Topological Numbers and Singularities

Abstract

Scale space approach provides a tool for studying a given image at all scales simultaneously. Features that can be detected at large scales can provide clues for tracing more detailed information at fine scales. For this ideology to be constructive, one needs to investigate which local properties (or local operators) are appropriate for quantifying the desired features and how these properties are changing from scale to scale. In various applications (Kass et al., 1987; Koenderink, 1990), specially those concerned with oriented structures, singular points are of particular interest. Singular points, or simply singularities below, are those points in a grayscale image, where the gradient vector field vanishes (Lindeberg, 1992; Johansen, 1994). Examples of such points in two dimensional images are extrema, saddle points, “monkey” saddles etc. Singular points can be characterized by their order. The order of a singular point is the lowest non-vanishing power in the Taylor expansion of the image field L(x ₁, x ₂, ..., x _d ) around this point. Obviously the order must be greater than 1, since the gradient vector L _i = ∂L(x ₁, x ₂, ..., x _d ) is equal to zero in this point.