AVA Christmas Meeting, London, UK 18 December 2012

Abstract

Cohen (IEEE Trans. Signal Processing, 41, 3275-3292, 1993) suggested that a signal’s scale can beregarded as a ‘physical attribute’ that decouples the size of a phenomenon from its shape. His idea, incombination with invariant signal representations, has clear ramifications for the phenomenon of sizeconstancy in vision science. In the visual system, it is hoped that size constancy might be derived fromthe collected responses of a distribution of isotropic spatial filters whose underlying spatial extentare systematically varied from coarse to fine scales according to a diffusion model of image blur: theso-called scale-space representation (eg Koenderink, Biol. Cyb. 50, 363-370, 1984). We demonstratethat this ‘blurring’ approach is flawed. The reason is because scaling and blurring are fundamentallydifferent image operations – application of linear filters whose degrees of blur are different beforeattempting to extract scale information can seriously impair one’s ability to extract the physical attributeof scale. We show that local scale (and position) invariant signal representations can be derived byfinding unknown coefficients that allow one to predict the image intensity signal from a power seriesexpansion. We further show that the inverse of this power series is a Taylor expansion of discretelocal image derivatives whose coefficients are invariant of position and scale. The expansion retainsthe benefit of efficiency when representing 2D shape. Finally, we show how our ideas link scaling,fractional orders of differentiation and pyramid sampling as a means for determining the scale of a 2Dshape. We suggest that similar computations underpin position and scale invariant computations in the visual system.