Nonseparable Shearlet Transform

Over the past few years, various representation systems which sparsely approximate functions governed by anisotropic features, such as edges in images, have been proposed. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms are provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In this paper, we introduce a new shearlet transform associated with a nonseparable shearlet generator, which improves the directional selectivity of previous shearlet transforms. Our approach is based on a discrete framework, which allows a faithful digitization of the continuum domain directional transform based on compactly supported shearlets introduced as means to sparsely encode anisotropic singularities of multivariate data. We show numerical experiments demonstrating the potential of our new shearlet transform in 2D and 3D image processing applications.