M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Could humans recognize odor by phonon assisted tunneling?

Our sense of smell relies on sensitive, selective atomic-scale processes that are initiated when a scent molecule meets specific receptors in the nose. However, the physical mechanisms of detection are not clear. While odorant shape and size are important, experiment indicates these are insufficient. One novel proposal suggests inelastic electron tunneling from a donor to an acceptor mediated by the odorant actuates a receptor, and provides critical discrimination. We test the physical viability of this mechanism using a simple but general model. Using values of key parameters in line with those for other biomolecular systems, we find the proposed mechanism is consistent both with the underlying physics and with observed features of smell, provided the receptor has certain general properties. This mechanism suggests a distinct paradigm for selective molecular interactions at receptors (the swipe card model): recognition and actuation involve size and shape, but also exploit other processes.Comment: 10 pages, 1 figur