Digital ridgelet transform based on true ridge functions

DONOHO, D.L.; FLESIA, A.G.

Beyond Wavelets

Academic Press

Año: 2003; p. 1 - 29

We study a notion of ridgelet transform for arraysof digital data in which the analysis operator uses trueridge functions, as does the synthesis operator.There are fast algorithms for analysis, for synthesis, and forpartial reconstruction. Associated with this is a transformwhich isa digital analog of the orthonormal ridgelettransform (but not orthonormal for finite n).  In either approach, we get anovercomplete frame; the result of ridgelet transforming an $n imes n$ arrayis a $2n imes 2n$ array.The analysis operator is invertible onits range; the appropriately preconditionedoperator has a tightly controlled spreadof singular values.  There is a near-parsevalrelationship.Our construction exploits the recentdevelopment by Averbuch et al. (2001) of theFast Slant Stack, a Radon transformfor digital image data; it may be viewedas following a Fast Slant Stack with fast 2-d wavelet transform.A consequence of this construction is that it offersdiscrete objects (discrete ridgelets, discrete Radon transform,discrete PseudopolarFourier domain)which obey inter-relationships paralleling those in the continuumridgelet theory(between ridgelets, Radon transform, and polar Fourier domain).We make comparisons with other notions of ridgelettransform, and we investigate what we view as thekey issue: the summability of the kernel underlyingthe constructed frame. The sparsity observedin our current implementation is not nearly asgood as the sparsity of the underlying continuumtheory, so there is room for substantial progressin future implementations.