For each pair ( n , k ) with 1 ≤ k < n , we construct a tight frame (ρ λ : λ ∈ Λ) for L2 (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 L2 (R n )—the k -plane ridgelets. This construction makes analysis of a function f ∈ L2 (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 χn,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.