Two theorems about similarity maps

Abstract

One of the problems arising when exploring toponome or other multivariate-image data is the following: Given a family of n gray-value images of, e.g., a given sample of cell tissue, indexed by a collection of n proteins under investigation (so-called MELK data) - each single image representing the varying local concentration of one of those n proteins at the various sites (pixels) of the given sample, how should one quantify, for any two pixels (or clusters of pixels), the (dis)similarity between the corresponding "vectors" of local protein concentrations in question. Some (dis)similarity mappings defined on ℝMn allowing for fast OpenGL texture mapping turned out to be useful in visual inspection of toponome data. Here, we derive two rather general results regarding similarity and dissimilarity mappings and, as a corollary, the fact that the functions that were used for visual inspection of MELK data are, indeed, metrics. We believe that our results are, however, also of more general interest within the ongoing program of elucidating the structure of metrics from a more abstract point of view. © 2008 Birkhaueser.