Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rⁿ

Abstract

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρ_λ : λ ∈ Λ) for L² (Rⁿ), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rⁿ. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χ_n,k—the fiber bundle having the Grassman manifold G_n,k of k-planes in Rⁿ for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χ_n,k, via the smooth local coordinates of G_n,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space R^k(n−k) × R^n−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L²(Rⁿ)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L²(Rⁿ) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χ_n,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.