Interpolatory refinement pairs with properties of symmetry and polynomial filling
Date
2008-03
Authors
Gavhi, Mpfareleni Rejoyce
Journal Title
Journal ISSN
Volume Title
Publisher
Stellenbosch : University of Stellenbosch
Abstract
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.
Description
Thesis (MSc (Mathematics))--University of Stellenbosch, 2008.
Keywords
Dissertations -- Mathematics, Theses -- Mathematics, Refinable functions, Computer-aided design, Interpolation, Polynomials, Symmetry (Mathematics)