A computational method for segmenting topological point-sets and application to image analysis

We propose a new computational method for segmenting topological subdimensi onal point-sets in scalar images of arbitrary spatial dimensions. The techn ique is based on calculating the homotopy class defined by the gradient vec tor in a subdimensional neighborhood around every image point. This neighbo rhood is defined as the linear envelope spawned over a given subdimensional 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 a nd saddle points, We consider, in particular, the important case of frames formed by an arbitrary number of the first largest by absolute value princi pal directions of the Hessian. The method then segments positive and and ne gative ridges as well as other types of critical surfaces of different dime nsionalities. The signs of the eigenvalues associated to the principal dire ctions provide a natural labeling of the critical subsets. The result, in g eneral, is a constructive definition of a hierarchy of point-sets of differ ent dimensionalities linked by inclusion relations. Because of its explicit computational nature, the method gives a fast way to segment height ridges or edges in different applications. The defined topological point-sets are connected manifolds and, therefore, our method provides a tool for geometr ical grouping using only local measurements. We have demonstrated the group ing properties of our construction by presenting two different cases where an extra image coordinate is introduced. In one of the examples, we conside red the image analysis in the framework of the linear scale-space concept, where the topological properties are gradually simplified through the scale parameter. This scale parameter can be taken as an additional coordinate. In the second example, a local orientation parameter was used for grouping and segmenting elongated structures.