650 research outputs found
On multi-degree splines
Multi-degree splines are piecewise polynomial functions having sections of
different degrees. For these splines, we discuss the construction of a B-spline
basis by means of integral recurrence relations, extending the class of
multi-degree splines that can be derived by existing approaches. We then
propose a new alternative method for constructing and evaluating the B-spline
basis, based on the use of so-called transition functions. Using the transition
functions we develop general algorithms for knot-insertion, degree elevation
and conversion to B\'ezier form, essential tools for applications in geometric
modeling. We present numerical examples and briefly discuss how the same idea
can be used in order to construct geometrically continuous multi-degree
splines
Efficient spatial modelling using the SPDE approach with bivariate splines
Gaussian fields (GFs) are frequently used in spatial statistics for their
versatility. The associated computational cost can be a bottleneck, especially
in realistic applications. It has been shown that computational efficiency can
be gained by doing the computations using Gaussian Markov random fields (GMRFs)
as the GFs can be seen as weak solutions to corresponding stochastic partial
differential equations (SPDEs) using piecewise linear finite elements. We
introduce a new class of representations of GFs with bivariate splines instead
of finite elements. This allows an easier implementation of piecewise
polynomial representations of various degrees. It leads to GMRFs that can be
inferred efficiently and can be easily extended to non-stationary fields. The
solutions approximated with higher order bivariate splines converge faster,
hence the computational cost can be alleviated. Numerical simulations using
both real and simulated data also demonstrate that our framework increases the
flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table
B-splines, Pólya curves, and duality
AbstractLocal duality between B-splines and Pólya curves is examined, mostly from the viewpoint of computer-aided geometric design. Certain known results for the two curve types are shown to be related. A few new results for Pólya curves and a curve scheme related to B-splines also follow from these investigations
A Framework for Unbiased Model Selection Based on Boosting
Variable selection and model choice are of major concern in many statistical applications, especially in high-dimensional regression models. Boosting is a convenient statistical method that combines model fitting with intrinsic model selection.
We investigate the impact of base-learner specification on the performance of boosting as a model selection procedure.
We show that variable selection may be biased if the covariates are of different nature.
Important examples are models combining continuous and categorical covariates, especially if the number of categories is large. In this case, least squares base-learners offer increased flexibility for the categorical covariate and lead to a preference even if the categorical covariate is non-informative.
Similar difficulties arise when comparing linear and nonlinear base-learners for a continuous covariate. The additional flexibility in the nonlinear base-learner again yields a preference of the more complex modeling alternative.
We investigate these problems from a theoretical perspective and suggest a framework for unbiased model selection based on a general class of penalized least squares base-learners.
Making all base-learners comparable in terms of their degrees of freedom strongly reduces the selection bias observed in naive boosting specifications. The importance of unbiased model selection is demonstrated in simulations and an application to forest health models
Recursive subdivision algorithms for curve and surface design
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented.
Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987
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