The Continuous Shearlet Transform in Arbitrary Space Dimensions

Abstract

Abstract:This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.

Introduction to Shearlets

“…One approach proposed in [8] and continued in [10] and [12] applies a powerful methodology called coorbit theory, which is used to derive different discretizations while ensuring frame properties. 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.…”

“…The theory of coorbit spaces was applied as a systematic approach towards the construction of shearlet spaces in the series of papers [8,10,11,12]. This ansatz leads to the so-called shearlet coorbit spaces, which are associated to decay properties of shearlet coefficients of discrete shearlet frames.…”

“…A new construction yielding smooth Parseval frames of discrete shearlets in any dimensions was recently introduced in [37]. Several other results have also recently appeared, including the extension of the optimally sparse approximation results and the analysis and detection of surface singularities [10,33,34,35,36,54]. In 3-dimensional data, different types of anisotropic features occur, namely, singularities on 1-dimensional and 2-dimensional manifolds.…”

Reproducing Subgroups of $Sp(2,\mathbb{R})$ . Part I: Algebraic Classification

“…The continuous wavelet transform [7,11,20,21] and its many variants, such as, for example, the shearlet transform [5,6,14,17], lie in the background of a growing body of techniques, that may be collectively referred to as signal analysis, whose common feature is perhaps the decomposition of functions, primarily in L 2 (R d ), by means of superpositions of projections along selected "directions". Symmetry and finite dimensional geometry often play a prominent rôle in the way in which these directions are generated or selected, and hence, with this notion of signal analysis, topological transformation groups and their representations provide a natural setup.…”

Data compression on the sphere

“…These discrete constructions allow for the exact reconstruction of a signal from its wavelet coefficients but they may not necessarily lead to a stable basis (see Sweldens 1997, and references therein). Other authors have focused on continuous wavelet methodologies on the sphere (Freeden & Windheuser 1997;Holschneider 1996;Torrésani 1995;Dahlke & Maass 1996;Antoine & Vandergheynst 1998A&A 531, A98 (2011) coefficients in these continuous methodologies in theory, the absence of an infinite range of dilations precludes exact reconstruction in practice. Approximate reconstruction formula may be developed by building discrete wavelet frames that are based on the continuous methodology (e.g.…”