  In this paper, we first introduce the concept of an adaptive MRA (AMRA)
structure which is a variant of the classical MRA structure suited to the main
goal of a fast flexible decomposition strategy adapted to the data at each
decomposition level. We then study this novel methodology for the general case
of affine-like systems, and derive a Unitary Extension Principle (UEP) for
filter design. Finally, we apply our results to the directional representation
system of shearlets. This leads to a comprehensive theory for fast
decomposition algorithms associated with shearlet systems which encompasses
tight shearlet frames with spatially compactly supported generators within such
an AMRA structure. Also shearlet-like systems associated with parabolic scaling
and unimodular matrices optimally close to rotation as well as 3D shearlet
systems are studied within this framework.
