  We study efficient and reliable methods of capturing and sparsely
representing anisotropic structures in 3D data. As a model class for
multidimensional data with anisotropic features, we introduce generalized
three-dimensional cartoon-like images. This function class will have two
smoothness parameters: one parameter \beta controlling classical smoothness and
one parameter \alpha controlling anisotropic smoothness. The class then
consists of piecewise C^\beta-smooth functions with discontinuities on a
piecewise C^\alpha-smooth surface. We introduce a pyramid-adapted, hybrid
shearlet system for the three-dimensional setting and construct frames for
L^2(R^3) with this particular shearlet structure. For the smoothness range
1<\alpha =< \beta =< 2 we show that pyramid-adapted shearlet systems provide a
nearly optimally sparse approximation rate within the generalized cartoon-like
image model class measured by means of non-linear N-term approximations.
