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 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) 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 χn,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 χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × 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 ∈ 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 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.

Eosinophil recruitment into the respiratory epithelium following antigenic challenge in hyper-IgE mice is accompanied by interleukin 5-dependent bronchial hyperresponsiveness.

A murine model for antigen-induced bronchial hyperreactivity (BHR) and airway eosinophilia, two hallmarks of asthma, was developed using ovalbumin-immunized mice, which produce large amounts of IgE (named BP2, "Bons Producteurs 2," for High Line of Selection 2). A single intranasal ovalbumin challenge failed to modify the bronchial responses, despite the intense eosinophil recruitment into the bronchoalveolar lavage fluid and airways. When mice were challenged twice a day for 2 days or once a day for 10 days, BHR in response to i.v. 5-hydroxytryptamine or to inhaled methacholine was induced in BP2 mice but not in BALB/c mice. Histological examination showed that eosinophils reached the respiratory epithelium after multiple ovalbumin challenges in BP2 mice but remained in the bronchial submucosa in BALB/c mice. Total IgE titers in serum were augmented significantly with immunization in both strains, but much more so in BP2 mice. Interleukin 5 (IL-5) titers in serum and bronchoalveolar lavage fluid of BP2 mice were augmented by the antigenic provocation, and a specific anti-IL5 neutralizing antibody suppressed altogether airway eosinophilia and BHR, indicating a participation of IL-5 in its development. Our results indicate that the recruitment of eosinophils to the airways alone does not induce BHR in mice and that the selective effect on BP2 mice is related to their increased IgE titers associated with antigen-driven eosinophil migration to the epithelium, following formation and secretion of IL-5.