Multivariate Balanced Vector-Valued Refinable Functions

Abstract

Vanishing moments of sufficiently high order and compact supports of reasonable size contribute to the great success of wavelets in various areas of applications, particularly in signal and image processing. However, for multi—wavelets, polynomial preservation of the refinable function vectors does not necessarily imply annihilation of discrete polynomials by the high—pass filters of the corresponding orthogonal or bi—orthogonal multi—wavelets. This led to the introduction of the notion of “balanced” multi—wavelets by Lebrun and Vetterli, and later, generalization to higher—order balancing by Selesnick. Selesnick’s work is concerned only with orthonormal refinable function vectors and orthonormal multi—wavelets. In this paper after giving a brief overview of the state—of—the—art of vector—valued refinable functions in the preliminary section, we will discuss our most recent contribution to this research area. Our goal is to derive a set of necessary and sufficient conditions that characterize the balancing property of any order for the general multivariate matrix—dilation setting. We will end the second section by demonstrating our theory with examples of univariate splines and bivariate splines on the four—directional mesh.