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.

Effects of extracellular Ca2+ concentration on hair-bundle stiffness and gating-spring integrity in hair cells

When a hair cell is stimulated by positive deflection of its hair bundle, increased tension in gating springs opens transduction channels, permitting cations to enter stereocilia and depolarize the cell. Ca2+ is thought to be required in mechanoelectrical transduction, for exposure of hair bundles to Ca2+ chelators eliminates responsiveness by disrupting tip links, filamentous interstereociliary connections that probably are the gating springs. Ca2+ also participates in adaptation to stimuli by controlling the activity of a molecular motor that sets gating-spring tension. Using a flexible glass fiber to measure hair-bundle stiffness, we investigated the effect of Ca2+ concentration on stiffness before and after the disruption of gating springs. The stiffness of intact hair bundles depended nonmonotonically on the extracellular Ca2+ concentration; the maximal stiffness of ≈1200 μN⋅m−1 occurred when bundles were bathed in solutions containing 250 μM Ca2+, approximately the concentration found in frog endolymph. For cells exposed to solutions with sufficient chelator capacity to reduce the Ca2+ concentration below ≈100 nM, hair-bundle stiffness fell to ≈200 μN⋅m−1 and no longer exhibited Ca2+-dependent changes. Because cells so treated lost mechanoelectrical transduction, we attribute the reduction in bundle stiffness to tip-link disruption. The results indicate that gating springs are not linearly elastic; instead, they stiffen with increased strain, which rises with adaptation-motor activity at the physiological extracellular Ca2+ concentration.