SCALE-SPACE PROPERTIES OF QUADRATIC FEATURE-DETECTORS

Feature detectors using a quadratic nonlinearity in the filtering stag e:are known to have some advantages over linear detectors; here, we co nsider their scale-space properties. in particular, we investigate whe ther, like linear detectors, quadratic feature detectors permit a scal e selection scheme with the ''causality property,'' which guarantees t hat features are never created as scale is coarsened. We concentrate o n the design most common in practice, i.e., one-dimensional detectors with two constituent filters, with scale selection implemented as conv olution with a scaling function. We consider two special cases of inte rest: constituent filter pairs related by the Hilbert transform, and b y the first spatial derivative. We show that, under reasonable assumpt ions, Hilbert-pair quadratic detectors cannot have the causality prope rty. In the case of derivative-pair detectors, we describe a family of scaling functions related to fractional derivatives of the Gaussian t hat are necessary and sufficient for causality. In addition, we report experiments that show the effects of these properties in practice. Th us we show that at least one class of quadratic feature detectors has the same desirable scaling property as the more familiar detectors bas ed on linear filtering.