Variational use of B-splines and Kernel Based Functions
Christophe Rabut, INSA Toulouse

Seminar • 27 October 2017

Abstract
Kernel Based Functions are generalizations of spline functions and radial basis functions. These R^d to R functions are in the form f = \sum_{i=1}^n λi φ(x−xi) or \sum_{i=1}^n λi φ(x−xi)+pk(x) where φ is called the kernel, (xi)_{i=1:n} ∈ (R^d)^n are the so called centers of f, (λi)_{i=1:n} are real coefficients, and pk is some degree k polynomial. When φ is a bell shaped function meeting some property (such as, in particular \sum_{i=1}^n φ(x) = 1 for any x ∈ R^d), we write it B and call it, for short, B-spline. In this talk we present two particular uses of these Kernel Based Functions, and a property of a specific polynomial interpolation. First, hierarchical B-splines: using B-splines of different scales, and a mean square optimization, we show how to approximate scattered data with possibility of zoom on some regions, adaptively from the data. We so obtain locally tensor product functions, where the grid of the centers is finer in some regions and coarser in other regions. Second, in a CAGD aim and using modified (variational) Bézier curves or surfaces, we show that it is possible to derive B-spline curves or surfaces being closer to (or further from) the control polygon, while being in the same vectorial space. This gives more flexibility to easily derive new forms. Third we present variational polynomial interpolation, which is true polynomial interpolation of any given data, and so obtain a polynomial interpolation without the famous Runge oscillations. These interpolating polynomials converge towards the interpolating polynomial spline of the data.