An exposition on the use of O’Sullivan penalized splines in contemporary semiparametric regression, including mixed model and Bayesian formulations, is presented. O’Sullivan penalized splines are similar to P-splines, but have the advantage of being a direct generalization of smoothing splines. Exact expressions for the O’Sullivan penalty matrix are obtained. Comparisons between the two types o...

A variety of techniques to facilitate fast graphical display of curves and surfaces for interactive CAGD purposes have been developed, in the past years. Many of these are commonly referred to as subdivision algorithms PI. In this paper we propose a method for subdividing simplex splines, i.e., splines that can be defined over non-regular grid partitions. There is a relatively exhaustive theory...

Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.

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With the aim of generalizing P-splines, we here define a special type penalized splines, called HP-splines, where polynomial splines are replaced by richer class hyperbolic-polynomial and suitably tailored discrete penalty term is used. Hyperbolic-polynomial important in several applications, natural generalization consisting piecewise-defined functions with segments spanned ‘atoms’ x r e ? = 0...

T-splines is a recently developed surface modelling technique which is a generalization of B-splines and allows true local refinement. Another well established method supporting local refinement is hierarchical B-splines. This paper presents algorithms for transformation between T-splines and hierarchical (rational) B-splines. With the transformation, a surface expressed in terms of T-splines c...

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, re...

The first type of pseudo-splines were introduced in [12, 24] to construct tight framelets with desired approximation orders via the unitary extension principle of [22]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provid...