Singularity theory and phantom edges in scale space

The process of detecting edges in a one-dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal. It is shown that the second derivative of the contrast function at the critical point is related to the classification of the associated edge as being phantom or authentic. The contrast of authentic edges decreases with filter scale, while the contrast of phantom edges are shown to increase with scale. As the filter scale increases, an authentic edge must either turn into a phantom edge or join with a phantom edge and vanish. The points in the scale space at which these events occur are seen to be singular points of the contrast function. Using ideas from singularity, or catastrophy theory, the scale map contours near these singular points are found to be either vertical or parabolic