Deep structure III. topological numbers

Abstract

In the previous chapters we detected and followed the singularity strings through scale-space in an ac hoc manner. In the scale selection section, the detection of maxima was done by simply looking for pixels with values larger than its neighbors. In the edge focusing section, minima and maxima were detected (and distinguished) for a 1D signal by looking at sign changes for the derivative signal. Furthermore these extrema were tracked down through scale-space by simply looking for extrema in a close neighborhood in successive scale levels below. Finally, in the multi-scale segmentation section, the dissimilarity minima were represented indirectly by the catchment basins, and the linking across scale was done robustly by matching regions instead of points. These approaches seem somewhat heuristic. However, they can be expressed in a more formal manner with solid theoretical foundations. Furthermore, the implementations can be refined in order to make them more robust. The approach presented in this section is therefore not to be considered superior to the previous. It does, however, have a number of properties that are appealing both in theory and implementation. The concept presented in the following can be studied in more detail in [Kalitzin1996a, Kalitzin1997b, Staal1999a].