ADAPTIVE DIRECTIONAL SUBDIVISION SCHEMES ANDSHEARLET MULTIRESOLUTION ANALYSIS

摘要

In this paper, we propose a solution for a fundamental problem in computationalharmonic analysis, namely, the construction of a multiresolution analysis with directional compo-nents. We will do so by constructing subdivision schemes which provide a means to incorporatedirectionality into the data and thus the limit function. We develop a new type of nonstationarybivariate subdivision scheme, which allows us to adapt the subdivision process depending on di-rectionality constraints during its performance, and we derive a complete characterization of thosemasks for which these adaptive directional subdivision schemes converge. In addition, we presentseveral numerical examples to illustrate how this scheme works. Secondly, we describe a fast decom-position associated with a sparse directional representation system for two-dimensional data, wherewe focus on the recently introduced sparse directional representation system of shearlets. In fact, weshow that the introduced adaptive directional subdivision schemes can be used as a framework forderiving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fastshearlet decomposition.