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Scale invariance.

A formulation by Florack [] and continued work by Pauwels et al. [] show that the class of allowable scale-space kernels can be restricted under weaker conditions, essentially by combining the conditions about linearity, shift invariance, rotational invariance and semi-group structure with scale invariance. It can be shown that for a scale invariant rotationally symmetric semi-group, the Fourier transform of the convolution kernel must be of the form
for some tex2html_wrap_inline929 and p > 0, which gives a one-parameter class of possible semi-groups. Florack [] proposed to use separability in Cartesian coordinates as an additional basic constraint. Except in the one-dimensional case, this fixates h to be a Gaussian. Pauwels et al. showed [] that the corresponding multi-scale representations have local infinitesimal generators (basically meaning that the operator tex2html_wrap_inline935 in (9) is a differential operator) if and only if the exponent p is an even integer. Out of this countable set of choices, p = 2 is the only choice that corresponds to a non-negative convolution kernel (recall from above that non-creation of local extrema implies that the kernel has to be non-negative). Koenderink and van Doorn [] carried out a closely related study, and showed that derivative operators are natural operators to derive from a scale-space representation given the assumption of scale invariance.

Tony Lindeberg
Tue Jul 1 14:57:47 MET DST 1997