Digital Implementation of Ridgelet Packets

FLESIA, A.G,; HEL-OR, H.; AVERBUCH, A.,; CAND\'ES, E.,; COIFMAN, R.,; DONOHO, D

Beyond Wavelets

Academic Press

Año: 2003; p. 31 - 60

The Ridgelet Packets libraryprovides a large family of orthonormal bases forfunctions $f(x,y)$ in $L^2(dx dy)$ which includesorthonormal ridgelets as well as basesderiving from tilings reminiscent from the theory of waveletsand the study of oscillatory Fourier integrals.An intuitively appealing feature:many of these bases have elementswhose envelope is strongly aligned along specified `ridges´while displaying oscillatory componentsacross the main `ridge´.There are twoapproaches to constructing ridgelet packets;the most direct is a frequency-domain viewpoint.We take a recursive dyadic partition of the polar Fourier domain intoa collection of rectangular tiles of variouswidths and lengths. Focusing attentionon each tile in turn,we take a tensor basis, usingwindowed sinusoids in $ heta$ times windowed sinusoids in$r$. There is also a Radon-domain approach to constructingridgelet packets, which involves applying the Radonisometry and then, in the Radon plane,using wavelets in $ heta$ times wavelet packets in $t$,with the scales of the wavelets in the two directionscarefully related.We discuss digital implementations of the two continuum approaches,yielding many new frames for representationof digital images $I(i,j)$.  These relyon two tools: the pseudopolar Fast Fourier Transform,and a pseudo Radon isometrycalled the normalized Slant Stack;these are described in Averbuch et al. (2001).In the Fourier approach, we mimic thecontinuum Fourier approachby partitioning the pseudopolar Fourierdomain, buildingan orthonormal basis in the image space subordinate toeach tile of the partition.  On each rectangle of the partition, we usewindowed sinusoids in $ heta$ times windowed sinusoids in$r$. In the Radon approach, we operateon the pseudo-Radon plane, and mimic the constructionof orthonormal ridgelets, but with differentscaling relationships between angular wavelets and ridgewavelets. Using wavelet packets in the ridge directionwould also be possible.Because of the wide range of possibleridgelet packet frames, the questionarises: what is the best frame for a given dataset?Because of the Cartesian format of our 2-Dpseudopolar domain, it ispossible to apply best-basis algorithmsfor best anisotropic cosine packetsbases;   this will rapidly search among allsuch frames for the best possible frame according toa sparsity criterion -- compare N. Bennett´s 1997 Yale Thesis.This automatically finds the best ridgelet packetframe for a given dataset.