The novel framework of parabolic molecules provides for the first time a unifying framework for (sparse) approximation properties of directional representation systems by, in particular, including curvelets and shearlets. However, the considered common bracket is parabolic scaling, which excludes systems such as ridgelets and wavelets. In this paper, we therefore provide a generalization of this framework, which we coin α-molecules, by introducing an additional parameter α, which specifies the extent of anisotropy in the scaling. We show that, for instance, both ridgelets and wavelets are in fact α-molecules. As an application of the concept, we then analyze the sparse approximation behavior of α-molecules. Utilizing the idea of sparsity equivalence, it is possible to identify large classes of α-molecules providing the same sparse approximation behavior.

Date Published: 26 September 2013
PDF: 12 pages
Proc. SPIE 8858, Wavelets and Sparsity XV, 885804 (26 September 2013); doi: 10.1117/12.2022665

Published in SPIE Proceedings Vol. 8858:
Wavelets and Sparsity XV
Dimitri Van De Ville; Vivek K. Goyal; Manos Papadakis, Editor(s)