Frame-Relative Critical Point Sets in Image Analysis

Abstract

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.

Detection of Critical Structures in Scale Space

“…One can show that relative critical points, detected in a frame of subdimension D CP , belong to a set which is locally isomorphic to a linear space of dimension C CP = D − D CP . A proof can be found in [8,9]. Note that C CP is the codimension of the dimension in which the relative critical point is detected.…”

“…The method is based on computing a surface integral of a functional of the gradient vector on the border of a closed neighborhood around every image point [8,9]. This integral evaluates to zero for regular image points and to an integer number for critical points.…”