Frame-relative critical point sets in image analysis

Abstract

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 computing the homotopy class defined by the gradient vector in a sub-dimensional neighborhood around every image point. The neighborhood is defined as the linear envelope spawned over a given sub-dimensional vector frame. In the paper we consider in particular the frame formed by an arbitrary number of the first largest principal directions of the Hessian. In general, the method segments ridges, valleys and other critical surfaces of different dimensionalities. Because of its explicit computational nature, the method gives a fast way to segment height ridges in different applications. The so defined topological point sets are connected manifolds and therefore our method provides a tool for feature grouping. We have demonstrated the grouping properties of our construction by introducing in two different cases an extra image coordinate. In one of the examples we considered the scale as an additional coordinate and in the second example, local orientation parameter was used for grouping and segmenting elongated structures.