Quincunx Fundamental Refinable Functions in Arbitrary Dimensions

Xiaosheng Zhuang

doi:10.3390/axioms6030020

Abstract

In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension d = 2 , we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks.

Highlights

Introduction and MotivationIn dimension d = 1, Deslauriers and Dubuc [11] proposed a family of interpolatory subdivision schemes associated with a family { a2n−1 : n ∈ N} of quincunx interpolatory masks (with respect to the dyadic dilation factor M = 2)
Introduction and MotivationWe say that a d × d integer matrix M is a dilation matrix if limn→∞ M−n = 0, that is, all the eigenvalues of M are greater than 1 in modulus
We assume that the refinement mask a is finitely supported and normalized; i.e., a ∈ l0 (Zd ) and ∑k∈Zd a(k) = 1, where by l (Zd ) we denote the linear space of all sequences v : Zd → C of complex numbers on Zd and by l0 (Zd ) we denote the linear space of all sequences v = {v(k)}k∈Zd ∈ l (Zd ) such that the cardinality of {k ∈ Zd : v(k) 6= 0} is finite

Summary

Introduction and Motivation

In dimension d = 1, Deslauriers and Dubuc [11] proposed a family of interpolatory subdivision schemes associated with a family { a2n−1 : n ∈ N} of quincunx interpolatory masks (with respect to the dyadic dilation factor M = 2). Such a family is unique in the following sense:. (3) a(m,n) satisfies the sum rules of order m + n + 1 with respect to the quincunx lattice Q2 defined as in (5) for d = 2 The uniqueness of such a family a(m,n) implies that a(m,n) is minimally supported among all the quincunx interpolatory masks which satisfies the sum rules of order m + n + 1.

Quincunx Interpolatory Masks in Arbitrary Dimensions

Explicit Form of the Bivariate Quincunx Interpolatory Masks

Conclusions