|
Abstract : |
Feature detectors using a quadratic nonlinearity in the filtering 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 filters, one even-symmetric and one odd-symetric. 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., |
