Scale-Space Properties of Quadratic Edge Detectors

Abstract

Edge detectors which use a quadratic nonlinearity in the filtering stage are attracting interest in machine vision applications because of several advantages they enjoy over linear edge detectors. However, many important properties of these quadratic or “energy” edge detectors remain unknown. In this paper, we investigate the behavior of quadratic edge detectors under scaling. We consider two cases important in practice: quadratic detectors with constituent filters related by the Hilbert transform, and with constituent filters related by the first spatial derivative. We prove that in one dimension, Hilbert-pair detectors with Gaussian scaling permit the creation of new features as scale is increased, but such causality failures cannot generically occur with derivative-pair detectors. In addition, we report experiments that show the effects of these properties in practice. Thus at least one class of quadratic edge detectors can have the same desirable scaling property as detectors based on linear differential filtering.