Digital Ridgelet Transform Based on True Ridge Functions

We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not orthonormal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately preconditioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the underlying continuum theory, so there is room for substantial progress in future implementations.