Polynomial cubic splines with tension properties

In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems

Scalable GPU acceleration of b-spline signal processing operations

B-Splines are a useful tool in signal processing, and are widely used in the analysis of two and three-dimensional images. B-Splines provide a continuous representation of the signal, image, or volume, which is useful for interpolation, resampling, noise removal, and differentiation - all important steps in many signal processing algorithms. These splines are defined entirely by an array of coefficients that is roughly the same size as the original signal and of values in the same order of magnitude, making storage and representation trivial. What is not trivial, however, is the quick calculation and processing of those coefficients, especially for very large data. As technology improves in fields such as medical imaging, algorithms that use B-Splines will need to process increasingly higher resolution images and voxel volumes. New implementations are needed to make use of modern parallel architectures to keep these algorithms practical. This thesis presents a library for performing many common B-Splines operations in CUDA, the parallel programming framework for NVIDIA GPUs, and analyzes the considerations necessary when implementing a large-scale parallel version of such a well-established sequential algorithm. This library is meant to be used both by C++ programs as well as algorithms implemented in MATLAB without requiring significant changes. Significant speedups are obtained using this library to perform various common B-Spline image processing operations (as much as 30x for some), and the scalability limitations of the GPU implementation are addressed

Spatially Varying Coefficient Regression with GAM Gaussian Process splines: GAM(e)-on

This paper describes initial work exploring GAM Gaussian Process (GP) splines parameterised by observation location, as a geographical varying coefficient model. Similar to GWR, this approach accommodates process spatial heterogeneity and generates spatially distributed, local coefficient estimates. These can be mapped to indicate the nature of the heterogeneity. The paper investigates the effect of the smoothing parameters used in the splines and how they alter the nature of the modelled heterogeneity. It optimises these in the GAM GP and the tuned model has subtle but important differences with the initial model. This has impacts on the nature of the process understanding (inference) that can be extracted from the model. This in turn suggest the need examine the underlying semantics of the resultant models in relation to the scale of process suggested by the smoothing parameters. A number of areas of further work are identified

B-Spline based uncertainty quantification for stochastic analysis

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided