Ridgelets: a key to higher-dimensional intermittency? (English) Zbl 1082.42503

Candès, Emmanuel J.; Donoho, David L.

Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1760, 2495-2509 (1999).

Summary: 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 \(\psi(u_1x_1+\cdots+u_nx_n)\) whose ridge profiles \(\psi\) are wavelets, or alternatively from performing a wavelet analysis in the Radon domain.
The paper reviews recent work on the continuous ridgelet transform, ridgelet frames, ridgelet orthonormal bases, ridgelets and edges, and describes a new notion of smoothness naturally attached to this new representation.

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces

Keywords:

continuous ridgelet transform (CRT); ridgelet frames; ridgelet orthonormal bases; ridgelets and edges