α-Molecules

Abstract

Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the L 2 -error of the best N -term approximation for a certain class of functions. In this paper, we introduce the general framework of α-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with α being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an α-scaled index distance, we first prove that two systems of α-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time-frequency localization conditions of the involved systems of α-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results for a specific class of cartoonlike functions by sufficient conditions on the 'control' parameters of a system of α-molecules.

Optimal adaptive ridgelet schemes for linear advection equations

“…7.1. This change does not affect the theoretical properties, as can be shown using [16]. As in Section 4, the ridgelets are constructed in the Fourier domain, for a sufficiently smooth transition function between zero and one.…”

“…This development started with wavelets [9] and is continuing with ridgelets [3], curvelets [6,5,8], shearlets [21,22], contourlets [12] etc. -the latter three of which fall into the framework of so-called "parabolic molecules" [15], while all of the mentioned systems are encompassed by the even broader framework of α-molecules [16].…”

Edge, Ridge, and Blob Detection with Symmetric Molecules

“…In spite of various differences in their construction and their approximation behavior for different classes of signals, many of the aforementioned function systems are constructed by applying translation, scaling, and rotation or shear operators to generator functions, which typically need to satisfy a certain admissibility condition. In [35], so-called α-molecules were introduced to provide a unifying framework that includes different constructions of multiscale representations in order to enable the transfer of results from one theory to another, to allow for a categorization of multiscale representations with respect to the approximation behavior and also to facilitate the construction of novel systems.…”

“…The authors of [35] aimed to provide a maximum amount of flexibility by allowing each molecule of an α-molecule system to be based on a different generator in order to allow shears to be replaced by rotations. In practice, however, systems of α-molecules are often constructed from only a small number of generators.…”

“…These constructions have been considered in a wide range of applications in digital image processing including edge and feature detection [74,50,31,43]. Grohs et al recently introduced the so-called α-molecules framework, which allows for a unified treatment of the analysis of most multiscale systems developed in the field of applied harmonic analysis [35]. In the present work, we derive measures for the detection of edges, ridges, and blobs that are inspired by the notion of phase congruency but realized by making full use of the flexibility of modern construction principles for anisotropic analyzing functions provided within the framework of α-molecules.…”