The Uncertainty Principle Associated With the Continuous Shearlet Transform

Abstract

Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

Introduction to Shearlets

“…This can be illustrated by looking at the behaviour of the isotropic Continuous Wavelet Transform of functions containing singularities. Indeed, consider a function or distribution f , which is regular everywhere except for a point singularity at x 0 , and let us examine the behaviour of W ψ f (a,t), given by (9). Provided ψ is smooth, a direct computations shows that W ψ f (a,t) has rapid asymptotic decay, as a → 0, for all values of t, unless t = x 0 .…”

“…Similar to the relation of wavelet systems to group representation theory discussed in Subsection 3.4, the theory of continuous shearlet systems can also be developed within the theory of unitary representations of the affine group and its generalizations [9].…”

Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets, Traces and Embeddings

“…is a unitary representation of S, see [5,6]. In the following, we use the abbreviation ψ a,s,t := π(a, s, t) ψ.…”

“…Among all these transforms, the shearlet transform stands out since it stems from a square-integrable group representation. This has been clarified in [5] where the underlying group, the full shearlet group, has been established. This pure group theoretical approach to shearlets has some important advantages.…”

The Construction of Smooth Parseval Frames of Shearlets

“…This is another fundamental difference with respect to curvelets, since the shear matrices preserve the integer lattice and this allows the shearlet systems to provide a unified treatment of the continuum and digital setting. These properties and the special flexibility of the shearlet framework have made this approach very successful both as a theoretical and an applicable tool (see, for example, publications [6,14,16,19,28,30] for the main theoretical results, and [5,9,10,11,30,33,34,35] for applications).…”