Simplifying Structures by Selectively Specifying Suited Scale Space Saddles

Blurring an image with a Gaussian of width sigma and considering sigma as an extra dimension, extends the image to an Gaussian scale space (GSS) image. In this GSS-image the iso-intensity manifolds behave in an nicely pre-determined manner. As a result of that, the GSS-image directly generates a hierarchy in the form of a binary ordered rooted tree, that can be used for segmentation, indexing, recognition and retrieval. Understanding the geometry of the manifolds allows fast implementational methods to derive the hierarchy. Scale space saddles form the pivot for this hierarchy. However, not all scale space saddles are relevant. The key to solve this ambiguity is the investigation of both the scale space saddles and the iso-intensity manifolds through them.In this paper the different situations that one can encounter in this investigation are described, the relevant scale space saddles are pointed out, examples are given, and the difference between selecting the relevant and the non-relevant (``void'') scale space saddles is shown. Next, the relevant geometric properties of GSS-images is discussed, as well as their implications for algorithms used for the tree extraction. It appears that one doesn't need to search through the whole GSS-image to find regions related to each relevant scale space saddle. Examples show the applicability and increased speed of the proposed method compared to traditional ones.