Scale-space for N-D discrete signals

Abstract

The extension of the previous one-dimensional scale-space theory to two and higher dimensions is not obvious, since it is possible to show that there are no non-trivial kernels on ℤ² or ℝ² with the property that they never introduce new local extrema. Lifshitz and Pizer (1987) have presented an illuminating counter-example (quoted freely):

Imagine a two-dimensional image function consisting of two hills, one of them somewhat higher than the other one (see figure 4.1). Assume that they are smooth, wide, rather bell-shaped surfaces situated some distance apart, clearly separated by a deep valley running between them. Connect the two tops by a narrow sloping ridge without any local extrema, so that the top point of the lower hill no longer is a local maximum. Let this configuration be the input image. When the operator corresponding to the diffusion equation is applied to the geometry, the ridge will erode much faster than the hills. After a while it has eroded so much that the lower hill appears as a local maximum again. Thus, a new local extremum has been created.