PNAS Inaugural Article: Tight Frames of k-Plane Ridgelets and the Problem of Representing Objects Which Are Smooth Away from d-Dimensional Singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L (R), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions which are smooth away from singularities along k-planes in R. We also develop tools to help decide whether in fact k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle Xn,k – the fiber bundle having the Grassman manifold Gn,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 Xn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space R ×R. We then use the X-ray isometry [Solmon, 1976] to map this tight frame isometrically to a tight frame for L(R) – the k-plane ridgelets. This construction makes analysis of a function f ∈ L(R) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for Xn,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.