- Contourlets
Contourlets were introduced by M. Do
and M Vetterli. Their approach
towards directional representations starts with a discrete-domain construction and then studies
its convergence to an expansion in the continuous domain. They construct a discrete-domain
multiresolution and multidirection expansion using non-separable filter banks, in much the same way
that wavelets were derived from filter banks.
- Curvelets
In a groundbreaking work, E. Candes
and D. L. Donoho introduced
the curvelets as an efficient tool to extract directional information from images, i.e., the
intrinsic geometrical structure that is key in visual information. The curvelets
are a system consisting of translations, and rotation of a sequence of basic functions depending
on a parabolic scaling parameter. The curvelet transform is first developed in the continuous domain and
then discretized for sampled data. In contract to the shearlets, these system does not form an
affine system. For further information we refer to the website
www.curvelet.org.
- Wavelets with Composite Dilations
This theory studies affine systems build by using a composition
of two sets of matrices as dilation matrices. In a series of several papers, K. Guo,
D. Labate,
W.-Q. Lim,
G. Weiss, and
E. Wilson
studied the behavior of those systems, thereby deriving an extensive theory. Shearlets
can be regarded as a special case of those systems exhibiting many additional
properties due to their specialized structure.