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.

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...

Broadly speaking, the term ‘exploitation’ can be applied to a number of different kinds of interaction. Mineral deposits, mistakes, workers, and wombs are all said to be exploited. One sense of exploitation, often applied to the extraction of resources, or capitalising on an opponent’s mistaken move in a board game, connotes mere use or advantage taking. Exploitation in this sense need not impl...

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...

Powell-Sabin splines are C-continuous quadratic splines defined on an arbitrary triangulation. Their construction is based on a particular split of each triangle in the triangulation into six smaller triangles. In this article we give an overview of the properties of Powell-Sabin splines in the context of computer aided geometric design. These splines can be represented in a compact normalized ...

We first review B-splines and splines constructed using B-splines as the basis functions.

Bivariate cubic L1 smoothing splines are introduced. The coefficients of a cubic L1 smoothing spline are calculated by minimizing the weighted sum of the L1 norms of second derivatives of the spline and the 1 norm of the residuals of the data-fitting equations. Cubic L1 smoothing splines are compared with conventional cubic smoothing splines based on the L2 and 2 norms. Computational results fo...