Finding an informative, structure-preserving map between two shapes has been a long-standing problem in geometry processing, involving a variety of solution approaches and applications. However, in many cases, we are given not only two related shapes, but a collection of them, and considering each pairwise map independently does not take full advantage of all existing information. For example, a notorious problem with computing shape maps is the ambiguity introduced by the symmetry problem -for two similar shapes which have reflectional symmetry there exist two maps which are equally favorable, and no intrinsic mapping algorithm can distinguish between them based on these two shapes alone. Another prominent issue with shape mapping algorithms is their relative sensitivity to how "similar" two shapes are -good maps are much easier to obtain when shapes are very similar. Given the context of additional shape maps connecting our collection, we propose to add the constraint of global map consistency, requiring that any composition of maps between two shapes should be independent of the path chosen in the network. This requirement can help us choose among the equally good symmetric alternatives, or help us replace a "bad" pairwise map with the composition of a few "good" maps between shapes that in some sense interpolate the original ones. We show how, given a collection of pairwise shape maps, to define an optimization problem whose output is a set of alternative maps, compositions of those given, which are consistent, and individually at times much better than the original. Our method is general, and can work on any collection of shapes, as long as a seed set of good pairwise maps is provided. We demonstrate the effectiveness of our method for improving maps generated by state-of-the-art mapping methods on various shape databases.

Planar cell polarity (PCP) signaling polarises cells along tissue axes. Although pathways involved are becoming better understood, outstanding issues include; (i) existence/identity of cues that orchestrate global polarisation in tissues, and (ii) the generality of the link between polarisation of primary cilia and asymmetric localisation of PCP proteins. Mammalian lenses are mainly comprised of epithelial-derived fiber cells. Concentrically arranged fibers are precisely aligned as they elongate along the anterior-posterior axis and orientate towards lens poles where they meet fibers from other segments to form characteristic sutures. We show that lens exhibits PCP, with each fiber cell having a apically situated cilium and in most cases this is polarised towards the anterior pole. Frizzled and other PCP proteins are also asymmetrically localised along the equatorial-anterior axis. Mutations in core PCP genes Van Gogh-like 2 and Celsr1 perturb oriented fiber alignment and suture formation. Suppression of the PCP pathway by overexpressing Sfrp2, shows that whilst local groups of fibers are often similarly oriented, they lack global orientation; consequently when local groups of fibers with different orientations meet they form multiple, small, ectopic suture-like configurations. This indicates that this extracellular inhibitor disrupts a global polarising signal that utilises a PCP-mediated mechanism to coordinate the global alignment and orientation of fibers to lens poles.

Figure 1: Shape interpolation from a cow to a duck to a torus via convolutional Wasserstein barycenters on a 100×100×100 grid, using the method at the beginning of §7. AbstractThis paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

The normal aging process is associated with stereotyped changes in gene expression, but the regulators responsible for these age-dependent changes are poorly understood. Using a novel genomics approach, we identified HOX co-factor unc-62 (Homothorax) as a developmental regulator that binds proximal to age-regulated genes and modulates lifespan. Although unc-62 is expressed in diverse tissues, its functions in the intestine play a particularly important role in modulating lifespan, as intestine-specific knockdown of unc-62 by RNAi increases lifespan. An alternatively-spliced, tissue-specific isoform of unc-62 is expressed exclusively in the intestine and declines with age. Through analysis of the downstream consequences of unc-62 knockdown, we identify multiple effects linked to aging. First, unc-62 RNAi decreases the expression of yolk proteins (vitellogenins) that aggregate in the body cavity in old age. Second, unc-62 RNAi results in a broad increase in expression of intestinal genes that typically decrease expression with age, suggesting that unc-62 activity balances intestinal resource allocation between yolk protein expression and fertility on the one hand and somatic functions on the other. Finally, in old age, the intestine shows increased expression of several aberrant genes; these UNC-62 targets are expressed predominantly in neuronal cells in developing animals, but surprisingly show increased expression in the intestine of old animals. Intestinal expression of some of these genes during aging is detrimental for longevity; notably, increased expression of insulin ins-7 limits lifespan by repressing activity of insulin pathway response factor DAF-16/FOXO in aged animals. These results illustrate how unc-62 regulation of intestinal gene expression is responsible for limiting lifespan during the normal aging process.