AN EXTENDED CLASS OF SCALE-INVARIANT AND RECURSIVE SCALE-SPACE FILTERS

In this paper we explore how the functional form of scale space filter s is determined by a number of a priori conditions. In particular, if we assume scale space filters to be linear, isotropic convolution filt ers, then two conditions (viz. recursivity and scale-invariance) suffi ce to narrow down the collection of possible filters to a family that essentially depends on one parameter which determines the qualitative shape of the filter. Gaussian filters correspond to one particular val ue of this shape-parameter. For other values the filters exhibit a mor e complicated pattern of excitatory and inhibitory regions. This might well be relevant to the study of the neurophysiology of biological vi sual systems, for recent research shows the existence of extensive dis inhibitory regions outside the periphery of the classical center-surro und receptive field of LGN and retinal ganglion cells (in cats). Such regions cannot be accounted for by models based on the second order de rivative of the Gaussian. Finally, we investigate how this work ties i n with another axiomatic approach of scale space operators (propounded by Lindeberg and Alvarez et al.) which focuses on the semigroup prope rties of the operator family. We show that only a discrete subset of f ilters gives rise to an evolution which can be characterized by means of a partial differential equation.