Sufficient Conditions for Composite Frames

Abstract

In many fields of science and technology, multiscale (or multiresolution) analysis is a useful tool to study data via efficient representation at increasingly precise resolution. In this paper, we present a result on composite wavelet frames (or composite frames), a relatively recent development in a long line of multiscale techniques including the celebrated Fourier and wavelet analyses. Composite frames generalize many traditional wavelet-based constructions and can be used to provide new effective schemes, including such successful examples as shearlets, that capture directionality in data, images in particular. Our construction is motivated by wavelets with composite dilations, which were introduced in Guo et al. (Appl Comput Harmon Anal 20:202–236, 2006). We focus on frames, a robust system that generalizes orthogonal bases and can include redundant elements, and show that we can construct composite frames for \(L^2(\mathbb {R}^n)\) using two main components: dilation operators from admissible groups and generating functions which are refinable with respect to these groups. We illustrate this theory with the construction of a composite dilation frame based on a Haar wavelet with quincunx dilation and local mollification.