ADAPTIVE DIRECTIONAL SUBDIVISION SCHEMES AND SHEARLET MULTIRESOLUTION ANALYSIS

Abstract

In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of nonstationary bivariate subdivision scheme, which allows us to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two-dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.