Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis

TL;DR

This paper introduces adaptive non-stationary subdivision schemes for directional multiresolution analysis, enabling efficient shearlet-based decomposition of 2D data with finite filters.

Contribution

It develops a new class of adaptive directional subdivision schemes and demonstrates their application to shearlet multiresolution analysis with fast decomposition algorithms.

Findings

Convergent adaptive directional subdivision schemes characterized.

Framework for shearlet multiresolution analysis with finitely supported filters.

Efficient shearlet decomposition demonstrated with numerical examples.

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 non-stationary bivariate subdivision schemes, which allow 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.