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

For each pair (n, k) with 1 less than or equal to k < n, we construct a tight frame (rho(lambda) : lambda is an element of Lambda) 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 chi(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 chi(n,k), via the smooth local coordinates of G(n,k), of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n-k) x Rn-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 is an element of L-2(R-n) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform off by an appropriate wavelet-like system for chi(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.