The Relevance of Non-Generic Events in Scale Space Models

In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe and model the behaviour of critical point trajectories when various different types of generic events, viz. annihilations and creations of pairs of spatial critical points, (almost) coincide. Although such events are non-generic in mathematical sense, they are not unlikely to be encountered in practice due to numerical limitations. Furthermore, the behaviour of these trajectories leads to the observation that fine-to-coarse tracking of critical points doesnʹʹt suffice, since they can form closed loops in scale space. The modelling of the trajectories include these loops. We apply the theory to an artificial image and a simulated MR image and show the occurrence of the described behaviour.