The shearlet representation has gained increasing recognition in recent years as a
framework for the efficient representation of multidimensional data. This representation
consists of a countable collection of functions defined at various locations, scales and
orientations, where the orientations are obtained through the use of shear matrices. While
shear matrices offer the advantage of preserving the integer lattice and being more
appropriate than rotations for digital implementations, the drawback is that the action of
the shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, in
the standard construction, a Parseval frame of shearlets is obtained by combining
different systems of cone-based shearlets which are projected onto certain subspaces of
L2(ℝD) with the consequence that
the elements of the shearlet system corresponding to the boundary of the cone regions lose
their good spatial localization property. In this paper, we present a new construction
yielding smooth Parseval frame of shearlets for
L2(ℝD). Specifically, all
elements of the shearlet systems obtained from this construction are compactly supported
and C∞ in the frequency domain, hence ensuring that the system
has also excellent spatial localization.