Deep Structure of Gaussian Scale Space

The deep structure of scale space is essential to the further processing of an image at different resolutions to extract certain features or to carry out other requirements such as image denosing. In this paper we study the deep structure of Gaussian scale space by virtue of singularity theory. Five normal forms of Gaussian scale space are provided, which is more accurate and rigorous than the results given by Damon and Kuijper. Then we analyze the creation and annihilation phenomena of extreme points and saddle points for the Gaussian scale space, which is the major part of deep structure of Gaussian scale space. We point out the creation phenomena conflicts with the intuitive result given by Koenderink. However, it does not influence the deep structure analysis of Gaussian scale space.