# Interpolatory refinement pairs with properties of symmetry and polynomial filling

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008.

Thesis

Subdivision techniques have, over the last two decades, developed into a powerful tool in computer-aided geometric design (CAGD). In some applications it is required that data be preserved exactly; hence the need for interpolatory subdivision schemes. In this thesis,we consider the fundamentals of themathematical analysis of symmetric interpolatory subdivision schemes for curves, also with the property of polynomial filling up to a given odd degree, in the sense that, if the initial control point sequence is situated on such a polynomial curve, all the subsequent subdivision iterates fills up this curve, for it to eventually also become also the limit curve. A subdivision scheme is determined by its mask coefficients, which we find convenient to mathematically describe as a bi-infinite sequnce a with finite support. This sequence is in one-to-one correspondence with a corresponding Laurent polynomial A with coefficients given by the mask sequence a. After an introductory Chapter 1 on notation, basic definitions, and an overview of the thesis, we proceed in Chapter 2 to separately consider the issues of interpolation, symmetry and polynomial filling with respect to a subdivision scheme, eventually leading to a definition of the class Am,n of mask symbols in which all of the above desired properties are combined. We proceed in Chapter 3 to deduce an explicit characterization formula for the classAm,n, in the process also showing that its optimally local member is the well-known Dubuc–Deslauriers (DD) mask symbol Dm of order m. In fact, an alternative explicit characterization result appears in recent work by De Villiers and Hunter, in which the authors characterized mask symbols A ∈Am,n as arbitrary convex combinations of DD mask symbols. It turns out that Am,m = {Dm}, whereas the class Am,m+1 has one degree of freedom, which we interpret here in the formof a shape parameter t ∈ R for the resulting subdivision scheme. In order to investigate the convergence of subdivision schemes associated with mask symbols in Am,n, we first introduce in Chapter 4 the concept of a refinement pair (a,φ), consisting of a finitely-supported sequence a and a finitelysupported function φ, where φ is a refinable function in the sense that it can be expressed as a finite linear combination, as determined by a, of the integer shifts of its own dilation by factor 2. After presenting proofs of a variety of properties satisfied by a given refinement pair (a,φ), we next introduce the concept of an interpolatory refinement pair as one for which the refinable function φ interpolates the delta sequence at the integers. A fundamental result is then that the existence of an interpolatory refinement pair (a,φ) guarantees the convergence of the interpolatory subdivision scheme with subdivision mask a, with limit function © expressible as a linear combination of the integer shifts of φ, and with all the subdivision iterates lying on ©. In Chapter 5, we first present a fundamental result byMicchelli, according to which interpolatory refinable function existence is obtained for mask symbols in Am,n if the mask symbol A is strictly positive on the unit circle in complex plane. After showing that the DD mask symbol Dm satisfies this sufficient property, we proceed to compute the precise t -interval for such positivity on the unit circle to occur for the mask symbols A = Am(t |·) ∈Am,m+1. Also, we compare our numerical results with analogous ones in the literature. Finally, in Chapter 6, we investigate the regularity of refinable functions φ = φm(t |·) corresponding to mask symbols Am(t |·). Using a standard result fromthe literature in which a lower bound on the Hölder continuity exponent of a refinable function φ is given explicitly in terms of the spectral radius of a matrix obtained from the corresponding mask sequence a, we compute this lower bound for selected values of m.