Necessary conditions for  shearlet  and cone-adapted shearlet systems to be frames are presented with respect to the admissibility condition of generators.

This chapter is devoted to the generalization of the continuous shearlet transform to higher dimensions as well as to the construction of associated smoothness spaces and to the analysis of their structural properties, respectively. To construct canonical scales of smoothness spaces , so-called shearlet coorbit spaces , and associated atomic decompositions and Banach frames we prove that the ge...

This papers examines structural properties of the recently developed shearlet coorbit spaces in higher dimensions. We prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone in three dimensions into Besov spaces. The results are based on general atomic decompositions of Besov spaces. Furthermore, we establish trace results for these subspaces with resp...

In this paper, we introduce pyramid-adapted shearlet systems for the three-dimensional setting, and show how one can construct frames for L(R) with this particular shearlet structure. We then introduce a generalized three-dimensional cartoon-like image model class of piecewise C smooth functions with discontinuities on a C smooth surface with 1 < α ≤ 2 and show that pyramid-adapted shearlet sys...

We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real a...

Abstract. 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 β controlling classical smoothness and one parameter α controlling a...

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 shearing matrices. While shearing matrices offer the advantage of p...

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. Fi...

In this paper, using shearlet frames, we present a numerical  method  for solving  the wave equation. We define a new shearlet system and by the Plancherel theorem, we calculate the shearlet coefficients.