGL to create a smoke trail effect with streamsurfaces in the dragon fly data set. Many aspects affecting performance are tested to determine the best options or approach

Constructing streak surfaces for 3D unsteady vector fields

Visualization of 3D, unsteady flow (4D) is very difficult due to both perceptual challenges and the large size of 4D vector field data. One approach to this challenge is to use integral surfaces to visualize the 4D properties of the field. However the construction of streak surfaces has remained elusive due to problems stemming from expensive computation and complex meshing schemes. We present a novel streak surface construction algorithm that generates the surface using a quadrangular mesh. In contrast to previous approaches the algorithm offers a combination of speed for exploration of 3D unsteady flow, high precision, and places less restriction on data or mesh size due to its CPU-based implementation compared to a GPU-based method. The algorithm can be applied to large data sets because it is based on local operations performed on the quad primitives. We demonstrate the technique on a variety of 3D, unsteady simulation data sets to show its speed and robustness. We also present both a detailed implementation and a performance evaluation. We show that a technique based on quad meshes handles large data sets and can achieve interactive frame rates