Using Catastrophe Theory to Derive Trees from Images

Abstract

In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe the different types of annihilations and the creation of pairs of critical points and how this knowledge can be exploited in a scale space hierarchy tree for the purpose of a topology based segmentation. A key role is played by scale space saddles and iso-intensity manifolds through them. We discuss the role of non-generic catastrophes and their influence on the tree and the segmentation. Furthermore it is discussed, based on the structure of iso-intensity manifolds, why creations of pairs of critical points don't influence the tree. We clarify the theory with an artificial image and a simulated MR image.