Scale-Space Properties of Quadratic Feature Detectors

Feature detectors using a quadratic nonlinearity in the ltering stage are known to have some advantages over linear detectors; here we consider how their scale-space properties compare. In particular, we investigate the question 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 quadratic detector designs most commonly used in practice, one-dimensional detectors with two constituent lters, one even-symmetric and one odd-symetric. We consider two special cases of interest: constituent lter pairs related by the Hilbert transform, and by the rst 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 suucient for causality. In addition, we report experiments that show the eeects 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 ltering.