On topological deep-structure segmentation

A hierarchical segmentation model is obtained by using linear scale evolution of gray-scale images. At each scale segments are generated as Voronoi diagrams with a distance measure defined on the image landscape. The set of centers of the Voronoi cells is the set of local extrema of the gray-scale image. This set is localized by using the winding number distribution of the gradient vector field. Scale evolution induces hierarchical structure of embedded segments. Objects defined at coarser scales "decompose" into sub-objects at finer scales. The process is naturally described in terms of singularity catastrophes within the smooth scale evolution. Alternatively, we present a purely topological segmentation procedure, based on singular isophotes. The last are generated by the set of saddle points in the image which are detected also with the topological winding-number method.