  Wavelets and their associated transforms are highly efficient when
approximating and analyzing one-dimensional signals. However, multivariate
signals such as images or videos typically exhibit curvilinear singularities,
which wavelets are provably deficient of sparsely approximating and also of
analyzing in the sense of, for instance, detecting their direction. Shearlets
are a directional representation system extending the wavelet framework, which
overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful
implementation and fast associated transforms. In this paper, we will introduce
a comprehensive carefully documented software package coined ShearLab 3D
(www.ShearLab.org) and discuss its algorithmic details. This package provides
MATLAB code for a novel faithful algorithmic realization of the 2D and 3D
shearlet transform (and their inverses) associated with compactly supported
universal shearlet systems incorporating the option of using CUDA. We will
present extensive numerical experiments in 2D and 3D concerning denoising,
inpainting, and feature extraction, comparing the performance of ShearLab 3D
with similar transform-based algorithms such as curvelets, contourlets, or
surfacelets. In the spirit of reproducible reseaerch, all scripts are
accessible on www.ShearLab.org.
