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.

Visual habit formation in monkeys with neurotoxic lesions of the ventrocaudal neostriatum

Visual habit formation in monkeys, assessed by concurrent visual discrimination learning with 24-h intertrial intervals (ITI), was found earlier to be impaired by removal of the inferior temporal visual area (TE) but not by removal of either the medial temporal lobe or inferior prefrontal convexity, two of TE's major projection targets. To assess the role in this form of learning of another pair of structures to which TE projects, namely the rostral portion of the tail of the caudate nucleus and the overlying ventrocaudal putamen, we injected a neurotoxin into this neostriatal region of several monkeys and tested them on the 24-h ITI task as well as on a test of visual recognition memory. Compared with unoperated monkeys, the experimental animals were unaffected on the recognition test but showed an impairment on the 24-h ITI task that was highly correlated with the extent of their neostriatal damage. The findings suggest that TE and its projection areas in the ventrocaudal neostriatum form part of a circuit that selectively mediates visual habit formation.

Island colonization and evolution of the insular woody habit in Echium L. (Boraginaceae).

Numerous island-inhabiting species of predominantly herbaceous angiosperm genera are woody shrubs or trees. Such "insular woodiness" is strongly manifested in the genus Echium, in which the continental species of circummediterranean distribution are herbaceous, whereas endemic species of islands along the Atlantic coast of north Africa are woody perennial shrubs. The history of 37 Echium species was traced with 70 kb of noncoding DNA determined from both chloroplast and nuclear genomes. In all, 239 polymorphic positions with 137 informative sites, in addition to 27 informative indels, were found. Island-dwelling Echium species are shown to descend from herbaceous continental ancestors via a single island colonization event that occurred < 20 million years ago. Founding colonization appears to have taken place on the Canary Islands, from which the Madeira and Cape Verde archipelagos were invaded. Colonization of island habitats correlates with a recent origin of perennial woodiness from herbaceous habit and was furthermore accompanied by intense speciation, which brought forth remarkable diversity of forms among contemporary island endemics. We argue that the origin of insular woodiness involved response to counter-selection of inbreeding depression in founding island colonies.