期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：1999
卷号：96
期号：5
页码：1828-1833
DOI：10.1073/pnas.96.5.1828
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：For each pair (n, k) with 1 [≤] k < n, we construct a tight frame ({rho}{lambda} : {lambda} [isin] {Lambda}) 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 {chi}n,k--the fiber bundle having the Grassman manifold Gn,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{chi} n,k, via the smooth local coordinates of Gn,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 L2(Rn)--the k-plane ridgelets. This construction makes analysis of a function f [isin] 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{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.
