Irregular Shearlet Frames: Geometry and Approximation Properties

Abstract

Abstract. Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far.In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L 2 (R 2 ), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such a oversampled shearlet systems and coshearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.

Introduction to Shearlets

“…In fact, the regular shearlet frame which will be introduced in the next subsection can be derived using this machinery, and this approach will be further discussed in Chapter 4 of this volume. A different path, which also relies on the group properties of continuous shearlet systems, is taken in [50]. In this paper, a quantitative density measure for discrete subsets of the shearlet group S is introduced, adapted to its group multiplication, which is inspired by the well-known Beurling density for subsets of the Abelian group R 2 .…”

Necessary conditions and sufficient conditions of irregular shearlet frames

“…In Section 4, we first derive general sufficient conditions for the irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. We show that functions from the class of shearlet generators B 0 introduced by P. Kittipoom et al in [22], [24] generate shearlet frames for every sufficiently dense sequence of well-spread space-scale-shear parameters. Secondly, we study the stability of the irregular shearlet systems and show a frame generated by certain shearlet function remains a frame when the space-scale-shear parameters and the generating function undergo small perturbations.…”

“…Similar to the Feichtinger's algebra S 0 studied in for Gabor analysis, B 0 contains nice shearlet atoms which make them useful in shearlet analysis. For example, B 0 is dense in , the irregular frames with generators from B 0 satisfy the strong version of the Homogeneous Approximation Property (HAP) (see , ).…”

“…In this section, we derive sufficient conditions and stability for irregular systems to be frames for . Many ideas and proofs of our work are inspired by , and . In order to obtain the irregular upper frame bound, we require the Lemma .…”

Shearlets on Bounded Domains

“…This allow the introduction of shearlet systems which treat different slopes equally in contrast to the shearlet group-based approach. We though wish to mention that historically the shearlet group-based approach was developed first due to very favorable theoretical properties and it still often serves as a system for developing novel analysis strategies (see, for instance, [11]). …”