The Construction of Smooth Parseval Frames of Shearlets

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 L-2(R-D) 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 L-2(R-D). Specifically, all elements of the shearlet systems obtained from this construction are compactly supported and C-infinity in the frequency domain, hence ensuring that the system has also excellent spatial localization.