Multi-valued geodesic tractography for diffusion weighted imaging

Diffusion-Weighted Imaging (DWI) is a Magnetic Resonance(MR) technique that measures water diffusion characteristics in tissue for a given direction. The diffusion profile in a specific location can be obtained by combining the DWI measurements of different directions. The diffusion profile gives information about the underlying fibrous structure, e.g., in human brain white matter, based on the assumption that water molecules are moving less freely perpendicularly to the fibrous structure. From the DW-MRI measurements often a positive definite second-order tensor is defined, the so-called diffusion tensor (DT). Neuroscientists have begun using diffusion tensor images (DTI) to study a host of various disorders and neurodegenerative diseases including Parkinson, Alzheimer and Huntington. The techniques for reconstructing the fiber tracts based on diffusion profiles are known as tractography or fiber tracking. There are several ways to extract fibers from the raw diffusion data. In this thesis, we explain and apply geodesic-based tractography techniques specifically, where the assumption is that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. A shortest path in this manifold is one with the strongest diffusion along this path. Therefore geodesics (i.e., shortest paths) on this manifold follow the most efficient diffusion paths. The geodesics are often computed from the stationary Hamilton-Jacobi equation (HJ). One characteristic of solving the HJ equation is that it gives only the single-valued viscosity solution corresponding to the minimizer of the length functional. It is also well known that the solution of the HJ equation can develop discontinuities in the gradient space, i.e., cusps. Cusps occur when the correct solution should become multi-valued. HJ methods are not able to handle this situation. To solve this, we developed a multi-valued solution algorithm for geodesic tractography in a metric space defined by given by diffusion tensor imaging data. The algorithm can capture all possible geodesics arriving at a single voxel instead of only computing the first arrival. Our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points. In this thesis, we first focus on the mathematical and numerical model for analytic and synthetic fields in twodimensional domains. Later, we present the algorithm in three-dimensions with examples of synthetic and brain data. Despite the simplicity of the DTI model, the tractography techniques using DT are shown to be very promising to reveal the structure of brain white matter. However, DTI assumes that each voxel contains fibers with only one main orientation and it is known that brain white matter has multiple fiber orientations, which can be arbitrary many in arbitrary directions. Recently, High Angular Resolution Diffusion Imaging (HARDI) acquisition and its modeling techniques have been developed to overcome this limitation. As a next contribution we propose an extension of the multi-valued geodesic algorithm to HARDI data. First we introduce the mathematical model for more complex geometries using Finsler geometry. Next, we propose, justify and exploit the numerical methods for computing the multi-valued solution of these equations

Exceptional Points in a Non-Hermitian Topological Pump

We investigate the effects of non-Hermiticity on topological pumping, and uncover a connection between a topological edge invariant based on topological pumping and the winding numbers of exceptional points. In Hermitian lattices, it is known that the topologically nontrivial regime of the topological pump only arises in the infinite-system limit. In finite non-Hermitian lattices, however, topologically nontrivial behavior can also appear. We show that this can be understood in terms of the effects of encircling a pair of exceptional points during a pumping cycle. This phenomenon is observed experimentally, in a non-Hermitian microwave network containing variable gain amplifiers.Comment: 7 pages, 7 figures. The first author did the experiment, and the second author did the theoretical stud

A Survey of Ekeland\u27s Variational Principle and Related Theorems and Applications

Ekeland\u27s Variational Principle has been a key result used in various areas of analysis such as fixed point analysis, optimization, and optimal control theory. In this paper, the application of Ekeland\u27s Variational Principle to Caristi\u27s Fixed Point Theorem, Clarke\u27s Fixed Point Theorem, and Takahashi\u27s Minimization theorem is the focus. In addition, Ekeland produced a version of the classical Pontryagin Mini- mum Principle where his variational principle can be applied. A further look at this proof and discussion of his approach will be contrasted with the classical method of Pontryagin. With an understanding of how Ekeland\u27s Variational Princple is used in these settings, I am motivated to explore a multi-valued version of the principle and investigate its equivalence with a multi-valued version of Caristi\u27s Fixed Point Theorem and Takahashi\u27s Minimization theorem

Central Limit Theory for Lipschitz Mappings

Central limit theorems are derived for mappings that are Lipschitzian at a given point. This theory results from a new perspective on first-order behavior -- the upper pseudo-derivative, the graph of which is the contingent cone to the graph of the mapping at a given point. We adopt the general setting of the convergence in distribution of measures induced by mappings that may be multi-valued on sets of measure zero. By requiring the upper pseudo-derivative to be single-valued a.s., we obtain a central limit theorem under distinctively weaker conditions than classical Frechet differentiability