Digital Implementation of Ridgelet Packets

The Ridgelet Packets library provides a large family of orthonormal bases for functions f(x, y) in L(dxdy) which includes orthonormal ridgelets as well as bases deriving from tilings reminiscent from the theory of wavelets and the study of oscillatory Fourier integrals. An intuitively appealing feature: many of these bases have elements whose envelope is strongly aligned along specified ‘ridges’ while displaying oscillatory components across the main ‘ridge’. There are two approaches to constructing ridgelet packets; the most direct is a frequency-domain viewpoint. We take a recursive dyadic partition of the polar Fourier domain into a collection of rectangular tiles of various widths and lengths. Focusing attention on each tile in turn, we take a tensor basis, using windowed sinusoids in θ times windowed sinusoids in r. There is also a Radon-domain approach to constructing ridgelet packets, which involves applying the Radon isometry and then, in the Radon plane, using wavelets in θ times wavelet packets in t, with the scales of the wavelets in the two directions carefully related. We discuss digital implementations of the two continuum approaches, yielding many new frames for representation of digital images I(i, j). These rely on two tools: the pseudopolar Fast Fourier Transform, and a pseudo Radon isometry called the normalized Slant Stack; these are described in Averbuch et al. (2001). In the Fourier approach, we mimic the continuum Fourier approach by partitioning the pseudopolar Fourier domain, building an orthonormal basis in the image space subordinate to each tile of the partition. On each rectangle of the partition, we use windowed sinusoids in θ times windowed sinusoids in r. In the Radon approach, we operate on the pseudo-Radon plane, and mimic the construction of orthonormal ridgelets, but with different scaling relationships between angular wavelets and ridge wavelets. Using wavelet packets in the ridge direction would also be possible. Because of the wide range of possible ridgelet packet frames, the question arises: what is the best frame for a given dataset? Because of the Cartesian format of our 2-D pseudopolar domain, it is possible to apply best-basis algorithms for best anisotropic cosine packets bases; this will rapidly search among all such frames for the best possible frame according to a sparsity criterion – compare N. Bennett’s 1997 Yale Thesis. This automatically finds the best ridgelet packet frame for a given dataset.