Radicals in arithmetic

Let K be a field. A radical is an element of the algebraic closure of K of which a power is contained in K. In this thesis we develop a method for determining what we call entanglement. This describes unexpected additive relations between radicals, and is encoded in an entanglement group. We give methods for computing the entanglement group, and show how to use these to compute field degrees of radical extensions over the field of rationals. Moreover, we show that these methods give rise to a new explicit method for computing the correction factor in Artin's primitive root conjecture, in a way that more readily admits different generalizations than traditional methods. In chapters 5 and 6 we show how our approach applies to a number of such generalizations of Artin's conjecture. Specifically, we study near-primitive roots, higher rank analogues, and the setting of rank one tori. The last chapter covers an entirely separate topic, and describes an algorithm for enumerating so-called ABC triples. It also reports results from the ABC@home project, a volunteer computing project that has used this algorithm to enumerate all ABC triples up to 10^1

Radicals in arithmetic

Let K be a field. A radical is an element of the algebraic closure of K of which a power is contained in K. In this thesis we develop a method for determining what we call entanglement. This describes unexpected additive relations between radicals, and is encoded in an entanglement group. We give methods for computing the entanglement group, and show how to use these to compute field degrees of radical extensions over the field of rationals. Moreover, we show that these methods give rise to a new explicit method for computing the correction factor in Artin's primitive root conjecture, in a way that more readily admits different generalizations than traditional methods. In chapters 5 and 6 we show how our approach applies to a number of such generalizations of Artin's conjecture. Specifically, we study near-primitive roots, higher rank analogues, and the setting of rank one tori. The last chapter covers an entirely separate topic, and describes an algorithm for enumerating so-called ABC triples. It also reports results from the ABC@home project, a volunteer computing project that has used this algorithm to enumerate all ABC triples up to 10^18UBL - phd migration 201

A distributed SIRT implementation for the ASTRA Toolbox

The ASTRA Toolbox is a software toolbox that enables rapid development of GPU accelerated tomography algorithms. It contains GPU implementations of forward and backprojection operations for common scanning geometries, as well as a set of algorithms for iterative reconstruction. These algorithms are currently limited to using a single GPU. A drawback of iterative reconstruction algorithms is that they are slow compared to classical backprojection algorithms. As a result, using only a single GPU can result in prohibitively long reconstruction times when working with large data volumes. In this paper, we present an extension of the ASTRA Toolbox with implementations of forward projection, backprojection and the SIRT algorithm that can be distributed over multiple GPUs and multiple workstations to make processing larger data sets with ASTRA feasible

EDS tomographic reconstruction regularized by total nuclear variation joined with HAADF-STEM tomography

Energy-dispersive X-ray spectroscopy (EDS) tomography is an advanced technique to characterize compositional information for nanostructures in three dimensions (3D). However, the application is hindered by the poor image quality caused by the low signal-to-noise ratios and the limited number of tilts, which are fundamentally limited by the insufficient number of X-ray counts. In this paper, we explore how to make accurate EDS reconstructions from such data. We propose to augment EDS tomography by joining with it a more accurate high-angle annular dark-field STEM (HAADF-STEM) tomographic reconstruction, for which usually a larger number of tilt images are feasible. This augmentation is realized through total nuclear variation (TNV) regularization, which encourages the joint EDS and HAADF reconstructions to have not only sparse gradients but also common edges and parallel (or antiparallel) gradients. Our experiments show that reconstruction images are more accurate compared to the non-regularized and the total variation regularized reconstructions, even when the number of tilts is small or the X-ray counts are low

Modeling blurring effects due to continuous gantry rotation: Application to region of interest tomography

Purpose: Projections acquired with continuous gantry rotation may suffer from blurring effects, depending on the rotation speed and the exposure time of each projection. This leads to blurred reconstructions if conventional reconstruction algorithms are applied. In this paper, the authors propose a reconstruction method for fast acquisitions based on a continuously moving and continuously emitting x-ray source. They study the trade-off between total acquisition time and reconstruction quality and compare with conventional reconstructions using projections acquired with a stepwise moving x-ray source. Methods: The authors introduce the algebraic reconstruction technique with angular integration concept, which models the angular integration due to the relative motion of the x-ray source during the projection. Results: Compared to conventional reconstruction from projections acquired with pulsed x-ray emission, the proposed method results in substantially improved reconstruction quality around the center of rotation. Outside this region, the proposed method results in improved radial resolution and a decreased tangential resolution. For a fixed reconstruction quality of this region of interest, the proposed method enables a lower number of projections and thus a faster acquisition. Conclusions: The modeling of the continuous gantry rotation in the proposed method substantially improves the reconstruction quality in a region of interest around the rotation center. The proposed method shows potential for fast region of interest tomography

Tomographic image reconstruction from continuous projections

An important design aspect in tomographic image reconstruction is the choice between a step-and-shoot protocol versus continuous X-ray tube movement for image acquisition. A step-and-shoot protocol implies a perfectly still tube during X-ray exposure, and hence involves moving the tube to its next position only in between exposures. In a continuous movement protocol, the tube is in a constant motion. The angular integration of the rays inherently produces blurred projections. Conventional reconstruction from such projections leads to blurred reconstructed images, and therefore the projection angles are kept small. Important advantages of a continuous scanning protocol are shorter acquisition times and less demands on modality construction from a mechanical point of view. In this work, the continuous protocol is extended with continuous projections, in which the X-ray source is continuously emitting X-rays over larger angles. The focal spot motion can no longer be ignored and is modeled in the reconstruction. The reconstruction quality is compared with the equivalent step-and-shoot counterpart showing improved results for region of interest tomography