Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is correcte

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

Distance function wavelets - Part I: Helmholtz and convection-diffusion transforms and series

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and the general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz-Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimension-dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge.Comment: Welcome any comments to [email protected]

Solid NURBS Conforming Scaffolding for Isogeometric Analysis

This work introduces a scaffolding framework to compactly parametrise solid structures with conforming NURBS elements for isogeometric analysis. A novel formulation introduces a topological, geometrical and parametric subdivision of the space in a minimal plurality of conforming vectorial elements. These determine a multi-compartmental scaffolding for arbitrary branching patterns. A solid smoothing paradigm is devised for the conforming scaffolding achieving higher than positional geometrical and parametric continuity. Results are shown for synthetic shapes of varying complexity, for modular CAD geometries, for branching structures from tessellated meshes and for organic biological structures from imaging data. Representative simulations demonstrate the validity of the introduced scaffolding framework with scalable performance and groundbreaking applications for isogeometric analysis

Geometric properties of electromagnetic waves

This work studies geometrical properties of electromagnetic wave propagation. The work starts by studying geometrical properties of electromagnetic Gaussian beams in inhomogeneous anisotropic media. These are asymptotical solutions to Maxwell's equations that have a very characteristic feature. Namely, at each time instant the entire energy of the solution is concentrated around one point in space. When time moves forward, a Gaussian beam propagates along a curve. In recent work by A. P. Kachalov, Gaussian beams have been studied from a geometrical point of view. Under suitable conditions on the media, Gaussian beams propagate along geodesics. Furthermore, the shape of a Gaussian beam is determined by a complex tensor Riccati equation. The first paper of this dissertation provides a partial classification of media where Gaussian beams geometrize. The second paper shows that the real part of a solution to the aforementioned Riccati equation is essentially the shape operator for the phase front for the Gaussian beam. An important phenomena for electromagnetic Gaussian beams is that their propagation depend on their polarization. The last paper studies this phenomena from a very general point of view in arbitrary media. It also studies a connection between contact geometry and electromagnetism.reviewe

Curve and surface framing for scientific visualization and domain dependent navigation

Thesis (Ph.D.) - Indiana University, Computer Science, 1996Curves and surfaces are two of the most fundamental types of objects in computer graphics. Most existing systems use only the 3D positions of the curves and surfaces, and the 3D normal directions of the surfaces, in the visualization process. In this dissertation, we attach moving coordinate frames to curves and surfaces, and explore several applications of these frames in computer graphics and scientific visualization. Curves in space are difficult to perceive and analyze, especially when they are densely clustered, as is typical in computational fluid dynamics and volume deformation applications. Coordinate frames are useful for exposing the similarities and differences between curves. They are also useful for constructing ribbons, tubes and smooth camera orientations along curves. In many 3D systems, users interactively move the camera around the objects with a mouse or other device. But all the camera control is done independently of the properties of the objects being viewed, as if the user is flying freely in space. This type of domain-independent navigation is frequently inappropriate in visualization applications and is sometimes quite difficult for the user to control. Another productive approach is to look at domain-specific constraints and thus to create a new class of navigation strategies. Based on attached frames on surfaces, we can constrain the camera gaze direction to be always parallel (or at a fixed angle) to the surface normal. Then users will get a feeling of driving on the object instead of flying through the space. The user's mental model of the environment being visualized can be greatly enhanced by the use of these constraints in the interactive interface. Many of our research ideas have been implemented in Mesh View, an interactive system for viewing and manipulating geometric objects. It contains a general purpose C++ library for nD geometry and supports a winged-edge based data structure. Dozens of examples of scientifically interesting surfaces have been constructed and included with the system

Fast exact and approximate geodesics on meshes

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.Engineering and Applied Science