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

Tony Lindeberg

IEEE Trans. on Pattern Analysis and Machine Intelligence, 15(10), 1068--1074, 1993.

Also in Technical report ISRN KTH/NA/P--94/01--SE.


This article shows how a notion of effective 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 infinite lifetime would be assigned to structures existing in the original signal. Here we show how such an effective scale parameter can be defined 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 different random noise signals, both in the continuous and discrete cases.

Keywords: scale-space, effective scale, scale-space lifetime, discrete smoothing transformations, density of local extrema, multi-scale representation, computer vision, digital signal processing

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