Optimising for Scale in Globally Multiply-Linked Gravitational Point Set Registration Leads to Singularities

The recently introduced gravitational point set registration methods can robustly align point sets with high noise ratios. While many results with rotation and translation resolution have been presented in the literature so far, the uncertainty about the scale resolving capability of globally multiply-linked gravitational point set registration methods is remaining. To address the uncertainty, we analyse the gravitational potential energy (GPE) functional of the system with two point sets with multivariate calculus and come to the conclusion that GPE of a singularity, i.e., the state when the template collapses to a single point, always has a lower energy than the state of the optimal alignment. Moreover, the GPE as a function of scale monotonically increases and does not have equienergetic states, which we prove for several hollow and volumetric geometric primitives in R 2 and R 3 including a unit circle, a unit sphere, a unit disk and a unit ball. We perform a series of experiments with various regular shapes and different combinations of references and templates to validate our findings. The consequence is that in practice, it is highly unlikely for globally multiply-linked gravitational alignment to resolve the scale accurately. We propose ways to overcome the limitation in scale resolution, which can be considered in the next generation of gravitational approaches.