Scale-Space Properties of Quadratic Feature Detectors

• Authors: Paul Kube;Pietro Perona
• Venue: IEEE Transactions on Pattern Analysis and Machine Intelligence
• Year: 1996

Feature detectors using a quadratic nonlinearity in the filtering stage are known to have some advantages over linear detectors; here, we consider their scale-space properties. In particular, we investigate whether, like linear detectors, quadratic feature detectors permit a scale selection scheme with the "causality property," which guarantees that features are never created as scale is coarsened. We concentrate on the design most common in practice, i.e., one dimensional detectors with two constituent filters, with scale selection implemented as convolution with a scaling function. We consider two special cases of interest: constituent filter pairs related by the Hilbert transform, and by the first spatial derivative. We show that, under reasonable assumptions, Hilbert-pair quadratic detectors cannot have the causality property. In the case of derivative-pair detectors, we describe a family of scaling functions related to fractional derivatives of the Gaussian that are necessary and sufficient for causality. In addition, we report experiments that show the effects of these properties in practice. Thus we show that at least one class of quadratic feature detectors has the same desirable scaling property as the more familiar detectors based on linear filtering.