NUMERICAL APPLICATIONS OF THE METHOD OF HURWITZ-RADON MATRICES

Computer sciences need suitable methods for numerical calculations of interpolation, extrapolation, quadrature, derivative and solution of nonlinear equation. Classical methods, based on polynomial interpolation, have some negative features: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomial considerably, also the Runge‘s phenomenon cannot be forgotten. To deal with numerical interpolation, extrapolation, integration and differentiation dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skewsymmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz- Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical differentiation. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the function point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes, interpolation of L points of the curve is connected with the computational cost of rank O(L), MHR interpolation is not a linear interpolation

NUMERICAL DIFFERENTIATION VIA THE INTERPOLATION METHOD OF HURWITZ-RADON MATRICES

Mathematics and computer science need suitable method for numerical calculation of derivative. Classical methods, based on polynomial interpolation, have some negative features: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomial considerably, also the Runge’s phenomenon cannot be forgotten. To deal with numerical interpolation and differentiation dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical differentiation. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the function point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes, interpolation of L points of the curve is connected with the computational cost of rank O(L), MHR interpolation is not a linear interpolation

Aerial Drone-based System for Wildfire Monitoring and Suppression

Wildfire, also known as forest fire or bushfire, being an uncontrolled fire crossing an area of combustible vegetation, has become an inherent natural feature of the landscape in many regions of the world. From local to global scales, wildfire has caused substantial social, economic and environmental consequences. Given the hazardous nature of wildfire, developing automated and safe means to monitor and fight the wildfire is of special interest. Unmanned aerial vehicles (UAVs), equipped with appropriate sensors and fire retardants, are available to remotely monitor and fight the area undergoing wildfires, thus helping fire brigades in mitigating the influence of wildfires. This thesis is dedicated to utilizing UAVs to provide automated surveillance, tracking and fire suppression services on an active wildfire event. Considering the requirement of collecting the latest information of a region prone to wildfires, we presented a strategy to deploy the estimated minimum number of UAVs over the target space with nonuniform importance, such that they can persistently monitor the target space to provide a complete area coverage whilst keeping a desired frequency of visits to areas of interest within a predefined time period. Considering the existence of occlusions on partial segments of the sensed wildfire boundary, we processed both contour and flame surface features of wildfires with a proposed numerical algorithm to quickly estimate the occluded wildfire boundary. To provide real-time situational awareness of the propagated wildfire boundary, according to the prior knowledge of the whole wildfire boundary is available or not, we used the principle of vector field to design a model-based guidance law and a model-free guidance law. The former is derived from the radial basis function approximated wildfire boundary while the later is based on the distance between the UAV and the sensed wildfire boundary. Both vector field based guidance laws can drive the UAV to converge to and patrol along the dynamic wildfire boundary. To effectively mitigate the impacts of wildfires, we analyzed the advancement based activeness of the wildfire boundary with a signal prominence based algorithm, and designed a preferential firefighting strategy to guide the UAV to suppress fires along the highly active segments of the wildfire boundary

Machine Learning

Machine Learning can be defined in various ways related to a scientific domain concerned with the design and development of theoretical and implementation tools that allow building systems with some Human Like intelligent behavior. Machine learning addresses more specifically the ability to improve automatically through experience

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Radiation doses and risks from paediatric computed tomography

The use of computed tomography (CT) worldwide has increased dramatically since its introduction. In Australia, more than two million CT services are funded by Medicare every year and the rate of imaging is increasing beyond population growth. CT imaging currently accounts for the largest source of ionising radiation exposure to the population from all diagnostic procedures. There is a small, theoretical risk of carcinogenesis attributable to low doses of ionising radiation based on epidemiological evidence at higher doses and dose rates. The doses from CT examinations fall into this low dose range. Recognition of the potential radiation risks combined with the high utilisation of CT imaging has led to greater awareness of population health risks. Furthermore, the exposure risks from radiation are higher for children than for adults due to their increased radiosensitivity and greater prospective life expectancy. There is only limited information on Australian paediatric CT imaging rates, doses and risks. This thesis aims to assess the medical radiation exposure of children in Australia from CT examinations. An experimental method for paediatric CT organ dosimetry was developed using a physical anthropomorphic phantom representing a child and high sensitivity thermoluminescent dosemeters. Radiation doses for typical paediatric CT clinical protocols performed at the Royal Children’s Hospital (RCH) in Melbourne were quantified. An analysis of indirect dose computation methods was undertaken to identify a robust and reliable method for paediatric CT organ dosimetry feasible for clinical implementation. A practical method for assessing doses and scan parameters from a local dose survey was developed to enable identification of areas for dose optimisation. Local diagnostic reference levels were established based on the dose distributions from RCH patient data across all paediatric age groups. The first comprehensive analysis of CT imaging frequency and trends for the Australian paediatric population was undertaken. Finally, cancer risk projections for incidence and mortality from paediatric CT scanning have been made for the Australian population