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

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. e-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. ntify the dividing scale-space saddles, a global investigation of the scale-space saddles and the iso-intensity manifolds through them is needed. s 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. n 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.