Ridgelets: a key to higher-dimensional intermittency?

Abstract

In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behaviour. In effect, wavelets are well adapted for point–like phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes and other non–point–like structures, for which wavelets are poorly adapted.We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with line–like phenomena in dimension 2, plane–like phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1 + ⃛ + unxn whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain.The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.