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

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρ(λ) : λ ∈ Λ) for L(2) (R(n)), 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 R(n). 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 R(n) 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 L(2)(R(n))—the k-plane ridgelets. This construction makes analysis of a function f ∈ L(2)(R(n)) 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.