The Analysis To Bifurcation Of Critical Points Of Gaussian Scale Space Model

With the development of the image Gaussian scale-space theory, its analysis method and relevant applications gradually become an active and challenging topic in the computer vision and image processing fields. The theory has been widely used in the areas such as image segmentation, feature detection, higher order invariants, deblurring, etc. Gaussian scale-space is a typical model in the scale-space theory. It is of great importance to analyze evolution properties of the model singularities for the research of the scale-space deep structure.We begin our research from the model singularities and discuss the evolution properties of the model singularities and their deep structure. Here model singularities denote those points with zero partial derivatives, which correspond to the local feature points of the image intensity for the practical image processing. In this paper, we introduce theκe - equivalence to discuss the normal forms of stable germs for the two dimensional Gaussian scale-space. Based on the above classification, local bifurcation phenomena of singularities for those normal forms are presented. In particular, we find that the creation phenomenon of extreme points only occurs at the origin of the fifth normal form, and then annihilate again in a short time. Thus we may claim that Gaussian scale space always suppress image details when t > 0 although there exists the creation phenomenon. In other words, there is no creation bifurcation phenomenon of singularities for the practical image processing by Gaussian scale space method, which provides a theoretical basis for the conclusion drawn by Koenderink about the scale space model.