Effective Scale: A Natural Unit for Measuring Scale-Space Lifetime

We develop how a notion of e ective scale can be introduced in a formal way. For continuous signals a scaling argument directly gives that a natural unit for measuring scale-space lifetime is in terms of the logarithm of the ordinary scale parameter. That approach is, however, not appropriate for discrete signals, since then an in nite lifetime would be assigned to structures existing in the original signal. Here we show how such an e ective scale parameter can be de ned as to give consistent results for both discrete and continuous signals. The treatment is based upon the assumption that the probability that a local extremum disappears during a short scale interval should not vary with scale. As a tool for the analysis we give estimates of how the density of local extrema can be expected to vary with scale in the scale-space representation of di erent random noise signals, both in the continuous and discrete cases.