comparing points to points

For a number of projects, my group has been trying to compare point sets to point sets, to determine transformations. Some contexts have been calibration (like photometric and astrometric calibration of images, where stars need to align, either on the sky or in magnitude space) and others have been in dynamics. Right now Suroor Gandhi (NYU), Adrian Price-Whelan (Flatiron), and I have been trying to find transformations that align phase-space structures (and especially the Snail) observed in different tracers: What transformation between tracers matches the phase-space structure? These projects are going by our code name MySpace.

Projects like these tend to have a pathology, however, related to a pathology that Robyn Sanderson (Flatiron) and I found in a different context in phase space: If you write down a naive objective for matching two point clouds, the optimal match often has one point cloud shrunk down to zero size and put on top of the densest location on the other point cloud! Indeed, Gandhi is finding this so we decided (today) to try symmetrizing the objective function to stop it. That is, don't just compare points A to points B, but also symmetrically compare points B to points A. Then (I hope) neither set can shrink to zero usefully. I hope this works! Now to make a symmetric objective function...