EFFECTIVE SCALE - A NATURAL UNIT FOR MEASURING SCALE-SPACE LIFETIME

The article develops a manner in which a notion of effective scale can be introduced in a formal way. For continuous signals, a scaling argu ment directly gives that a natural unit for measuring scale-space life time is in terms of the logarithm of the ordinary scale parameter. Tha t approach is, however, not appropriate for discrete signals since the n, an infinite lifetime would be assigned to structures existing in th e original signal. Here, it is shown how such an effective scale param eter can be defined to give consistent results for both discrete and c ontinuous signals. The treatment is based on the assumption that the p robability that a local extremum disappears during a short-scale inter val should not vary with scale. As a tool for the analysis, estimates are given of how the density of local extrema can be expected to vary with scale in the scale-space representation of different random noise signals both in the continuous and discrete cases.