  Over the past years, various representation systems which sparsely
approximate functions governed by anisotropic features such as edges in images
have been proposed. We exemplarily mention the systems of contourlets,
curvelets, and shearlets. Alongside the theoretical development of these
systems, algorithmic realizations of the associated transforms were 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 fact, shearlet systems are the only systems so far which satisfy this
property, yet still deliver optimally sparse approximations of cartoon-like
images. In this chapter, we provide an introduction to digital shearlet theory
with a particular focus on a unified treatment of the continuum and digital
realm. In our survey we will present the implementations of two shearlet
transforms, one based on band-limited shearlets and the other based on
compactly supported shearlets. We will moreover discuss various quantitative
measures, which allow an objective comparison with other directional transforms
and an objective tuning of parameters. The codes for both presented transforms
as well as the framework for quantifying performance are provided in the Matlab
toolbox ShearLab.
