A Computational Method for Segmenting Topological Point Sets and Application to Image Analysis

We propose a new computational method for segmenting topological sub-dimensional point-sets in scalar images of arbitrary spatial dimensions. The technique is based on calculating the homotopy class defined by the gradient vector in a sub-dimensional neighborhood around every image point. This neighborhood is defined as the linear envelope spawned over a given sub-dimensional vector frame. In the simplest case where the rank of this frame is maximal, we obtain a technique for localizing the critical points, i.e. extrema and saddle points. We consider in particular the important case of frames formed by an arbitrary number of the first largest by absolute value principal directions of the Hessian. The method then segments positive and and negative ridges as well as other types of critical surfaces of different dimensionalities. The signs of the eigenvalues associated to the principal directions provide a natural labeling of the critical sub-sets