Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Modelling and control of a floating oscillating water column

A novel numerical model of a Bent Backwards Duct Buoy (BBDB) Oscillating Water Column (OWC) Wave Energy Converter was created based on existing isolated numerical models of the different energy conversion systems utilised by an OWC. The novel aspect of this numerical model is that it incorporates the interdependencies of the different power conversion systems rather than modelling each system individually. This was achieved by accounting for the dynamic aerodynamic damping caused by the changing turbine rotational velocity by recalculating the turbine damping for each simulation sample and applying it via a feedback loop. The accuracy of the model was validated using experimental data collected during the Components for Ocean Renewable Energy Systems (CORES) EU FP-7 project that was tested in Galway Bay, Ireland. During the verification process, it was discovered that the model could also be applied as a valuable tool when troubleshooting device performance. A new turbine was developed and added to a full scale model after being investigated using Computational Fluid Dynamics. The energy storage capacity of the impulse turbine was investigated by modelling the turbine with both high and low inertia and applying three turbine control theories to the turbine using the full scale model. A single Maximum Power Point Tracking algorithm was applied to the low-inertia turbine, while both a fixed and dynamic control algorithm was applied to the high-inertia turbine. These results suggest that the highinertia turbine could be used as a flywheel energy storage device that could help minimize output power variation despite the low operating speed of the impulse turbine. This research identified the importance of applying dynamic turbine damping to a BBDB OWC numerical model, revealed additional value of the model as a device troubleshooting tool, and found that an impulse turbine could be applied as an energy storage system.

Iterative reconstruction as a novel method of radiation dose reduction at computed tomography in patients with Crohn's disease

Prior work of our research group, that quantified the alarming levels of radiation dose to patients with Crohn’s disease from medical imaging and the notable shift towards CT imaging making these patients an at risk group, provided context for this work. CT delivers some of the highest doses of ionising radiation in diagnostic radiology. Once a medical imaging examination is deemed justified, there is an onus on the imaging team to endeavour to produce diagnostic quality CT images at the lowest possible radiation dose to that patient. The fundamental limitation with conventional CT raw data reconstruction was the inherent coupling of administered radiation dose with observed image noise – the lower the radiation dose, the noisier the image. The renaissance, rediscovery and refinement of iterative reconstruction removes this limitation allowing either an improvement in image quality without increasing radiation dose or maintenance of image quality at a lower radiation dose compared with traditional image reconstruction. This thesis is fundamentally an exercise in optimisation in clinical CT practice with the objectives of assessment of iterative reconstruction as a method for improvement of image quality in CT, exploration of the associated potential for radiation dose reduction, and development of a new split dose CT protocol with the aim of achieving and validating diagnostic quality submillisiever t CT imaging in patients with Crohn’s disease. In this study, we investigated the interplay of user-selected parameters on radiation dose and image quality in phantoms and cadavers, comparing traditional filtered back projection (FBP) with iterative reconstruction algorithms. This resulted in the development of an optimised, refined and appropriate split dose protocol for CT of the abdomen and pelvis in clinical patients with Crohn’s disease allowing contemporaneous acquisition of both modified and conventional dose CT studies. This novel algorithm was then applied to 50 patients with a suspected acute complication of known Crohn’s disease and the raw data reconstructed with FBP, adaptive statistical iterative reconstruction (ASiR) and model based iterative reconstruction (MBIR). Conventional dose CT images with FBP reconstruction were used as the reference standard with which the modified dose CT images were compared in terms of radiation dose, diagnostic findings and image quality indices. As there are multiple possible user-selected strengths of ASiR available, these were compared in terms of image quality to determine the optimal strength for this modified dose CT protocol. Modified dose CT images with MBIR were also compared with contemporaneous abdominal radiograph, where performed, in terms of diagnostic yield and radiation dose. Finally, attenuation measurements in organs, tissues, etc. with each reconstruction algorithm were compared to assess for preservation of tissue characterisation capabilities. In the phantom and cadaveric models, both forms of iterative reconstruction examined (ASiR and MBIR) were superior to FBP across a wide variety of imaging protocols, with MBIR superior to ASiR in all areas other than reconstruction speed. We established that ASiR appears to work to a target percentage noise reduction whilst MBIR works to a target residual level of absolute noise in the image. Modified dose CT images reconstructed with both ASiR and MBIR were non-inferior to conventional dose CT with FBP in terms of diagnostic findings, despite reduced subjective and objective indices of image quality. Mean dose reductions of 72.9-73.5% were achieved with the modified dose protocol with a mean effective dose of 1.26mSv. MBIR was again demonstrated superior to ASiR in terms of image quality. The overall optimal ASiR strength for the modified dose protocol used in this work is ASiR 80%, as this provides the most favourable balance of peak subjective image quality indices with less objective image noise than the corresponding conventional dose CT images reconstructed with FBP. Despite guidelines to the contrary, abdominal radiographs are still often used in the initial imaging of patients with a suspected complication of Crohn’s disease. We confirmed the superiority of modified dose CT with MBIR over abdominal radiographs at comparable doses in detection of Crohn’s disease and non-Crohn’s disease related findings. Finally, we demonstrated (in phantoms, cadavers and in vivo) that attenuation values do not change significantly across reconstruction algorithms meaning preserved tissue characterisation capabilities with iterative reconstruction. Both adaptive statistical and model based iterative reconstruction algorithms represent feasible methods of facilitating acquisition diagnostic quality CT images of the abdomen and pelvis in patients with Crohn’s disease at markedly reduced radiation doses. Our modified dose CT protocol allows dose savings of up to 73.5% compared with conventional dose CT, meaning submillisievert imaging is possible in many of these patients.

The development of an algorithm to track blood pressure accurately based on the pulse transit time method

Oscillometric blood pressure (BP) monitors are currently used to diagnose hypertension both in home and clinical settings. These monitors take BP measurements once every 15 minutes over a 24 hour period and provide a reliable and accurate system that is minimally invasive. Although intermittent cuff measurements have proven to be a good indicator of BP, a continuous BP monitor is highly desirable for the diagnosis of hypertension and other cardiac diseases. However, no such devices currently exist. A novel algorithm has been developed based on the Pulse Transit Time (PTT) method, which would allow non-invasive and continuous BP measurement. PTT is defined as the time it takes the BP wave to propagate from the heart to a specified point on the body. After an initial BP measurement, PTT algorithms can track BP over short periods of time, known as calibration intervals. After this time has elapsed, a new BP measurement is required to recalibrate the algorithm. Using the PhysioNet database as a basis, the new algorithm was developed and tested using 15 patients, each tested 3 times over a period of 30 minutes. The predicted BP of the algorithm was compared to the arterial BP of each patient. It has been established that this new algorithm is capable of tracking BP over 12 minutes without the need for recalibration, using the BHS standard, a 100% improvement over what has been previously identified. The algorithm was incorporated into a new system based on its requirements and was tested using three volunteers. The results mirrored those previously observed, providing accurate BP measurements when a 12 minute calibration interval was used. This new system provides a significant improvement to the existing method allowing BP to be monitored continuously and non-invasively, on a beat-to-beat basis over 24 hours, adding major clinical and diagnostic value.