Exploring and exploiting the structure of saddle points in Gaussian scale space

摘要

When an image is filtered with a Gaussian of width σ and σ is considered as an extra dimension, the image is extended to a Gaussian scale-space (GSS) image. In earlier work it was shown that the GSS-image contains an intensity-based hierarchical structure that can be represented as a binary ordered rooted tree. Key elements in the construction of the tree are iso-intensity manifolds and scale-space saddles.A scale-space saddle is a critical point in scale space. When it connects two different parts of an iso-intensity manifold, it is called “dividing”, otherwise it is called “void”. Each dividing scale-space saddle is connected to an extremum in the original image via a curve in scale space containing critical points. Using the nesting of the iso-intensity manifolds in the GSS-image and the dividing scale-space saddles, each extremum is connected to another extremum. In the tree structure, the dividing scale-space saddles form the connecting elements in the hierarchy: they are the nodes of the tree. The extrema of the image form the leaves, while the critical curves are represented as the edges.To identify the dividing scale-space saddles, a global investigation of the scale-space saddles and the iso-intensity manifolds through them is needed.In this paper an overview of the situations that can occur is given. In each case it is shown how to distinguish between void and dividing scale-space saddles. Furthermore, examples are given, and the difference between selecting the dividing and the void scale-space saddles is shown. Also relevant geometric properties of GSS images are discussed, as well as their implications for algorithms used for the tree extraction.As main result, it is not necessary to search through the whole GSS image to find regions related to each relevant scale-space saddle. This yields a considerable reduction in complexity and computation time, as shown in two examples.