Scale space hierarchy

We investigate the deep structure of a scale space image. We concentrate on scale space critical points—points with vanishing gradient with respect to both spatial and scale direction. We show that these points are always saddle points. They turn out to be extremely useful, since the iso-intensity manifolds through these points provide a scale space hierarchy tree and induce a pre-segmentation: a segmentation without a priori knowledge. Furthermore, both these scale space saddles and the so-called catastrophe points form the critical points of the parameterised critical curves—the curves along which the spatial critical points move in scale space. This enables one to localise these two types of special points relatively easy and automatically. Experimental results concerning the hierarchical representation and pre-segmentation are given and show results that correspond to a fair degree to both the mathematical and the intuitive forecast.